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MICROECONOMICS
FOURTH EDITION
DAV I D A . B E S A N KO
Northwestern University,
Kellogg School of Management
RONALD R. BRAEUTIGAM
Northwestern University,
Department of Economics
with Contributions from
Michael J. Gibbs
The University of Chicago,
Booth School of Business
J O H N W I L E Y & S O N S, I N C .
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Library of Congress Cataloging-in-Publication Data
Besanko, David, 1955Microeconomics / David Besanko, Ronald Braeutigam. —4th ed.
p. cm.
Includes index.
ISBN 978-0-470-56358-8 (hardback)
1. Microeconomics. I. Braeutigam, Ronald R. (Ronald Ray) II. Title.
HB172.B49 2010
338.5—dc22
2010032314
To order books or for customer service, please call 1-800-CALL WILEY (225-5945).
Main Book ISBN:
978-0-470-56358-8
Binder-Ready Version ISBN: 978-0-470-91756-5
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
ABOUT THE AUTHORS
DAV I D B E S A N KO is the Alvin J. Huss Distinguished Professor of Management and
Strategy at the Kellogg School of Management at Northwestern University. From 2007 to 2009
he served as Senior Associate Dean for Academic Affairs: Strategy and Planning and from 2001
to 2003 served as Senior Associate Dean for Academic Affairs: Curriculum and Teaching.
Professor Besanko received his AB in Political Science from Ohio University in 1977, his MS
in Managerial Economics and Decision Sciences from Northwestern University in 1980, and
his PhD in Managerial Economics and Decision Sciences from Northwestern University in
1982. Before joining the Kellogg faculty in 1991, Professor Besanko was a member of the faculty of the School of Business at Indiana University from 1982 to 1991. In addition, in 1985, he
held a postdoctorate position on the Economics Staff at Bell Communications Research.
Professor Besanko teaches courses in the fields of Management and Strategy, Competitive
Strategy, and Managerial Economics. In 1995 and 2010, the graduating classes at Kellogg
awarded Professor Besanko the L.G. Lavengood Professor of the Year, the highest teaching
honor a faculty member at Kellogg can receive. He is only one of two faculty members of
Kellogg to have received this award twice. At the Kellogg School, he has also received the
Alumni Choice Teaching Award in 2006, the Sidney J. Levy Teaching Award (1998, 2000, 2009)
and the Chair’s Core Teaching Award (1999, 2001, 2003, 2005).
Professor Besanko does research on topics relating to competitive strategy, industrial organization, the theory of the firm, and economics of regulation. He has published two books
and over 40 articles in leading professional journals in economics and business, including the
American Economic Review, Econometrica, the Quarterly Journal of Economics, the RAND Journal of
Economics, the Review of Economic Studies, and Management Science. Professor Besanko is a
co-author of Economics of Strategy with David Dranove, Mark Shanley, and Scott Schaefer.
R O N A L D R . B R A E U T I G A M is the Harvey Kapnick Professor of Business Institutions
in the Department of Economics at Northwestern University. He is currently Associate Provost
for Undergraduate Education, and he has served as Associate Dean for Undergraduate Studies
in the Weinberg College of Arts and Sciences. He received a BS in Petroleum Engineering
from the University of Tulsa in 1970 and then attended Stanford University, where he received
an MS in engineering and a PhD in Economics in 1976. He has taught at Stanford University
and the California Institute of Technology, and he has also held an appointment as a Senior
Research Fellow at the Wissenschaftszentrum Berlin (Science Center Berlin). He has also
worked in both government and industry, beginning his career as a petroleum engineer with
Standard Oil of Indiana. He served as research economist in The White House Office of
Telecommunications Policy and as an economic consultant to Congress, many government
agencies, and private firms on matters of pricing, costing, managerial strategy, antitrust, and
regulation.
Professor Braeutigam has received many teaching awards, including the Northwestern
University Alumni Association Excellence in Teaching Award (1991), and recognition as a
Charles Deering McCormick Professor of Teaching Excellence at Northwestern (1997–2000),
the highest teaching award that can be received by a faculty member at Northwestern.
Professor Braeutigam’s research interests are in the field of microeconomics and industrial
organization. Much of his work has focused on the economics of regulation and regulatory
reform, particularly in the telephone, transportation, and energy sectors. He has published
many articles in leading professional journals in economics, including the American Economic
Review, the RAND Journal of Economics, the Review of Economics and Statistics, and the
International Economic Review. Professor Braeutigam is a co-author of The Regulation Game with
Bruce Owen, and Price Level Regulation for Diversified Public Utilities with Jordan J. Hillman. He
has also served as President of the European Association for Research in Industrial Economics.
v
vi
A B O U T T H E AU T H O R S
M I C H A E L G I B B S is Clinical Professor of Economics and Human Resources at the
University of Chicago Booth School of Business, and a Research Fellow at the Institute for the
Study of Labor. He received a BA, MA and PhD in Economics from the University of Chicago.
Professor Gibbs has also taught at Harvard, the University of Michigan, USC, Sciences Po
(Paris), and conducted research at the Aarhus School of Business (Denmark). Professor Gibbs
is a leading scholar in personnel economics. He has received several teaching and research
awards. Professor Gibbs is a co-author of Personnel Economics in Practice with Edward Lazear.
PREFACE
After many years of experience teaching microeconomics at the undergraduate and MBA
levels, we have concluded that the most effective way to teach it is to present the content
with a variety of engaging applications, coupled with an ample number of practice
problems and exercises. The applications ground the theory in the real world, and the
exercises and problems sets enable students to master the tools of economic analysis and
make them their own. The applications and the problems are combined with verbal intuition and graphs, so that they are reinforced and amplified. This approach enables students
to see clearly the interplay of key concepts, to thoroughly grasp these concepts through
abundant practice, and to see how they apply in actual markets and business firms.
Our reviewers and adopters of the first edition have told us that this approach
worked for them and their students. In the second edition, we built on this approach,
adding even more applications and problems and revisiting every explanation, every
graph, and every Learning-By-Doing example to make sure the text was as clear as possible. In the third edition, we continued in the spirit of the second edition, adding more
current applications and problems. In fact, we added at least five problems to each
chapter (nearly 90 new problems in all). In the fourth edition, we added still more new
problems, and we put in over 30 new current applications. In addition, we added a new
Appendix to Chapter 4 that introduces the basic concepts of time value of money, such
as present and future value. Finally, every chapter now begins with a set of concrete,
actionable learning goals based on Bloom’s Taxonomy of Educational Objectives.
• The Solution Is in the Problems. Our emphasis on practice exercises and numerous, varied problems sets this book apart from others. Based on our experience, students
need drill in order to internalize microeconomic theory. They need to work through many
problems that are tangible, problems that have specific equations and numbers in them.
Anyone who has mastered a skill or a sport, whether it be piano, ballet, or golf,
understands that a fundamental part of the learning process involves repetitive drills that
seemingly bear no relation to how one would actually execute the skill under “real” conditions. We feel that drill problems in microeconomics serve the same purpose. A student
may never have to do a numerical comparative statics analysis after completing the microeconomics course. However,
having seen concretely, through
the use of numbers and equaL E A R N I N G - B Y- D O I N G E X E R C I S E 2 . 6
tions, how a shift in demand or
Elasticities along Special Demand Curves
supply affects the equilibrium, a
Problem
10 and Q 400 10P,
( b)(P Q). Since b
Q, P
when P 30,
student will have a deeper ap(a) Suppose a constant elasticity demand curve is given
by the formula Q 200P . What is the price elasticity
30
preciation for comparative statb
3
10 a
Q,P
of demand?
400 10(30)
ics analysis and will be better
(b) Suppose a linear demand curve is given by the
and when P 10,
prepared to interpret events in
formula Q 400 10P. What is the price elasticity of
10
demand at P 30? At P 10?
b
0.33
10 a
real markets.
Q, P
400 10(10)
Learning-By-Doing
Solution
Note that demand is elastic at P 30, but it is inelastic at
Exercises, embedded in the text
P 10 (in other words, P 30 is in the elastic region of
(a) Since this is a constant elasticity demand curve, the
the demand curve, while P 10 is in the inelastic region).
price elasticity of demand is equal to 1 2 everywhere
of each chapter, guide the stualong the demand curve.
dent through specific numerical
Similar Problems: 2.5, 2.6, 2.12
(b) For this linear demand curve, we can find the price
problems. We use three to ten
elasticity of demand by using equation (2.4):
Learning-By-Doing exercises
S
E
D
1
2
vii
viii
P R E FAC E
in each chapter and have designed them to illustrate the core ideas of the chapter. They
are integrated with the graphical and verbal exposition, so that students can clearly see,
through the use of numbers and tangible algebraic relationships, what the graphs and
words are striving to teach. These exercises set the student up to do similar practice
problems as well as more difficult analytical problems at the end of each chapter and in
the study guide that accompanies this text.
As noted above, we have added to the already complete end-of-chapter problem
sets to give students and instructors more opportunity to assess student understanding. Chapters 1–13 have between 25 and 35 end-of-chapter exercises, while Chapters
14–17 have between 20 and 25 exercises. There is at least one exercise for each of the
topics covered in the chapter, and the topics covered by the exercises generally follow
the order of topics in the chapter. At the end of the book, there are fully worked-out
solutions to selected exercises.
A P P L I C A T I O N
2.7
What Hurricane Katrina Tells Us
about the Price Elasticity of Demand
for Gasoline
prices usually rise in the spring through late summer, due to warmer weather, closed schools, and
summer vacations. They are usually lower in winter.
Gasoline prices can also fluctuate due to changes
in crude oil prices, since gasoline is refined from
crude oil.
In addition to these factors, gasoline prices are
highly responsive to changes in supply. Prices may
change dramatically if there are disruptions to
the supply chain. Typical inventory levels of commercial gasoline usually amount to only a few days of
• It Works in Theory, but
Does It Work in the Real
“realWorld? Numerous
world” examples illustrate how
microeconomics applies to
business decision-making and
Gasoline prices tend to be highly volatile. Figure 2.23
illustrates this by plotting the average retail gasoline
public policy issues. We begin
price in the United States in 2005.21 Large swings in
each chapter with an extended
price in short periods of time are common, as are
seasonal fluctuations. The seasonal changes are
example that introduces the
largely attributable to shifts in demand. Gasoline
key themes of the chapter and
uses real markets and companies to reinforce particular concepts and tools. Each chapter contains, on average,
seven examples, called Applications, woven into the narrative or highlighted in sidebars. In this fouth edition, we have taken care to update our applications and to add
to them, so that we now have over 120 Applications. A full list may be found on the
front endpapers of this text. New applications include health care reform in the U.S.,
the collapse of AIG, parking meter privatization in Chicago, and the bailout of the
Parmesan cheese industry in Italy.
• Graphs Tell the
Story. We use graphs
FIGURE 2.5
Excess Demand and Excess
Supply in Market for Corn
If the price of corn were $3, per bushel, excess
demand would result because 14 billion bushels
would be demanded, but only 9 billion bushels
would be supplied. If the price of corn were $5
per bushel, excess supply would result because
13 billion bushels would be supplied but only 8
billion bushels would be demanded.
Price (dollars per bushel)
S
Excess supply
when price
is $5
$5
E
$4
$3
Excess
demand
when price is $3
D
8 9 11 13 14
Quantity (billions of bushels per year)
and tables more abundantly than most texts,
because they are central
to economic analysis, enabling us to depict complex interactions simply.
In economics, a picture
truly is worth a thousand
words. In each new edition we have worked to
make the graphs even
clearer and more useful
for students.
ix
P R E FAC E
• Get to the Point. All too often, verbal explanations of economic ideas and concepts seem convoluted and unintuitive. Tables and graphs are powerful economic
tools, but many students cannot interpret them readily at first. We believe our exposition of the economic intuition underlying the graphs is clear and easy to follow. We
have worked through every line to streamline the exposition. Patient step-by-step explanations with examples enable even nonvisual learners to understand how graphs
are constructed and what they mean.
O R G A N I Z AT I O N A N D C OV E R AG E
This book is traditional in its coverage and organization. To the extent that we have made
a trade-off, it is to cover traditional topics more thoroughly, as opposed to adding a broad
range of additional topics that might not easily fit into a one-quarter or one-semester
microeconomics course. Thus an instructor teaching a one-semester microeconomics
course could use all or nearly all of the chapters in the book, and an instructor teaching
a one-quarter microeconomics or managerial economics course could use more than
two-thirds of the chapters. The following chart shows how the book is organized.
Introduction to
Microeconomics
1
Overview and
introduction
to constrained
optimization,
equilibrium
analysis, and
comparative
statics analysis
2
Introduction
to demand
curves, supply
curves, market
equilibrium,
and elasticity
Consumer
Theory
3
Introduction
to consumer
choice
Production and
Cost Theory
Perfectly
Competitive
Markets
6
Production
function,
marginal
and average
product, and
returns
to scale
4
7
Budget lines,
utility maximization, and
analysis of
revealed
preference
Concept of
cost, input
choice and
cost minimization
5
8
Comparative
statics of
consumer
choice and
consumer
surplus
Construction
of total,
average, and
marginal cost
curves
Monopoly and
Monopsony
Imperfectly
Competitive
Markets and
Strategic Behavior
Special Topics
9
11
13
15
Profit-maximizing
output choice by a
price-taking firm
and prices in shortrun and long-run
equilibrium
Theories of
monopoly and
monopsony
price setting
Price determination in imperfectly
competitive
markets
Risk,uncertainty,
and information,
including a
utility-theoretic
approach to
uncertainty and
decision tree
analysis, Insurance
markets and
asymmatric information, and
auctions
10
Using the
competitive
market model
to analyze
public policy
interventions
12
Price discrimination
14
Simultaneousmove games
and sequential
move games
A LT E R N AT I V E C O U R S E D E S I G N S
In writing this book, we have tried to serve the needs of instructors teaching microeconomics in a variety of different formats and time frames.
16
Overview of
general equilibrium theory
and economic
efficiency
17
Externalities
and public
goods
x
P R E FAC E
• One-quarter course (10 weeks): An instructor teaching a one-quarter undergraduate microeconomics course that fully covers all of the traditional topics
(including consumer theory and production and cost theory) would probably
assign Chapters 1–11. If the instructor prefers to deemphasize consumer theory
or production theory, he or she might also be able to cover Chapters 13 and 14.
• One-semester course (15 weeks): In a one-semester undergraduate course, an
instructor should be able to cover Chapters 1–15. If the course must include
general equilibrium theory, public goods, and externalities, then Chapter 15
could be dropped and the instructor could assign Chapters 1–14, 16, and 17.
• Two-quarter course (20 weeks): For a two-quarter sequence (the structure
we have at Northwestern), the first quarter could cover Chapters 1–11, and
the second quarter could pick up where the first quarter left off and cover
Chapters 12–17.
• MBA-level managerial economics course (10 weeks or 15 weeks): For a
one-quarter course, the instructor would probably want to skip the chapters on
consumer theory, production functions, and cost minimization (Chapters 3–6
and the second half of Chapter 7) and cover Chapters 1–2, the first half of
Chapter 7—economic concepts of cost—Chapter 8, and Chapters 9–14.
Extending such a course to a full semester would allow the instructor to include
the material on production and cost minimization as well as Chapter 15.
S U P P L E M E N TA RY R E S O U R C E S
Thank you to Katharine Rockett, University of Essex; Dorothea Herreiner, Loyola
Marymount University; David Spigelman, University of Miami; Daya Muralidharan,
University of California, Riverside; Brian Kench, University of Tampa; Douglas
W. Copeland, Kansas State University; and Lanny Arvan for preparing the following
resources:
COMPANION WEB SITE (www.wiley.com/college/besanko) includes resources for
both students and instructors. Provides many of the resources listed here as well as
Lecture Outline PowerPoint presentations, and Excel-based problems that provide
graphical illustrations related to key concepts within the text.
INSTRUCTOR’S MANUAL includes additional examples related to the chapter
topics, references to relevant written works, Web site addresses, and so on, which
enhance the material within each chapter of the text, additional problem sets, and
sample exams. Found on the companion Web site.
SOLUTIONS MANUAL provides answers to end-of-chapter material and worked out
solutions to any additional material not already provided within the text. Found on the
companion Web site.
TEST BANK contains nearly 1,000 multiple-choice and short answer questions as well
as a set of problems varying in level of difficulty and correlated to all learning objectives. Found on the companion Web site.
COMPUTERIZED TEST BANK consists of content from the Test Bank provided
within a test-generating program that allows instructors to customize their exams.
Found on the companion Web site.
P R E FAC E
STUDY GUIDE includes a Chapter Summary, Exercises with multiple-choice answers
(answers provided at the end of the chapter), Chapter Review Questions with
Answers, Problems with Answers, and Practice Exam Questions with Answers for
each chapter.
STUDENT PRACTICE QUIZZES contain at least 10–15 practice questions per chapter.
Multiple choice and short answer questions, of varying difficulty, help students evaluate individual progress through a chapter.
BUSINESS EXTRA SELECT Wiley’s Business Extra Select program is a simple, integrated, online custom-publishing process that allows you to combine content from
Wiley’s leading business publications with copyright-cleared content from such respected sources as INSEAD, Fortune, The Economist, The Wall Street Journal, Harvard
Business School cases, and much more. In just a few simple steps you can help your students make the connection between the concepts you teach in your class and their
real-world applications! Contact your Wiley representative for more information.
APLIA Aplia is a basic course management system which includes a gradebook, and
offers additional text-correlated tutorials, problems, graphing tools, news analysis,
and experiments. An electronic version of Microeconomics, 4e is also included. For
more information, visit www.aplia.com/wiley or ask your local Wiley respresentative.
WILEYPLUS is an innovative, research-based, online environment for effective teaching and learning.
What do Students receive with WileyPLUS?
A Research-based Design. Provides an online environment that integrates relevant
resources, including the entire digital textbook, in an easy-to-navigate framework that
helps students study more effectively.
• WileyPLUS adds structure by organizing textbook content into smaller, more
manageable “chunks”.
• Related media, examples, and sample practice items reinforce the learning
objectives.
• Innovative features such as calendars, visual progress tracking, and self-evaluation
tools improve time management and strengthen areas of weakness.
One-on-one Engagement. Students receive 24/7 access to resources that promote
positive learning outcomes. Students engage with related examples (in various media)
and sample practice items, including:
• Animated Learning-By-Doing Exercises
• Excel Templates
• Concept Questions
Measurable Outcomes. Throughout each study session, students can assess their
progress and gain immediate feedback. WileyPLUS provides precise reporting of
strengths and weaknesses, as well as individualized quizzes, so that students are confident they are spending their time on the right things. With WileyPLUS, students always
know the exact outcome of their efforts.
xi
xii
P R E FAC E
What do Instructors receive with WileyPLUS?
WileyPLUS provides reliable, customizable resources that reinforce course goals inside and outside of the classroom as well as visibility into individual student progress.
Pre-created materials and activities help instructors optimize their time:
Customizable Course Plan: WileyPLUS comes with a pre-created Course Plan
designed by a subject matter expert uniquely for this course. Simple drag-anddrop tools make it easy to assign the course plan as-is or modify it to reflect your
course syllabus.
Pre-created Activity Types Include:
•
•
•
•
•
Questions
Readings and resources
Presentations
Print Tests
Concept Mastery
Course Materials and Assessment Content:
• Instructor’s Manual
• Solutions Manual
• Test Bank
• PowerPoint Presentation Slides
• Classroom Response System (Clicker) Questions
• Gradable Reading Assignment Questions (embedded with online text)
• Question Assignments: all end-of-chapter problems coded with hints, links to
text, whiteboard/show work feature and instructor controlled problem-solving
help.
Gradebook: WileyPLUS provides instant access to reports on trends in class performance, student use of course materials and progress towards learning objectives, helping
inform decisions and drive classroom discussions.
ACKNOWLEDGMENTS
While the book was in development, we benefited enormously from the guidance of
a host of individuals both from within John Wiley & Sons and outside. We appreciate
the vision and guidance of the economics team at Wiley. Their commitment to this
book has remained strong from the beginning of the first edition. We are grateful for
their support.
We would like to thank Lacey Vitetta, Acquisitions Editor, for guiding, encouraging, and supporting us throughout this fourth edition. Jennifer Manias, Project
Editor, provided editorial support and kept us on track with deadlines on this and previous editions. Jeanine Furino of Furino Production, handled the production of the
book in a meticulous and constructive fashion. Amy Scholz, our Marketing Manager,
worked tirelessly to reach our markets. Others at Wiley who contributed to the beautiful production and design include Dorothy Sinclair, Janet Foxman, Maddy Lesure,
Sheena Goldstein, and Anna Melhorn.
We are extraordinarily grateful to Michael Gibbs, who made significant contributions to the Applications in the Fourth edition. He updated existing Applications and
added many new Applications. In so doing, he has helped us keep the book fresh and
up to date. Mike’s work was creative, thoughtful, well-organized, and conscientious. It
has been a pleasure to work with him.
The clarity of the presentation and organization in this book owes a great deal
to the efforts of Leonard Neufeld, who provided a close and insightful line and art
edit. Len carefully worked through every line of the manuscript and made numerous
thoughtful suggestions for sharpening and streamlining the exposition. Melissa
Hayes, a Northwestern undergraduate, made extensive suggestions for making the
first edition of the book readable from a student’s point of view. We owe a special
debt to Nick Kreisle. Nick has worked with us as a colleague, as a teaching assistant
in our courses, and as an instructor using our text in his own course in intermediate
microeconomics. He carefully reviewed drafts of the manuscripts of the first and second editions, and provided many valuable suggestions. We are pleased that he is now
Dr. Kreisle.
We would also like especially to thank Eric Schulz, who offered suggestions for
the book while at Williams College and has used the book in his classes at
Northwestern. Ken Brown and Matthew Eichner also tested the manuscript in their
classes prior to publication. Ken also put together a thorough and extremely useful
diary that related his experiences in using the first edition and offered many constructive suggestions for improving the presentation of key topics in the book. We have
also benefited from many helpful suggestions from Yossi Spiegel, Mort Kamien, Nabil
Al-Najjar, Ambarish Chandra, Justin Braeutigam, and Kate Rockett.
Finally, we owe a large debt of gratitude to the students in Ron Braeutigam’s sections of Economics 310-1 at Northwestern and to the students in Microeconomics
430 at the Kellogg School at Northwestern. These students have helped us eliminate
some of the rough edges as the book has evolved over time. Their experience of learning from the book helped make our chapters clearer and more accessible.
The development of this book was aided by colleagues who participated in focus
groups or reviewed early drafts of the manuscript. Our thanks go to all of the individuals listed below.
xiii
xiv
AC K N OW L E D G M E N T S
We are grateful for the comments we received from those who reviewed for the
Fourth Edition of this book:
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AC K N OW L E D G M E N T S
Coughlin, University of Maryland; Steven Craig, University of Houston; Mike
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Donald Keenan, University of Georgia; Mark Killingsworth, Rutgers University;
Philip King, San Francisco State University; Charles Lamberton, South Dakota State
University; Donald Lien, University of Kansas; Leonard Loyd, University of
Houston; Mark Machina, University of California at San Diego; Mukul K.
Manjumdar, Cornell University; Gilbert Mathis, Murray State University; Michael
Mckee, University of New Mexico; Claudio Mezzetti, University of North Carolina;
Peter Morgan, University of Michigan; John Moroney, Texas A & M; Wilhelm
Neuefeind, Washington University; Peter Norman, University of Wisconsin,
Madison; Mudziviri Nziramasanga, Washington State University; Iyatokunbo
Okediji, University of Oklahoma; Ken Parzych, Eastern Connecticut State
University; Donald Pursell, University of Nebraska-Lincoln; Michael Raith,
University of Chicago; Sunder Ramaswamy, Middlebury College; Jeanne Ringel,
Louisiana State University; Robert Rosenman, Washington State University; Santanu
Roy, Florida International University; Jolyne Sanjak, SUNY-Albany; David Schmidt,
Indiana University; Mark Schupack, Brown University; Richard Sexton, University of
California, Davis; Jason Shachat, University of California, San Diego; Maxwell
Stinchcombe, University of Texas, Austin; Beck Taylor, Baylor University; Curtis
Taylor, Texas A & M University; Thomas TenHoeve, Iowa State University; John
Thompson, Louisiana State University; Paul Thistle, Western Michigan University;
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Pennsylvania; Mark Walbert, Illinois University; Mark Walker, University of Arizona;
Larry Westphal, Swarthmore College; Kealoha Widdows, Wabash College;
Chiounan Yeh, Alabama State University.
xv
BRIEF CONTENTS
PART 1 INTRODUCTION TO MICROECONOMICS
CHAPTER 1 Analyzing Economic Problems 1
CHAPTER 2 Demand and Supply Analysis 26
APPENDIX: Price Elasticity of Demand Along a Constant Elasticity Demand Curve
72
PART 2 CONSUMER THEORY
CHAPTER 3 Consumer Preferences and the Concept of Utility
CHAPTER 4 Consumer Choice 103
APPENDIX 1: The Mathematics of Consumer Choice
APPENDIX 2: The Time Value of Money 144
CHAPTER 5 The Theory of Demand
73
143
150
PART 3 PRODUCTION AND COST THEORY
CHAPTER 6 Inputs and Production Functions
200
APPENDIX: The Elasticity of Substitution for a Cobb–Douglas Production Function
CHAPTER 7 Costs and Cost Minimization
245
APPENDIX: Advanced Topics in Cost Minimization
CHAPTER 8 Cost Curves
281
285
APPENDIX: Shephard’s Lemma and Duality
323
PART 4 PERFECT COMPETITION
CHAPTER 9 Perfectly Competitive Markets
327
APPENDIX: Profit Maximization Implies Cost Minimization
CHAPTER 10 Competitive Markets: Applications
384
386
PART 5 MARKET POWER
CHAPTER 11 Monopoly and Monopsony
CHAPTER 12 Capturing Surplus 485
438
PART 6 IMPERFECT COMPETITION AND STRATEGIC BEHAVIOR
CHAPTER 13 Market Structure and Competition
528
APPENDIX: The Cournot Equilibrium and the Inverse Elasticity Pricing Rule
CHAPTER 14 Game Theory and Strategic Behavior
570
571
PART 7 SPECIAL TOPICS
CHAPTER 15 Risk and Information 604
CHAPTER 16 General Equilibrium Theory
648
APPENDIX: Deriving the Demand and Supply Curves for General Equilibrium
CHAPTER 17 Externalities and Public Goods
Mathematical Appendix 729
Solutions to Selected Problems
Glossary 771
Photo Credits 781
Index 783
xvi
749
697
692
243
CONTENTS
PART 1 INTRODUCTION TO MICROECONOMICS
CHAPTER 1 Analyzing Economic Problems
1
Microeconomics and Climate Change
Market-Level versus Brand-Level Price Elasticities
of Demand 49
1.1 Why Study Microeconomics?
2.3 Other Elasticities
1.2 Three Key Analytical Tools
Constrained Optimization
Equilibrium Analysis 12
Comparative Statics 13
4
6
1.3 Positive and Normative Analysis
2.4 Elasticity in the Long Run versus
the Short Run 54
18
LEARNING-BY-DOING EXERCISES
1.1 Constrained Optimization: The Farmer’s Fence 7
1.2 Constrained Optimization: Consumer Choice 8
1.3 Comparative Statics with Market Equilibrium in the
U.S. Market for Corn 16
1.4 Comparative Statics with Constrained
Optimization 18
CHAPTER 2 Demand and Supply Analysis
26
What Gives with the Price of Corn?
2.1 Demand, Supply, and Market Equilibrium
Demand Curves 30
Supply Curves 32
Market Equilibrium 33
Shifts in Supply and Demand
51
Income Elasticity of Demand 51
Cross-Price Elasticity of Demand 52
Price Elasticity of Supply 54
5
29
Greater Elasticity in the Long Run Than in
the Short Run 54
Greater Elasticity in the Short Run Than in the
Long Run 56
2.5 Back-of-the-Envelope Calculations
57
Fitting Linear Demand Curves Using Quantity, Price,
and Elasticity Information 58
Identifying Supply and Demand Curves on the Back of an
Envelope 59
Identifying the Price Elasticity of Demand from
Shifts in Supply 61
APPENDIX Price Elasticity of Demand along a
Constant Elasticity Demand Curve 72
LEARNING-BY-DOING EXERCISES
2.1
2.2
2.3
2.4
35
2.2 Price Elasticity of Demand
43
Elasticities along Specific Demand Curves 45
Price Elasticity of Demand and Total Revenue 47
Determinants of the Price Elasticity of Demand 48
Sketching a Demand Curve 31
Sketching a Supply Curve 33
Calculating Equilibrium Price and Quantity 34
Comparative Statics on the Market
Equilibrium 37
2.5 Price Elasticity of Demand 45
2.6 Elasticities along Special Demand Curves 47
PART 2 CONSUMER THEORY
CHAPTER 3 Consumer Preferences and
the Concept of Utility 73
Why Do You Like What You Like?
3.1 Representations of Preferences
75
Assumptions about Consumer Preferences
Ordinal and Cardinal Ranking 76
3.2 Utility Functions
75
77
Preferences with a Single Good: The Concept of
Marginal Utility 77
Preferences with Multiple Goods: Marginal Utility, Indifference
Curves, and the Marginal Rate of Substitution 81
3.3 Special Preferences
92
Perfect Substitutes 92
Perfect Complements 93
The Cobb–Douglas Utility Function
Quasilinear Utility Functions 95
94
LEARNING-BY-DOING EXERCISES
3.1 Marginal Utility 82
3.2 Marginal Utility That Is Not Diminishing 83
3.3 Indifference Curves with Diminishing
MRSx,y 90
3.4 Indifference Curves with Increasing MRSx,y 91
xvii
xviii
CONTENTS
CHAPTER 4 Consumer Choice
The Effects of a Change in Price or Income: An Algebraic
Approach 160
103
How Much of What You Like Should You Buy?
4.1 The Budget Constraint
5.2 Change in the Price of a Good: Substitution
Effect and Income Effect 162
105
How Does a Change in Income Affect the
Budget Line? 107
How Does a Change in Price Affect the
Budget Line? 107
4.2 Optimal Choice
The Substitution Effect 163
The Income Effect 163
Income and Substitution Effects When Goods Are Not
Normal 165
110
5.3 Change in the Price of a Good: The Concept
of Consumer Surplus 173
Using the Tangency Condition to Understand When
a Basket Is Not Optimal 114
Finding an Optimal Consumption Basket 115
Two Ways of Thinking About Optimality 116
Corner Points 118
4.3 Consumer Choice with Composite Goods
Application:
Application:
Application:
Application:
121
Coupons and Cash Subsidies 121
Joining a Club 125
Borrowing and Lending 126
Quantity Discounts 131
4.4 Revealed Preference
5.4 Market Demand
5.5 The Choice of Labor and Leisure
APPENDIX 1 The Mathematics of Consumer
Choice 143
5.6 Consumer Price Indices
187
192
LEARNING-BY-DOING EXERCISES
144
LEARNING-BY-DOING EXERCISES
4.1
4.2
4.3
4.4
Good News/Bad News and the Budget Line 110
Finding an Interior Optimum 115
Finding a Corner Point Solution 119
Corner Point Solution with Perfect
Substitutes 120
4.5 Consumer Choice That Fails to Maximize Utility 134
4.6 Other Uses of Revealed Preference 136
150
Why Understanding the Demand for Cigarettes Is
Important for Public Policy
5.1 Optimal Choice and Demand
184
As Wages Rise, Leisure First Decreases, Then
Increases 187
The Backward-Bending Supply of Labor 189
Are Observed Choices Consistent with Utility
Maximization? 133
CHAPTER 5 The Theory of Demand
182
Network Externalities 184
Market Demand with Network Externalities
132
APPENDIX 2 The Time Value of Money
Understanding Consumer Surplus from the
Demand Curve 173
Understanding Consumer Surplus from the Optimal
Choice Diagram: Compensating Variation
and Equivalent Variation 175
152
The Effects of a Change in Price 152
The Effects of a Change in Income 155
5.1 A Normal Good Has a Positive Income Elasticity
of Demand 159
5.2 Finding a Demand Curve (No Corner Points) 160
5.3 Finding a Demand Curve (with a Corner
Point Solution) 161
5.4 Finding Income and Substitution Effects
Algebraically 168
5.5 Income and Substitution Effects with a
Price Increase 170
5.6 Income and Substitution Effects with a Quasilinear
Utility Function 171
5.7 Consumer Surplus: Looking at the Demand Curve 174
5.8 Compensating and Equivalent Variations with No
Income Effect 178
5.9 Compensating and Equivalent Variations with
an Income Effect 180
5.10 The Demand for Leisure and the Supply of Labor 191
PART 3 PRODUCTION AND COST THEORY
CHAPTER 6 Inputs and Production
Functions 200
Total Product Functions 205
Marginal and Average Product 206
Relationship Between Marginal and Average Product
Can They Do It Better and Cheaper?
6.3 Production Functions with More Than
One Input 210
6.1 Introduction to Inputs and Production
Functions 202
6.2 Production Functions with a Single Input
210
204
Total Product and Marginal Product with Two Inputs
Isoquants 212
210
xix
CONTENTS
Economic and Uneconomic Regions
of Production 216
Marginal Rate of Technical Substitution
6.4 Substitutability among Inputs
Comparative Statics: Short-Run Input Demand
versus Long-Run Input Demand 273
More Than One Variable Input in the
Short Run 274
217
219
APPENDIX Advanced Topics in Cost
Minimization 281
Describing a Firm’s Input Substitution Opportunities
Graphically 220
Elasticity of Substitution 222
Special Production Functions 225
6.5 Returns to Scale
LEARNING-BY-DOING EXERCISES
230
Definitions 230
Returns to Scale versus Diminishing Marginal
Returns 233
6.6 Technological Progress
233
APPENDIX The Elasticity of Substitution for a
Cobb–Douglas Production Function 243
LEARNING-BY-DOING EXERCISES
6.1 Deriving the Equation of an Isoquant 216
6.2 Relating the Marginal Rate of Technical Substitution
to Marginal Products 219
6.3 Calculating the Elasticity of Substitution from
a Production Function 223
6.4 Returns to Scale for a Cobb–Douglas Production
Function 232
6.5 Technological Progress 235
CHAPTER 7 Costs and Cost Minimization
What’s Behind the Self-Service Revolution?
7.1 Cost Concepts for Decision Making
247
Opportunity Cost 247
Economic versus Accounting Costs 250
Sunk (Unavoidable) versus Nonsunk (Avoidable)
Costs 251
7.2 The Cost-Minimization Problem
253
Long Run versus Short Run 253
The Long-Run Cost-Minimization Problem 254
Isocost Lines 255
Graphical Characterization of the Solution to the
Long-Run Cost-Minimization Problem 256
Corner Point Solutions 258
7.3 Comparative Statics Analysis of the
Cost-Minimization Problem 260
Comparative Statics Analysis of Changes in
Input Prices 260
Comparative Statics Analysis of Changes
in Output 264
Summarizing the Comparative Statics Analysis:
The Input Demand Curves 265
The Price Elasticity of Demand
for Inputs 267
7.4 Short-Run Cost Minimization
269
Characterizing Costs in the Short Run 270
Cost Minimization in the Short Run 272
245
7.1 Using the Cost Concepts for a College Campus
Business 252
7.2 Finding an Interior Cost-Minimization
Optimum 258
7.3 Finding a Corner Point Solution with Perfect
Substitutes 259
7.4 Deriving the Input Demand Curves from a Production
Function 267
7.5 Short-Run Cost Minimization with One
Fixed Input 274
7.6 Short-Run Cost Minimization with Two Variable
Inputs 275
CHAPTER 8 Cost Curves
285
How Can HiSense Get a Handle on Costs?
8.1 Long-Run Cost Curves
287
Long-Run Total Cost Curve 287
How Does the Long-Run Total Cost Curve Shift When
Input Prices Change? 289
Long-Run Average and Marginal Cost Curves 292
8.2 Short-Run Cost Curves
302
Short-Run Total Cost Curve 302
Relationship Between the Long-Run and the Short-Run
Total Cost Curves 303
Short-Run Average and Marginal Cost Curves 305
Relationships Between the Long-Run and the Short-Run
Average and Marginal Cost Curves 306
When Are Long-Run and Short-Run Average and Marginal
Costs Equal, and When Are They Not? 307
8.3 Special Topics in Cost
310
Economies of Scope 310
Economies of Experience: The Experience
Curve 313
8.4 Estimating Cost Functions
Constant Elasticity Cost Function
Translog Cost Function 316
315
316
APPENDIX Shephard’s Lemma and Duality
323
LEARNING-BY-DOING EXERCISES
8.1 Finding the Long-Run Total Cost Curve from a
Production Function 288
8.2 Deriving Long-Run Average and Marginal Cost
Curves from a Long-Run Total Cost Curve 294
8.3 Deriving a Short-Run Total Cost Curve 303
8.4 The Relationship between Short-Run and Long-Run
Average Cost Curves 308
xx
CONTENTS
PART 4 PERFECT COMPETITION
CHAPTER 9 Perfectly Competitive Markets
327
LEARNING-BY-DOING EXERCISES
332
9.1 Deriving the Short-Run Supply Curve for a
Price-Taking Firm 341
9.2 Deriving the Short-Run Supply Curve for a Price-Taking
Firm with Some Nonsunk Fixed Costs 343
9.3 Short-Run Market Equilibrium 349
9.4 Calculating a Long-Run Equilibrium 356
9.5 Calculating Producer Surplus 375
A Rose Is a Rose Is a Rose
9.1 What Is Perfect Competition?
330
9.2 Profit Maximization by a Price-Taking Firm
Economic Profit versus Accounting Profit 333
The Profit-Maximizing Output Choice for a
Price-Taking Firm 334
9.3 How the Market Price Is Determined:
Short-Run Equilibrium 337
CHAPTER 10 Competitive Markets:
Applications 386
The Price-Taking Firm’s Short-Run Cost Structure 337
Short-Run Supply Curve for a Price-Taking Firm
When All Fixed Costs Are Sunk 339
Short-Run Supply Curve for a Price-Taking Firm When
Some Fixed Costs Are Sunk and Some
Are Nonsunk 341
Short-Run Market Supply Curve 344
Short-Run Perfectly Competitive Equilibrium 348
Comparative Statics Analysis of the Short-Run
Equilibrium 349
Is Support a Good Thing?
10.1 The Invisible Hand, Excise Taxes and
Subsidies 388
The Invisible Hand 389
Excise Taxes 390
Incidence of a Tax 394
Subsidies 397
10.2 Price Ceilings and Floors
9.4 How the Market Price Is Determined:
Long-Run Equilibrium 352
10.3 Production Quotas
Long-Run Output and Plant-Size Adjustments by
Established Firms 352
The Firm’s Long-Run Supply Curve 353
Free Entry and Long-Run Perfectly Competitive
Equilibrium 354
Long-Run Market Supply Curve 356
Constant-Cost, Increasing-Cost, and Decreasing-Cost
Industries 358
What Does Perfect Competition Teach Us? 363
9.5 Economic Rent and Producer Surplus
413
10.4 Price Supports in the Agricultural Sector
417
Acreage Limitation Programs 418
Government Purchase Programs 418
10.5 Import Quotas and Tariffs 422
Quotas 422
Tariffs 425
LEARNING-BY-DOING EXERCISES
367
Economic Rent 367
Producer Surplus 370
Economic Profit, Producer Surplus, Economic Rent
400
Price Ceilings 400
Price Floors 408
376
APPENDIX Profit Maximization Implies Cost
Minimization 384
10.1
10.2
10.3
10.4
10.5
Impact of an Excise Tax 393
Impact of a Subsidy 400
Impact of a Price Ceiling 407
Impact of a Price Floor 412
Comparing the Impact of an Excise Tax, a Price Floor,
and a Production Quota 417
10.6 Effects of an Import Tariff 428
PART 5 MARKET POWER
CHAPTER 11 Monopoly and Monopsony
438
How Do Firms Play Monopoly?
11.1 Profit Maximization by a Monopolist
440
The Profit-Maximization Condition 440
A Closer Look at Marginal Revenue: Marginal Units and
Inframarginal Units 444
Average Revenue and Marginal Revenue 445
The Profit-Maximization Condition Shown
Graphically 447
A Monopolist Does Not Have a Supply Curve 449
11.2 The Importance of Price Elasticity
of Demand 450
Price Elasticity of Demand and the Profit-Maximizing
Price 450
Marginal Revenue and Price Elasticity of Demand 451
Marginal Cost and Price Elasticity of Demand: The Inverse
Elasticity Pricing Rule 453
The Monopolist Always Produces on the Elastic Region
of the Market Demand Curve 454
The IEPR Applies Not Only to Monopolists 456
Quantifying Market Power: The Lerner Index 457
xxi
CONTENTS
11.3 Comparative Statics for Monopolists
458
Shifts in Market Demand 458
Shifts in Marginal Cost 461
11.9 Applying the Inverse Elasticity Rule for a
Monopsonist 478
CHAPTER 12 Capturing Surplus
11.4 Monopoly with Multiple Plants
and Markets 463
485
Why Did Your Ticket Cost So Much Less Than Mine?
Output Choice with Two Plants 463
Output Choice with Two Markets 465
Profit Maximization by a Cartel 466
12.1 Capturing Surplus
11.5 The Welfare Economics of Monopoly
469
487
12.2 First-Degree Price Discrimination: Making the
Most from Each Consumer 490
The Monopoly Equilibrium Differs from the Perfectly
Competitive Equilibrium 469
Monopoly Deadweight Loss 471
Rent-Seeking Activities 471
12.3 Second-Degree Price Discrimination:
Quantity Discounts 495
11.6 Why Do Monopoly Markets Exist?
12.4 Third-Degree Price Discrimination: Different
Prices for Different Market Segments 501
Block Pricing 495
Subscription and Usage Charges
471
Natural Monopoly 472
Barriers to Entry 473
11.7 Monopsony
475
The Monopsonist’s Profit-Maximization Condition 475
An Inverse Elasticity Pricing Rule for Monopsony 477
Monopsony Deadweight Loss 478
LEARNING-BY-DOING EXERCISES
11.1 Marginal and Average Revenue for a Linear Demand
Curve 447
11.2 Applying the Monopolist’s Profit-Maximization
Condition 449
11.3 Computing the Optimal Monopoly Price for a
Constant Elasticity Demand Curve 453
11.4 Computing the Optimal Monopoly Price for a Linear
Demand Curve 454
11.5 Computing the Optimal Price Using the Monopoly
Midpoint Rule 460
11.6 Determining the Optimal Output, Price, and Division
of Production for a Multiplant Monopolist 465
11.7 Determining the Optimal Output and Price for a
Monopolist Serving Two Markets 466
11.8 Applying the Monopsonist’s Profit-Maximization
Condition 476
498
Two Different Segments, Two Different Prices 501
Screening 504
Third-Degree Price Discrimination with Capacity
Constraints 506
Implementing the Scheme of Price Discrimination:
Building “Fences” 508
12.5 Tying (Tie-In Sales)
Bundling 513
Mixed Bundling
512
515
12.6 Advertising
518
LEARNING-BY-DOING EXERCISES
12.1 Capturing Surplus: Uniform Pricing versus
First-Degree Price Discrimination 492
12.2 Where Is the Marginal Revenue Curve with
First-Degree Price Discrimination? 493
12.3 Increasing Profits with a Block Tariff 497
12.4 Third-Degree Price Discrimination in Railroad
Transport 503
12.5 Third-Degree Price Discrimination for Airline Tickets 505
12.6 Price Discrimination Subject to Capacity Constraints 507
12.7 Markup and Advertising-to-Sales Ratio 520
PART 6 IMPERFECT COMPETITION AND STRATEGIC BEHAVIOR
CHAPTER 13 Market Structure and
Competition 528
13.3 Dominant Firm Markets
Is Competition Always the Same? If Not, Why Not?
13.1 Describing and Measuring Market
Structure 530
13.2 Oligopoly with Homogeneous Products
The Cournot Model of Oligopoly 533
The Bertrand Model of Oligopoly 541
Why Are the Cournot and Bertrand Equilibria
Different? 543
The Stackelberg Model of Oligopoly 544
533
546
13.4 Oligopoly with Horizontally Differentiated
Products 549
What Is Product Differentiation? 549
Bertrand Price Competition with Horizontally
Differentiated Products 553
13.5 Monopolistic Competition
558
Short-Run and Long-Run Equilibrium in Monopolistically
Competitive Markets 558
Price Elasticity of Demand, Margins, and Number of Firms
in the Market 560
Do Prices Fall When More Firms Enter? 560
xxii
CONTENTS
APPENDIX The Cournot Equilibrium and the Inverse
Elasticity Pricing Rule 570
LEARNING-BY-DOING EXERCISES
13.1 Computing a Cournot Equilibrium 536
13.2 Computing the Cournot Equilibrium for Two or
More Firms with Linear Demand 540
13.3 Computing the Equilibrium in the Dominant Firm
Model 548
13.4 Computing a Bertrand Equilibrium with Horizontally
Differentiated Products 556
CHAPTER 14 Game Theory and Strategic
Behavior 571
What’s in a Game?
14.1 The Concept of Nash Equilibrium
A Simple Game
573
573
The Nash Equilibrium 574
The Prisoners’ Dilemma 574
Dominant and Dominated Strategies 575
Games with More Than One Nash Equilibrium 579
Mixed Strategies 583
Summary: How to Find All the Nash Equilibria in a
Simultaneous-Move Game with Two Players 584
14.2 The Repeated Prisoners’ Dilemma
584
14.3 Sequential-Move Games and Strategic
Moves 590
Analyzing Sequential-Move Games 590
The Strategic Value of Limiting One’s Options
593
LEARNING-BY-DOING EXERCISES
14.1 Finding the Nash Equilibrium: Coke versus Pepsi 578
14.2 Finding All of the Nash Equilibria in a Game 582
14.3 An Entry Game 592
PART 7 SPECIAL TOPICS
CHAPTER 15 Risk and Information
604
What Are My Chances of Winning?
15.1 Describing Risky Outcomes
Lotteries and Probabilities
Expected Value 608
Variance 608
606
606
15.4 The Willingness to Pay for Insurance 621
15.5 Verifying the Nash Equilibrium in a First-Price
Sealed-Bid Auction with Private Values 636
CHAPTER 16 General Equilibrium Theory
How Do Gasoline Taxes Affect the Economy?
15.2 Evaluating Risky Outcomes
16.1 General Equilibrium Analysis: Two Markets
611
Utility Functions and Risk Preferences 611
Risk-Neutral and Risk-Loving Preferences 614
16.2 General Equilibrium Analysis: Many
Markets 654
15.3 Bearing and Eliminating Risk
The Origins of Supply and Demand in a Simple
Economy 654
The General Equilibrium in Our Simple Economy
Walras’ Law 664
617
Risk Premium 617
When Would a Risk-Averse Person Choose to Eliminate
Risk? The Demand for Insurance 620
Asymmetric Information in Insurance Markets: Moral
Hazard and Adverse Selection 622
15.4 Analyzing Risky Decisions
627
Decision Tree Basics 627
Decision Trees with a Sequence of Decisions
The Value of Information 631
15.5 Auctions
648
633
Types of Auctions and Bidding Environments 634
Auctions When Bidders Have Private Values 635
Auctions When Bidders Have Common Values:
The Winner’s Curse 639
LEARNING-BY-DOING EXERCISES
15.1 Computing the Expected Utility for Two Lotteries
for a Risk-Averse Decision Maker 614
15.2 Computing the Expected Utility for Two Lotteries:
Risk-Neutral and Risk-Loving Decision Makers 616
15.3 Computing the Risk Premium from a Utility
Function 620
660
16.3 General Equilibrium Analysis: Comparative
Statics 665
16.4 The Efficiency of Competitive Markets
629
650
669
What Is Economic Efficiency? 669
Exchange Efficiency 670
Input Efficiency 676
Substitution Efficiency 678
Pulling the Analysis Together: The Fundamental Theorems
of Welfare Economics 681
16.5 Gains from Free Trade
Free Trade Is Mutually Beneficial
Comparative Advantage 686
682
682
APPENDIX Deriving the Demand and Supply Curves
for General Equilibrium in Figure 16.9 and Learningby-Doing Exercise 16.2 692
LEARNING-BY-DOING EXERCISES
16.1 Finding the Prices at a General Equilibrium with
Two Markets 654
xxiii
CONTENTS
16.2 Finding the Conditions for a General Equilibrium
with Four Markets 663
16.3 Checking the Conditions for Exchange
Efficiency 674
CHAPTER 17 Externalities and Public Goods
Efficient Provision of a Public Good
The Free Rider Problem 722
720
LEARNING-BY-DOING EXERCISES
697
17.1
17.2
17.3
17.4
The Efficient Amount of Pollution 704
Emissions Fee 707
The Coase Theorem 717
Optimal Provision of a Public Good 721
When Does the Invisible Hand Fail?
17.1 Introduction
699
17.2 Externalities
700
Negative Externalities and Economic Efficiency 702
Positive Externalities and Economic Efficiency 711
Property Rights and the Coase Theorem 716
17.3 Public Goods
719
Mathematical Appendix 729
Solutions to Selected Problems
Glossary 771
Photo Credits 781
Index 783
749
1
ANALYZING ECONOMIC
PROBLEMS
1 . 1 W H Y S T U DY M I C R O E C O N O M I C S ?
1.1
W H Y S T U DY M I C R O E C O N O M I C S ?
1.2
T H R E E K E Y A N A LY T I C A L TO O L S
Generating Electricity: 8,760
Decisions per Year
APPLICATION 1.2 The Toughest Ticket in Sports
APPLICATION 1.1
1.3
P O S I T I V E A N D N O R M AT I V E A N A LYS I S
Positive and Normative
Analyses of the Minimum Wage
APPLICATION 1.3
Microeconomics and Climate Change
By the late 2000s, the scientific consensus had formed: climate change is for real, and it cannot be
explained entirely by natural forces:
• There is compelling scientific evidence that concentrations of greenhouse gasses—compounds such as
carbon dioxide and methane whose properties work to warm surface temperatures on the Earth—
have accumulated to levels substantially higher than those that prevailed at any time during the last
500,000 years.
• There is strong evidence that the climate is warming. According to the Fourth Assessment of the
Intergovernmental Panel on Climate Change (IPCC) issued in 2007—the best representation of the
scientific consensus on climate change—“Warming of the climate system is unequivocal, as is now
evident from observations of increases in global average air and ocean temperatures, widespread
melting of snow and ice, and rising global average sea level.”1
1
“Summary for Policymakers” in Climate Change 2007: The Physical Science Basis. Contributions of Working
Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, S. Soloman,
D. Qin, M. Manning, Z. Chen, M. Marquis, K. B. Avery, M. Tignor, and H. L. Mikllers (eds.) (Cambridge:
Cambridge University Press 2007), p. 5. http://www.ipcc.ch/ipccreports/ar4-wg1.htm (accessed April 3,
2009).
1
• There is persuasive evidence that climate change
has been induced, in part, by humans. According
to the IPCC: “The common conclusion of a wide
range of fingerprint studies conducted over the
last 15 years is that observed climate changes cannot
be explained by natural factors alone.”2
But if the diagnosis of climate change is unequivocal, what to do about it is less obvious.
Greenhouse gas emissions come from power plants,
factories, and automobiles all over the world. The
number of pollution sources that potentially need to
be controlled is mind-boggling. And large countries such as China and the United States, the two countries accounting for the largest share of greenhouse gas emissions, might balk at the enormous price
tag associated with curtailing their emissions. In light of these issues, the challenge of combating
global climate change would appear to be insurmountable.
Microeconomics offers powerful insights into why climate change is such a difficult problem and
what to do about it. Climate change is a tough problem to deal with because the parties that cause
greenhouse gas emissions are unlikely to take into account the environmental harm that their decisions cause for others. For example, economists estimate that in the mid-2000s, the typical American
household caused about $150 annually in environmental damage by consuming products or services
that caused greenhouse gas emissions.3 Did you or your family take this into account when you
made decisions about how much electricity to use or how much to drive? Probably not. After all, you
did not have to pay this cost, either directly (because no one directly charged you for this cost) or
indirectly (because it was not reflected in the price of the products you consumed because the
producers of those products were not charged for this cost). New York Times columnist Tom
Friedman puts it this way:
[I]f I had my wish, the leaders of the world’s 20 top economies would commit themselves to a new
standard of accounting—call it “Market to Mother Nature” accounting. Why? Becouse it’s now obvious that the reason we’re experiencing a simultaneous meltdown in the financial system and the
climate system is because we have been mispricing risk in both arenas—producing a huge excess of
both toxic assets and toxic air that now threatens the stability of the whole planet.
Just as A.I.G. sold insurance derivatives at prices that did not reflect the real costs and the real risks
of massive defaults (for which we the taxpayers ended up paying the difference), oil companies, coal
companies and electric utilities today are selling energy products at prices that do not reflect the
real costs to the environment and real risks of disruptive climate change (so future taxpayers will
end up paying the difference).4
2
H. R. Le Treut, R. Somerville, U. Cubasch, Y. Ding, C. Mauritzen, A. Mokssit, T. Peterson, and M. Prather,
“Historical Overview of Climate Change,” in Climate Change 2007: The Physical Science Basis, p. 103.
3
The estimate of the social cost of electricity usage comes from W. Nordhaus, A Question of Balance:
Weighing the Options on Global Warming Policies (New Haven, CT: Yale University Press, 2008), p. 11.
4“
The Price Is Not Right,” New York Times (March 31, 2009).
2
But Friedman’s diagnosis of the problem is also suggestive of a solution: to induce parties to make
decisions that reflect the real costs of climate change, find a way to put a price on the harm that
greenhouse gas emissions cause to the climate and the economy. Basic ideas from microeconomics are
being applied today to help do this. Consider, for example, the European Union (EU) Emissions Trading
System. Under the provisions of the Kyoto Treaty, the countries of the EU must reduce their emissions
of greenhouse gases 8 percent below their emissions in 1990. To do so, the EU has adopted what is
called a cap-and-trade system.5
A cap-and-trade system applies microeconomics to achieve a given amount of pollution reduction at
a cost as low as possible. Here’s how it works. Caps are placed on how much of a greenhouse gas, say
carbon dioxide (CO2), can be emitted from specific sources (e.g., power plants or factories). At the same
time, CO2 permits are granted to the firms that own those sources of CO2 pollution, allowing them to
emit a given amount of CO2 within a given period of time. Firms are then free to trade these permits in
an open market. The idea behind this scheme is that a firm that can cheaply reduce its CO2 emissions
below its cap (e.g., by installing pollution control equipment) can sell its allowances to other firms for
whom pollution control would be more expensive. The beauty of this system—which follows directly
from the fact that it is market-based—is that reductions in emissions of a given amount are achieved as
cheaply as possible. Moreover, a government (or group of governments as in the case of the EU) does
not need to know which firms can reduce pollution more cheaply. The free market identifies those firms
through the purchase and sale of permits: firms with low costs of compliance sell permits; firms with
high costs of compliance buy them. And by reducing the supply of allowances over time, the government can reduce pollution, all the while being assured that the reduction is done at as low a cost as is
possible.
Microeconomics is a field of study that has broad applicability. It can help public policy makers deal
with difficult issues such as climate change, and it can help those same public officials anticipate the
unintended consequences of the policies they adopt. For example, microeconomic analyses of cap-andtrade systems reveal that while a cap-and-trade system offers the potential to correctly price greenhouse
gas emissions, there are circumstances under which this system can result in significant underpricing
or overpricing of those emissions if policy makers make even small mistakes in setting the cap.6
Microeconomics can also help business firms better understand their competitive environments, and it can
give them concrete tools that can be used to unlock additional profitability through pricing strategies. It
can help us understand how households’ consumption decisions are shaped by the fundamentals (e.g.,
tastes and price levels) they face, and it can shed light on why prices in competitive markets fluctuate as
they do. Microeconomics can even help us understand social phenomena such as crime and marriage
(yes, economists have even studied these). What’s remarkable is that nearly all phenomena studied by
5
The Kyoto Treaty was adopted in the late 1990s, and it called for industrialized countries to scale back
the amount of greenhouse gases. The treaty was ratified by EU counties, but not by the United States.
6
See, for example, W. J. McKibbin and P. J. Wilcoxen, “The Role of Economics in Climate Change
Policy,” Journal of Economic Perspectives, 16, no. 2 (Spring 2002): 107–129.
3
4
CHAPTER 1
A N A LY Z I N G E C O N O M I C P R O B L E M S
economists rely on three powerful analytical tools: constrained optimization, equilibrium analysis, and
comparative statics.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Contrast the two main branches of economics—microeconomics and macroeconomics.
• Describe the three main analytical tools of microeconomics—constrained optimization, equilibrium
analysis, and comparative statics—and recognize examples of each of these tools.
• Explain the difference between positive and normative analysis.
1.1
W H Y S T U DY
MICROECONOM I C S ?
E
conomics is the science that deals with the allocation of limited resources to satisfy unlimited human wants. Think of human wants as being all the goods and
services that individuals desire, including food, clothing, shelter, and anything else
that enhances the quality of life. Since we can always think of ways to improve our
well-being with more or better goods and services, our wants are unlimited.
However, to produce goods and services, we need resources, including labor, managerial talent, capital, and raw materials. Resources are said to be scarce because their
supply is limited. The scarcity of resources means that we are constrained in the
choices we can make about the goods and services we produce, and thus also about
which human wants we will ultimately satisfy. That is why economics is often
described as the science of constrained choice.
Broadly speaking, economics is composed of two branches, microeconomics and
macroeconomics. The prefix micro is derived from the Greek word mikros, which
means “small.” Microeconomics therefore studies the economic behavior of individual economic decision makers, such as a consumer, a worker, a firm, or a manager. It
also analyzes the behavior of individual households, industries, markets, labor unions,
or trade associations. By contrast, the prefix macro comes from the Greek word
makros, which means “large.” Macroeconomics thus analyzes how an entire national
economy performs. A course in macroeconomics would examine aggregate levels of
income and employment, the levels of interest rates and prices, the rate of inflation,
and the nature of business cycles in a national economy.
Constrained choice is important in both macroeconomics and microeconomics.
For example, in macroeconomics we would see that a society with full employment
could produce more goods for national defense, but it would then have to produce
fewer civilian goods. It might use more of its depletable natural resources, such as
natural gas, coal, and oil, to manufacture goods today, in which case it would conserve less of these resources for the future. In a microeconomic setting, a consumer
might decide to allocate more time to work, but would then have less time available
for leisure activities. The consumer could spend more income on consumption
today, but would then save less for tomorrow. A manager might decide to spend more
of a firm’s resources on advertising, but this might leave less available for research
and development.
Every society has its own way of deciding how to allocate its scarce resources.
Some resort to a highly centralized organization. For example, during the Cold War,
governmental bureaucracies heavily controlled the allocation of resources in the
5
1 . 2 T H R E E K E Y A N A LY T I C A L TO O L S
economies of Eastern Europe and the Soviet Union. Other countries, such as those in
North America or Western Europe, have historically relied on a mostly decentralized
market system to allocate resources. Regardless of its market system, every society
must answer these questions:
• What goods and services will be produced, and in what quantities?
• Who will produce the goods and services, and how?
• Who will receive the goods and services?
Microeconomic analysis attempts to answer these questions by studying the behavior of individual economic units. By answering questions about how consumers
and producers behave, microeconomics helps us understand the pieces that collectively make up a model of an entire economy. Microeconomic analysis also provides
the foundation for examining the role of the government in the economy and the effects of government actions. Microeconomic tools are commonly used to address
some of the most important issues in contemporary society. These include (but are
not limited to) pollution, rent controls, minimum wage laws, import tariffs and quotas, taxes and subsidies, food stamps, government housing and educational assistance
programs, government health care programs, workplace safety, and the regulation of
private firms.
To study real phenomena in a world that is exceedingly complex, economists con- 1.2
struct and analyze economic models, or formal descriptions, of the problems they
are addressing. An economic model is like a roadmap. A roadmap takes a complex
physical reality (terrain, roads, houses, stores, parking lots, alleyways, and other features) and strips it down to bare essentials: major streets and highways. The
roadmap is an abstract model that serves a particular purpose—it shows us where we
are and how we can get where we want to go. To provide a clear representation of
reality, it “ignores” or “abstracts from” much of the rich detail (the location of beautiful elm trees or stately homes, for example) that makes an individual town unique
and charming.
Economic models operate in much the same way. For example, to understand
how a drought in Colombia might affect the price of coffee in the United States, an
economist might employ a model that ignores much of the rich detail of the industry, including some aspects of its history or the personalities of the people who work
in the fields. These details might make an interesting article in Business Week, but
they do not help us understand the fundamental forces that determine the price of
coffee.
Any model, whether it is used to study chemistry, physics, or economics, must
specify what variables will be taken as given in the analysis and what variables are
to be determined by the model. This brings us to the important distinction between exogenous and endogenous variables. An exogenous variable is one whose
value is taken as given in a model. In other words the value of an exogenous variable is determined by some process outside the model being examined. An endogenous variable is a variable whose value is determined within the model being
studied.
THREE KEY
A N A LY T I C A L
TO O L S
exogenous variable
A variable whose value is
taken as given in the analysis of an economic system.
endogenous variable
A variable whose value is
determined within the economic system being studied.
6
CHAPTER 1
A N A LY Z I N G E C O N O M I C P R O B L E M S
To understand the distinction, suppose you want to build a model to predict how
far a ball will fall after it is released from the top of a tall building. You might assume
that certain variables, such as the force of gravity and the density of the air through
which the ball must pass, are taken as given (exogenous) in your analysis. Given the
exogenous variables, your model will describe the relationship between the distance
the ball will drop and the time elapsed after it is released. The distance and time predicted by your model are endogenous variables.
Nearly all microeconomic models rely on just three key analytical tools. We believe
this makes microeconomics unique as a field of study. No matter what the specific issue
is—coffee prices in the United States, or decision making by firms on the Internet—
microeconomics uses the same three analytical tools:
• Constrained optimization
• Equilibrium analysis
• Comparative statics
Throughout this book, we will apply these tools to microeconomic problems.
This section introduces these three tools and provides examples of how they can be
employed. Do not expect to master these tools just by reading this chapter. Rather,
you should learn to recognize them when we apply them in later chapters.
C O N S T R A I N E D O P T I M I Z AT I O N
constrained optimization An analytical tool for
making the best (optimal)
choice, taking into account
any possible limitations or
restrictions on the choice.
objective function
The relationship that a
decision maker seeks to
maximize or minimize.
constraints The restrictions or limits imposed on
a decision maker in a
constrained optimization
problem.
As we noted earlier, economics is the science of constrained choice. The tool of constrained optimization is used when a decision maker seeks to make the best (optimal) choice, taking into account any possible limitations or restrictions on the choices.
We can therefore think about constrained optimization problems as having two parts,
an objective function and a set of constraints. An objective function is the relationship that the decision maker seeks to “optimize,” that is, either maximize or minimize.
For example, a consumer may want to purchase goods to maximize her satisfaction. In
this case, the objective function would be the relationship that describes how satisfied
she will be when she purchases any particular set of goods. Similarly, a producer may
want to plan production activities to minimize the costs of manufacturing its product.
Here the objective function would show how the total costs of production depend on
the various production plans available to the firm.
Decision makers must also recognize that there are often restrictions on the choices
they may actually select. These restrictions reflect the fact that resources are scarce, or
that for some other reason only certain choices can be made. The constraints in a constrained optimization problem represent restrictions or limits that are imposed on the
decision maker.
Examples of Constrained Optimization
To make sure that the difference between an objective function and a constraint is clear,
let’s consider two examples. See if you can identify the objective function and the constraint in each example. (Do not attempt to solve the problems. We will present techniques for solving them in later chapters. At this stage the important point is simply
to understand examples of constrained optimization problems.)
1 . 2 T H R E E K E Y A N A LY T I C A L TO O L S
S
E
7
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 . 1
D
Constrained Optimization: The Farmer’s Fence
Suppose a farmer plans to build a rectangular fence as a pen for his sheep. He has F feet of fence
and cannot afford to purchase more. However, he can
choose the dimensions of the pen, which will have a
length of L feet and a width of W feet. He wants to
choose the dimensions L and W that will maximize the
area of the pen. He must also make sure that the total
amount of fencing he uses (the perimeter of the pen)
does not exceed F feet.
Problem
(a) What is the objective function for this problem?
(b) What is the constraint?
(c) Which of the variables in this model (L, W, and F )
are exogenous? Which are endogenous? Explain.
In other words, the farmer will choose L and W to maximize the objective function LW.
(b) The constraint will describe the restriction imposed
on the farmer. We are told that the farmer has only
F feet of fence available for the rectangular pen. The
constraint will describe the restriction that the perimeter of the pen 2L ⫹ 2W must not exceed the amount of
fence available, F. Therefore, the constraint can be written
as 2L ⫹ 2W ⱕ F.
(c) The farmer is given only F feet of fence to work with.
Thus, the perimeter F is an exogenous variable, since it
is taken as given in the analysis. The endogenous variables are L and W, since their values can be chosen by
the farmer (determined within the model).
Similar Problems:
1.4, 1.16, 1.17
Solution
(a) The objective function is the relationship that the
farmer is trying to maximize—in this case, the area LW.
By convention, economists usually state a constrained optimization problem
like the one facing the farmer in Learning-By-Doing Exercise 1.1 in the following way:
max LW
(L,W )
subject to: 2L ⫹ 2W ⱕ F
The first line identifies the objective function, the area LW, and tells whether it
is to be maximized or minimized. ( If the objective function were to be minimized,
“max” would be “min.’’) Underneath the “max” is a list of the endogenous variables
that the decision maker (the farmer) controls; in this example, “(L, W )” indicates that
the farmer can choose the length and the width of the pen.
The second line represents the constraint on the perimeter. It tells us that the
farmer can choose L and W as long as (“subject to” the constraint that) the perimeter
does not exceed F. Taken together, the two lines of the problem tell us that the farmer
will choose L and W to maximize the area, but those choices are subject to the constraint on the amount of fence available.
We now illustrate the concept of constrained optimization with a famous problem
in microeconomics, consumer choice. (Consumer choice will be analyzed in depth in
Chapters 3, 4, and 5.)
8
CHAPTER 1
S
E
A N A LY Z I N G E C O N O M I C P R O B L E M S
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 . 2
D
Constrained Optimization: Consumer Choice
Suppose a consumer purchases only two
types of goods, food and clothing. The consumer has to
decide how many units of each good to purchase each
month. Let F be the number of units of food that she
purchases each month, and C the number of units of clothing. She wants to maximize her satisfaction with the two
goods. Suppose the consumer’s level of satisfaction when
she purchases F units of food and C units of clothing is
measured by the product FC, but she can purchase only
limited amounts of goods per month because she must live
within her budget. Goods cost money, and the consumer
has a limited income. To keep the example simple, suppose
the consumer has a fixed monthly income I, and she must
not spend more than I during the month. Each unit of
food costs PF and each unit of clothing costs PC.
Problem
(a) What is the objective function for this problem?
(b) The constraint represents the amounts of food and
clothing that she may choose while living within her income. If she buys F units of food at a price of PF per unit,
her total expenditure on food will be (PF)(F ). If she buys
C units of clothing at a price of PC per unit, her total expenditure on clothing will be (PC)(C ). Therefore, her
total expenditure will be (PF)(F ) ⫹ (PC)(C ). Since her
total expenditure must not exceed her total income I, the
constraint is (PF)(F ) ⫹ (PC)(C ) ⱕ I.
(c) The exogenous variables are the ones the consumer
takes as given when she makes her purchasing decisions. Since her monthly income is fixed, I is exogenous. The prices of food PF and clothing PC are also
exogenous, since she cannot control these prices. The
consumer’s only choices are the amounts of food and
clothing to buy; hence, F and C are the endogenous
variables.
(d) The statement of the constrained optimization problem is
(b) What is the constraint?
(c) Which variables (PF, F, PC, C, and I ) are exogenous?
Which are endogenous? Explain.
(d) Write a statement of the constrained optimization
problem.
Solution
(a) The objective function is the relationship that the consumer seeks to maximize. In this example she will choose
the amount of food and clothing to maximize her satisfaction, measured by FC. Thus, the objective function is FC.
A P P L I C A T I O N
max FC
(F,C)
subject to: (PF)(F ) ⫹ (PC)(C ) ⱕ I
The first line shows that the consumer wants to maximize
FC and that she can choose F and C. The second line describes the constraint: Total expenditure cannot exceed
total income.
Similar Problems:
1.4, 1.16, 1.17
1.1
Generating Electricity: 8,760 Decisions
per Year
Examples of constrained optimization are all around
us. Electric power companies typically own and operate plants that produce electricity. A company must
decide how much electricity to produce at each plant
to meet the needs of its customers.
The constrained optimization problem for a power
company can be complex:
• The company needs to generate enough power
to ensure that its customers receive service during each hour of the day.
• To make good production decisions, the company must forecast the demand for electricity.
The demand for electricity varies from one hour
to another during the day, as well as across seasons
of the year. For example, in the summer the highest
demand may occur in the afternoon when customers use air conditioners to cool offices and
1 . 2 T H R E E K E Y A N A LY T I C A L TO O L S
homes. The demand for power may decline
considerably in the evening as the temperature falls.
• Some of the company’s plants are relatively expensive to operate. For example, it is more expensive to produce electricity by burning oil than
by burning natural gas. Plants using nuclear fuel
are even less costly to run. If the company wants
to produce power at the lowest possible cost, its
objective function must take these cost differences
into account.
• If the company expects the demand for electricity
to be low for a long period of time, it may want
to shut down production at some of its plants.
But there are substantial costs to starting up and
shutting down plants. Thus, if the company expects the demand for electricity to be low for
only a short time (e.g., a few hours), it might not
want to shut down a plant that will be needed
again when the demand goes up.
• The company must also take into account the
costs of transmitting power from the generators
to its customers.
• There is a spot market for electricity during each
hour of the day. A company may buy or sell
power from other electric power companies. If
the company can purchase electricity at a low
enough price, it may be able to lower the costs of
service by buying some electricity from other
producers, instead of generating all of the required electricity itself. If it can sell electricity at
a high enough price, the company may find it
profitable to generate more electricity than its
customers need. It can then sell the extra electricity to other power companies.
Electric power companies typically make production
decisions on an hourly basis—that’s 8,760 (365 days
times 24 hours per day) production decisions a year!7
Marginal Reasoning and Constrained Optimization
Constrained optimization analysis can reveal that the “obvious’’ answers to economic
questions may not always be correct. We will illustrate this point by showing how constrained optimization problems can be solved using marginal reasoning.
Imagine that you are the product manager for a small beer company that produces a high-quality microbrewed ale. You have a $1 million media advertising
budget for the next year, and you have to allocate it between local television and
radio spots. Although radio spots are cheaper, television spots reach a far wider
audience. Television spots are also more persuasive and thus on average stimulate
more new sales.
To understand the impact of a given amount of money spent on radio and TV advertisements, you have studied the market. Your research findings, presented in Table 1.1,
estimate the new sales of your beer when a given amount of money is spent on TV advertising and on radio advertising. For example, if you spent $1 million on TV advertising, you would generate 25,000 barrels of new beer sales per year. By contrast, if
you spent $1 million on radio advertising, you would generate 5,000 barrels of new
sales per year. Of course, you could also split your advertising budget between the two
media, and Table 1.1 tells you the impact of that decision, too. For example, if you
spent $400,000 on TV and $600,000 on radio, you would generate 16,000 barrels of
new sales from the TV ads and 4,200 barrels in new sales from the radio ads, for a total
of 16,000 ⫹ 4,200 ⫽ 20,200 barrels of beer overall.
7
9
For a good discussion of the structure of electricity markets, see P. Joskow and R. Schmalensee, Markets
for Power: An Analysis of Electric Utility Deregulation (Cambridge, MA: MIT Press, 1983).
10
CHAPTER 1
A N A LY Z I N G E C O N O M I C P R O B L E M S
TABLE 1.1 New Beer Sales Resulting from Amounts Spent on TV and
Radio Advertising
New Beer Sales Generated
(in barrels per year)
Total Spent
TV
Radio
$
0
$ 100,000
$ 200,000
$ 300,000
$ 400,000
$ 500,000
$ 600,000
$ 700,000
$ 800,000
$ 900,000
$1,000,000
0
4,750
9,000
12,750
16,000
18,750
21,000
22,750
24,000
24,750
25,000
0
950
1,800
2,550
3,200
3,750
4,200
4,550
4,800
4,950
5,000
In light of the information in Table 1.1, how would you allocate your advertising
budget if your objective is to maximize the new sales of beer?
This is a constrained optimization problem. You want to allocate spending on
TV and radio in a way that maximizes an objective (new sales of beer) subject to the
constraint that the total amount spent on TV and radio must not exceed your
$1 million advertising budget. Using notation similar to that introduced in the previous section, if B(T, R) represents the amount of new beer sales when you spend
T dollars on television advertising and R dollars on radio advertising, your constrained optimization problem is
max B(T, R)
(T,R)
subject to: T ⫹ R ⫽ 1 million
A quick reading of Table 1.1 might suggest an “obvious” answer to this problem:
Allocate your entire $1 million budget to TV spots and spend nothing on radio. After
all, as Table 1.1 suggests, a given amount of money spent on TV always generates
more new sales than the same amount of money spent on radio advertising. ( In fact,
a given amount of TV advertising is five times as productive in generating new sales
as is the same amount of radio advertising.) However, this answer is incorrect. And the
reason that it is incorrect illustrates the power and importance of constrained optimization analysis in economics.
Suppose you contemplate spending your entire budget on TV ads. Under that
plan, you would expect to get 25,000 barrels of new sales. But consider, now, what
would happen if you spent only $900,000 on TV ads and $100,000 on radio ads. From
Table 1.1, we see that your TV ads would then generate 24,750 barrels of new beer
sales, and your radio ads would generate 950 barrels of new beer sales. Thus, under
this plan your $1 million budget generates new beer sales equal to 25,700 barrels. This
is 700 barrels higher than before. In fact, you can do even better. By spending
$800,000 on TV and $200,000 on radio, you can generate 25,800 barrels of new beer
sales. Even though Table 1.1 seems to imply that radio ads are far less powerful than
1 . 2 T H R E E K E Y A N A LY T I C A L TO O L S
TV ads, it makes sense in light of your objective to split your budget between radio
and TV advertising.
This example highlights a theme that comes up repeatedly in microeconomics:
The solution to any constrained optimization problem depends on the marginal impact of the decision variables on the value of the objective function. The marginal impact of money spent on TV advertising is how much new beer sales go up for every
additional dollar spent on TV advertising. The marginal impact of money spent on
radio advertising is the rate at which new beer sales go up for every additional dollar
spent on radio advertising. You want to allocate some money to radio advertising because once you have allocated $800,000 of the $1,000,000 budget to TV, the marginal
impact of an additional $100,000 spent on TV advertising is less than the marginal impact of an additional $100,000 spent on radio advertising. Why? Because the rate at
which new beer sales increase when we allocate that next $100,000 to TV advertising
is (24,750 ⫺ 24,000) Ⲑ100,000, or 0.0075 barrels per additional dollar spent on TV advertising. But the rate at which new beer sales increase when we allocate the next
$100,000 to radio advertising is (24,000 ⫹ 950 ⫺ 24,000) Ⲑ100,000 or 0.0095 barrels
per additional dollar spent on radio advertising. Thus, the marginal impact of radio
advertising exceeds the marginal impact of TV advertising. In light of that, we now
want to allocate this additional $100,000 of our advertising budget to radio, rather
than TV. (In fact, as we already saw, you would want to go even further and allocate
the last $200,000 in your budget to radio spots.)
In our advertising story, marginal reasoning leads to a not-so-obvious conclusion
that might make you uncomfortable, or perhaps even skeptical. That’s fine—that’s
how students often react when they first encounter marginal reasoning in microeconomics classes. But whether or not you realize it, we all use marginal reasoning in our
daily lives. For example, even though pizza may be your favorite food and you may
prefer to eat it rather than vegetables like carrots and broccoli, you probably don’t
spend all of your weekly food budget on pizza. Why not? The reason must be that at
some point (perhaps after having eaten pizza for dinner Monday through Saturday
nights), the additional pleasure or satisfaction that you get from spending another $10
of your food budget on a pizza is less than what you would get from spending that $10
of your budget on something else. Although you may not realize it, this is marginal
reasoning in a constrained optimization problem.
The term marginal in microeconomics tells us how a dependent variable changes as
a result of adding one unit of an independent variable. The terms independent variable
and dependent variable may be new to you. To understand them, think of a relationship between two variables, such as between production volume (what economists call output)
and the total cost of manufacturing a product. We would expect that as a firm produces more, its total cost goes up. In this example, we would classify total cost as the
dependent variable because its value depends on the volume of production, which we
refer to as the independent variable.
Marginal cost measures the incremental impact of the last unit of the independent
variable (output) on the dependent variable (total cost). For example, if it costs an
extra $5 to increase production by one unit, the marginal cost will be $5. Equivalently,
marginal cost can be thought of as a rate of change of the dependent variable (again,
total cost) as the independent variable (output) changes. If the marginal cost is $5,
total cost is rising at a rate of $5 when a new unit of output is produced.
We will use marginal measures throughout this book. For example, we will use it
in Chapters 4 and 5 to find the solution to the consumer choice problem described in
Learning-By-Doing Exercise 1.2.
11
12
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A N A LY Z I N G E C O N O M I C P R O B L E M S
FIGURE 1.1
Equilibrium with a Ball
and Cup
This physical system is in equilibrium when
the ball is resting at point B at the bottom of
the cup. The ball could remain there indefinitely. The system will not be in equilibrium
when the ball is at point A because the force
of gravity would pull the ball toward B.
A
B
Force of gravity
E Q U I L I B R I U M A N A LYS I S
equilibrium A state or
condition that will continue
indefinitely as long as
factors exogenous to the
system remain unchanged.
A second important tool in microeconomics is the analysis of equilibrium, a concept
found in many branches of science. An equilibrium in a system is a state or condition
that will continue indefinitely as long as exogenous factors remain unchanged—that
is, as long as no outside factor upsets the equilibrium. To illustrate an equilibrium,
imagine a physical system consisting of a ball in a cup, as is depicted in Figure 1.1.
Here the force of gravity pulls the ball downward toward the bottom of the cup. A ball
initially held at point A will not remain at point A when the ball is released. Rather, it
will rock back and forth until it settles at point B. Thus, the system is not in equilibrium when the ball is released at A because the ball will not remain there. It would be
in equilibrium if the ball were released at B. The system will remain in equilibrium
when the ball is at B until some exogenous factor changes; for example, if someone
were to tip the cup, the ball would move from B to another point.
You may have encountered the notion of an equilibrium in competitive markets
earlier in an introductory course in economics. In Chapter 2 we will provide a more detailed treatment of markets, supply, and demand. But for now let’s briefly review how
the analysis of supply and demand can illustrate the concept of equilibrium in a market.
Consider the worldwide market for coffee beans. Suppose the demand and supply
curves for coffee beans are as depicted in Figure 1.2. The demand curve tells us what
quantity of coffee beans (Q) would be purchased in that market at any given price. Think
of a demand curve as representing the answer to a set of “what if ” questions. For example, what quantity of coffee beans would be demanded if the price were $2.50 per pound?
The demand curve in Figure 1.2 tells us that Q2 pounds would be purchased if the price
of coffee beans were $2.50 per pound. The demand curve also shows us that Q4 pounds
would be purchased if the price were $1.50 per pound. The negative or downward slope
of the demand curve shows that higher prices tend to reduce the consumption of coffee.
The supply curve shows what quantity of coffee beans would be offered for sale in the
market at any given price. You can also view a supply curve as representing the answer to
a set of “what if ” questions. For example, what quantity of coffee beans would be offered
for sale if the price were $1.50 per pound? The supply curve in Figure 1.2 shows us that
Q1 pounds would be offered for sale at that price. The supply curve also indicates that if
the price were $2.50 per pound, Q5 pounds would be offered for sale. The positive (or upward) slope of the supply curve suggests that higher prices tend to stimulate production.
How is the concept of equilibrium related to this discussion of supply and
demand? In a competitive market, equilibrium is achieved at a price at which the
market clears—that is, at a price at which the quantity offered for sale just equals the
quantity demanded by consumers. The coffee bean market depicted in Figure 1.2 will
13
Price per pound (dollars)
1 . 2 T H R E E K E Y A N A LY T I C A L TO O L S
Excess
supply
Supply (S)
$2.50
$2.00
$1.50
FIGURE 1.2
Excess
demand
Q1
Demand (D)
Q3
Q5
Q2 Q4
Quantity (pounds)
Equilibrium in the Market for
Coffee Beans
The equilibrium price of coffee beans is $2.00 per pound.
At that price the market clears (the quantity supplied and
the quantity demanded are equal at Q3 pounds). The
market would not be in equilibrium at a price above
$2.00 because there would be excess supply. The market
would also not be in equilibrium at a price below $2.00,
since there would be excess demand.
clear when the price is $2 per pound. At that price the producers will want to offer Q3
pounds for sale, and consumers will want to buy just that amount. (In graphical terms,
as illustrated by Figure 1.2, equilibrium occurs at the point where the demand curve and
the supply curve intersect.) All consumers who are willing to pay $2 per pound are
able to buy it, and all producers willing to sell at that price can find buyers. The price
of $2, therefore, could stay the same indefinitely because there is no upward or downward pressure on price. There is, in other words, an equilibrium.
To understand why one state of a system is in equilibrium, it helps to see why
other states are not in equilibrium. If the ball in Figure 1.1, were released at some position other than at the bottom of the cup, gravity would move it to the bottom. What
happens in the competitive market at nonequilibrium prices? For example, why would
the coffee market not be in equilibrium if the price of coffee were $2.50 per pound?
At that price, only Q2 pounds would be demanded, but Q5 pounds would be offered
for sale. Thus, there would be an excess supply of coffee in the market. Some sellers
would not find buyers for their coffee beans. To find buyers, these disappointed
producers would be willing to sell for less than $2.50. The market price would need
to fall to $2.00 to eliminate the excess supply.
Similarly, one might ask why a price below $2.00 is not an equilibrium price.
Consider a price of $1.50. At this price the quantity demanded would be Q4 pounds,
but only Q1 pounds would be offered for sale. There would then be excess demand in
the market. Some buyers would be unable to obtain coffee beans. These disappointed
buyers will be willing to pay more than $1.50 per pound. The market price would
need to rise to $2.00 to eliminate the excess demand and the upward pressure that it
generates on the market price.
C O M PA R AT I V E S TAT I C S
Our third key analytical tool, comparative statics analysis, is used to examine how
a change in an exogenous variable will affect the level of an endogenous variable in
an economic model. (See the discussion of exogenous and endogenous variables on
comparative statics
Analysis used to examine
how a change in some
exogenous variable will
affect the level of some
endogenous variable in an
economic system.
CHAPTER 1
FIGURE 1.3
A N A LY Z I N G E C O N O M I C P R O B L E M S
Comparative Statics in the Market for
Pistachio Nuts
The drought and cold weather in Iran in the spring of
2008 caused a leftward shift in the world’s supply curve
for pistachio nuts from S1 to S2. The equilibrium price of
pistachio nuts rose from 4,200 toman per kilogram to
5,300 toman per kilogram. The equilibrium quantity of
pistachio nuts decreased from Q1 to Q2.
Price (toman per kilogram)
14
S2
S1
5,300
4,200
D
Q2 Q1
Quantity (tons)
page 5.) Comparative statics analysis can be applied to constrained optimization
problems or to equilibrium analyses. Comparative statics allows us to do a “beforeand-after” analysis by comparing two snapshots of an economic model. The first
snapshot tells us the levels of the endogenous variables given a set of initial values of
exogenous variables. The second snapshot tells us how an endogenous variable we
care about has changed in response to an exogenous shock—that is, a change in the
level of some exogenous variable.
Let’s consider an example of how comparative statics might be applied to a
model of equilibrium: the market for pistachio nuts. The world’s largest producer
of pistachio nuts is Iran. Pistachio nuts are an extremely important product for
Iran: after oil, pistachio nuts are its largest export commodity, generating more
than $1 billion in earnings in 2007. In the spring of 2008, a combination of a severe drought and unusually cold weather caused Iran’s production of pistachio nuts
to decrease to one-third of what it had been in 2007.8 As a result of this exogenous
shock, the price of pistachio nuts rose from 4,200 toman per kilogram in 2007 to
5,300 toman per kilogram in 2008, an increase of 26 percent (approximately 900
toman equals 1 U.S. dollar).
We can use comparative statics analysis to illustrate what happened in the world
market for pistachio nuts. In a typical year such as 2007, the supply curve would have
been S1 and the demand curve would have been D, as shown in Figure 1.3. Under
these circumstances, the equilibrium price (an endogenous variable) would be 4,200
toman per kilogram, and the equilibrium quantity (also an endogenous variable)
would be Q1. The drought and cold weather in Iran in 2008 led to a leftward shift in
the world’s supply curve for pistachio nuts from S1 to S2. Because worldwide consumer
demand for pistachio nuts is likely to be unaffected by the presence of drought and
cold weather in Iran, it is reasonable to assume that the demand curve for pistachio
nuts did not change as a result of these weather shocks. As Figure 1.3 shows, the shift
8
“Iran Pistachio Prices Soar in Wake of Frost Damage,” BBC Monitoring Middle East (May 1, 2008).
1 . 2 T H R E E K E Y A N A LY T I C A L TO O L S
15
in the supply curve results in an increase in the equilibrium price of pistachio nuts
from 4,200 toman to 5,300 toman per kilogram and a decrease in the global equilibrium
quantity of pistachios from Q1 to Q2.
Almost every day you can find examples of comparative statics in The Wall Street
Journal or in the business section of your local newspaper. Typical items deal with exogenous events that influence the prices of agricultural commodities, livestock, and
metals. It is not unusual to see headlines such as “Coffee Prices Jump on News of
Colombian Labor Strike” or “Corn Prices Surge as Export Demand Increases.” When
you see headlines such as these, think about them in terms of comparative statics. As
Application 1.2 shows, we can even use comparative statics analysis to illustrate the
impact of an economic downturn on the price of tickets to a major sporting event.
The two Learning-by-Doing exercises that follow Application 1.2 show you how you
can perform a comparative statics analysis of a model of market equilibrium and a
model of constrained optimization.
A P P L I C A T I O N
1.2
The Toughest Ticket in Sports
The Masters, held every year in Augusta, Georgia, is arguably the most prestigious professional golf tournament in the world. (It is one of professional golf’s four
“Majors”). But Masters tickets (actually known as
“Masters badges”) are like season tickets to a football
team—if you have obtained them in the past, you can
continue to obtain them. And they are so prized that
the individuals who have obtained them in the past
continue to obtain them. As a result, tickets to the
Masters have not been sold to the general public since
1972. Even the waiting list has been closed off because
it is so long. For this reason, a ticket to the Masters is
known as the “toughest ticket in sports.” According to
one ticket broker, Masters badges are “among the most
coveted tickets for any event, sporting or otherwise.”9
If you want a Masters badge, you must obtain it
from a ticket broker such as Stubhub or on an
Internet auction site such as eBay. Even though the
face price of a Masters badge is in the hundreds of
dollars, people who obtain Masters badges on the
Internet or from a broker typically pay a price in the
thousands. Effectively, the price of Masters badges is
set in the marketplace.
In 2009, something happened that had not happened in several years: The price of Masters badges
9
went down. On April 10, 2009, Stubhub reported that
the price of Masters badges to the second round of
the tournament had fallen from $1,073 in 2008 to
$612 in 2009, a decline of 43 percent.10
The most important difference between 2008 and
2009 was that in the spring of 2009, the United States
was in the midst of a deep recession that affected the
demand for many goods that consumers viewed as luxuries. It seems likely that some people concluded that a
trip to watch the Masters golf tournament in person
was a luxury they could do without.
Figure 1.4 shows a comparative statics analysis
that illustrates the impact of recession on the market
for Masters badges. In a given year, the supply of
Masters badges is fixed, so the supply curve S is vertical, indicating that the supply of available badges
does not vary with the price. The demand curve in a
typical year is D1. A typical price (e.g., in a year such
as 2007 or 2008) for a Masters ticket would be, say,
$1,100, which occurs at the intersection of S1 and D1.
But the recession of 2009 caused a leftward shift in
the demand curve from D1 to D2, indicating that at
various possible prices of Masters badges, the quantity that consumers were willing to purchase was less
in 2009 than in 2008. The result of this change in the
market for Masters badges is a drop in price from
$1,100 to $600.
“How to Get Masters Tickets,” http://golf.about.com/od/majorchampionships/a/masters_tickets.htm
(accessed April 10, 2009).
10
“$612: Friday Masters Badges on Stubhub,” http://online.wsj.com/article/SB123932360425607253.
html#mod=article-outset-box (accessed April 10, 2009).
16
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A N A LY Z I N G E C O N O M I C P R O B L E M S
FIGURE 1.4
Comparative Statics in the Market for
Masters Badges
In a normal year, the market equilibrium occurs at the
intersection of D1 and S, and the equilibrium price for
Masters badges is $1,100. The recession of 2009 caused
a leftward shift in the demand curve from D1 to D2,
and the market equilibrium price of Masters badges
fell to $600.
S
E
Price per badge (dollars)
S
$1,100
$600
D2
D1
Quantity (number of Masters badges)
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 . 3
D
Comparative Statics with Market Equilibrium in the
U.S. Market for Corn
Suppose that in the United States the quantity of corn
demanded Q d depends on two things: the price of corn
P and the level of income in the nation I. Assume that
the demand curve for corn is downward sloping, so
that more corn will be demanded when the price of
corn is lower. Assume also that the demand curve
shifts to the right if income rises (i.e., higher income
increases the demand for corn). The dependence of
the quantity of corn demanded on the price of corn
and income is represented by the demand function
Q d(P, I ).
Suppose the quantity of corn offered for sale, Q s,
also depends on two things: the price of corn, P, and
the amount of rain that falls during the growing season, r. The supply curve is upward sloping, so that as
the price of corn rises, more corn will be offered for
sale. Assume that the supply curve shifts to the right
(more corn is produced) if there is more rain. The relationship showing the quantity of corn supplied at
any price and amount of rainfall is the supply function
Q s(P, r).
In equilibrium the price of corn will adjust so that
the market will clear (Q d ⫽ Q s ). Let’s call the equilibrium quantity exchanged Q * and the equilibrium price
P *. We can assume that the market for corn is only a
small part of the U.S. economy, so that national income is not noticeably affected by events in the market
for corn.
Problem
(a) Suppose that income rises from I1 to I2. On a
clearly labeled graph, illustrate how the change in this
exogenous variable affects each of the endogenous
variables.
(b) Suppose that income remains at I1 but that the
amount of rainfall increases from r1 to r2. On a second
clearly labeled graph, illustrate how the change in this
exogenous variable affects each of the endogenous
variables.
Solution
(a) As shown in Figure 1.5, the change in income
shifts the demand curve to the right (increases demand), from D1 to D2. The location of the supply
curve, S1, is unaffected because Q s does not depend on I.
The equilibrium price therefore rises from P 1* to P 2*. So
the change in income leads to a change in equilibrium
price.
1 . 2 T H R E E K E Y A N A LY T I C A L TO O L S
17
S1: supply of corn
Price of corn
D1: demand for corn
when income is I1
D2: demand for corn
when income is I2
S1
P2*
P1*
FIGURE 1.5
D1
Q1*
D2
Q2*
Quantity of corn
Comparative Statics:
Increase in Income
When income rises from l1 to l2 , the demand curve shifts from D1 to D2 (demand
increases). The equilibrium market price
will rise from P*
1 to P*
2. The equilibrium market quantity will rise from Q*
1 to Q*
2.
S1: supply of corn when
rainfall is r1
Price of corn
S2: supply of corn when
rainfall is r2
S1
S2
P1*
FIGURE 1.6 Comparative Statics:
P2*
D1, Demand for corn
Q1*
Q2*
Quantity of corn
The equilibrium quantity also rises, from Q1* to
Q2*. So the change in income also leads to a change in
quantity.
(b) As shown in Figure 1.6, the increase in rainfall shifts
the supply curve to the right (increases supply), from S1
to S2. The location of the demand curve, D1, is unaf-
Increase in Rainfall
When rainfall increases from r1 to r2,
the supply curve shifts from S1 to S2
(supply increases). The equilibrium
market price will fall from P*
1 to P*
2.
The equilibrium market quantity will
rise from Q*
1 to Q*
2.
fected because Q d does not depend on r. The equilibrium price therefore falls from P 1* to P 2*. So the change
in rainfall leads to a change in equilibrium price.
The equilibrium quantity rises, from Q*1 to Q*2. So
the change in rainfall also leads to a change in quantity.
Similar Problems:
1.2, 1.5, 1.6, 1.7, 1.12, 1.13
18
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S
E
A N A LY Z I N G E C O N O M I C P R O B L E M S
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 . 4
D
Comparative Statics with Constrained Optimization
In the farmer’s fencing problem (LearningBy-Doing Exercise 1.1), the exogenous variable is the
perimeter of the fence F, and the endogenous variables
are the length L and width W of the pen. You may have
solved a problem like this one before: The area is maximized when the farmer builds a square pen. (You do not
need to know how to arrive at that conclusion in this exercise. Just trust that it is correct.)
Problem If the farmer is given an extra length of
fence ⌬F (where ⌬, the Greek letter delta, means “the
change in”), how will the dimensions of the pen change? In
other words, how will a change in the exogenous variable
1.3
POSITIVE AND
N O R M AT I V E
A N A LYS I S
positive analysis
Analysis that attempts to
explain how an economic
system works or to predict
how it will change over
time.
normative analysis
Analysis that typically
focuses on issues of social
welfare, examining what
will enhance or detract
from the common good.
⌬F be reflected by changes in the endogenous variables
⌬L and ⌬W ?
Solution Since the optimal configuration of the
pen is a square, we know that the length and width of
the pen will each be one-fourth of the perimeter, so L ⫽
FⲐ4 and W ⫽ F Ⲑ4. Therefore, ⌬L ⫽ ⌬F Ⲑ4 and ⌬W ⫽
⌬F Ⲑ4. This comparative statics result tells us, for example, that if the farmer is given an extra 4 feet of fence,
the length and the width of the pen will each be increased by one foot.
Similar Problem: 1.20
Microeconomic analysis can be used to study both positive and normative questions.
Positive analysis attempts to explain how an economic system works or to predict how
it will change over time. Positive analysis asks explanatory questions such as “What has
happened?” or “What is happening?” It may also ask a predictive question: “What will
happen if some exogenous variable changes?” In contrast, normative analysis asks prescriptive questions, such as “What should be done?” Normative studies typically focus
on issues of social welfare, examining what will enhance or detract from the common
good. In so doing, they often involve value judgments. For example, policy makers may
want to consider whether we should raise the minimum wage to benefit the least skilled
and least experienced workers.
We have seen illustrations of positive questions throughout this chapter. In the
farmer’s fencing problem (Learning-By-Doing Exercise 1.1), one positive question is,
“What dimensions of the sheep’s pen will the farmer choose to maximize the area of the
pen?” Another is, “How will the area of the pen change if the farmer is given one more
foot of fence?” In the consumer choice problem (Learning-By-Doing Exercise 1.2), positive analysis will tell us how the consumer’s purchases of each good will depend on the
prices of all goods and on the level of her income. Positive analysis will help the manager
of the electricity generator (Application 1.1) to produce any given level of service with the
lowest possible cost. Finally, positive analysis enables us to understand why a particular
price of a commodity such as coffee beans is in equilibrium and why other prices are not.
It also explains why heavy rains, strikes, and frost result in higher commodity prices.
As all of these examples suggest, applying microeconomic principles for predictive
purposes is important for consumers and for managers of enterprises. Positive analysis
is also useful in the study of public policy. For example, policy makers might like to understand the effect of new taxes in a market, government subsidies to producers, or tariffs or quotas on imports. They may also want to know how producers and consumers
are affected, as well as the size of the impact on the government budget.
Normative studies might examine how to achieve a goal that some people consider
socially desirable. Suppose policy makers want to make housing more affordable to lowincome families. They may ask whether it is “better” to accomplish this by issuing these
1 . 3 P O S I T I V E A N D N O R M AT I V E A N A LYS I S
19
families housing vouchers that they can use on the open housing market or by implementing rent controls that prevent landlords from charging any renter more than an amount
controlled by law. Or, if government finds it desirable to reduce pollution, should it introduce taxes on emissions or strictly limit the emissions from factories and automobiles?
These examples illustrate that it is important to do positive analysis before normative analysis. A policy maker may want to ask the normative question, “Should we implement a program of rent controls or a program of housing vouchers?” To understand
the options fully, the policy maker will first need to do positive analysis to understand
what will happen if rent controls are imposed and to learn about the consequences of
housing vouchers. Positive analysis will tell us who is affected by each policy, and how.
Microeconomics can help policy makers understand and compare the impacts of
alternative policies on consumers and producers. It can therefore help sharpen debates
and lead to more enlightened public policy.
A P P L I C A T I O N
1.3
Positive and Normative Analyses
of the Minimum Wage
Over 100 countries around the world, including the
United States, set a minimum wage. (In 2009, the U.S.
minimum wage was $7.25 per hour.) The minimum
wage has been extensively studied and debated by
economists, and economists differ in their views about
it. For example, a 2006 survey by Robert Whaples of
210 economists belonging to the American Economic
Association found that nearly 47 percent of the economists surveyed believed that the federal minimum
wage in the United States should be eliminated, while
nearly 38 percent believed that the minimum wage
should be increased.11
Perhaps not surprisingly, one can find examples
of both positive analyses and normative analyses of
the minimum wage. Consider, for example, David
Card and Alan Krueger’s study of the impact on employment resulting from an increase in New Jersey’s
minimum wage in the early 1990s.12 Contrasting
changes in employment in fast-food restaurants in
New Jersey with changes in employment in fast-food
restaurants in an adjacent state (Pennsylvania) in
which there was no increase in the minimum wage,
11
Card and Krueger found that the increase in New
Jersey’s minimum wage did not decrease employment. Though provocative—Card and Krueger’s
study presents a finding that is at odds with the
implications of the analysis of the minimum wage
usually presented in microeconomics textbooks13—it
is nevertheless an example of a positive analysis. Its
purpose was to answer an explanatory question:
What happened to employment when the minimum
wage in a state increased?
By contrast, consider a piece written in 2004 by
the economist Steven Landsburg that makes a forceful case against the minimum wage:14
In fact, the minimum wage is very good for unskilled
workers. It transfers income to them. And therein
lies the right argument against the minimum wage.
Ordinarily, when we decide to transfer income to
some group or another—whether it be the working
poor, the unemployed, the victims of a flood, or the
stockholders of American Airlines—we pay for the
transfer out of general tax revenue. That has two
advantages: It spreads the burden across all taxpayers, and it makes politicians accountable for their
actions. It’s easy to look up exactly how much the
government gave American, and it’s easy to look up
exactly which senators voted for it.
Robert Whaples, “Do Economists Agree on Anything? Yes!” Economist’s Voice 3, no. 9 (November 2006),
http://www.bepress.com/ev/vol3/iss9/art1 (accessed September 1, 2009).
12
David Card and Alan Krueger, “Miniumum Wages and Employment: A Case Study of the Fast Food
Industry in New Jersey and Pennsylvania, American Economic Review, 84, no. 4 (September 1994): 772–793.
13
Including this one! See Section 10.6.
14
Steven Landsburg, “The Sin of Wages: The Real Reason to Oppose the Minimum Wage,” Slate ( July
9, 2004), http://slate.msn.com/id/2103486/ (accessed September 1, 2009).
20
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A N A LY Z I N G E C O N O M I C P R O B L E M S
By contrast, the minimum wage places the entire burden on one small group: the employers of
low-wage workers and, to some extent, their customers. Suppose you’re a small entrepreneur with,
say, 10 full-time minimum-wage workers. Then a 50
cent increase in the minimum wage is going to cost
you about $10,000 a year. That’s no different from a
$10,000 tax increase. But the politicians who imposed the burden get to claim they never raised anybody’s taxes.
If you want to transfer income to the working
poor, there are fairer and more honest ways to do it.
The Earned Income Tax Credit, for example, accomplishes pretty much the same goals as the minimum
wage but without concentrating the burden on a tiny
minority. For that matter, the EITC also does a better
job of helping the people you’d really want to help, as
opposed to, say, middle-class teenagers working summer jobs. It’s pretty hard to argue that a minimumwage increase beats an EITC increase by any criterion.
Landsburg’s article is an example of a normative
analysis. It addresses a prescriptive question: Should
the minimum wage be replaced by other policies
(such as the earned income tax credit) that transfer in-
come to low-wage workers? Note that Landsburg’s
piece involves two important value judgments: First,
it is generally better for the burden of income redistribution to be borne by a larger group of the citizenry than by a smaller group. Second, more transparent policies for redistributing income (like the
Earned Income Tax Credit) are better than policies
that are less transparent (like the minimum wage)
because it is easier to hold politicians accountable for
more transparent policies.
Examples of both positive analyses and normative analyses of economic issues abound, though one
generally finds them in different places. Positive
analyses tend to be found in professional academic
journals such as the American Economic Review
(where the Card and Krueger study was published),
Journal of Political Economy, or Econometrica.
Normative analyses are often presented in op-ed
pieces or columns (Landsburg’s article appeared in
the “Everyday Economics” column in Slate), policyoriented outlets, such as the Economists’ Voice, or (increasingly these days) blogs, such as the ones written
by economists Paul Krugman, Greg Mankiw, Brad
DeLong, or Gary Becker and Richard Posner.
CHAPTER SUMMARY
• Economics is the study of the allocation of limited
resources to satisfy unlimited human wants. It is often
described as the science of constrained choice.
• Microeconomics examines the economic behavior of
individual economic decision units, such as a consumer
or a firm, as well as groups of economic agents, such as
households or industries.
• Economic studies are often conducted by constructing and analyzing models of a particular problem.
Because the real world is complex, an economic model
represents an abstraction from reality.
• In analyzing any model, one needs to understand
what variables will be taken as given (exogenous variables), as well as what variables will be determined
within the model (endogenous variables).
• Three essential tools of microeconomic analysis are
(1) constrained optimization, a tool that decision makers
use to maximize or minimize some objective function
subject to a constraint (LBD Exercises 1.1 and 1.2);
(2) equilibrium analysis, used to describe a condition or
state that could continue indefinitely in a system, or at
least until there is a change in some exogenous variable;
and (3) comparative statics, used to examine how a change
in some exogenous variable will affect the level of some
endogenous variable in an economic model, including
equilibrium (LBD Exercise 1.3) and constrained optimization. (LBD Exercise 1.4)
• The term marginal in microeconomics measures the
amount by which a dependent variable changes as the result of adding one more unit of an independent variable.
• Microeconomics provides tools we can use to examine
positive and normative issues. Positive analysis attempts to
explain how an economic system works and to predict how
the endogenous variables will change as exogenous variables change. Normative analysis considers prescriptive
questions such as “What should be done?” Normative
studies introduce value judgments into the analysis.
PROBLEMS
21
REVIEW QUESTIONS
1. What is the difference between microeconomics and
macroeconomics?
2. Why is economics often described as the science of
constrained choice?
5. What is the difference between an exogenous variable and an endogenous variable in an economic
model? Would it ever be useful to construct a model
that contained only exogenous variables (and no endogenous variables)?
3. How does the tool of constrained optimization help
decision makers make choices? What roles do the objective function and constraints play in a model of constrained optimization?
6. Why do economists do comparative statics analysis?
What role do endogenous variables and exogenous variables play in comparative statics analysis?
4. Suppose the market for wheat is competitive, with
an upward-sloping supply curve, a downward-sloping
demand curve, and an equilibrium price of $4.00 per
bushel. Why would a higher price (e.g., $5.00 per
bushel) not be an equilibrium price? Why would a
lower price (e.g., $2.50 per bushel) not be an equilibrium price?
7. What is the difference between positive and normative analysis? Which of the following questions would
entail positive analysis, and which normative analysis?
a) What effect will Internet auction companies have on
the profits of local automobile dealerships?
b) Should the government impose special taxes on sales
of merchandise made over the Internet?
PROBLEMS
1.1. Discuss the following statement: “Since supply and
demand curves are always shifting, markets never actually
reach an equilibrium. Therefore, the concept of equilibrium is useless.”
1.2. In an article entitled, “Corn Prices Surge on
Export Demand, Crop Data,” the Wall Street Journal
identified several exogenous shocks that pushed U.S.
corn prices sharply higher.15 Suppose the U.S. market for
corn is competitive, with an upward-sloping supply curve
and a downward-sloping demand curve. For each of the
following scenarios, illustrate graphically how the exogenous event described will contribute to a higher price of
corn in the U.S. market.
a) The U.S. Department of Agriculture announces
that exports of corn to Taiwan and Japan were “surprisingly bullish,” around 30 percent higher than had been
expected.
b) Some analysts project that the size of the U.S. corn
crop will hit a six-year low because of dry weather.
c) The strengthening of El Niño, the meteorological
trend that brings warmer weather to the western coast of
South America, reduces corn production outside the
United States, thereby increasing foreign countries’ dependence on the U.S. corn crop.
15
See the article by Aaron Lucchetti, August 22, 1997, p. C17.
1.3. In early 2008, the price of oil on the world market
increased, hitting a peak of about $140 per barrel in July
2008. In the second half of 2008, the price of oil declined,
ending the year at just over $40 per barrel. Suppose that
the global market for oil can be described by an upwardsloping supply curve and a downward-sloping demand
curve. For each of the following scenarios, illustrate
graphically how the exogenous event contributed to a
rise or a decline in the price of oil in 2008:
a) A booming economy in China raised the global demand
for oil to record levels in 2008.
b) As a result of the financial crisis of 2008, the United
States and other developed economies plunged into a severe
recession in the latter half of 2008.
c) Reduced sectarian violence in Iraq in 2008 enabled
Iraq to increase its oil production capacity.
1.4. A firm produces cellular telephone service using
equipment and labor. When it uses E machine-hours of
equipment and hires L person-hours of labor, it can
provide up to Q units of telephone service. The relationship between Q, E, and L is as follows: Q ⫽ 2EL.
The firm must always pay PE for each machine-hour of
equipment it uses and PL for each person-hour of labor
it hires. Suppose the production manager is told to produce Q ⫽ 200 units of telephone service and that she
wants to choose E and L to minimize costs while
achieving that production target.
22
CHAPTER 1
A N A LY Z I N G E C O N O M I C P R O B L E M S
a) What is the objective function for this problem?
b) What is the constraint?
c) Which of the variables (Q, E, L, PE, and PL) are exogenous? Which are endogenous? Explain.
d) Write a statement of the constrained optimization
problem.
1.5. The supply of aluminum in the United States depends on the price of aluminum and the average price of
electricity (a critical input in the production of aluminum). Assume that an increase in the price of electricity shifts the supply curve for aluminum to the left (i.e.,
a higher average price of electricity decreases the supply
of aluminum). The demand for aluminum in the United
States depends on the price of aluminum and on national income. Assume that an increase in national income shifts the demand curve for aluminum to the right
(i.e., higher income increases the demand for aluminum). In 2004, national income in the United States
increased, while the price of electricity fell, as compared
to 2003. How would the equilibrium price of aluminum
in 2004 compare to the equilibrium price in 2003? How
would the equilibrium quantity in 2004 compare to the
equilibrium quantity in 2003?
1.6. Ethanol (i.e., ethyl alcohol) is a colorless, flammable liquid that, when blended with gasoline, creates a
motor fuel that can serve as an alternative to gasoline.
The quantity of ethanol motor fuel that is demanded depends on the price of ethanol and the price of gasoline.
Because ethanol fuel is a substitute for gasoline, an increase in the price of gasoline shifts the demand curve
for ethanol rightward. The quantity of ethanol supplied
depends on the price of ethanol and the price of corn
(since the primary input used to produce ethanol in the
United States is corn). An increase in the price of corn
shifts the supply curve of ethanol leftward. In the first
half of 2008, the price of gasoline in the United States
increased significantly as compared to 2007, and the
price of corn increased as well. How would the equilibrium price of ethanol motor fuel in the first half of 2008
compare to the price in 2007?
1.7. The price of gasoline in the United States depends
on the supply of gasoline and the demand for gasoline.
Gasoline is supplied by oil companies that sell it on several markets. Hence the supply of gasoline in the United
States depends on the price of gasoline in the United
States and its price on other markets. When the price of
gasoline outside the United States increases, the U.S.
supply decreases because firms prefer to sell the gasoline
elsewhere. How would an increase in the price of gasoline abroad affect the equilibrium price of gasoline in the
United States?
1.8. The demand for computer monitors is given by the
equation Q d ⫽ 700 ⫺ P, while the supply is given by the
equation Q s ⫽ 100 ⫹ P. In both equations P denotes the
market price. Fill in the following table. For what price is
the market in equilibrium—supply equals to the demand?
P
200
250
300
350
400
Qd
Qs
1.9. The demand for computer memory chips is given by
the equation Q d ⫽ 500 ⫺ 2P, while the supply is given by
the equation Q s ⫽ 50 ⫹ P. In both equations P denotes the
market price. For what price is the market in equilibrium—
supply equals demand? What is the equilibrium quantity?
P
50
100
150
200
250
Qd
Qs
1.10. The demand for sunglasses is given by equation
Q d ⫽ 1000 ⫺ 4P, where P denotes the market price. The
supply of sunglasses is given by equation Q s ⫽ 100 ⫹ 6P.
Fill in the following table and find the equilibrium price.
P
80
90
100
110
120
Qd
Qs
1.11. This year’s summer is expected to be very sunny.
Hence the demand for sunglasses increased and now is
given by equation Q d ⫽ 1200 ⫺ 4P. How is the equilibrium price going to change compared with the scenario
described in Problem 1.10? Explain and then fill in the
following table to verify your explanation.
P
80
90
100
110
120
Qd
Qs
1.12. Suppose the supply curve for wool is given by Qs ⫽ P,
where Q s is the quantity offered for sale when the price is P.
Also suppose the demand curve for wool is given by Q d ⫽
10 ⫺ P ⫹ I, where Q d is the quantity of wool demanded
when the price is P and the level of income is I. Assume I
is an exogenous variable.
23
PROBLEMS
a) Suppose the level of income is I ⫽ 20. Graph the supply
and demand relationships, and indicate the equilibrium
levels of price and quantity on your graph.
b) Explain why the market for wool would not be in
equilibrium if the price of wool were 18.
c) Explain why the market for wool would not be in
equilibrium if the price of wool were 14.
1.13. Consider the market for wool described by the
supply and demand equations in Problem 1.12. Suppose
income rises from I1 ⫽ 20 to I2 ⫽ 24.
a) Using comparative statics analysis, find the impact
of the change in income on the equilibrium price of
wool.
b) Using comparative statics analysis, find the impact of
the change in income on the equilibrium quantity of
wool.
1.14. You are the video acquisitions officer for your residence hall. The other officers of your hall will tell you
how many videos they would like to rent during the year.
Your job is to find the least expensive way of renting the
required number of videos. After researching the options, you have found that there are three rental plans
from which you can choose.
Plan A: Pay $3 per video, with no additional fees.
Plan B: Join the Frequent Viewer Club. Here you pay a
yearly membership fee of $50, with an additional charge
of $2 for each video rented.
Plan C: Join the Very Frequent Viewer Club. In this
club you pay a yearly membership fee of $150, with an
additional charge of $1 for each video rented.
a) Which plan would you select if your instructions are
to rent 75 movies a year at the lowest possible cost?
b) Which plan would you select if your instructions are
to rent 125 movies a year at the lowest possible cost?
c) In this exercise, is the number of videos rented endogenous or exogenous? Explain.
d) Is the choice of plan (A, B, or C) endogenous or
exogenous? Explain.
e) Are total expenditures on videos endogenous or exogenous? Explain.
1.15. Reconsider the problem of the video acquisitions
officer in Problem 1.14. Suppose the officers of your
residence hall give you a specified amount of money to
spend, and want you to maximize the number of videos
you can rent with that budget. You can choose from the
same three plans (A, B, and C) available in Problem 1.14.
a) Which plan would you select if your instructions are
to rent the most movies possible while spending $125 per
year?
b) Which plan would you select if your instructions are
to rent the most movies possible while spending $300
per year?
c) In this exercise, is the number of videos rented
endogenous or exogenous? Explain.
d) Is the choice of plan (A, B, or C) endogenous or
exogenous? Explain.
e) Are total expenditures on videos endogenous or exogenous? Explain.
1.16. A major automobile manufacturer is considering
how to allocate a $2 million advertising budget between
two types of television programs: NFL football games
and PGA tour professional golf tournaments. The following table shows the new sports utility vehicles (SUVs)
that are sold when a given amount of money is spent on
advertising during an NFL football game and a PGA
tour golf event.
Total
Spent (millions)
$0
$0.5
$1.0
$1.5
$2.0
New SUV Sales Generated
(thousands of vehicles per year)
NFL Football
PGA Tour Golf
0
10
15
19
20
0
4
6
8
9
The manufacturer’s goal is to allocate its $2 million advertising budget to maximize the number of SUVs sold.
Let F be the amount of money devoted to advertising on
NFL football games, G the amount of money spent on
advertising on PGA tour golf events, and C(F,G) the
number of new vehicles sold.
a) What is the objective function for this problem?
b) What is the constraint?
c) Write a statement of the constrained optimization
problem.
d) In light of the information in the table, how should the
manufacturer allocate its advertising budget?
1.17. An electricity producer has two power plants,
each of which emits carbon dioxide (CO2), a greenhouse
gas. Each plant is currently emitting 1 million metric
tons of CO2 per year. However, new emissions rules restrict the firm’s emissions to 1 million metric tons of
CO2 per year from both plants combined. The cost of operating a power plant goes up as it curtails its emissions.
The following table shows the cost of operating each
plant for different emissions levels:
24
CHAPTER 1
Emissions of
CO2 by a Plant
(metric tons
per year)
Annual Cost
of Operating
Plant 1
(millions)
A N A LY Z I N G E C O N O M I C P R O B L E M S
Annual Cost
of Operating
Plant 2
(millions)
0
$490
$250
250,000
$360
$160
500,000
$250
$ 90
750,000
$160
$ 40
1,000,000
$ 90
$ 10
The firm’s goal is to choose emissions levels at each plant
that minimize its total cost of operating its plants, subject to meeting its emissions target of 1 million metric
tons of CO2 per year from both plants combined. Let X
denote the quantity of emissions from plant 1 and Y denote the quantity of emissions from plant 2. Let TC(X, Y )
denote the total operating cost of the firm when the
quantity of emissions from plant 1 is X and the quantity
of emissions from plant 2 is Y.
a) What is the objective function for this problem?
b) What is the constraint?
c) Write a statement of the constrained optimization
problem.
d) In light of the information in the table, what emissions levels from each plant should the firm choose?
1.18. The demand curve for peaches is given by the
equation Q d ⫽ 100 ⫺ 4P, where P is the price of peaches
expressed in cents per pound and Q d is the quantity of
peaches demanded (expressed in thousands of bushels per
year). The supply curve for peaches is given by Q s ⫽ RP,
where R is the amount of rainfall (inches per month during the growing season) and Q s is the quantity of peaches
supplied (expressed in thousands of bushels per year). Let
P* denote the market equilibrium price and Q* denote
the market equilibrium quantity. Complete the following
table showing how the equilibrium quantity and price
vary with the amount of rainfall. Verify that when R ⫽ 1,
the equilibrium price is 20 cents per pound and the equilibrium quantity is 20,000 bushels per year.
R
1
Q*
20
P*
20
2
4
8
16
16.67
1.19. The worldwide demand curve for pistachios is
given by Qd ⫽ 10 ⫺ P, where P is the price of pistachios
in U.S. dollars and Qd is the quantity in millions of kilograms per year. The world supply curve for pistachios is
9P
given by Q s ⫽
, where T is the average
1 ⫹ .05(T ⫺ 70)2
temperature (measured in degrees Fahrenheit) in
pistachio-growing regions such as Iran. The supply
curve implies that as the temperature deviates from
the ideal growing temperature of 70o, the quantity of
pistachios supplied goes down. Let P* denote the
equilibrium price and Q* denote the equilibrium
quantity. Complete the following table showing how
the equilibrium quantity and price vary with the average temperature. Verify that when T ⫽ 70, the equilibrium price is $1 per kilogram and the equilibrium
quantity is 9 million kilograms per year.
30
T
50
65
70
Q*
(millions of
kilograms
per year)
9
P* ($ per
kilogram)
1
80
1.20. Consider the comparative statics of the farmer’s
fencing problem in Learning-By-Doing Exercise 1.4,
where L is the length of the pen, W is the width, and
A ⫽ LW is the area.
a) Suppose the number of feet of fence given to the farmer
was initially F1 ⫽ 200. Complete the following table. Verify
that the optimal design of the fence (the one yielding the
largest area with a perimeter of 200 feet) would be a square.
L
10
20
W
90
80
A
900
30
40
50
60
70
80
90
b) Now suppose the farmer is instead given 240 feet of fence
(F ⫽ 240). Complete the following table. By how much
would the length L of the optimally designed pen increase?
L
20
30
W
100
90
A
2000
40
50
60
70
80
90
100
c) When the amount of fence is increased from 200 to 240
(⌬F ⫽ 40), what is the change in the optimal length (⌬L)?
d) When the amount of fence is increased from 200 to 240
(⌬F ⫽ 40), what is the change in the optimal area (⌬ A)? Is
the area A endogenous or exogenous in this example?
Explain.
PROBLEMS
1.21. Which of the following statements suggest a positive analysis and which a normative analysis?
a) If the United States lifts the prohibition on imports of
Cuban cigars, the price of cigars will fall.
b) A freeze in Florida will lead to an increase in the price
of orange juice.
c) To provide revenues for public schools, taxes on alcohol,
tobacco, and gambling casinos should be raised instead of
increasing income taxes.
25
d) Telephone companies should be allowed to offer cable
TV service as well as telephone service.
(e) If telephone companies are allowed to offer cable TV
service, the price of both types of service will fall.
f ) Government subsidies to farmers are too high and
should be phased out over the next decade.
g) If the tax on cigarettes is increased by 50 cents per
pack, the equilibrium price of cigarettes will rise by 30
cents per pack.
2
DEMAND AND SUPPLY
ANALYSIS
2.1
D E M A N D, S U P P LY, A N D M A R K E T
EQUILIBRIUM
The Valentine’s Day Effect
APPLICATION 2.2 A Computer on Every Desk
and in Every Home
APPLICATION 2.1
2.2
PRICE ELASTICITY OF DEMAND
How People Buy Cars: The
Importance of Brands
APPLICATION 2.3
2.3
OT H E R E L A S T I C I T I E S
How People Buy Cars: The
Importance of Price
A P P L I C AT I O N 2 . 5 Coke versus Pepsi
APPLICATION 2.4
2.4
E L A S T I C I T Y I N T H E L O N G RU N
V E R S U S T H E S H O RT RU N
APPLICATION 2.6
Crude Oil: Price and Demand
2.5
BAC K - O F - T H E - E N V E L O P E
C A L C U L AT I O N S
What Hurricane Katrina Tells Us
about the Price Elasticity of Demand for Gasoline
APPLICATION 2.8 The California Energy Crisis
APPLICATION 2.7
APPENDIX
PRICE ELASTICITY OF DEMAND
A L O N G A C O N S TA N T E L A S T I C I T Y
D E M A N D C U RV E
What Gives with the Price of Corn?
Corn is one of the most important agricultural products in the United States. It is used to make many
food and industrial products we encounter in our daily lives, such as corn oil, sweeteners, and alcohol.
In recent years, especially with increasing prices of gasoline and oil, it has attracted increasing attention
because it may be used to produce the fuel ethanol.
26
In the late 1990s and early 2000s, the price of corn hovered around $2 per bushel. But in the last
half of the 2000s, the scenario changed dramatically, as Figure 2.1 shows. In late 2006, the price of corn
began to rise, and by mid-2008, it exceeded $5 per bushel. Even though the price fell in the last half of
2008, by mid-2009, it was still around $4 per bushel, well above the historical norm. And even in the first
half of the decade, when prices were closer to the historical norm, there was still variation. In 2003–2004
the price of corn increased to almost $3 bushel, while in 2004–2005, the price dropped, falling below
$2 per bushel in late 2005.
Figure 2.1 illustrates the vagaries of prices in a competitive market. Prices rise and fall in seemingly
random ways, and there is little that individual market participants (e.g., corn farmers, operators of
grain elevators, commodity traders) can do about it. However, we can understand why prices in a market change as they do. In the case of corn, the pattern of prices shown in Figure 2.1 can be traced to the
interaction of some important changes in supply and demand conditions in the corn market during the
2000s. The slight increase in the price of corn in 2002 and early 2003 reflect a decrease in the supply of
corn due to a drought in the corn-growing states in the United States in the summer of 2002. The falling
prices in 2004 and 2005 resulted from unexpectedly large U.S. corn crops during those years.
$6
U.S. price of corn (dollars per bushel)
$5
$4
$3
$2
$1
$2000
2001
2002
2003
2004
2005
Year
2006
2007
2008
2009
FIGURE 2.1
The Price of Corn in the United States, 2000–2009
The monthly price of corn received by farmers in the United States between January 2000
and June 2009; prices reached a peak of $5.47 per bushel in June 2008. Source: Economic
Research Service, Feed Grains Database, U.S. Department of Agriculture,
http://www.ers.usda.gov/data/feedgrains/ (accessed July 9, 2009).
27
The sustained increase in the price of corn beginning in late 2006 has its roots in a number of
changes in U.S. government policy. In the early 2000s, a number of states began to ban the use of MTBE
(methyl tertiary butyl ether), a compound used as an additive in gasoline to enhance octane ratings and
engine performance, because of concerns that it was carcinogenic. Ethanol, a colorless flammable liquid
that is used in a variety of applications including alcoholic beverages, solvents, and scents, began
increasingly to be used as a substitute for MBTE, and nearly all ethanol made in the United States is
produced from corn. The move toward corn-based ethanol as an additive in gasoline accelerated in
2005, when the Congress removed liability protection from refining companies that added MTBE to the
gasoline they produced. In the mid-2000s, the switch from MTBE to ethanol increased the demand for
corn-based ethanol and thus increased the demand for corn.
In addition, in 2005 and again in 2007, Congress passed energy bills that contained schedules of
“renewable fuel mandates,” requirements that called for minimum levels of consumptions of renewable
fuels used in the United States between 2009 and 2022. The mandates called for a sharp increase in the
amount of corn-based ethanol consumed until 2015, at which point the growth in renewable fuel consumption would come from other renewable fuels. The renewable fuel mandates resulted in an increase
in the amount of ethanol-based fuel produced in the United States (such as E85, a blend consisting of
85 percent ethanol and 15 percent gasoline) and thus increased the demand for corn even more. The
increased demand for corn that resulted from the growing use of ethanol in the United States is a key
reason why the U.S. corn price rose sharply in 2007
and 2008. The Congressional Budget Office estimates
that of the $1.75 per bushel increase in the price of
corn from April 2007 to April 2008 (i.e., from $3.39 to
$5.14), 28 percent to 47 percent of the increase can be
attributed to the increased demand from U.S. ethanol
producers.1
So what accounted for the remaining portion of
the large increase? Part of the increase was due a
growth in demand for corn resulting from the rapid
expansion of the U.S. and global economies that
took place during the “bubble” years of 2005–2008.
Another part of the increase was due to changes on
the supply side of the corn market. Increases in the
price of oil increased farmers’ production costs.
Furthermore, heavy rains and flooding in the U.S.
Corn Belt in early 2008 caused fear that a large
portion of the 2008 corn harvest would be wiped out.
1
“The Impact of Ethanol Use on Food Prices and Greenhouse-Gas
Emissions,” Congressional Budget Office (April 2009).
28
29
2 . 1 D E M A N D, S U P P LY, A N D M A R K E T E Q U I L I B R I U M
All of these factors driving the price of corn upward went away in the latter half of 2008 and 2009:
The economic crisis reduced the global demand for corn. Oil prices fell, giving farmers some relief from
high fuel prices. Fear of a greatly reduced corn harvest in 2008 proved to be exaggerated. And weather
conditions returned to normal in 2009. As a result, in the second half of 2008, the price of corn fell
from its June 2008 peak of $5.47 per bushel to about $3.90 per bushel in early 2009. Because the shifts
in ethanol demand continue to affect the market for corn, this price exceeds the $2.00 per bushel level
of the early 2000s.
The tools of supply and demand analysis that we introduced in Chapter 1 can help us understand
the story that unfolded in the corn market over the past decade. In fact, they can help us understand
the pattern of prices that prevail in many markets, ranging from fresh-cut roses to electricity to pepper.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Describe the three main building blocks of supply and demand analysis––demand curves, supply
curves, and the concept of market equilibrium.
• Analyze how changes in exogenous variables shift the demand and supply curves and thus change the
equilibrium price and quantity.
• Explain the concept of price elasticity.
• Calculate the price elasticity of demand for specific demand curves.
• Explain how price elasticity of demand is related to total revenue.
• Discuss the factors that determine the price elasticity of demand.
• Contrast the market-level price elasticity of demand with the brand-level price elasticity of demand.
• Explain and contrast other elasticities: the income elasticity of demand, the cross-price elasticity of
demand, and the price elasticity of supply.
• Indicate why the short-run price elasticities of demand and supply may differ from the long-run price
elasticities of demand and supply.
• Use “back-of-the-envelope” techniques to determine key properties of demand and supply curves
with only fragmentary data on prices, quantities, or elasticities.
Chapter 1 introduced equilibrium and comparative statics analysis. In this chapter, we 2.1
apply those tools to the analysis of perfectly competitive markets. Perfectly competitive
markets comprise large numbers of buyers and sellers. The transactions of any individual buyer or seller are so small in comparison to the overall volume of the good or service traded in the market that each buyer or seller “takes” the market price as given
when making purchase or production decisions. For this reason, the model of perfect
competition is often cited as a model of price-taking behavior.
Figure 2.2 illustrates the basic model of a perfectly competitive market. The
horizontal axis depicts the total quantity Q of a particular good—in this case corn—
that is supplied and demanded in this market. The vertical axis depicts the price P at
which this good is sold. A market can be characterized along three dimensions:
D E M A N D,
S U P P LY, A N D
MARKET
EQUILIBRIUM
30
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
Price (dollars per bushel)
S
E
$4
$3
FIGURE 2.2
The Market for Corn in the
United States in 2009
The curve labeled D is the demand curve for
corn. The curve labeled S is the supply curve
for corn. Point E, at which the two curves
intersect, is the market equilibrium.
D
9 11
14
Quantity (billions of bushels per year)
commodity––the product bought and sold (in Figure 2.2 this is corn); geography––the
location in which purchases are being made (in Figure 2.2 this is the United States);
and time––the period of time during which transactions are occurring (in Figure 2.2,
this is the year 2009, when corn prices were about $4 per bushel).
D E M A N D C U RV E S
market demand curve
A curve that shows us the
quantity of goods that consumers are willing to buy
at different prices.
derived demand
Demand for a good that is
derived from the production and sale of other
goods.
direct demand
Demand for a good that
comes from the desire of
buyers to directly consume
the good itself.
The curve D in Figure 2.2 is the market demand curve for corn. It tells us the quantity of corn that buyers are willing to purchase at different prices. For example, the
demand curve tells us that at a price of $3 per bushel, the annual demand for corn
would be 14 billion bushels, while at a price of $4 per bushel, the annual demand for
corn would be only 11 billion bushels.
Corn supplies are bought by companies (such as Archer Daniels Midland and
General Mills) that process the corn into intermediate products (e.g., high fructose
corn syrup or corn grits), which in turn are used to make final products (e.g., soft
drinks or breakfast cereal). Part of the demand depicted in Figure 2.2 is derived
demand––that is, it is derived from the production and sale of other goods. For example, the demand for high-fructose corn syrup is derived from the demand for soft
drinks in which it is used as a sweetener (instead of sugar). Corn is also purchased by
brokers and wholesale distributors, who then sell it to retailers who then resell it to
final consumers. Thus, another part of the demand for corn depicted in Figure 2.2
is direct demand––demand for the good itself. The demand curve D is a market
demand curve in that it represents the aggregate demand for corn from all the corn
purchasers in the U.S. market.
In Figure 2.2, we have drawn the demand curve with price on the vertical axis
and quantity on the horizontal axis. This representation emphasizes another useful
31
2 . 1 D E M A N D, S U P P LY, A N D M A R K E T E Q U I L I B R I U M
interpretation of the demand curve that we will return to in later chapters. The demand
curve tells us the highest price that the “market will bear” for a given quantity or supply of output. Thus, in Figure 2.2, if suppliers of corn offered, in total, 14 billion
bushels for sale, the highest price that the corn would fetch would be $3 per bushel.
Other factors besides price affect the quantity of a good demanded. The prices of
related goods, consumer incomes, consumer tastes, and advertising are among the factors that we expect would influence the demand for a typical product. However, the
demand curve focuses only on the relationship between the price of the good and the
quantity of the good demanded. When we draw the demand curve, we imagine that
all other factors that affect the quantity demanded are fixed.
The demand curve in Figure 2.2 slopes downward, indicating that the lower the
price of corn, the greater the quantity of corn demanded, and the higher the price of
corn, the smaller the quantity demanded. The inverse relationship between price and
quantity demanded, holding all other factors that influence demand fixed, is called the law
of demand. Countless studies of market demand curves confirm the inverse relationship between price and quantity demanded, which is why we call the relationship a
law. Still, you might wonder about so-called luxury goods, such as perfume, designer
labels, or crystal. It is alleged that some consumers purchase more of these goods
at higher prices because a high price indicates superior quality.2 However, these examples do not violate the law of demand because all of the other factors influencing
demand for these goods are not held fixed while the price changes. Consumers’ perceptions of the quality of these goods have also changed. If consumers’ perceptions of
quality could be held constant, then we would expect that consumers would purchase
less of these luxury goods as the price goes up.
D
Sketching a Demand Curve
Suppose the demand for new automobiles
in the United States is described by the equation
Qd 5.3 0.1P
Solution
(a) To find the yearly demand for automobiles, given the
average price per car, use equation (2.1):
(2.1)
d
where Q is the number of new automobiles demanded
per year (in millions) when P is the average price of an
automobile (in thousands of dollars). (At this point, don’t
worry about the meaning of the constants in equations
for demand or supply curves––in this case, 5.3 and 0.1.)
Problem
(a) What is the quantity of automobiles demanded per
year when the average price of an automobile is $15,000?
When it is $25,000? When it is $35,000?
(b) Sketch the demand curve for automobiles. Does this
demand curve obey the law of demand?
2
inverse relationship between
the price of a good and the
quantity demanded, when
all other factors that influence demand are held
fixed.
L E A R N I N G - B Y- D O I N G E X E R C I S E 2 . 1
S
E
law of demand The
Average Price
per Car (P )
Using Equation (2.1)
Quantity
Demanded (Q d )
$15,000
$25,000
$35,000
Qd 5.3 0.1(15) 3.8
Qd 5.3 0.1(25) 2.8
Qd 5.3 0.1(35) 1.8
3.8 million cars
2.8 million cars
1.8 million cars
(b) Figure 2.3 shows the demand curve for automobiles.
To sketch it, you can plot the combinations of prices and
quantities that we found in part (a) and connect them
with a line. The downward slope of the demand curve in
Figure 2.3 tells us that as the price of automobiles goes
up, consumers demand fewer automobiles.
Similar Problems:
2.1, 2.2, 2.4
Michael Schudson, Advertising, The Uneasy Persuasion: Its Dubious Impact on American Society (New York:
Basic Books, 1984), pp. 113–114.
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
Price (thousands of dollars)
32
$35
$25
$15
FIGURE 2.3
The U.S.
Demand Curve for Automobiles
The law of demand holds in this
market because the demand curve
slopes downward.
D
0
1.8
2.8
3.8
Quantity (millions of automobiles per year)
S U P P LY C U RV E S
market supply curve
A curve that shows us the
total quantity of goods that
their suppliers are willing
to sell at different prices.
law of supply The positive relationship between
price and quantity supplied,
when all other factors that
influence supply are held
fixed.
factors of production
Resources such as labor
and raw materials that are
used to produce a good.
The curve labeled S in Figure 2.2 is the market supply curve for corn. It tells us the
total quantity of corn that suppliers of corn are willing to sell at different prices. For
example, the supply curve tells us that at a price of $3 per bushel, 9 billion bushels of
corn would be supplied in 2009, while at a price of $4 per bushel, 11 billion bushels
would be supplied in that year.
The supply of corn in the United States comes primarily from corn farmers
around the country. The available supply in a given year consists of corn that is harvested in that year plus corn that has been stored from previous harvests. We should
think of the supply curve S as being constructed from the sum of the supply curves of
all individual suppliers of corn in the United States.
The supply curve slopes upward, indicating that at higher prices, suppliers of corn
are willing to offer more corn for sale than at lower prices. The positive relationship
between price and quantity supplied is known as the law of supply. Studies of market
supply curves confirm the positive relationship between the quantity supplied and the
price, which is why we call the relationship a law.
As with demand, other factors besides price affect the quantity of a good that producers will supply to the market. For example, the prices of factors of production—
resources such as labor and raw materials that are used to produce the good—will
affect the quantity of the good that sellers are willing to supply. The prices of other
goods that sellers produce could also affect the quantity supplied. For example, the
supply of natural gas goes up when the price of oil goes up, because higher oil prices
spur more oil production, and natural gas is a by-product of oil. When we draw a supply curve like the one in Figure 2.2, we imagine that all these other factors that affect
the quantity supplied are held fixed.
33
2 . 1 D E M A N D, S U P P LY, A N D M A R K E T E Q U I L I B R I U M
L E A R N I N G - B Y- D O I N G E X E R C I S E 2 . 2
S
E
D
Sketching a Supply Curve
Suppose the yearly supply of wheat in
Canada is described by the equation
Q s 0.15 P
(2.2)
where Q s is the quantity of wheat produced in Canada
per year (in billions of bushels) when P is the average
price of wheat (in dollars per bushel).
Problem
(a) What is the quantity of wheat supplied per year when
the average price of wheat is $2 per bushel? When the
price is $3? When the price is $4?
(b) Sketch the supply curve for wheat. Does it obey the
law of supply?
Solution
(a) To find the yearly supply of wheat, given the average
price per bushel, use equation (2.2):
Average Price
per Bushel (P)
Using Equation (2.2)
Quantity
Supplied (Q s )
$2
$3
$4
Q s 0.15 2 2.15
Q s 0.15 3 3.15
Q s 0.15 4 4.15
2.15 million bushels
3.15 million bushels
4.15 million bushels
(b) Figure 2.4 shows the graph of this supply curve. We
find it by plotting the prices and associated quantities
from part (a) and connecting them with a line. The fact
that the supply curve in Figure 2.4 slopes upward indicates that the law of supply holds.
Price (dollars per bushel)
S
$4
$3
$2
FIGURE 2.4
0
2.15
3.15
4.15
Quantity (billions of bushels per year)
The Supply Curve for
Wheat in Canada
The law of supply holds in this market because the supply curve slopes
upward.
MARKET EQUILIBRIUM
In Figure 2.2, the demand and supply curves intersect at point E, where the price is
$4 per bushel and the quantity is 11 billion bushels. At this point, the market is in
equilibrium (the quantity demanded equals the quantity supplied, so the market
clears). As we discussed in Chapter 1, an equilibrium is a point at which there is no
equilibrium A point at
which there is no tendency
for the market price to
change as long as exogenous variables remain
unchanged.
34
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
FIGURE 2.5
Excess Demand and Excess
Supply in Market for Corn
If the price of corn were $3, per bushel, excess
demand would result because 14 billion bushels
would be demanded, but only 9 billion bushels
would be supplied. If the price of corn were $5
per bushel, excess supply would result because
13 billion bushels would be supplied but only 8
billion bushels would be demanded.
excess supply A situation in which the quantity
supplied at a given price
exceeds the quantity
demanded.
excess demand A situation in which the quantity
demanded at a given price
exceeds the quantity
supplied.
S
E
Price (dollars per bushel)
S
Excess supply
when price
is $5
$5
E
$4
$3
Excess
demand
when price is $3
D
8 9 11 13 14
Quantity (billions of bushels per year)
tendency for the market price to change as long as exogenous variables (e.g., rainfall,
national income) remain unchanged. At any price other than the equilibrium price,
pressures exist for the price to change. For example, as Figure 2.5 shows, if the price
of corn is $5 per bushel, there is excess supply—the quantity supplied at that price
(13 billion bushels) exceeds the quantity demanded (8 billion bushels). The fact that
suppliers of corn cannot sell as much as they would like creates pressure for the price
to go down. As the price falls, the quantity demanded goes up, the quantity supplied
goes down, and the market moves toward the equilibrium price of $4 per bushel. If
the price of corn is $3 per bushel, there is excess demand—the quantity demanded
at that price (14 billion bushels) exceeds the quantity supplied (9 billion bushels). Buyers
of corn cannot procure as much corn as they would like, and so there is pressure for the
price to rise. As the price rises, the quantity supplied also rises, the quantity demanded
falls, and the market moves toward the equilibrium price of $4 per bushel.
L E A R N I N G - B Y- D O I N G E X E R C I S E 2 . 3
D
Calculating Equilibrium Price and Quantity
Suppose the market demand curve for cranberries is given by the equation Q d 500 4P, while the
market supply curve for cranberries (when
P 50) is described by the equation Q s 100 2P,
where P is the price of cranberries expressed in dollars per
barrel, and quantity (Q d or Q s ) is in thousands of barrels
per year.
Problem At what price and quantity is the market
for cranberries in equilibrium? Show this equilibrium
graphically.
Solution
At equilibrium, the quantity supplied equals
the quantity demanded, and we can use this relationship
to solve for P: Q d Q s, or 500 4P 100 2P,
2 . 1 D E M A N D, S U P P LY, A N D M A R K E T E Q U I L I B R I U M
which implies P 100. Thus, the equilibrium price is
$100 per barrel. We can then find the equilibrium quantity by substituting the equilibrium price into the equation for either the demand curve or the supply curve:
Thus, the equilibrium quantity is 100,000 barrels per
year. Figure 2.6 illustrates this equilibrium graphically.
Similar Problem:
2.3
d
Q 500 4(100) 100
Q s 100 2(100) 100
S
Price (dollars per barrel)
$120
$100
E
$80
$60
$40
FIGURE 2.6
$20
D
0
100
200
300
400
Quantity (thousands of barrels per year)
500
Equilibrium in the Market
for Cranberries
The market equilibrium occurs at point E,
where the demand and supply curves intersect. The equilibrium price is $100 per barrel,
and the equilibrium quantity is 100,000 barrels of cranberries per year.
S H I F T S I N S U P P LY A N D D E M A N D
Shifts in Either Supply or Demand
The demand and supply curves discussed so far in this chapter were drawn under the
assumption that all factors, except for price, that influence the quantity demanded and
quantity supplied are fixed. In reality, however, these other factors are not fixed, and
so the position of the demand and supply curves, and thus the position of the market
equilibrium, depend on their values. Figures 2.7 and 2.8 illustrate how we can enrich
our analysis to account for the effects of these other variables on the market equilibrium. These figures illustrate comparative statics analysis, which we discussed in
Chapter 1. In both cases, we can explore how a change in an exogenous variable (e.g.,
consumer income or wage rates) changes the equilibrium values of the endogenous
variables (price and quantity).
To do a comparative statics analysis of the market equilibrium, you first must determine how a particular exogenous variable affects demand or supply or both. You
then represent changes in that variable by a shift in the demand curve, in the supply
curve, or in both. For example, suppose that higher consumer incomes increase the
demand for a particular good. The effect of higher disposable income on the market
equilibrium is represented by a rightward shift in the demand curve (i.e., a shift away
from the vertical axis), as shown in Figure 2.7.3 This shift indicates that at any price
3
The shift does not necessarily have to be parallel, as it is in Figure 2.7.
35
D E M A N D A N D S U P P LY A N A LYS I S
Price
CHAPTER 2
FIGURE 2.7
Shift
in Demand Due to an
Increase in Disposable
Income
If an increase in
consumers’ disposable
incomes increases
demand for a particular
good, the demand
curve shifts rightward
(i.e., away from the vertical axis) from D1 to D2,
and the market equilibrium moves from point
A to point B. Equilibrium
price goes up, and
equilibrium quantity
goes up.
S
B
Equilibrium
price goes
up
A
D1
D2
Quantity
Equilibrium
quantity goes up
the quantity demanded is greater than before. This shift moves the market equilibrium
from point A to point B. The shift in demand due to higher income thus increases both
the equilibrium price and the equilibrium quantity.
For another example, suppose wage rates for workers in a particular industry go
up. Some firms might then reduce production levels because their costs have risen
with the cost of labor. Some firms might even go out of business altogether. An
increase in labor costs would shift the supply curve leftward (i.e., toward the vertical
S2
Price
36
FIGURE 2.8
Shift in
Supply Due to an
Increase in the Price of
Labor
An increase in the price
of labor shifts the supply curve leftward (i.e.,
toward the vertical axis)
from S1 to S2. The market equilibrium moves
from point A to point B.
Equilibrium price goes
up, but equilibrium
quantity goes down.
S1
B
Equilibrium
price goes
up
A
D
Quantity
Equilibrium
quantity goes down
2 . 1 D E M A N D, S U P P LY, A N D M A R K E T E Q U I L I B R I U M
37
axis), as shown in Figure 2.8. This shift indicates that less product would be supplied
at any price, and the market equilibrium would move from point A to point B. The
increase in the price of labor increases the equilibrium price and decreases the equilibrium quantity.
Figure 2.7 shows us that an increase in demand, coupled with an unchanged supply curve, results in a higher equilibrium price and a larger equilibrium quantity.
Figure 2.8 shows that a decrease in supply, coupled with an unchanged demand curve,
results in a higher equilibrium price and a smaller equilibrium quantity. By going
through similar comparative statics analyses for a decrease in demand and an increase
in supply, we can derive the four basic laws of supply and demand:
1. Increase in demand unchanged supply curve higher equilibrium price and
larger equilibrium quantity.
2. Decrease in supply unchanged demand curve higher equilibrium price and
smaller equilibrium quantity.
3. Decrease in demand unchanged supply curve lower equilibrium price and
smaller equilibrium quantity.
4. Increase in supply unchanged demand curve lower equilibrium price and
larger equilibrium quantity.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 2 . 4
D
Comparative Statics on the Market Equilibrium
Suppose that the U.S. demand for aluminum is given by the equation Q d 500 50P 10I,
where P is the price of aluminum expressed in dollars per
kilogram and I is the average income per person in the
United States (in thousands of dollars per year). Average
income is an important determinant of the demand for
automobiles and other products that use aluminum, and
hence is a determinant of the demand for aluminum
itself. Further suppose that the U.S. supply of aluminum
(when P 8) is given by the equation Qs 400 50P.
In both the demand and supply functions, quantity is
measured in millions of kilograms of aluminum per year.
Problem
(a) What is the market equilibrium price of aluminum
when I 10 (i.e., $10,000 per year)?
(b) What happens to the demand curve if average
income per person is only $5,000 per year (i.e., I 5
rather than I 10). Calculate the impact of this demand shift on the market equilibrium price and
quantity and then sketch the supply curve and the demand curves (when I 10 and when I 5) to illustrate
this impact.
Solution
(a) We substitute I 10 into the demand equation to get
the demand curve for aluminum: Q d 600 50P.
We then equate Q d to Q s to find the equilibrium
price: 600 50P 400 50P, which implies P 10.
The equilibrium price is thus $10 per kilogram. The equilibrium quantity is Q 600 50(10), or Q 100. Thus,
the equilibrium quantity is 100 million kilograms per year.
(b) The change in I creates a new demand curve that we
find by substituting I 5 into the demand equation shown
above: Q d 550 50P. Figure 2.9 shows this demand
curve as well as the demand curve for I 10. As before, we
equate Q d to Q s to find the equilibrium price: 550 50P
400 50P, which implies P 9.5. The equilibrium
price thus decreases from $10.00 per kilogram to $9.50 per
kilogram. The equilibrium quantity is Q 550 50(9.50),
or Q 75. Thus, the equilibrium quantity decreases from
100 million kilograms per year to 75 million kilograms.
Figure 2.9 shows this impact. Note that it is consistent with
the third law of supply and demand: A decrease in demand
coupled with an unchanged supply curve results in a lower
equilibrium price and a smaller equilibrium quantity.
Similar Problems:
2.11, 2.18
38
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
Price (dollars per kilograms)
S
$10.00
$9.50
D1 (I = 10)
D2 (I = 5)
0
75 100
Quantity (millions of kilograms per year)
FIGURE 2.9
Equilibrium in the Market for Aluminum
The market equilibrium initially occurs at a price of $10 per kilogram and a quantity of
100 million kilograms. When average income goes down (i.e., when we move from I ⫽ 10
to I ⫽ 5), the demand curve for aluminum shifts leftward. The new equilibrium price is
$9.50 per kilogram, and the new equilibrium quantity is 75 million kilograms.
A P P L I C A T I O N
2.1
The Valentine’s Day Effect
If you have ever bought fresh-cut roses, you may have
noticed that their price varies considerably during the
year. In particular, the price you pay for fresh-cut
roses––especially red roses––around Valentine’s Day is
usually three to five times higher than at other times
during the year. Figure 2.10 illustrates this pattern by
showing the prices and quantities of fresh-cut roses at
two different times of the year: February and August
in each of three years, 1991, 1992, and 1993.4 Are the
high prices of roses at Valentine’s Day a result of a conspiracy among florists and rose growers to gouge romantic consumers? Probably not. This pricing behavior
can best be understood as an application of comparative statics analysis.
4
Figure 2.11 depicts the market equilibrium in
the U.S. market for fresh-cut roses in the early
1990s. During this period, wholesale prices for red
hybrid tea roses were ordinarily about $0.20 per
stem.5 Every year, though, the market changes
around Valentine’s Day. During the days before
Valentine’s Day, demand for red roses increases dramatically, resulting in a rightward shift in the demand curve for roses from D1 to D2. This rightward
shift occurs because around Valentine’s Day, people
who do not ordinarily purchase roses want to buy
them for their spouses or sweethearts. The rightward shift in demand increases the equilibrium price
to about $0.50 per stem. Even though the price is
higher, the equilibrium quantity is also higher than it
was before. This outcome does not contradict the
The data in Figure 2.10 are derived from Tables 12 and 17 of “Fresh Cut Roses from Colombia and
Ecuador,” Publication 2766, International Trade Commission (March 1994). The data for February
actually consist of the last two weeks of January and the first two weeks of February.
5
These are wholesale prices (i.e., the prices that retail florists pay their suppliers), not the retail prices
paid by the final consumer.
2 . 1 D E M A N D, S U P P LY, A N D M A R K E T E Q U I L I B R I U M
39
$0.60
Price (dollars per stem)
$0.50
February 1991–1993
$0.40
$0.30
$0.20
FIGURE 2.10 Prices and
Quantities of Fresh-Cut Roses
Prices and quantities of roses
during 1991–1993 for the
months of August and
February––both are much
higher in February than they
are in August.
August 1991–1993
$0.10
0
2
4
6
8
10
Quantity (millions of stems per month)
law of demand. It reflects the fact that the
Valentine’s Day equilibrium occurs along a demand
curve that is different from the demand curve before
or after Valentine’s Day.
Figure 2.11 explains why we would expect the prices
of red roses to peak around Valentine’s Day (the occurrence of Valentine’s Day is an exogenous variable that
strongly impacts the demand for red roses). The logic
of Figure 2.11 also helps explain another aspect of the
rose market: the prices of white and yellow roses. Their
prices also go up around Valentine’s Day, but by less
than the prices of red roses. Overall, their prices show
more stability than the prices of red roses because
white and yellow roses are less popular on Valentine’s
Day and are used more for weddings and other special
events. These events are spread more evenly throughout the year, so the demand curves for white and yellow
roses fluctuate less dramatically than the demand
curve for red roses. As a result, their equilibrium prices
are more stable.
S
Price (dollars per stem)
$0.60
$0.50
$0.40
$0.30
$0.20
$0.10
D1
0
2
4
6
8
Quantity (millions of stems per month)
D2
10
FIGURE 2.11 The
Market for Fresh-Cut Roses
During “usual” months, the
market for fresh-cut roses
attains equilibrium at a price
of about $0.20 per stem.
However, during the weeks
around Valentine’s Day, the
demand curve for roses
shifts rightward, from D1 to
D 2, and the equilibrium
price and quantity go up.
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
Shifts in Both Supply and Demand
So far, we have focused on what happens when either the supply curve or the demand curve shifts. But sometimes we can better understand the dynamics of prices
and quantities in markets by exploring what happens when both supply and demand
shift.
We return to the example of the U.S. corn market in the 2000s to illustrate this
point. Figure 2.12 shows the difference between the equilibrium in the corn market
in 2006, when the price was around $2 per bushel (point A) and in 2008, when the
price had risen to $5 per bushel (point B). As we discussed in the Introduction, the
change in the price of corn over this period can be attributed to an increase in demand (driven, in particular, by the growth in the market for corn-based ethanol in
the United States) and a decrease in supply (due, in particular, to heavy rains and
flooding in the U.S. Corn Belt in 2008). The combined impact of both shifts was to
increase the equilibrium price. By contrast, the effect of these changes on equilibrium quantity is not clear. The increase in demand tends to push the equilibrium
quantity upward, while the increase in supply tends to push the equilibrium quantity
downward. The net impact on the equilibrium quantity would depend on the magnitude of those shifts, as well as the shapes of the demand and supply curves themselves.
S2008 S2006
Price (dollars per bushel)
40
B
$5
A
$2
D2006
D2008
10 12
Quantity (billions of bushels per year)
FIGURE 2.12 The U.S. Corn Market, 2006–2008
The increase in price can be explained by the combined effect of a shift in supply and a
shift in demand. In particular, the demand curve shifted rightward from D2006 to D2008,
while the supply curve shifted leftward from S2006 to S2008, moving the equilibrium from
point A to point B. The result was an increase in the equilibrium price from $2 per bushel
to $5 per bushel.
41
2 . 1 D E M A N D, S U P P LY, A N D M A R K E T E Q U I L I B R I U M
magnitude of those shifts, as well as the shapes of the demand and supply curves
themselves. Figure 2.12 shows an increase in the equilibrium quantity (from 10 billion
bushels to 12 billion bushels), which is what happened in the United States between
2006 and 2008.
A P P L I C A T I O N
2.2
A Computer on Every Desk
and in Every Home
In 1975 Bill Gates and Paul Allen founded Microsoft,
famously declaring that the company’s mission was
“a computer on every desk and in every home.” At
the time only a handful of personal computer models
had been sold in small quantities to hobbyists. Those
computers could do very little. Now, of course,
Microsoft’s goal has largely been realized in advanced
economies worldwide. The primary reason for this is the
dramatically falling price of computers, peripherals, and
software. Figure 2.13 illustrates how the cost of
computers fell in the last 20 years.
The data in the figure are a price index showing
how the average price of a computer of similar capability changed over time. The index is scaled to equal
100 at the end of 1988. Values of the index are calculated as a computer’s price that month as a percentage of the price of a comparable computer at the end
of 1988. For example, suppose that the computer
priced in December 1988 was $5,000. The index’s
value at the end of 1990 was about 90, so a comparable computer would have cost about $4,500 (90 percent of $5,000) that month. The price estimates are
constructed by the Bureau of Labor Statistics (BLS).
Quality and price of computer components changed
so rapidly in recent decades that the BLS had to develop special methods to estimate computer prices
100
80
60
40
20
FIGURE 2.13 Quality-Adjusted Prices of Computers and Peripheral Equipment, 1988–2008
This is the graph of a price index showing how the average price of a computer of similar
capability changed over time. The index is scaled to equal 100 at the end of 1988. By
2008, the price index had fallen to about 10.
Dec–2008
Dec–2006
Dec–2004
Dec–2002
Dec–2000
Dec–1998
Dec–1996
Dec–1994
Dec–1992
Dec–1990
Dec–1988
0
42
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
over time.6 Briefly, every six months the BLS finds new
computer components or peripherals with functionality similar to those used to construct the most recent
computer price estimate. The price of the new components is then used to produce a new estimated computer price.
Figure 2.13 shows an incredible decline in computer prices over time. A computer bought in mid1990 would cost about one-tenth of what a computer with similar capabilities would have cost 20
years before! If data on quality-adjusted prices were
available going back to when Microsoft was
founded in 1975, we would see similar trends. At the
same time, the total quantity of computers sold
grew many times over. What explains this pattern of
prices and quantities?
Figure 2.14 illustrates what was happening.
Since personal computers appeared in the 1970s, the
demand curve for computers shifted rightward. A
combination of factors drove this shift. As computers
became more powerful, companies started developing a vast array of software and peripherals to work
with them. For consumers, these new complementary products increased the value of owning a com-
P
D1975
puter. Along similar lines, many new uses for computers were introduced over time. In addition, consumers became more educated in how to use computers, increasing their productivity from using
them.
We know that an increase in demand, holding
the supply curve fixed, should cause the equilibrium
price to rise. That computer prices fell indicates that
something other than the demand curve must have
shifted. Figure 2.14 shows that the pattern of observed priced and quantities is consistent with a
simultaneous rightward shift of both the demand
and supply curves.
What caused the increase in supply for computers? The most important effect was “Moore’s Law”
(named after Intel co-founder Gordon Moore, who
first described it).7 Moore’s Law states that the number of transistors that can be fit on an integrated
circuit doubles every two years. This has been approximately true for several decades. This exponential growth has led to vastly faster and less expensive computer chips. Many other computer
components also saw rapid improvements in quality
and declines in price over same period. These
S1975
FIGURE 2.14
Supply and
Demand for Computers,
1975–2009
The pattern of prices in Figure
2.13, as well as rapid growth in
quantities over the same period,
can be explained by rightward
shifts over time in both the
demand and supply curves for
computers. The supply curve
shifted from S1975 to S2009, while
the demand curve shifted from
D1975 to D2009.
6
Price (quality adjusted)
D2009
S2009
Path of computer
prices and quantities
over time
Quantity (computers sold per year)
Q
“How BLS Measures Price Change for Personal Computers and Peripheral Equipment in the
Consumer Price Index.” U.S Bureau of Labor Statistics, June 2008,
http://www.bls.gov/cpi/cpifaccomp.htm.
7
“Cramming More Components onto Integrated Circuits.” Gordon Moore, Electronics Magazine,1965.
43
2.2 PRICE ELASTICITY OF DEMAND
advances made it possible for computer manufacturers to produce computers of given capability much
more cheaply. As we will see later in this book, when
a firm’s costs fall in this way, the supply curve shifts
rightward. Finally, the supply curve also shifted rightward because many new computer firms entered the
market. The combined effect of technological advances
and new entry pushed the supply curve for computers
rightward by an amount that equaled or exceeded the
rightward shift in demand. The result is the long-term
path for prices and quantities represented by the
dashed line in Figure 2.14.
The price elasticity of demand measures the sensitivity of the quantity demanded to 2.2
price. The price elasticity of demand (denoted by ⑀Q,P) is the percentage change in quantity demanded (Q) brought about by a 1 percent change in price (P), which means that
⑀Q,P
percentage change in quantity
PRICE
ELASTICITY
OF DEMAND
percentage change in price
If Q is the change in quantity and P is the change in price, then
percentage change in price
price elasticity of
demand A measure of
the rate of percentage
change of quantity demanded with respect to
price, holding all other
determinants of demand
constant.
¢Q
100%
Q
and
percentage change in quantity
¢P
100%
P
Thus, the price elasticity of demand is
⑀Q,P
¢Q
Q
100%
¢P
P
100%
or
⑀Q,P
¢Q P
¢P Q
(2.3)
For example, suppose that when the price of a good is $10 (P 10), the quantity
demanded is 50 units (Q 50), and that when the price increases to $12 (P 2), the
quantity demanded decreases to 45 units (Q 5). If we plug these numbers into
equation (2.3), we find that in this case the price elasticity of demand is
⑀Q,P
¢Q P
5 10
0.5
¢P Q
2 50
As illustrated by this example, the value of ⑀Q,P must always be negative, reflecting the
fact that demand curves slope downward because of the inverse relationship of price
and quantity: When price increases, quantity decreases, and vice versa. The following
table shows how economists classify the possible range of values for ⑀Q,P.
44
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
Value of ⑀Q,P
Classification
perfectly inelastic
demand Price elasticity
0
Perfectly inelastic demand
of demand equal to 0.
between 0 and 1
Inelastic demand
1
Unitary elastic demand
between 1 and q
Elastic demand
q
Perfectly elastic demand
inelastic demand
Price elasticity of demand
between 0 and 1.
unitary elastic demand
Price elasticity of demand
equal to 1.
elastic demand Price
elasticity of demand
between 1 and q.
perfectly elastic
demand Price elasticity
of demand equal to q.
Meaning
Quantity demanded is
completely insensitive to price.
Quantity demanded is
relatively insensitive to price.
Percentage increase in quantity
demanded is equal to
percentage decrease in price.
Quantity demanded is
relatively sensitive to price.
Any increase in price results in
quantity demanded decreasing
to zero, and any decrease in
price results in quantity
demanded increasing to infinity.
To see the relationship between the price elasticity of demand and the shape of
the demand curve, consider Figure 2.15. In this figure, demand curves D1 and D2 cross
at point A, where the price is P and the quantity is Q. (For the moment ignore the
demand curve D3.) For a given percentage increase in price P P from point A, the
percentage decrease in quantity demanded, Q2 Q, along D2 is larger than the percentage decrease in the quantity demanded, Q1 Q, along demand curve D1. Thus, at
point A, demand is more elastic on demand curve D2 than on demand curve D1––that
is, at point A, the price elasticity of demand is more negative for D2 than for D1. This
shows that for any two demand curves that cross at a particular point, the flatter of the
two curves is more elastic at the point where they cross.
Comparing the Price
Elasticity of Demand on
Different Demand Curves
If we start at point A, a
given percentage increase
in price, ¢PP, along
demand curve D1 results in
a relatively small percentage drop in quantity
demanded, Q1Q, while
the same percentage
change in price results in a
relatively large percentage
drop in quantity demanded, Q2Q, along
demand curve D2. Thus, at
point A, demand is more
elastic on demand curve D2
than on demand curve D1.
The demand curve D3 is
perfectly elastic. Along this
demand curve, the price
elasticity of demand is
equal to minus infinity.
Price (dollars per unit)
FIGURE 2.15
P
A
∆P
D3
D2
∆Q2
∆Q 1
D1
Q
Quantity (thousands of units per year)
45
2.2 PRICE ELASTICITY OF DEMAND
The demand curve D3 in Figure 2.15 shows what happens in the extreme as
demand becomes increasingly elastic. The demand curve D3 illustrates perfectly elastic demand (i.e., ⑀Q,P q). Along the perfectly elastic demand curve D3, any positive quantity can be sold at the price P, so the demand curve is a horizontal line. The
opposite of perfectly elastic demand is perfectly inelastic demand (i.e., ⑀Q, P 0), when
the quantity demanded is completely insensitive to price.8
The price elasticity of demand can be an extremely useful piece of information for
business firms, nonprofit institutions, and other organizations that are deciding how
to price their products or services. It is also an important determinant of the structure
and nature of competition within particular industries. Finally, the price elasticity of
demand is important in determining the effect of various kinds of governmental
interventions, such as price ceilings, tariffs, and import quotas. In later chapters, we
explore the analysis of these questions using price elasticities of demand.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 2 . 5
D
Price Elasticity of Demand
Suppose price is initially $5.00, and the
corresponding quantity demanded is 1,000 units.
Suppose, too, that if the price rises to $5.75, the quantity demanded will fall to 800 units.
Problem What is the price elasticity of demand over
this region of the demand curve? Is demand elastic or
inelastic?
Thus, over the range of prices between $5.00 and $5.75,
quantity demanded falls at a rate of 1.33 percent for
every 1 percent increase in price. Because the price
elasticity of demand is between 1 and q, demand is
elastic over this price range (i.e., quantity demanded is
relatively sensitive to price).
Similar Problem:
2.4
Solution In this case, P 5.75 5 $0.75, and
Q 800 1000 200, so
⑀Q, P
¢Q P
200 $5
1.33
¢P Q
$0.75 1000
E L A S T I C I T I E S A L O N G S P E C I F I C D E M A N D C U RV E S
Linear Demand Curves
A commonly used form of the demand curve is the linear demand curve, represented
by the equation Q a b P, where a and b are positive constants. In this equation, the
constant a embodies the effects of all the factors (e.g., income, prices of other goods)
other than price that affect demand for the good. The coefficient b reflects how the
price of the good affects the quantity demanded.9
Any downward-sloping demand curve has a corresponding inverse demand
curve that expresses price as a function of quantity. We can find the inverse demand
8
In Problem 2.12 at the end of the chapter, you will be asked to sketch the graph of a demand curve
that is perfectly inelastic.
9
However, as you will see soon, the term b is not the price elasticity of demand.
linear demand curve
A demand curve in the
form Q a bP.
inverse demand curve
An equation for the demand
curve that expresses price
as a function of quantity.
46
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
curve by taking the equation for the demand curve and solving it for P in terms of Q.
The inverse demand curve for the linear demand curve is given by
P
choke price The price
at which quantity
demanded falls to 0.
a
1
Q
b
b
The term ab is called the choke price. This is the price at which the quantity
demanded falls to 0.10
Using equation (2.3), we see that the price elasticity of demand for the linear
demand curve in Figure 2.16 is given by the formula
⑀Q,P
¢Q P
P
b
¢P Q
Q
(2.4)
This formula tells us that for a linear demand curve, the price elasticity of demand
varies as we move along the curve. Between the choke price a b (where Q 0) and a
price of a2b at the midpoint M of the demand curve, the price elasticity of demand is
between q and 1. This is known as the elastic region of the demand curve. For
prices between a2b and 0, the price elasticity of demand is between 1 and 0. This
is the inelastic region of the demand curve.
a
b
εQ,P = –∞
ε
Q
or
E
,P b la
et stic
w
ee
FIGURE 2.16 Price Elasticity of
Demand along a Linear Demand Curve
In the region to the northwest of the midpoint M, demand is elastic, with the price
elasticity of demand between minus infinity
and 1. In the region to the southeast of
the midpoint M, demand is inelastic, with
the price elasticity of demand between 1
and 0.
10
Price (dollars per unit)
n
a
2b
Q = a – bP
Q
P= a –
b
b
re
gi
–∞ on
an
d
–1
εQ,P = –1
M
ε
Q
I
,P b nel
et ast
w
ic
ee
n
0
re
–1 gion
an
d
0
a
D
εQ,P = 0
a
2
Quantity (units per year)
You can verify that quantity demanded falls to 0 at the choke price by substituting P
equation of the demand curve:
a
Q a ba b
b
aa
0
ab into the
47
2.2 PRICE ELASTICITY OF DEMAND
Equation (2.4) highlights the difference between the slope of the demand curve,
b, and the price elasticity of demand, b(PⲐQ). The slope measures the absolute
change in quantity demanded (in units of quantity) brought about by a one-unit change
in price. By contrast, the price elasticity of demand measures the percentage change in
quantity demanded brought about by a 1 percent change in price.
You might wonder why we do not simply use the slope to measure the sensitivity
of quantity to price. The problem is that the slope of a demand curve depends on the
units used to measure price and quantity. Thus, comparisons of slope across different
goods (whose quantity units would differ) or across different countries (where prices
are measured in different currency units) would not be very meaningful. By contrast,
the price elasticity of demand expresses changes in prices and quantities in common
terms (i.e., percentages). This allows us to compare the sensitivity of quantity
demanded to price across different goods or different countries.
Constant Elasticity Demand Curves
constant elasticity
demand curve A
Another commonly used demand curve is the constant elasticity demand curve,
given by the general formula: Q aPb, where a and b are positive constants. For the
constant elasticity demand curve, the price elasticity is always equal to the exponent
b.11 For this reason, economists frequently use the constant elasticity demand curve
to estimate price elasticities of demand using statistical techniques.
demand curve of the form
Q ⫽ aPb where a and b
are positive constants. The
term b is the price elasticity of demand along this
curve.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 2 . 6
D
Elasticities along Special Demand Curves
Problem
(a) Suppose a constant elasticity
demand curve is given
1
by the formula Q 200P 2. What is the price elasticity
of demand?
(b) Suppose a linear demand curve is given by the
formula Q ⫽ 400 10P. What is the price elasticity of
demand at P 30? At P 10?
Solution
(a) Since this is a constant elasticity demand curve, the
price elasticity of demand is equal to 1Ⲑ2 everywhere
along the demand curve.
(b) For this linear demand curve, we can find the price
elasticity of demand by using equation (2.4):
⑀Q, P (b)(PⲐQ) Since b 10 and Q 400 10P,
when P 30,
⑀Q,P 10 a
30
b 3
400 10(30)
and when P 10,
⑀Q, P 10 a
10
b 0.33
400 10(10)
Note that demand is elastic at P 30, but it is inelastic at
P 10 (in other words, P 30 is in the elastic region of
the demand curve, while P 10 is in the inelastic region).
Similar Problems:
2.5, 2.6, 2.13
P R I C E E L A S T I C I T Y O F D E M A N D A N D TOTA L R E V E N U E
Businesses, management consultants, and government bodies use price elasticities of
demand a lot. To see why a business might care about the price elasticity of demand,
let’s consider how an increase in price might affect a business’s total revenue, that is,
the selling price times the quantity of product it sells, or PQ. You might think that
11
We prove this result in the appendix to this chapter.
total revenue Selling
price times the quantity of
product sold.
48
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
when the price rises, so will the total revenue, but a higher price will generally reduce
the quantity demanded. Thus, the “benefit” of the higher price is offset by the “cost”
due to the reduction in quantity, and businesses must generally consider this trade-off
when they think about raising a price. If the demand is elastic (the quantity demanded
is relatively sensitive to price), the quantity reduction will outweigh the benefit of the
higher price, and total revenue will fall. If the demand is inelastic (the quantity
demanded is relatively insensitive to price), the quantity reduction will not be too
severe, and total revenue will go up. Thus, knowledge of the price elasticity of demand
can help a business predict the revenue impact of a price increase.
DETERMINANTS OF THE PRICE ELASTICITY OF DEMAND
Price elasticities of demand have been estimated for many products using statistical techniques. Table 2.1 presents these estimates for a variety of food, liquor, and tobacco products
in the United States, while Table 2.2 presents estimates for various modes of transportation. What determines these elasticities? Consider the estimated elasticity of 0.107 for
cigarettes in Table 2.1, which indicates that a 10 percent increase in the price of cigarettes
would result in a 1.07 percent drop in the quantity of cigarettes demanded. This tells us
that cigarettes have an inelastic demand: When the prices of all the individual brands of
cigarettes go up (perhaps because of an increase in cigarette taxes), overall consumption of
cigarettes is not likely to be affected very much. This conclusion makes sense. Even
though consumers might want to cut back their consumption when cigarettes become
more expensive, most would find it difficult to do so because cigarettes are habit forming.
In many circumstances, decision makers do not have precise numerical estimates
of price elasticities of demand based on statistical techniques. Consequently, they have
to rely on their knowledge of the product and the nature of the market to make educated conjectures about price sensitivity.
TABLE 2.1 Estimates of the Price Elasticity of Demand for Selected Food,
Tobacco, and Liquor Products
Product
Cigars
Canned and cured seafood
Fresh and frozen fish
Cheese
Ice cream
Beer and malt beverages
Bread and bakery products
Wine and brandy
Cookies and crackers
Roasted coffee
Cigarettes
Chewing tobacco
Pet food
Breakfast cereal
Estimated
Q,P
0.756
0.736
0.695
0.595
0.349
0.283
0.220
0.198
0.188
0.120
0.107
0.105
0.061
0.031
Source: Emilio Pagoulatos and Robert Sorensen, “What Determines the Elasticity of Industry
Demand,” International Journal of Industrial Organization, 4 (1986): 237–250.
2.2 PRICE ELASTICITY OF DEMAND
TABLE 2.2 Estimates of the Price Elasticity of Demand for Selected Modes
of Transportation
Category
Airline travel, leisure
Rail travel, leisure
Airline travel, business
Rail travel, business
Urban transit
Estimated
between
Q,P
1.52
1.40
1.15
0.70
0.04 and
0.34
Source: Elasticities from the cross-sectional studies summarized in Tables 2, 3, 4 in Tae Hoon Oum,
W. G. Waters II, and Jong-Say Yong, “Concepts of Price Elasticities of Transport Demand and
Recent Empirical Estimates,” Journal of Transport Economics and Policy (May 1992): 139–154.
Here are some factors that determine a product’s price elasticity of demand––that
is, the extent to which demand is relatively sensitive or insensitive to price.
• Demand tends to be more price elastic when there are good substitutes for a product (or,
alternatively, demand tends to be less price elastic when the product has few or
not very satisfactory substitutes). One reason that the demand for airline travel
by leisure travelers is price elastic (as Table 2.2 shows) is that leisure travelers
usually perceive themselves as having reasonably good alternatives to traveling
by air; for example, they can often travel by automobile instead. For business
travelers, automobile travel is usually a less desirable substitute because of the
time-sensitive nature of much business travel. This explains why, as Table 2.2
shows, the price elasticity of demand for business travel is smaller (in absolute
magnitude) than that for leisure travel.
• Demand tends to be more price elastic when a consumer’s expenditure on the product
is large (either in absolute terms or as a fraction of total expenditures). For example,
demand is more elastic for products such as refrigerators or automobiles. By
contrast, demand tends to be less price elastic when a consumer’s expenditure
on the product is small, as is the case for many of the individual grocery items
in Table 2.1. When a consumer must spend a lot of money to buy a product,
the gain from carefully evaluating the purchase and paying close attention to
price is greater than it is when the item does not entail a large outlay of money.
• Demand tends to be less price elastic when the product is seen by consumers as being a
necessity. For example, household demand for water and electricity tends to be
relatively insensitive to price because virtually no household can do without
these essential services.
M A R K E T- L E V E L V E R S U S B R A N D - L E V E L P R I C E
ELASTICITIES OF DEMAND
A common mistake in the use of price elasticities of demand is to suppose that just
because the demand for a product is inelastic, the demand each seller of that product
faces is also inelastic. Consider, for example, cigarettes. As already discussed, the demand
for cigarettes is not especially sensitive to price: an increase in the price of all brands
of cigarettes would only modestly affect overall cigarette demand. However, if the
price of only a single brand of cigarettes (e.g., Salem) went up, the demand for that
brand would probably drop substantially because consumers would switch to the now
49
50
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
lower-priced brands whose prices did not change. Thus, even if demand is inelastic at
the market level, it can be highly elastic at the individual brand level.
The distinction between market-level and brand-level elasticities reflects the
impact of substitution possibilities on the degree to which consumers are sensitive
to price. In the case of cigarettes, for example, a typical smoker needs cigarettes because there are no good alternatives. But that smoker doesn’t necessarily need Salem
cigarettes because, when the price of Salem goes up, switching to another brand will
provide more or less the same degree of satisfaction.
What determines whether a firm should use market-level or brand-level elasticity in assessing the effect of a price change? The answer depends on what the firm
expects its competitors to do. If a firm expects its rivals to quickly match its price
change, then the market-level elasticity will provide the appropriate measure of how
the demand for the firm’s product is likely to change with price. If, by contrast, a firm
expects its rivals not to match its price change (or to do so only after a long time lag),
then the brand-level elasticity is appropriate.
A P P L I C A T I O N
2.3
How People Buy Cars: The
Importance of Brands
Using modern statistical techniques, Steven Berry,
James Levinsohn, and Ariel Pakes estimated price elasticities of demand for numerous makes of automobiles.12 Table 2.3 shows some of their estimates. These
estimates illustrate that demands for individual models of automobiles are highly elastic (between 3.5
and 6.5). By contrast, estimates of the market-level
price elasticity of demand for automobiles generally
fall between 0.8 and 1.5.13 This highlights the distinction between brand-level price elasticity of demand and market-level price elasticity of demand.
Brand-level price elasticities of demand are more
negative than market-level price elasticities of demand
because consumers have greater substitution possibilities when only one firm raises its price. This suggests
that the most negative brand-level elasticities for automobiles should be in those market segments in which
consumers have the greatest substitution possibilities.
The data in Table 2.3 bear this out. The most elastic demands are generally for automobiles in the compact
and subcompact market segments (Mazda 323, Nissan
12
Sentra), which are the most crowded. By contrast, demands for cars in the luxury segment (Lexus LS400,
BMW 735i) are somewhat less price elastic because
there are fewer substitutes for them.
TABLE 2.3 Estimates of Price Elasticities of
Demand for Selected Makes of Automobiles, 1990
Model
Mazda 323
Nissan Sentra
Ford Escort
Chevrolet Cavalier
Honda Accord
Ford Taurus
Buick Century
Nissan Maxima
Acura Legend
Lincoln Town Car
Cadillac Seville
Lexus LS400
BMW 735i
Price
$ 5,039
$ 5,661
$ 5,663
$ 5,797
$ 9,292
$ 9,671
$ 10,138
$ 13,695
$ 18,944
$ 21,412
$24,544
$27,544
$37,490
Estimated
Q,P
6.358
6.528
6.031
6.433
4.798
4.220
6.755
4.845
4.134
4.320
3.973
3.085
3.515
Source: Table V in S. Berry, J. Levinsohn, and A. Pakes,
“Automobile Prices in Market Equilibrium,” Econometrica,
63 (July 1995): 841–890.
S. Berry, J. Levinsohn, and A. Pakes, “Automobile Prices in Market Equilibrium,” Econometrica, 63
( July 1995): 841–890.
13
See, for example, McCarthy, Patrick, “Market Price and Income Elasticities of New Vehicle Demands,”
Review of Economics and Statistics, 78 (August 1996): 543–547.
51
2 . 3 OT H E R E L A S T I C I T I E S
We can use elasticity to characterize the responsiveness of demand to any of the 2.3
determinants of demand. Two of the more common elasticities in addition to the price
elasticity of demand are the income elasticity of demand and the cross-price elasticity
of demand.
OT H E R
ELASTICITIES
INCOME ELASTICITY OF DEMAND
The income elasticity of demand is the ratio of the percentage change of quantity
demanded to the percentage change of income, holding price and all other determinants of demand constant:
⑀Q,I
¢Q
Q
100%
¢I
I
100%
¢Q I
¢I Q
or, after rearranging terms,
Q, I
(2.5)
Table 2.4 shows estimated income elasticities of demand for two different types of
U.S. households: those whose incomes place them below the poverty line and those
whose incomes place them above it. For both types of households, the estimated income
elasticities of demand are positive, indicating that the quantity demanded of the good
increases as income increases. However, it is also possible that income elasticity of demand can be negative. Some studies suggest that in economically advanced countries in
Asia, such as Japan and Taiwan, the income elasticity of demand for rice is negative.14
TABLE 2.4 Income Elasticity of Demand for Selected Food Products
According to Household Status
Product
Beef
Pork
Chicken
Fish
Cheese
Milk
Fruits
Vegetables
Breakfast cereals
Bread
Fats and oils
Food away from home
Estimated Income
Elasticity: Nonpoverty
Status Households
Estimated Income
Elasticity: Poverty Status
Households
0.4587
0.4869
0.3603
0.4659
0.3667
0.4247
0.3615
0.3839
0.3792
0.3323
0.4633
1.1223
0.2657
0.2609
0.2583
0.3167
0.2247
0.2650
0.2955
0.2593
0.2022
0.1639
0.2515
0.6092
Source: Tables 7 and 8, John L. Park, Rodney B. Holcomb, Kellie Curry Raper, and Oral Capps Jr.,
“A Demand Systems Analysis of Food Commodities by U.S. Households Segmented by Income,”
American Journal of Agricultural Economics, 78, no. 2 (May 1996): 290–300.
14
See Shoichi Ito, E. Wesley, F. Peterson, and Warren R. Grant. “Rice in Asia: Is it Becoming an Inferior
Good?,” American Journal of Agricultural Economics, 71 (1989): 32–42.
income elasticity of
demand The ratio of the
percentage change of
quantity demanded to the
percentage change of
income, holding price and
all other determinants of
demand constant.
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CROSS-PRICE ELASTICITY OF DEMAND
cross-price elasticity
of demand The ratio of
the percentage change of
the quantity of one good
demanded with respect to
the percentage change in
the price of another good.
The cross-price elasticity of demand for good i with respect to the price of good j
is the ratio of the percentage change of the quantity of good i demanded to the percentage change of the price of good j:
⑀Qi, Pj ⫽
¢Qi
Qi
⫻ 100%
¢Pj
Pj
⫻ 100%
or, after rearranging terms,
⑀Qi, Pj ⫽
demand substitutes
Two goods related in such
a way that if the price of
one increases, demand for
the other increases.
demand complements
Two goods related in such
a way that if the price of
one increases, demand for
the other decreases.
¢Qi Pj
¢Pj Qi
(2.6)
where Pj denotes the initial price of good j and Qi denotes the initial quantity of
good i demanded. Table 2.5 shows cross-price elasticities of demand for selected
fruit products.
Cross-price elasticity can be positive or negative. If ⑀Qi Pj 7 0, a higher price
for good j increases the quantity of good i demanded. In this case, goods i and j are
demand substitutes. Table 2.5 indicates that apples and peaches are demand substitutes: As the price of peaches increases, the quantity of apples demanded
increases (cross-price elasticity of the demand for apples with respect to the price
of peaches ⫽ 0.118). Likewise, as the price of apples increases, the quantity of
peaches demanded increases (cross-price elasticity of the demand for peaches with
respect to the price of apples ⫽ 0.015).
If ⑀Qi , Pj 6 0, a higher price for good j decreases the quantity of good i demanded.
In this case, goods i and j are demand complements. Table 2.5 indicates that apples
and bananas are demand complements: As the price of bananas increases, the quantity of apples demanded decreases (cross-price elasticity of demand for apples with respect to the price of bananas ⫽ ⫺0.207). Likewise, as the price of apples increases, the
quantity of bananas demanded decreases (cross-price elasticity of demand for bananas
with respect to the price of apples ⫽ ⫺0.409).
TABLE 2.5 Cross-Price Elasticities of Demand for Selected
Fresh Fruits Products
Price of apples
Price of bananas
Price of peaches
a
Demand for
Apples
Demand for
Bananas
Demand for
Peaches
⫺0.586a
⫺0.207b
0.118
⫺0.409
⫺1.199
0.546
0.015
1.082
⫺1.105
This is the price elasticity of demand of apples.
This is the cross-price elasticity of demand of apples with respect to the price of peaches.
Source: Elasticities taken from Table 5 in S. R. Henneberry, K. P. Piewthongngam, and H. Qiang.
“Consumer Safety Concerns and Fresh Produce Consumption,” Journal of Agricultural Resource
Economics, 24 (July 1999): 98–113.
b
53
2 . 3 OT H E R E L A S T I C I T I E S
A P P L I C A T I O N
2.4
How People Buy Cars: The
Importance of Price
Table 2.6 presents estimates of the cross-price elasticities of demand for some of the makes of automobiles
shown in Table 2.3. (The table contains the price elasticities of demand for these makes as well.) The table
shows, for example, that the cross-price elasticity of
demand for Ford Escort with respect to the price of a
Nissan Sentra is 0.054, indicating that the demand for
Ford Escorts goes up at a rate of 0.054 percent for each
1 percent increase in the price of a Nissan Sentra.
Although all of the cross-price elasticities are fairly
small, note that the cross-price elasticities between
compact cars (Sentra, Escort) and luxury cars (Lexus
LS400, BMW 735i) are zero or close to zero. This
makes sense: Compacts and luxury cars are distinct
market segments. Different people buy BMWs than
buy Ford Escorts, so the demand for one should not
be much affected by the price of the other. By contrast, the cross-price elasticities within the compact
segment are relatively higher. This suggests that consumers within this segment view Sentras and Escorts
as substitutes for one another.
TABLE 2.6 Cross-Price Elasticities of Demand for Selected Makes
of Automobiles
Price of Sentra
Price of Escort
6.528a
0.054
0.000
0.000
0.078b
6.031
0.001
0.001
Demand for Sentra
Demand for Escort
Demand for LS400
Demand for 735i
Price of LS400
0.000
0.001
3.085
0.032
Price of 735i
0.000
0.000
0.093
3.515
a
This is the price elasticity of demand for a Sentra.
This is the cross-price elasticity of demand for a Sentra with respect to the price of an Escort.
Sources: Adapted from Table VI in S. Berry, J. Levinsohn, and A. Pakes, “Automobile Prices in
Market Equilibrium,” Econometrica, 63 (July 1995): 841–890.
b
A P P L I C A T I O N
2.5
Coke versus Pepsi15
If the price of Coke goes down, what is the effect on
the demand for Pepsi? And if Pepsi’s price goes
down, how is Coke’s demand affected? Farid Gasmi,
Quang Vuong, and Jean-Jacques Laffont (GVL) studied competitive interactions in the U.S. soft drink
market and estimated demand equations for Coca-
15
Cola and Pepsi.16 Using the average values of prices
and other variables in their study, we can infer the
price elasticity, cross-price elasticity, and income
elasticities of demand for Coke and Pepsi shown in
Table 2.7.17
As you can see in Table 2.7, the cross-price elasticities of demand are positive numbers (0.52 and
0.64). This tells us that a decrease in Coke’s price
This example is based on F. Gasmi, J. J. Laffont, and Q. Vuong, “Econometric Analysis of Collusive
Behavior in a Soft Drink Market,” Journal of Economics and Management Strategy, 1 (Summer 1992):
278–311. It was inspired by the classroom notes of our former colleague Matthew Jackson.
16
In Chapter 13, we will use these demand functions to study price competition between Coke and Pepsi.
17
GVL estimated these demand functions under several different assumptions about market behavior.
The ones reported here correspond to what the authors believe is the best model.
54
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
TABLE 2.7 Price, Cross-Price, and Income
Elasticities of Demand for Coca-Cola and Pepsi
Elasticity
Price elasticity of demand
Cross-price elasticity of demand
Income elasticity of demand
Coca-Cola
Pepsi
–1.47
0.52
0.58
–1.55
0.64
1.38
and a decrease in the price of one brand would
hurt demand for the other. In addition, the demand for both products goes up when consumer
income goes up, indicating that increases in consumer incomes benefit both brands. Finally, the
price elasticity of demand for each brand falls in
the range between 1 and q. Thus, the brandlevel demand for both Coke and Pepsi is elastic.
will decrease the demand for Pepsi, and a decrease
in Pepsi’s price will decrease the demand for Coke.
Thus, consumers view these products as substitutes,
P R I C E E L A S T I C I T Y O F S U P P LY
price elasticity of
supply The percentage
change in quantity supplied
for each percent change in
price, holding all other
determinants of supply
constant.
The price elasticity of supply measures the sensitivity of quantity supplied Q s to
price. The price elasticity of supply––denoted by ⑀Q s,P ––tells us the percentage
change in quantity supplied for each percent change in price:
⑀Q s, P
¢Q s
s
Q
100%
¢P
P
100%
¢Q s P
¢P Q s
This formula applies to both the firm level and the market level. The firm-level
price elasticity of supply tells us the sensitivity of an individual firm’s supply to price,
while the market-level price elasticity of supply tells us the sensitivity of market
supply to price.
2.4
ELASTICITY
IN THE LONG
RUN VERSUS
THE SHORT
RUN
long-run demand
curve The demand curve
that pertains to the period
of time in which consumers
can fully adjust their purchase decisions to changes
in price.
G R E AT E R E L A S T I C I T Y I N T H E L O N G R U N
THAN IN THE SHORT RUN
Consumers cannot always adjust their purchasing decisions instantly in response to a
change in price. For example, a consumer faced with an increase in the price of natural
gas can, in the short run, turn down the thermostat, which will reduce consumption.
But over time, this consumer can reduce natural gas consumption even more by replacing the old furnace with an energy-efficient model. Thus, it is useful to distinguish
between the long-run demand curve for a product––the demand curve that pertains
to the period of time in which consumers can fully adjust their purchase decisions to
changes in price––and the short-run demand curve—the demand curve that pertains
to the period of time in which consumers cannot fully adjust their purchasing decisions
to changes in price. We would expect that for products, such as natural gas, for which
consumption is tied to physical assets whose stocks change slowly, long-run demand
would be more price elastic than short-run demand. Figure 2.17 illustrates this possibility. The long-run demand curve is “flatter” than the short-run demand curve.
Price (dollars per thousand cubic feet)
2 . 4 E L A S T I C I T Y I N T H E L O N G RU N V E R S U S T H E S H O RT RU N
$6
$4
Short-run
demand curve
15
0
Long-run
demand curve
38 40
Quantity (trillions of cubic feet per year)
FIGURE 2.17 ShortRun and Long-Run Demand
Curves for Natural Gas
In the short run, an
increase in the price of
natural gas from $4 to $6
(per thousand cubic feet)
induces consumers to
reduce their quantity
demanded from a rate of
40 trillion cubic feet per
year to 38 trillion cubic
feet per year. In the long
run, though, when
consumers can fully adjust to the price increase
from $4 to $6, the quantity demanded falls to a
rate of 15 trillion cubic
feet per year.
Similarly, firms sometimes cannot fully adjust their supply decisions in response
to changes in price. For example, in the short run, a producer of semiconductors
might not be able to increase its supply of chips in response to an increase in price by
very much because it faces a capacity constraint––a facility can only produce so many
chips, even if extra workers are hired. However, if the price increase is expected to be
permanent, then the firm can expand the capacity of its existing facilities or build new
ones. The increase in the quantity supplied as a result of the price increase will thus
be greater in the long run than in the short run. Figure 2.18 illustrates the distinction
Short-run
supply curve
55
short-run demand
curve The demand curve
that pertains to the period
of time in which consumers
cannot fully adjust their
purchase decisions to
changes in price.
Long-run
supply curve
Price (dollars per megabyte)
$20
$10
0
100 120
250
Quantity (million megabytes per year)
FIGURE 2.18 Short-Run and LongRun Supply Curves for Semiconductors
In the short run, an increase in the price of
semiconductors from $10 to $20 per
megabyte induces a small increase in the
quantity supplied (from 100 million to 120
million megabytes of chips per year). In the
long run, though, when producers can
fully adjust to the price increase, the longrun supply curve applies and the quantity
supplied rises to a rate of 250 million
megabytes of chips per year.
56
CHAPTER 2
long-run supply curve
between the long-run supply curve––the supply curve that pertains to the period of
time in which sellers can fully adjust their supply decisions in response to changes in
price, and the short-run supply curve––the supply curve that pertains to the period
of time in which sellers cannot fully adjust their supply decisions in response to a
change in price. Figure 2.18 shows that for a good such as semiconductors the longrun supply curve is flatter than the short-run supply curve.
The supply curve that pertains to the period of time
in which producers can fully
adjust their supply decisions
to changes in price.
short-run supply curve
The supply curve that
pertains to the period of
time in which sellers cannot
fully adjust their supply
decisions in response to
changes in price.
durable goods Goods,
such as automobiles or
airplanes, that provide
valuable services over
many years.
D E M A N D A N D S U P P LY A N A LYS I S
G R E AT E R E L A S T I C I T Y I N T H E S H O R T R U N
THAN IN THE LONG RUN
For certain goods, long-run market demand can be less elastic than short-run demand.
This is particularly likely to be true for goods such as automobiles or airplanes––
durable goods––that provide valuable services over many years. To illustrate this point,
consider the demand for commercial airplanes. Suppose that Boeing and Airbus (the
world’s two producers of commercial aircraft) are able to raise the prices of new commercial aircraft. It seems unlikely that this would dramatically affect the demand for
aircraft in the long run: Airlines, such as United and British Airways, need aircraft to do
their business. There are no feasible substitutes.18 But in the short run, the impact of
higher aircraft prices might be dramatic. Airlines that might have operated an aircraft
for 15 years might now try to get an extra 2 or 3 years out of it before replacing it. Thus,
A P P L I C A T I O N
2.6
Crude Oil: Price and Demand
Using data on oil prices and oil consumption over the
years 1970 through 2000, John C. B. Cooper estimated
short-run and long-run price elasticities of demand
for crude oil for 23 different countries.19 Table 2.8
shows estimates for some of the countries he studied.
For example, the short-run price elasticity of demand
for oil in Japan was estimated to be 0.071, while the
long-run price elasticity of demand was estimated to
be 0.357.
For all countries, demand in the short run is highly
price inelastic. Even though demand in the long run is
also price inelastic, it is less so than in the short run.
This is consistent with the idea that, in the long run,
buyers of oil make adjustments to their consumption
in response to higher or lower prices but do not make
such adjustments in the short run.
18
TABLE 2.8 Long-Run and Short-Run Price
Elasticities of Demand for Crude Oil in Selected
Countries
Price Elasticity
Country
Australia
France
Germany
Japan
Korea
Netherlands
Spain
United Kingdom
United States
Short-Run
Long-Run
–0.034
–0.069
–0.024
–0.071
–0.094
–0.057
–0.087
–0.068
–0.061
–0.068
–0.568
–0.279
–0.357
–0.178
–0.244
–0.146
–0.182
–0.453
That is not to say there would be no impact on demand. Higher aircraft prices may raise the costs of
entering the airline business sufficiently that some prospective operators of airlines would choose to stay
out of the business.
19
John C. B. Cooper, “Price Elasticity of Demand for Crude Oil: Estimates for 23 Countries,” OPEC
Review (March 2003): 3–8.
57
Price (milions of dollars per airplane)
2 . 5 BAC K - O F - T H E - E N V E L O P E C A L C U L AT I O N S
1.25
1.00
Long-run
demand curve
0
Short-run
demand curve
180 360 400
Quantity (airplanes per year)
FIGURE 2.19 Short-Run and Long-Run
Demand Curves for Commercial Aircraft
An increase in the price of a commercial
aircraft from $1 million to $1.25 million per
airplane is likely to reduce the long-run rate
of demand only modestly, from 400 to 360
aircraft per year, as illustrated by the long-run
demand curve. However, in the short run
(e.g., the first year after the price increase),
the rate of demand will fall more dramatically,
from 400 aircraft per year to just 180 aircraft
per year, as shown by the short-run demand
curve. Eventually, though, as existing aircraft
wear out, the rate of demand will rise to the
long-run level (360 aircraft per year), corresponding to the new price of $1.25 million
per airplane.
while demand for new commercial aircraft in the long run might be relatively price
inelastic, in the short run (within 2 or 3 years of the price change), demand would be
relatively more elastic. Figure 2.19 shows this possibility. The steeper demand curve
corresponds to the long-run effect of the price increase in the total size of aircraft
fleets worldwide; the flatter demand curve shows the effect of the price increase on
orders for new aircraft in the first year after the price increase.
For some goods, long-run market supply can be less elastic than short-run market
supply. This is especially likely to be the case for goods that can be recycled and resold
in the secondary market (i.e., the market for used or recycled goods). For example, in
the short run an increase in the price of aluminum would elicit an increased supply
from two sources: additional new aluminum and recycled aluminum made from scrap.
However, in the long run, the stock of scrap aluminum will diminish, and the increase
in quantity supplied induced by the increased price will mainly come from the production of new aluminum.
S
o where do demand curves come from, and how do you derive the equation of a
demand function for a real product in a real market? One approach to determining
demand curves involves collecting data on the quantity of a good purchased in a
market, the prices of that good, and other possible determinants of that good’s demand
and then applying statistical methods to estimate an equation for the demand function
that best fits the data. This broad approach is data-intensive: the analyst has to collect
enough data on quantities, prices, and other demand drivers, so that the resulting statistical estimates are sensible. However, analysts often lack the resources to collect
enough data for a sophisticated statistical analysis, so they need some techniques that
allow them, in a conceptually correct way, to infer the shape or the equation of a demand curve from fragmentary information about prices, quantities, and elasticities.
These techniques are called back-of-the-envelope calculations because they are simple
enough to do on the back of an envelope.
2.5
BAC K - O F - T H E ENVELOPE
CALCULATIONS
58
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
F I T T I N G L I N E A R D E M A N D C U RV E S U S I N G Q UA N T I T Y,
P R I C E , A N D E L A S T I C I T Y I N F O R M AT I O N
Often, you can obtain information on the prevailing or typical prices and quantities
within a particular market as well as estimates of the price elasticity of demand in that
market. These estimates might come from statistical studies (this is where the elasticities
in Tables 2.1, 2.2, and 2.3 came from) or the judgments of informed observers (e.g.,
industry participants, investment analysts, consultants). If you assume as a rough approximation that the equation of the demand curve is linear (i.e., Q a b P), you can then
derive the equation of this linear demand (i.e., the values of a and b) from these three
pieces of information (prevailing price, prevailing quantity, and estimated elasticity).
The approach to fitting a linear demand curve to quantity, price, and elasticity
data proceeds as follows. Suppose Q* and P * are the known values of quantity and
price in this market, and ⑀Q, P is the estimated value of the price elasticity of demand.
Recall the formula for the price elasticity of demand for a linear demand function.
⑀Q, P b
P*
Q*
(2.7)
b ⑀Q, P
Q*
P*
(2.8)
Solving equation (2.7) for b yields
To solve for the intercept a, we note that Q* and P* must be on the demand curve.
Thus, it must be that Q* a bP*, or a Q* bP*.
Substituting the expression in equation (2.8) for b gives
a Q* a⑀Q, P
Q*
b P*
P*
Then, by canceling P* and factoring out Q*, we get
a (1 ⑀Q, P)Q*
(2.9)
Taken together, equations (2.8) and (2.9) provide a set of formulas for generating
the equation of a linear demand curve.
We can illustrate the fitting process using data on the price and consumption of
chicken in the United States. In 1990, the per capita consumption of chicken in the
United States was about 70 pounds per person, while the average inflation-adjusted
retail price of chicken was about $0.70 per pound. Demand for chicken is relatively
price inelastic, with estimates in the range of 0.5 to 0.6.20 Thus,
Q* 70
P* 0.70
⑀Q, P 0.55 (splitting the difference)
20
All data are from Richard T. Rogers, “Broilers: Differentiating a Commodity,” in Larry Duetsch, ed.,
Industry Studies (Englewood Cliffs, NJ: Prentice Hall, 1993), pp. 3–32. See especially the data summarized on pp. 4–6.
2 . 5 BAC K - O F - T H E - E N V E L O P E C A L C U L AT I O N S
Price (dollars per pound)
$2.00
$1.50
$1.00
Observed price
and quantity
$0.70
$0.50
D
0
20
100
40
60
70
80
Quantity (pounds per person per year)
120
FIGURE 2.20 Fitting a
Linear Demand Curve to
Observed Market Data
A linear demand curve D has
been fitted to the observed
data in the U.S. market for
chicken.
Applying equations (2.8) and (2.9), we get
b (0.55)
70
55
0.70
a [1 (0.55)]70 108.5
Thus, the equation of our demand curve for chicken in 1990 is Q 108.5 55P.
This curve is depicted in Figure 2.20.
I D E N T I F Y I N G S U P P LY A N D D E M A N D C U RV E S
O N T H E BAC K O F A N E N V E L O P E
Earlier in this chapter, we discussed how exogenous factors can cause shifts in demand
and supply that alter the equilibrium prices and quantities in a market. In this section,
we show how information about such shifts and observations of the resulting market
prices can be used to do back-of-the-envelope derivations of supply and demand
curves.
We will use a specific example to illustrate the logic of the analysis. Consider the
market for crushed stone in the United States in the late 2000s. Let’s suppose that the
market demand and supply curves for crushed stone are linear: Q d a b P and
Q s f h P. Since we expect the demand curve to slope downward and the supply
curve to slope upward, we expect that b 0 and h 0.
Now, suppose that we have the following information about the market for
crushed stone between 2006 and 2010:
• Between 2006 and 2008, the market was uneventful. The market price was $9
per ton, and 30 million tons were sold each year.
• In 2009, there was a 1-year burst of highway building as a result of the Obama
administration’s economic stimulus plan. The market price of crushed stone
rose to $10 per ton, and 33 million tons were sold.
59
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
D2008
D2009
S2010
S2008
Price ($ per ton)
60
$10
$9
0
28 30
Quantity (millions of tons per year)
33
FIGURE 2.21 Identifying Demand and Supply Curves from Observed Price
and Quantity Changes
The market for crushed stone is in equilibrium during the years 2006 through 2008. In
2009, a one-year burst of highway construction activity shifts the demand curve rightward
to D2009. The market moves along the supply curve S2008, so the change in equilibrium
price and quantity identifies the slope of the supply curve S2008. In 2010, the demand curve
shifts back to D2008, but the supply curve shifts leftward to S2010 due to an increase in the
wages of workers in the crushed stone industry. The market thus moves along the demand
curve D2008, so the change in the equilibrium price and quantity identifies the slope of the
demand curve D2008.
• By 2010, the burst of new construction had ended. A new union contract raised
the wages of workers in the crushed stone industry. The market price of crushed
stone was $10 per ton, and 28 million tons were sold.
Let’s now put this information to work. The 1-year burst of highway building in
2009 most likely resulted in a rightward shift in the demand curve for crushed stone.
Let’s assume that shift is parallel, as shown in Figure 2.21. On the assumption that
there was no reason for any appreciable shift in the supply curve during the period
2006–2009, the rightward shift in demand allows us to compute the slope of the supply curve because the 2006–2008 and the 2009 market equilibria both fall along the
initial supply curve, labeled S2008 in Figure 2.21.
h slope of S2008
¢Q*
33 million 30 million
3 million
¢P *
10 9
Therefore, the shift in demand identifies the slope of the supply curve. It may seem
curious that it takes a shift in demand to provide information about the supply curve,
but on reflection, it really isn’t that surprising. The shift in demand moves the market
along a particular supply curve and thus tells us how sensitive the quantity supplied is
2 . 5 BAC K - O F - T H E - E N V E L O P E C A L C U L AT I O N S
to the price. Similarly, the shift in the market supply of crushed stone caused by the
rise in wage rates identifies the slope of the demand curve, labeled D2008 in Figure 2.21. Note that the burst of highway construction subsided in 2010, so in that year
the demand curve for crushed stone reverted to its initial position, and the shift in supply (also assumed to be parallel) thus moved the market along the demand curve D2008.
b slope of D2008
¢Q * 28 million 30 million
2 million
¢P *
10 9
Note the unifying logic that was used in both calculations. Knowing that one curve
shifted while the other did not allowed us to calculate the slope of the curve that did
not shift.
Having calculated the slopes of the demand and supply curves, we can now work
backward to calculate the intercepts a and f of the demand and supply curves for 2010.
Since we know that 28 million tons were sold at $10 per ton during that year, the following equations must hold:
28 a (2 10)
28 f (3 10)
(demand)
(supply)
Solving these equations gives a 48 and f 2. Thus, the demand and supply curves
for this market in 2010 were Q d 48 2P and Q s 2 3P.
Having identified equations for the demand and supply curves, we can now use
them to forecast how changes in demand or supply will affect the equilibrium price
and quantity. For example, suppose we expected that in the year 2011 another burst
of new road construction would increase the demand for crushed stone by 15 million
tons per year no matter what the price. Suppose, further, that supply conditions were
expected to resemble those in 2010. At equilibrium, Q d Q s, so we could forecast the
equilibrium price by solving the equation 48 2P 15 2 3P, which gives P
$13 per ton. The equilibrium quantity in the year 2011 would be expected to equal
2 3(13) 37 million tons. Our back-of-the-envelope analysis provides us with a
“quick and dirty” way to forecast future price and quantity movements in this market.
There is an important limitation to this analysis. We can identify the slope of the
demand curve by a shift in supply only if the demand curve remains fixed, and we can
identify the slope of the supply curve by a shift in demand only if the supply curve
stays fixed. If both curves shift at the same time, then we are moving along neither a
given demand curve nor a given supply curve, so changes in the equilibrium quantity
and the equilibrium price cannot identify the slope of either curve.
IDENTIFYING THE PRICE ELASTICITY OF DEMAND
F R O M S H I F T S I N S U P P LY
In the preceding section, we used actual changes in prices and quantities to identify
the equations of supply or demand curves. In some instances, however, we might not
know the change in the equilibrium quantity for a product, but we might have a good
idea about the extent to which its supply curve has shifted. (Business-oriented newspapers such as The Wall Street Journal or the Financial Times often carry reports about
supply conditions in markets for agricultural products, metals, and energy products.)
If we also know the extent to which the market price has changed (which is also widely
61
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
Price (dollars per unit)
S1
S1
S2
P1
P2
D
Quantity (units per year)
(a) Effect on price of shift in supply when
demand is relatively elastic
Price (dollars per unit)
62
S2
P1
P2
D
Quantity (units per year)
(b) Effect on price of shift in supply when
demand is relatively inelastic
FIGURE 2.22 Effect of Supply Shift on Price Depends on the Price Elasticity of Demand
In (a) demand is relatively elastic, and a shift in supply would have a modest impact on price. In
(b) demand is relatively inelastic, and the identical shift in supply has a more dramatic impact
on the equilibrium price.
reported for many markets), we can use this information to assess the degree to which
the demand for the product is price elastic or inelastic.
Figure 2.22 illustrates this point. Panel (a) in Figure 2.22 shows that when
demand is relatively elastic, a given shift in supply (from S1 to S2) would have a modest
impact on the equilibrium price. But when demand is relatively inelastic, as in panel
(b) in Figure 2.22, the same shift in supply would have a more pronounced impact on
the equilibrium price. Figure 2.22 teaches us that when a modest change in supply has
a large impact on the market price of a product, the demand for that product is most
likely price inelastic. By contrast, when a large shift in supply for a product has a relatively small impact on the market price, demand for the product is likely to be relatively
elastic.
A P P L I C A T I O N
2.7
What Hurricane Katrina Tells Us
about the Price Elasticity of Demand
for Gasoline
Gasoline prices tend to be highly volatile. Figure 2.23
illustrates this by plotting the average retail gasoline
price in the United States in 2005.21 Large swings in
price in short periods of time are common, as are
seasonal fluctuations. The seasonal changes are
largely attributable to shifts in demand. Gasoline
21
prices usually rise in the spring through late summer, due to warmer weather, closed schools, and
summer vacations. They are usually lower in winter.
Gasoline prices can also fluctuate due to changes
in crude oil prices, since gasoline is refined from
crude oil.
In addition to these factors, gasoline prices are
highly responsive to changes in supply. Prices may
change dramatically if there are disruptions to
the supply chain. Typical inventory levels of commercial gasoline usually amount to only a few days of
These data are from the Energy Information Administration of the U.S. government.
2 . 5 BAC K - O F - T H E - E N V E L O P E C A L C U L AT I O N S
63
$3.15
Price of gasoline (per gallon)
$3.00
$2.85
$2.70
$2.55
$2.40
$2.25
$2.10
$1.95
$1.80
Jan 01 Jan 31 Mar 02 Apr 01 May 01 May 31 Jun 30 Jul 30 Aug 29 Sep 28 Oct 28 Nov 27 Dec 27
Date during 2005
FIGURE 2.23 U.S. Price of Gasoline, 2005
During 2005, the price of gasoline in the United States fluctuated greatly, reaching a high
of over $3 per gallon in early September 2005.
consumption. If a refinery or pipeline goes offline,
gasoline prices can spike quickly.
This was especially evident in the aftermath of
Hurricane Katrina, which hit Louisiana and the Gulf
Coast on August 29, 2005.22 This region plays a large
role in the U.S. oil and gasoline industries in several
ways. Oil rigs in the gulf produce roughly 25 percent
of total U.S. crude oil. The Louisiana Offshore Oil Port
(LOOP) receives delivery from oil tankers bringing additional supply to the United States. Many oil refineries operate in Louisiana, Mississippi, or Texas. Finally,
pipelines run from this region to the East Coast and
Midwest of the country.
Damage to an oil rig, refinery, pipeline, or LOOP
could cause a spike in oil prices, but Katrina affected
all of them simultaneously. Immediately after the
storm, nearly all petroleum production in the Gulf of
Mexico halted temporarily. LOOP closed for several
22
days. Pipeline capacities fell as well. Many refineries
were damaged or cut off from power and staff,
and were taken offline. Refining capacity fell by
approximately 2 million barrels per day. According to
government figures, supply fell by approximately
8.3 percent in August 2005.
From August 29 to September 5, retail gasoline
prices rose 17.5 percent. That increase was on top of
an additional price increase in late August in anticipation of Katrina’s being a major hurricane. In total,
gasoline prices were about 33.5 percent higher than
they had been a month before. Prices soon began to
decline again as supply increased to more normal
levels. This increase in supply partly reflected gradual repairing of the oil and gasoline supply chain,
and partly temporary government policies to increase short-term supply. LOOP and the pipelines returned to nearly full capacity quickly. On August 31,
“Oil and Gas: Supply Issues after Katrina,” Congressional Research Service, Library of Congress,
September 2005.
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the U.S. government authorized loans of crude oil
from the Strategic Petroleum Reserve totaling about
12.5 million barrels. The International Energy
Agency coordinated a similar global response. The
Environmental Protection Agency temporarily
waived some gasoline and diesel fuel standards that
applied to some regions, allowing the industry to
better balance supply and demand across the country.
By mid-November 2005, gasoline prices returned to
pre-Katrina levels.
Why do changes in supply have such a large
impact on the price of gasoline? The logic of the preceding section tells us that the demand for gasoline is
probably quite inelastic. In fact, we can use data on
gasoline supply and prices to determine approximately how inelastic short-run demand for gasoline is.
Figure 2.24 shows how.
The decrease in supply of gasoline following
Katrina is depicted as a leftward shift in the supply
curve, from S0 to S1. If the supply curve shifts leftward
by 8.3 percent, the equilibrium quantity demanded
must decrease, but by less than the amount of the
P
supply shift, as Figure 2.24 shows. We can conclude
the following:
• Percent change in equilibrium price of gasoline
(%P) 17.5% to 33.5%, depending on
whether we include the price rise in anticipation
of Katrina.
• Percent change in equilibrium quantity of gasoline demanded (%Q) is between 0% and
8.3%.
Taken together, these numbers imply that the price
elasticity of demand for gasoline (%Q)/ (%P) is
between 0 and (8.3)/ 17.5 0.47. If we include
the anticipatory price increase, the price elasticity is
between 0 and (8.3)/ 33.5 0.24. This tells us that
short-term demand for gasoline is inelastic. This conclusion makes sense. In the short run it is difficult for
consumers to change commuting methods or cancel
summer vacations, so that consumption does not
change much when the price of gasoline goes up.
D
S1
S0
8.3%
decrease
in supply
Increase in
price = 17.5%
to 33.5%
Decrease in
equilibrium
quantity < 8.3%
Q
FIGURE 2.24 The Gasoline Market after Hurricane Katrina
Immediately after Hurricane Katrina in 2005, gasoline supply fell by approximately 8.3 percent.
This is reflected by the leftward shift in supply from S0 to S1. Assuming that demand remains
fixed, this supply shift translates into a decreased equilibrium quantity of less than 8.3 percent.
Retail gasoline prices rose 17.5 percent in the week after Katrina, and 33.5 percent including
the price rise in anticipation right before Katrina. This implies a price elasticity of demand
between 0 and 0.47.
65
2 . 5 BAC K - O F - T H E - E N V E L O P E C A L C U L AT I O N S
A P P L I C A T I O N
2.8
The California Energy Crisis 23
The California energy crisis of 2000 and 2001 attracted
attention from around the world. During the first four
months of 2001, the average wholesale price of electricity was about 10 times the price in 1998 and 1999.
Even at these high prices, many customers were forced
to cut back on their consumption of electricity because
of supply shortages. California’s two largest electric
utilities, Pacific Gas & Electric and Southern California
Edison, were buying electricity at wholesale prices that
were higher than the retail prices they were allowed to
charge. The electric utility industry was threatened
with bankruptcy. How did the crisis arise?
Figure 2.25 provides a simplified illustration of the
structure of the electric power industry. Electricity is
typically generated at plants that convert other forms
of energy (such as nuclear power, hydroelectric power,
natural gas, oil, coal, solar power, and wind) to electricity. In California, there were four large firms generating electricity, along with a number of smaller firms.
The generators sell electricity at wholesale prices. It
flows through the transmission grid, a large network
that delivers electricity to local electric utilities and
some large industrial users. Electric utilities then
Generation
4 large firms
and smaller
firms
Sell at
wholesale
price
Transmission
Grid
distribute the power to retail customers, including
residential and business customers.
In the early 1990s the California electric power industry was heavily regulated. The California Public
Utilities Commission (PUC) set electricity prices after
reviewing production costs. Because production costs
and prices were among the highest in the country, the
PUC began a major review of the industry in 1993.
After four years of highly politicized debate, a new
set of complex rules emerged for California’s electricity market. Wholesale prices were deregulated, but
the PUC continued to set retail prices, holding them
essentially fixed. Before the reform, investor-owned
electric utilities produced electricity from generating
plants they owned. Following the restructuring, the
utilities were required to sell most of their generating
plants and then obliged to buy power at the unregulated wholesale prices.
The reforms seemed to be working well until several events simultaneously shocked the electricity
market between 1999 and early 2001. The supply of
electricity in wholesale markets shifted to the left as
the amount of power from hydroelectric generators
fell by 50 percent, the price of natural gas rose sixfold, and power outages removed some generators
Distribution
of power by
electric utilities
to retail
customer
Utilities pay
wholesale price
plus price of
transmission
Sell to retail
customers
Retail price
was fixed by
regulators
FIGURE 2.25 Structure of the Electric Power Industry in California
Electricity flows from firms that generate electric power; those firms sell the electricity at
the wholesale price to large industrial users and to local electric power utilities. The utilities
distribute the electricity to retail customers at the retail price.
23
This discussion draws from Paul Joskow, “California’s Energy Crisis,” Oxford Review of Economic Policy,
17, no. 3 (2001): 365–388.
Retail
Customer
Buy at retail
price
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FIGURE 2.26
The California Energy
Crisis: The Wholesale Market
Between 1999 and early 2001, the supply of
electricity in wholesale markets shifted to
the left as the supply of power from hydroelectric sources fell by 50 percent, the price
of natural gas rose by 600 percent, and
power outages increased by a factor of 10.
The demand for electricity also shifted to
the right. Because the supply and demand
curves were steeply sloped, the shifts in both
curves led to a sharp increase in the price of
electricity in the wholesale market.
Price ($ per megawatt-hour)
from production. The amount of power that
California could import from neighboring states also
declined. The demand for electricity also shifted to
the right, increasing by about 12 percent.
The steep slopes of the supply and demand curves
in Figure 2.26 help to explain why wholesale prices
rose so dramatically during the crisis. The supply of
electricity is relatively inelastic because California had
severely limited the construction of new generating
capacity over the past two decades. When generators
needed to produce more electricity, they had to
utilize older, less efficient plants, many of which were
fueled by natural gas. The demand for electricity is
also relatively inelastic because electricity is essential
for many consumers and producers. Because the supply and demand curves were steeply sloped, the shifts
in both curves led to a sharp increase in the price of
electricity in wholesale markets in early 2001.
As the crisis unfolded, the state of California
sought to ensure that the shortages experienced during the crisis would not occur in the future. It made a
decision that threatened its financial viability, entering into long-term contracts to purchase electricity at
very high prices, a move that it soon regretted. By the
latter part of 2001, wholesale prices had returned to
the levels prevailing before the crisis. The decline in
prices in part reflected several developments that
shifted the supply curve back to the right, as natural
gas prices fell, several new plants began to produce
in the summer of 2001, and significant generating
capacity that had been unavailable earlier in the year
returned to service. In addition, measures to conserve
electricity during the crisis may have shifted the
demand curve to the left, contributing further to a
decline in prices.
While there were several flaws in the design of
the public policy shaping the industry, two stand out
above the others. First, because the PUC held retail
prices at a low level, customers had little incentive to
cut back their consumption of electricity, even
though wholesale prices rose substantially. As Paul
Joskow observed, “Competitive electricity markets
will not work well if consumers are completely insulated from wholesale market price. . . . Not only did
this drive the utilities to the point of insolvency after
wholesale prices rose above the fixed retail price in
June 2000, but it also made it very difficult for competing retail suppliers to attract customers or for consumers to respond to high prices by reducing
consumption.” Second, in the wake of the crisis there
have been allegations that, with four large suppliers,
wholesale markets might not have been competitive
and that some producers might have strategically
withdrawn capacity to drive prices higher. Some analysts have suggested that, prior to deregulation, the
generating sector of the industry should have been
restructured to have more, smaller generating firms
to ensure that producers acted as price takers, and
not price makers.
Searly 2001
Pearly 2001
S1999
P1999
D1999 Dearly 2001
Quantity (megawatthours per month)
REVIEW QUESTIONS
67
CHAPTER SUMMARY
• The market demand curve shows the quantity that
consumers are willing to purchase at different prices.
The market supply curve shows the quantity that producers are willing to sell at different prices. (LBD
Exercises 2.1 and 2.2)
• Market equilibrium occurs at the price at which
quantity supplied equals quantity demanded. At this
price, the supply curve and the demand curve intersect.
(LBD Exercise 2.3)
• Comparative statics analysis on the market equilibrium involves tracing through the effect of a change in
exogenous variables, such as consumer income, the
prices of other goods, or the prices of factors of production, on the market equilibrium price and quantity.
(LBD Exercise 2.4)
• The price elasticity of demand measures the sensitivity of quantity demanded to price. It is the percentage
change in quantity demanded per percentage change in
price. (LBD Exercise 2.5)
• Commonly used demand curves include the constant
elasticity demand curve and the linear demand curve.
The price elasticity of demand is constant along a constant elasticity demand curve, while it varies along a linear demand curve. (LBD Exercise 2.6)
• A product’s demand tends to be more price elastic
when good substitutes are available and when the product represents a significant fraction of buyers’ total expenditures. A product’s demand tends to be less price
elastic when it has few good substitutes, when it represents a small fraction of buyers’ total expenditures, and
when it is seen as a necessity by buyers.
• It is important to distinguish between market-level
price elasticities of demand and brand-level price elasticities of demand. Demand can be price inelastic
at the market level but highly price elastic at the
brand level.
• Other key elasticities include the income elasticity of
demand and the cross-price elasticity of demand.
• For many products, long-run demand is likely to be
more price elastic than short-run demand. However,
for durable goods, such as commercial aircraft, long-run
demand is likely to be less price elastic than short-run
demand.
• Similarly, long-run supply for many goods is likely to
be more price elastic than short-run supply. However,
for products that can be recycled, long-run supply can
be less price elastic than short-run supply.
• Several back-of-the-envelope techniques can be used
to fit demand and supply curves to observed market
data. If you have price, quantity, and price elasticity
of demand data, you can fit a demand curve to observed
data. Information on price movements, coupled with
knowledge that the demand curve has shifted, can be
used to identify a stationary supply curve. Knowledge
that the supply curve has shifted can be used to identify
a stationary demand curve.
REVIEW QUESTIONS
1. Explain why a situation of excess demand will result
in an increase in the market price. Why will a situation of
excess supply result in a decrease in the market price?
2. Use supply and demand curves to illustrate the impact of the following events on the market for coffee:
a) The price of tea goes up by 100 percent.
b) A study is released that links consumption of caffeine
to the incidence of cancer.
c) A frost kills half of the Colombian coffee bean crop.
d) The price of styrofoam coffee cups goes up by 300
percent.
3. Suppose we observe that the price of soybeans goes
up, while the quantity of soybeans sold goes up as well.
Use supply and demand curves to illustrate two possible
explanations for this pattern of price and quantity changes.
4. A 10 percent increase in the price of automobiles
reduces the quantity of automobiles demanded by 8 percent. What is the price elasticity of demand for automobiles?
5. A linear demand curve has the equation Q ⫽
50 ⫺ 100P. What is the choke price?
6. Explain why we might expect the price elasticity of
demand for speedboats to be more negative than the
price elasticity of demand for light bulbs.
7. Many business travelers receive reimbursement
from their companies when they travel by air, whereas
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vacation travelers typically pay for their trips out of their
own pockets. How would this affect the comparison between the price elasticity of demand for air travel for
business travelers versus vacation travelers?
8. Explain why the price elasticity of demand for an entire product category (such as yogurt) is likely to be less
negative than the price elasticity of demand for a typical
brand (such as Dannon) within that product category.
9. What does the sign of the cross-price elasticity of
demand between two goods tell us about the nature of
the relationship between those goods?
10. Explain why a shift in the demand curve identifies
the supply curve and not the demand curve.
PROBLEMS
2.1. The demand for beer in Japan is given by the following equation: Q d ⫽ 700 ⫺ 2P ⫺ PN 0.1I, where P is
the price of beer, PN is the price of nuts, and I is average
consumer income.
a) What happens to the demand for beer when the price
of nuts goes up? Are beer and nuts demand substitutes or
demand complements?
b) What happens to the demand for beer when average
consumer income rises?
c) Graph the demand curve for beer when and PN ⫽ 100
and I ⫽ 10,000.
2.5. The demand curve for ice cream in a small town
has been stable for the past few years. In most months,
when the equilibrium price is $3 per serving for the most
popular ice cream, customers buy 300 servings per month.
For one month the price of materials used to make ice
cream increased, shifting the supply curve to the left.
The equilibrium price in that month increased to $4, and
customers bought only 200 portions in the month. With
these data draw a graph of a linear demand curve for ice
cream in the town. Find price elasticity of demand for
prices equal to $3 and $4. At what price would the demand be unitary elastic?
2.2. Suppose the demand curve in a particular market is
given by Q ⫽ 5 ⫺ 0.5P.
a) Plot this curve in a graph.
b) At what price will demand be unitary elastic?
2.6. Granny’s Restaurant sells apple pies. Granny knows
that the demand curve for her pies does not shift over
time, but she wants to learn more about that demand. She
has tested the market for her pies by charging different
prices. When she charges $4 per pie, she sells 30 pies per
week. When she charges $5, she sells 24 pies per week. If
she charges $4.50, she sells 27 apple pies per week.
a) With these data draw a graph of the linear demand
curve for Granny’s apple pies.
b) Find the price elasticity of demand at each of the
three prices.
2.3. The demand and supply curves for coffee are given
by Q d ⫽ 600 ⫺ 2P and Q s ⫽ 300 4P.
a) Plot the supply and demand curves on a graph and
show where the equilibrium occurs.
b) Using algebra, determine the market equilibrium
price and quantity of coffee.
2.4. Suppose that demand for bagels in the local store is
given by equation Q d ⫽ 300 ⫺ 100P. In this equation, P
denotes the price of one bagel in dollars.
a) Fill in the following table:
0.10
P
0.45
0.50
0.55
d
Q
⑀Q, P
b)
c)
d)
e)
Plot this curve in a graph. Is it linear?
At what price is demand unitary elastic?
At what price is demand inelastic?
At what price is demand elastic?
2.50
2.7. Every year there is a shortage of Super Bowl tickets at the official prices P0. Generally, a black market
(known as scalping) develops in which tickets are sold for
much more than the official price. Use supply and demand analysis to answer these questions:
a) What does the existence of scalping imply about the
relationship between the official price P0 and the equilibrium price?
b) If stiff penalties were imposed for scalping, how
would the average black market price be affected?
2.8. You have decided to study the market for freshpicked cherries. You learn that over the last 10 years,
cherry prices have risen, while the quantity of cherries
purchased has also risen. This seems puzzling because
you learned in microeconomics that an increase in price
PROBLEMS
usually decreases the quantity demanded. What might
explain this seemingly strange pattern of prices and consumption levels?
2.9. Suppose that, over a period of 6 months, the price
of corn increased. Yet, the quantity of corn sold by producers decreased. Does this contradict the law of supply?
If not, why not?
2.10. Explain why a good with a positive price elasticity of demand must violate the law of demand.
2.11. Suppose that the quantity of corn supplied
depends on the price of corn (P) and the amount of rainfall (R). The demand for corn depends on the price of
corn and the level of disposable income (I). The equations describing the supply and demand relationships are
Q s 20R 100P and Q d 4000 100P 10I.
a) Sketch a graph of demand and supply curves that
shows the effect of an increase in rainfall on the equilibrium price and quantity of corn.
b) Sketch a graph of demand and supply curves that
shows the effect of a decrease in disposable income on the
equilibrium price and quantity of corn.
2.12. Recall that when demand is perfectly inelastic,
⑀Q, P 0.
a) Sketch a graph of a perfectly inelastic demand curve.
b) Suppose the supply of 1961 Roger Maris baseball
cards is perfectly inelastic. Suppose, too, that renewed
interest in Maris’s career caused by Mark McGwire and
Sammy Sosa’s quest to break his home run record in
1998 caused the demand for 1961 Maris cards to go up.
What will happen to the equilibrium price? What will
happen to the equilibrium quantity of Maris baseball
cards bought and sold?
2.13. Consider a linear demand curve, Q 350 7P.
a) Derive the inverse demand curve corresponding to
this demand curve.
b) What is the choke price?
c) What is the price elasticity of demand at P 50?
2.14. Suppose that the quantity of steel demanded in
France is given by Qs 100 2Ps 0.5Y 0.2PA, where
Qs is the quantity of steel demanded per year, Ps is the
market price of steel, Y is real GDP in France, and PA is
the market price of aluminum. In 2011, Ps 10, Y 40,
and PA 100. How much steel will be demanded in
2011? What is the price elasticity of demand, given
market conditions in 2011?
2.15. A firm currently charges a price of $100 per unit
of output, and its revenue (price multiplied by quantity)
is $70,000. At that price it faces an elastic demand
(⑀Q,P 1). If the firm were to raise its price by $2 per
69
unit, which of the following levels of output could the
firm possibly expect to see? Explain.
a) 400
b) 600
c) 800
d) 1000
2.16. Gina usually pays a price between $5 and $7 per
gallon of ice cream. Over that range of prices, her
monthly total expenditure on ice cream increases as the
price decreases. What does this imply about her price
elasticity of demand for ice cream?
2.17. Consider the following demand and supply relationships in the market for golf balls: Q d 90 2P 2T
and Q s 9 5P 2.5R, where T is the price of
titanium, a metal used to make golf clubs, and R is the
price of rubber.
a) If R 2 and T 10, calculate the equilibrium price
and quantity of golf balls.
b) At the equilibrium values, calculate the price elasticity
of demand and the price elasticity of supply.
c) At the equilibrium values, calculate the cross-price
elasticity of demand for golf balls with respect to the
price of titanium. What does the sign of this elasticity tell
you about whether golf balls and titanium are substitutes
or complements?
2.18. In Metropolis only taxicabs and privately owned
automobiles are allowed to use the highway between the
airport and downtown. The market for taxi cab service is
competitive. There is a special lane for taxicabs, so taxis
are always able to travel at 55 miles per hour. The demand for trips by taxi cabs depends on the taxi fare P, the
average speed of a trip by private automobile on the
highway E, and the price of gasoline G. The number of
trips supplied by taxi cabs will depend on the taxi fare and
the price of gasoline.
a) How would you expect an increase in the price of
gasoline to shift the demand for transportation by taxi
cabs? How would you expect an increase in the average
speed of a trip by private automobile to shift the demand
for transportation by taxi cabs? How would you expect an
increase in the price of gasoline to shift the demand for
transportation by taxi cabs?
b) Suppose the demand for trips by taxi is given by the
equation Q d 1000 50G 4E 400P. The supply of
trips by taxi is given by the equation Q s 200 30G
100P. On a graph draw the supply and demand curves for
trips by taxi when G 4 and E 30. Find equilibrium
taxi fare.
c) Solve for equilibrium taxi fare in a general case, that
is, when you do not know G and E. Show how the equilibrium taxi fare changes as G and E change.
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2.19. For the following pairs of goods, would you expect the cross-price elasticity of demand to be positive,
negative, or zero? Briefly explain your answers.
a) Tylenol and Advil
b) DVD players and VCRs
c) Hot dogs and buns
2.20. For the following pairs of goods, would you
expect the cross-price elasticity of demand to be positive,
negative, or zero? Briefly explain your answer.
a) Red umbrellas and black umbrellas
b) Coca-Cola and Pepsi
c) Grape jelly and peanut butter
d) Chocolate chip cookies and milk
e) Computers and software
2.21. Suppose that the market for air travel between
Chicago and Dallas is served by just two airlines, United
and American. An economist has studied this market and
has estimated that the demand curves for round-trip
tickets for each airline are as follows:
Q dU 10,000 100PU 99PA (United’s demand)
Q dA 10,000 100PA 99PU (American’s demand)
where PU is the price charged by United, and PA is the
price charged by American.
a) Suppose that both American and United charge a
price of $300 each for a round-trip ticket between
Chicago and Dallas. What is the price elasticity of
demand for United flights between Chicago and
Dallas?
b) What is the market-level price elasticity of demand
for air travel between Chicago and Dallas when both
airlines charge a price of $300? (Hint: Because United
and American are the only two airlines serving the
Chicago–Dallas market, what is the equation for the
total demand for air travel between Chicago and
Dallas, assuming that the airlines charge the same
price?)
2.22. You are given the following information:
• Price elasticity of demand for cigarettes at current
prices is 0.5.
• Current price of cigarettes is $0.05 per cigarette.
• Cigarettes are being purchased at a rate of 10 million per year.
Find a linear demand that fits this information, and graph
that demand curve.
2.23. For each of the following, discuss whether you
expect the elasticity (of demand or of supply, as specified)
to be greater in the long run or the short run.
a) The supply of seats in the local movie theater.
b) The demand for eye examinations at the only
optometrist in town.
c) The demand for cigarettes.
2.24. Suppose that in 2011, the global market for
hard drives for notebook computers consists of a large
number of producers. It is relatively easy for new producers to enter the industry, and when the market for
notebook hard drives is booming, new producers do, in
fact, enter.
In February 2011, there is an unexpected temporary
surge in the demand for notebook hard drives, increasing the monthly demand for hard drives by 25 percent
at any possible price. As a result, the price of notebook
hard drives increased by $5 per megabyte by the end
of February. This surge in demand ended in March
2011, and the price of notebook hard drives fell back
to its level just before the temporary demand surge
occurred.
Later that year, in August 2011, a permanent increase
in the demand for notebook computers occurs, increasing the monthly demand for hard drives by 25
percent per month at any possible price. Nine months
later, the price of notebook hard drives had increased
by $1 per unit.
In both circumstances, the market experienced a shift in
demand of exactly the same magnitude. Yet, the change
in the equilibrium price appears to have been different.
Why?
2.25. The demand for dinners in the only restaurant in
town has a unitary price elasticity of demand when the
current average price of a dinner is $8. At that price 120
people eat dinners at the restaurant every evening.
a) Find a linear demand curve that fits this information
and draw it on a clearly labeled graph.
b) Do you need the information on the price elasticity of
demand to find the curve? Why?
2.26. In each of the following pairs of goods, identify
the one that you would expect to have a greater price
elasticity of demand. Briefly explain your answers.
a) Butter versus eggs
b) Trips by your congressman to Washington (say, to
vote in the House) versus vacation trips by you to Hawaii
c) Orange juice in general versus the Tropicana brand of
orange juice
PROBLEMS
2.27. In a city, the price for a trip on local mass transit
(such as the subway or city buses) has been 10 pesos for a
number of years. Suppose that the market for trips is
characterized by the following demand curves: in the
long run: Q ⫽ 30 ⫺2P; in the short run: Q ⫽ 15 ⫺ PⲐ2.
Verify that the long-run demand curve is “flatter” than
the short-run curve. What does this tell you about the
sensitivity of demand to price for this good? Discuss why
this is the case.
2.28. Consider the following sequence of events in
the U.S. market for strawberries during the years
1998–2000:
• 1998: Uneventful. The market price was $5 per
bushel, and 4 million bushels were sold.
• 1999: There was a scare over the possibility of
contaminated strawberries from Michigan. The
market price was $4.50 per bushel, and 2.5 million
bushels were sold.
• 2000: By the beginning of the year, the scare over
contaminated strawberries ended when the media
reported that the initial reports about the contamination were a hoax. A series of floods in the
Midwest, however, destroyed significant portions
of the strawberry fields in Iowa, Illinois, and
Missouri. The market price was $8 per bushel, and
3.5 million bushels were sold.
Find linear demand and supply curves that are consistent
with this information.
January: Initial demand and supply are given by the equations Q s ⫽ 30P ⫺ 30 (when P ⱖ 1), and Q d ⫽ 120 ⫺ 20P
February: Due to higher prices of gasoline, the supply of
cab service changed to Q s ⫽ 30P ⫺ 60 (when P ⱖ 2).
March: Over the spring break, the demand for taxi service was higher and therefore the demand curve was given
by the equation Q d ⫽ 140 ⫺ 20P.
a) For each month find the equilibrium price and quantity.
b) Illustrate your answer with a graph. Illustrate the
equilibrium prices and quantities on the graph.
2.30. Consider the demand curve for pomegranates
in two countries. In one country, pomegranates are a
critical part of the diet and are central to the preparation of many popular food recipes. For most of these
dishes, there is no feasible substitute for pomegranates.
In the second country, households will purchase pomegranates if the price is right, but consumers do not consider them to be particularly special or unique, and few
popular dishes rely on pomegranates in their recipes.
Suppose pomegranates are native to both countries.
Suppose, further, that due to inherent limitations of
shipping options, there is no intercountry trade in
pomegranates. Each country’s market for pomegranates
is independent of that of the other countries. Finally,
suppose that in both countries, droughts and other
weather-related shocks periodically cause unexpected
changes in supply conditions.
The following graphic shows the time paths of pomegranate prices over a 10-year period in each country (the
solid line is the time path in one country; the dashed line
is the time path in the other country). Based on the information provided, which is the time path for each
country?
Price of pomegranates
2.29. Consider the following sequence of changes in
the demand and supply for cab service in some city. The
price P is a price per mile, while quantity is the total
length of cab rides over a month (in thousands of miles).
71
Time
72
CHAPTER 2
D E M A N D A N D S U P P LY A N A LYS I S
A P P E N D I X : Price Elasticity of Demand along a Constant Elasticity Demand Curve
In this section, we show that the point price elasticity of demand is the same along a
constant elasticity demand curve of the form Q ⫽ aP⫺b. For this demand curve,
dQ
⫽ ⫺baP⫺(b1)
dP
Forming the expression for the point elasticity of demand, we have
⑀Q, P ⫽
dQ P
dP Q
⫽ ⫺baP⫺(b1) ⫻
P
(substituting in the expression for Q)
aP⫺b
⫽ ⫺b (after canceling terms)
This shows that the price elasticity of demand for the constant elasticity demand curve
is simply the exponent in the equation of the demand curve, ⫺b. (For more on the use
of derivatives, see the Mathematical Appendix at the end of the book.)
3
CONSUMER PREFERENCES
AND THE CONCEPT
OF UTILITY
3.1
R E P R E S E N TAT I O N S O F P R E F E R E N C E S
3.2
UTILITY FUNCTIONS
Influencing Your Preferences
How People Buy Cars: The
Importance of Attributes
APPLICATION 3.1
APPLICATION 3.2
3.3
SPECIAL PREFERENCES
Taste Tests
APPLICATION 3.4 Hula Hoops and Beanie Babies
APPLICATION 3.5 Does More Make You
Happier? Reference-Dependent Preferences
APPLICATION 3.3
Why Do You Like What You Like?
The economic recession that swept across the globe in 2008 and 2009 led to remarkable adjustments in
consumer behavior, with changes especially noticeable in sectors like the automobile industry. Several
factors contributed to these changes. Declining stock prices and incomes meant that consumers generally
had less money to spend on goods and services. Higher fuel prices and increased consumer interest in the
environment led many consumers to purchase more fuel-efficient vehicles. Government programs also
influenced consumer behavior. In the summer of 2009, the United States government introduced a “Cashfor-Clunkers” program (officially called the Car Allowance Rebate System) that offered consumers as much
as $4,500 to trade in an old car for a more fuel-efficient new model. This subsidy to consumers led to
increased vehicle sales, at least temporarily aiding an industry in financial difficulty. Starting in late 2008,
some European countries also offered similar cash incentives to induce consumers to trade in their old cars
for new ones.
As a consumer, you make choices every day of your life. Besides choosing among automobiles, you
must decide what kind of housing to rent or purchase, what food and clothing to buy, how much education
to acquire, and so on. Consumer choice provides an excellent example of constrained optimization, one of
the key tools discussed in Chapter 1. People have unlimited desires but limited resources. The theory of
consumer choice focuses on how consumers with limited resources choose goods and services.
73
In the next three chapters, we will learn about consumer choice. In this chapter we will learn about consumer preferences. We study consumer preferences to understand how a consumer compares (or ranks) the
desirability of different sets of goods. For this discussion we ignore the costs of purchasing the goods. Thus,
consumer preferences indicate whether the consumer likes one particular set of goods better than another,
assuming that all goods can be “purchased” at no cost. For example, putting operating and purchase costs
aside, a consumer may prefer a fuel-efficient car to a less efficient one out of concern for the environment.
Of course, in the real world it does cost the consumer something to purchase goods, and a consumer
has limited income. This reality leads us to the second part of our discussion of consumer choice in Chapter 4.
When goods are costly, a consumer’s income limits the set of goods she can purchase. In Chapter 4 we
will show how to describe the set of goods that is affordable given a consumer’s income and the prices of
goods. Then we will use consumer preferences to answer the following question: Which goods among
those that are affordable will the consumer choose?
Why should we study consumer choice in such depth? Consumers are not the only parties interested in
consumer choice, and in Chapter 5 we will use the theory of consumer choice to derive a consumer’s demand curve for any good or service. Businesses care about consumer demand curves because they reveal
how much a consumer is willing to pay for a product. Governments also care about consumer preferences
and demands. For example, if a government is interested in helping low-income families buy food, policy
makers must decide how to do it. Should the government simply give the families a cash supplement and
let them spend the money in any way they wish? Or should the aid be in the form of certificates, such as
food stamps, that can only be used to buy food? One might also ask if a Cash-for-Clunkers program is the
best way to stimulate consumer purchases of fuel-efficient automobiles. As we will see, the effectiveness
and costliness of such government programs will very much depend on consumers’ preferences.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Represent consumer preferences in terms of market baskets of goods and services.
• Apply three basic assumptions about consumer preferences: Preferences are complete, preferences
are transitive, and more is better.
• Distinguish between ordinal and cardinal ranking of preferences.
• Apply utility functions as a tool for representing
preferences and analyze the concept of marginal utility and the principle of diminishing marginal utility.
• Apply utility functions in the analysis of preferences
with a single good and with multiple goods.
• Construct indifference curves as a way of representing utility functions in simplified form.
• Analyze the concept of the marginal rate of substitution of one good for another.
• Describe and compare some special utility functions.
74
75
3 . 1 R E P R E S E N TAT I O N S O F P R E F E R E N C E S
I
n a modern economy, consumers can purchase a vast array of goods and services. We
begin by considering a market basket (sometimes called a bundle), defined as a collection of goods and services that an individual might consume. For example, one basket
of goods might include a pair of jeans, two pairs of shoes, and 5 pounds of chocolate
candy. A second basket might contain two pairs of jeans, one pair of shoes, and 2
pounds of chocolate candy. More generally, a basket may contain specified amounts of
not only jeans, shoes, and chocolate candy, but also housing, electronic goods, tickets
for theatrical and sporting events, and many other items.
To illustrate the idea of a basket, consider a simplified example in which a consumer can purchase only two goods, food and clothing. Seven possible consumption
baskets are illustrated in Figure 3.1. A consumer who buys basket E consumes 20 units
of food and 30 units of clothing per week. One who chooses basket B instead consumes 60 units of food and 10 units of clothing weekly. A basket might contain only
one good, such as basket J (only food) or basket H (only clothing).
Consumer preferences tell us how an individual would rank (i.e., compare the
desirability of ) any two baskets, assuming the baskets were available at no cost. Of course,
a consumer’s actual choice will ultimately depend on a number of factors in addition
to preferences, including income and what the baskets cost. But for now we will consider only consumer preferences for different baskets.
3.1
REPRESENTATI O N S O F
PREFERENCES
basket A combination of
goods and services that an
individual might consume.
consumer preferences
Indications of how a consumer would rank (compare
the desirability of) any two
possible baskets, assuming
the baskets were available
to the consumer at no cost.
ASSUMPTIONS ABOUT CONSUMER PREFERENCES
Our study of consumer preferences begins with three basic assumptions that underlie
the theory of consumer choice. In making these assumptions, we take it for granted
that consumers behave rationally under most circumstances. Later we will discuss situations in which these assumptions might not be valid.
Units of clothing
1. Preferences are complete. That is, the consumer is able to rank any two baskets.
For baskets A and B, for example, the consumer can state her preferences
according to one of the following possibilities:
30
H
E
G
20
10
A
D
B
J
0
20
40
60
Units of food
FIGURE 3.1
Weekly Baskets of Food and Clothing
Seven possible weekly baskets of food and clothing that
consumers might purchase are illustrated by points A, B,
D, E, G, H, and J.
76
CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
She prefers basket A to basket B (written A Ɑ B).
She prefers basket B to basket A (written B Ɑ A).
She is indifferent between, or equally happy with, baskets A and B (written A B).
2. Preferences are transitive. By this we mean that the consumer makes choices
that are consistent with each other. Suppose that a consumer tells us that she
prefers basket A to basket B, and basket B to basket E. We can then expect her
to prefer basket A to basket E. Using the notation we have just introduced to
describe preferences, we can represent transitivity as follows: If A Ɑ B and if
B Ɑ E, then A Ɑ E.
3. More is better. In other words, having more of a good is better for the consumer.
Suppose the consumer is considering the baskets in Figure 3.1. If more is better, she likes more food better than less food and prefers to have more clothing
rather than less clothing. In that case, she would prefer basket A to E or H
because she receives the same amount of clothing with these three baskets, but
more food at A. She would prefer basket A to B or J because she receives the
same amount of food in these three baskets, but more clothing at A. She will
also prefer A to G or D because she receives more food and more clothing
at A than at either of the other two baskets. Therefore, among the seven
baskets, her most preferred basket is A. However, without further information
about the consumer’s preferences, we do not know how she would rank every
pair of baskets. For example, without further information we do not know
whether she prefers E to G because she would receive more food but less
clothing at G.
L
ORDINAL AND CARDINAL RANKING
ordinal ranking
Ranking that indicates
whether a consumer prefers
one basket to another, but
does not contain quantitative information about the
intensity of that preference.
cardinal ranking A
quantitative measure of the
intensity of a preference for
one basket over another.
In this book we will refer to two types of rankings: ordinal and cardinal. Ordinal
rankings give us information about the order in which a consumer ranks baskets. For
example, for basket A in Figure 3.1 the consumer buys three times as much food and
three times as much clothing as she does for basket D. We know that the consumer
prefers basket A to D because more is better. However, an ordinal ranking would not
tell us how much more she likes A than D.
Cardinal rankings give us information about the intensity of a consumer’s preferences. With a cardinal ranking, we not only know that she prefers basket A to
basket D, but we can also measure the strength of her preference for A over D. We
can make a quantitative statement, such as “The consumer likes basket A twice as
much as basket D.”1 A cardinal ranking therefore contains more information than
an ordinal ranking.
It is usually easy for consumers to answer a question about an ordinal ranking,
such as “Would you prefer a basket with a hamburger and french fries or a basket with
a hot dog and onion rings?” However, consumers often have more difficulty describing
how much more they prefer one basket to another because they have no natural
1
As noted in the text, the consumer buys three times as much food and clothing at basket A as at D.
However, this does not necessarily mean that the consumer likes basket A exactly three times more than
basket D. Would your own satisfaction triple if you bought three times as much of all goods as you now
do? For most consumers satisfaction would rise, but by less than three times.
77
3.2 UTILITY FUNCTIONS
measure of the amount of pleasure they derive from different baskets. Fortunately, as
we develop the theory of consumer behavior, you will see that it is not important for
us to measure the amount of pleasure a consumer receives from a basket. Although we
often use a cardinal ranking to facilitate exposition, an ordinal ranking will normally
give us enough information to explain a consumer’s decisions.
T
he three assumptions––preferences are complete, they are transitive, and more is
better––allow us to represent preferences with a utility function. A utility function
measures the level of satisfaction that a consumer receives from any basket of goods.
We can represent the utility function with algebra or a graph.
PREFERENCES WITH A SINGLE GOOD:
THE CONCEPT OF MARGINAL UTILITY
3.2
UTILITY
FUNCTIONS
utility function
To illustrate the concept of a utility function, let’s begin with a simple scenario in
which a consumer, Sarah, purchases only one good, hamburgers. Let y denote the
number of hamburgers she purchases each week, and let U( y) measure the level of satisfaction (or utility) that Sarah derives from purchasing y hamburgers.
Figure 3.2(a) depicts Sarah’s utility function for hamburgers. The equation of the
utility function that gives rise to this graph is U( y) ⫽ 1y. We observe that Sarah’s
preferences satisfy the three assumptions just described. They are complete because
she can assign a level of satisfaction to each value of y. The assumption that more is
better is also satisfied because the more hamburgers consumed, the higher her utility.
For example, suppose the number of hamburgers in basket A is 1, the number in basket B is 4, and the number in basket C is 5. Then Sarah ranks the baskets as follows:
C Ɑ B and B Ɑ A, which we can see from the fact that Sarah’s utility at point C is
higher than it is at point B, and her utility at point B is higher than her utility at point A.
Finally, Sarah’s preferences are transitive: Since she prefers basket C to basket B and
basket B to basket A, she also prefers basket C to basket A.
A
function that measures
the level of satisfaction a
consumer receives from
any basket of goods and
services.
Marginal Utility
While studying consumer behavior, we will often want to know how the level of satisfaction will change (⌬U ) in response to a change in the level of consumption (⌬y,
where ⌬ is read as “the change in”). Economists refer to the rate at which total utility
changes as the level of consumption rises as the marginal utility (MU ). The marginal
utility of good y (MUy) is thus:
MUy ⫽
¢U
¢y
(3.1)
Graphically, the marginal utility at a particular point is represented by the slope of a
line that is tangent to the utility function at that point. For example, in Figure 3.2(a),
Sarah’s marginal utility for hamburgers at y ⫽ 4 is the slope of the tangent line RS.
Since the slopes of the tangents change as we move along the utility function U( y),
Sarah’s marginal utility will depend on the quantity of hamburgers she has already
purchased. In this respect, Sarah is like most people: The additional satisfaction that
she receives from consuming more of a good depends on how much of the good she
has already consumed.
marginal utility The
rate at which total utility
changes as the level of
consumption rises.
78
CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
U( y), total utility of hamburgers
(a)
S
U( y) = √ y
2.24
2.00
B
C
R
1.00
0
A
1
2
3
4
5
6
y, weekly consumption of hamburgers
FIGURE 3.2
Total and Marginal Utility
with a Single Good (Hamburgers)
The utility function U (y) ⫽ 1y is shown in
the top panel, and the corresponding
marginal utility is illustrated in the bottom
panel. The slope of the utility function in
the top panel is the marginal utility. For
example, when y ⫽ 4, the slope of the utility
function is 0.25 (represented by the slope of
the tangent RS at point B). Therefore, when
y ⫽ 4, the marginal utility is 0.25.
MUy, marginal utility of hamburgers
(b)
A'
1.00
B'
0.25
0.22
C'
MUy = 1
2√ y
0
1
2
3
4
5
6
y, weekly consumption of hamburgers
In Figure 3.2, where the utility function is U( y) ⫽ 1y, as shown in panel (a), the
marginal utility is MUy ⫽ 1Ⲑ(2 1y ), as shown in panel (b).2 This equation reflects
the precise way in which marginal utility depends on the quantity y.
Learning-By-Doing Exercise A.4 in the Mathematical Appendix shows how to derive the equation of
marginal utility when you know the formula for total utility. To show that the formula MUy ⫽ 1Ⲑ(2 1y )
makes sense in this example, let’s verify the value of marginal utility numerically. Suppose consumption
increases from y ⫽ 4 to y ⫽ 4.01, so that ⌬y ⫽ 0.01. Then the level of utility increases from U(4) ⫽
14 ⫽ 2 to U(4.01) ⫽ 24.01 L 2.0025. Therefore, utility has increased by ¢U L 0.0025. So marginal
utility is ¢UⲐ ¢y ⫽ 0.0025Ⲑ0.01 ⫽ 0.25. This is the number we would get if we substituted y ⫽ 4 into the
formula MUy ⫽ 1Ⲑ(2 1y ).
2
3.2 UTILITY FUNCTIONS
79
Principle of Diminishing Marginal Utility
When drawing total utility and marginal utility curves, you should keep the following
points in mind:
• Total utility and marginal utility cannot be plotted on the same graph. The horizontal
axes in the two panels of Figure 3.2 are the same (both representing the number
of hamburgers consumed each week, y), but the vertical axes in the two graphs
are not the same. Total utility has the dimensions of U (whatever that may be),
while marginal utility has the dimensions of utility per hamburger (⌬U divided
by ⌬y). Therefore, the curves representing total utility and marginal utility must
be drawn on two different graphs.
• The marginal utility is the slope of the (total) utility function. The slope at any point
on the total utility curve in panel (a) of Figure 3.2 is ¢UⲐ ¢y, the rate of change
in total utility at that point as consumption rises or falls, which is what marginal
utility measures (note that ¢UⲐ ¢y at any point is also the slope of the line segment tangent to the utility curve at that point). For example, at point B in panel
(a) of Figure 3.2: slope of utility curve U( y) ⫽ 0.25 (i.e., ¢U Ⲑ ¢y ⫽ 0.25 when
y ⫽ 4) ⫽ slope of tangent line segment RS ⫽ marginal utility at that point ⫽
value of vertical coordinate at point B¿ on marginal utility curve MUy in panel (b).
• The relationship between total and marginal functions holds for other measures in economics. The value of a marginal function is often simply the slope of the corresponding total function. We will explore this relationship for other functions
throughout this book.
In Figure 3.2(b), Sarah’s marginal utility declines as she eats more hamburgers.
This trend illustrates the principle of diminishing marginal utility: After some point,
as consumption of a good increases, the marginal utility of that good will begin to fall.
Diminishing marginal utility reflects a common human trait. The more of something
we consume, whether it be hamburgers, candy bars, shoes, or baseball games, the less
additional satisfaction we get from additional consumption. Marginal utility may not
decline after the first unit, the second unit, or even the third unit. But it will normally
fall after some level of consumption.
To understand the principle of diminishing marginal utility, think about the additional satisfaction you get from consuming another hamburger. Suppose you have
already eaten one hamburger this week. If you eat a second hamburger, your utility will
go up by some amount. This is the marginal utility of the second hamburger. If you
have already consumed five hamburgers this week and are about to eat a sixth hamburger, the increase in your utility will be the marginal utility of the sixth hamburger.
If you are like most people, the marginal utility of your sixth hamburger will be less
than the marginal utility of the second hamburger. In that case, your marginal utility
of hamburgers is diminishing.
Is More Always Better?
What does the assumption that more is better imply about marginal utility? If more of
a good is better, then total utility must increase as consumption of the good increases.
In other words, the marginal utility of that good must always be positive.
In reality this assumption is not always true. Let’s return to the example of consuming hamburgers. Sarah may find that her total utility increases as she eats the first,
second, and third hamburgers each week. For these hamburgers, her marginal utility
principle of diminishing marginal utility
The principle that after
some point, as consumption of a good increases,
the marginal utility of that
good will begin to fall.
CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
U( y), total utility of hamburgers
80
(a)
M
S
B
K
L
C
U = ( y)
A
R
0
N
2
7
9
y, weekly consumption of hamburgers
Marginal Utility May Be Negative
The utility curve U(y) is shown in panel (a), and the
corresponding marginal utility curve is illustrated in
panel (b). The slope of the utility curve in the top panel
is positive at A; thus, the marginal utility is positive, as
indicated at point A⬘ in panel (b). At point B the slope
of the utility curve is zero, meaning that the marginal
utility is zero, as shown at point B ⬘. At point C the
slope of the utility function is negative; therefore, the
marginal utility is negative, as indicated at point C ⬘.
MUy , marginal utility of hamburgers
FIGURE 3.3
(b)
A'
B'
0
2
9
7
C'
MUy
y, weekly consumption of hamburgers
is positive, even though it may be diminishing with each additional hamburger she
eats. But presumably at some point she will find that an additional hamburger will
bring her no more satisfaction. For example, she might find that the marginal utility
of the seventh hamburger per week is zero, and the marginal utility of the eighth or
ninth hamburgers might even be negative.
Figure 3.3 depicts the total and marginal utility curves for this case. Initially (for
values of y ⬍ 7 hamburgers), total utility rises as consumption increases, and the slope
of the utility curve is positive (e.g., note that the segment RS, which is tangent to the
utility curve at point A when Sarah is purchasing her second hamburger, has a positive slope); thus, the marginal utility is positive (as depicted at point A⬘). However, the
marginal utility is diminishing as consumption increases, and at a consumption level
of seven hamburgers, Sarah has purchased so much of the good that the marginal utility is zero (point B⬘). Since the marginal utility is zero, the slope of the total utility
curve is zero. (The segment MN, which is tangent to the utility curve at point B, has a
slope of zero.) If Sarah were to buy more than seven hamburgers, her total satisfaction
3.2 UTILITY FUNCTIONS
81
would decline (e.g., the slope of the total utility curve at point C is negative, and thus
the marginal utility is negative, as indicated at point C⬘).
Although more may not always be better, it is nevertheless reasonable to assume
that more is better for amounts of a good that a consumer might actually purchase.
For example, in Figure 3.3 we would normally only need to draw the utility function
for the first seven hamburgers. The consumer would never consider buying more than
seven hamburgers because it would make no sense for her to spend money on hamburgers that reduce her satisfaction.
P R E F E R E N C E S W I T H M U LT I P L E G O O D S :
M A R G I N A L U T I L I T Y, I N D I F F E R E N C E C U RV E S,
A N D T H E M A R G I N A L R AT E O F S U B S T I T U T I O N
Let’s look at how the concepts of total utility and marginal utility might apply to a
more realistic scenario. In real life, consumers can choose among myriad goods and
services. To study the trade-offs a consumer must make in choosing his optimal basket, we must examine the nature of consumer utility with multiple products.
We can illustrate many of the most important aspects of consumer choice
among multiple products with a relatively simple scenario in which a consumer,
Brandon, must decide how much food and how much clothing to purchase in a
given month. Let x measure the number of units of food and y measure the number
of units of clothing purchased each month. Further, suppose that Brandon’s utility
for any basket (x, y) is measured by U ⫽ 1xy. A graph of this consumer’s utility
function is shown in Figure 3.4. Because we now have two goods, a graph of
Brandon’s utility function must have three axes. In Figure 3.4 the number of units
of food consumed, x, is shown on the right axis, and the number of units of clothing consumed, y, is represented on the left axis. The vertical axis measures
Brandon’s level of satisfaction from purchasing any basket of goods. For example,
U = 10
12
U, level of utility
10
U=8
8
U=6
6
U=4
4
2
B
C
12
10
y,
Graph of the
Utility Function U ⫽ 1xy
The level of utility is shown on the
vertical axis, and the amounts of
food (x) and clothing (y) are shown,
respectively, on the right and left
axes. Contours representing lines of
constant utility are also shown. For
example, the consumer is indifferent
between baskets A, B, and C
because they all yield the same level
of utility (U ⫽ 4).
FIGURE 3.4
A
8
its 6
4
of
clo
2
thi
ng
U=2
un
0
2
4
8
6
f food
x, units o
10
12
82
CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
basket A contains two units of food (x 2) and eight units of clothing ( y 8).
Thus, Brandon realizes a level of utility of U ⫽ 1(2)(8) ⫽ 4 with basket A. As the
graph indicates, Brandon can achieve the same level of utility by choosing other baskets, such as basket B and basket C.
The concept of marginal utility is easily extended to the case of multiple goods.
The marginal utility of any one good is the rate at which total utility changes as the
level of consumption of that good rises, holding constant the levels of consumption of all
other goods. For example, in the case in which only two goods are consumed and the
utility function is U(x, y), the marginal utility of food (MUx) measures how the level
of satisfaction will change (⌬U ) in response to a change in the consumption of food
(⌬x), holding the level of y constant:
MUx ⫽
¢U
2
¢x y is held constant
(3.2)
Similarly, the marginal utility of clothing (MUy) measures how the level of satisfaction will change (⌬U ) in response to a small change in the consumption of clothing
(⌬y), holding constant the level of food (x):
MUy ⫽
¢U
2
¢y x is held constant
(3.3)
One could use equations (3.2) and (3.3) to derive the algebraic expressions for MUx and
MUy from U(x, y).3 When the total utility from consuming a bundle (x, y) is U ⫽ 1xy,
the marginal utilities are MUx ⫽ 1yⲐ(2 1x) and MUy ⫽ 1xⲐ(2 1y). So, at basket A (with
x ⫽ 2 and y ⫽ 8), MUx ⫽ 18Ⲑ(2 12) ⫽ 1 and MUy ⫽ 12Ⲑ(2 18) ⫽ 1Ⲑ4.
Learning-By-Doing Exercise 3.1 shows that the utility function U ⫽ 1xy satisfies
the assumptions that more is better and that marginal utilities are diminishing. Because
these are widely regarded as reasonable characteristics of consumer preferences, we will
often use this utility function to illustrate concepts in the theory of consumer choice.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 3 . 1
D
Marginal Utility
Let’s look at a utility function that satisfies
the assumptions that more is better and that marginal
utilities are diminishing. Suppose a consumer’s preferences between food and clothing can be represented by
the utility function U ⫽ 1xy, where x measures the
number of units of food and y the number of units of
clothing, and the marginal utilities for x and y are expressed by the following equations: MUx ⫽ 1yⲐ(2 1x)
and MUy ⫽ 1xⲐ(2 1y).
3
Problem
(a) Show that a consumer with this utility function
believes that more is better for each good.
(b) Show that the marginal utility of food is diminishing
and that the marginal utility of clothing is diminishing.
Learning-By-Doing Exercise A.7 in the Mathematical Appendix shows how to derive the equations of
MUx and MUy in this case.
3.2 UTILITY FUNCTIONS
Solution
(a) By examining the utility function, we can see that U
increases whenever x or y increases. This means that the
consumer likes more of each good. Note that we can
also see that more is better for each good by looking at
the marginal utilities MUx and MUy, which must always
be positive because the square roots of x and y must
always be positive (all square roots are positive numbers).
83
This means the consumer’s utility always increases when
he purchases more food and/or clothing.
(b) In both marginal utility functions, as the value of the
denominator increases (holding the numerator constant), the marginal utility diminishes. Thus, MUx and
MUy are both diminishing.
Similar Problem: 3.4
Learning-By-Doing Exercise 3.2 shows the two ways to determine whether the
marginal utility of a good is positive. First, you can look at the total utility function.
If it increases when more of the good is consumed, marginal utility is positive. Second,
you can look at the marginal utility of the good to see if it is a positive number. When
the marginal utility is a positive number, the total utility will increase when more of
the good is consumed.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 3 . 2
D
Marginal Utility That Is Not Diminishing
Some utility functions satisfy the assumption that more is better, but with a marginal utility that
is not diminishing. Suppose a consumer’s preferences for
hamburgers and root beer can be represented by the
utility function U 2H ⫹ R, where H measures the
number of hamburgers consumed and R the number of
root beers. The marginal utilities are
2 2H
MUR 1
MUH
1
Problem
(a) Does the consumer believe that more is better for
each good?
Solution
(a) U increases whenever H or R increases, so more
must be better for each good. Also, MUH and MUR are
both positive, again indicating that more is better.
( b) As H increases, MUH falls, so the consumer’s marginal utility of hamburgers is diminishing. However,
MUR ⫽ 1 (no matter what the value of R), so the consumer has a constant (rather than a diminishing) marginal
utility of root beer (i.e., the consumer’s utility always increases by the same amount when he purchases another
root beer).
Similar Problem:
3.5
(b) Does the consumer have a diminishing marginal
utility of hamburgers? Is the marginal utility of root beer
diminishing?
Indifference Curves
To illustrate the trade-offs involved in consumer choice, we can reduce the threedimensional graph of Brandon’s utility function in Figure 3.4 to a two-dimensional
graph like the one in Figure 3.5. Both graphs illustrate the same utility function
U ⫽ 1xy. In Figure 3.5 each curve represents baskets yielding the same level of utility
to Brandon. Each curve is called an indifference curve because Brandon would be
indifference curve A
curve connecting a set of
consumption baskets that
yield the same level of satisfaction to the consumer.
84
CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
12
y, units of clothing
10
Indifference Curves for
the Utility Function U ⫽ 1xy
The utility is the same for all baskets
on a given indifference curve. For
example, the consumer is indifferent
between baskets A, B, and C in the
graph because they all yield the same
level of utility (U 4).
A
8
6
U=8
B
4
FIGURE 3.5
U=6
C
2
U=4
0
U=2
2
4
6
8
10
12
x, units of food
equally satisfied with (or indifferent in choosing among) all baskets on that curve. For
example, Brandon would be equally satisfied with baskets A, B, and C because they all
lie on the indifference curve with the value U 4. (Compare Figures 3.4 and 3.5 to see
how the indifference curve U 4 looks in a three-dimensional and a two-dimensional
graph of the same utility function.) A graph like Figure 3.5 is sometimes referred to
as an indifference map because it shows a set of indifference curves.
Indifference curves on an indifference map have the following four properties.
1. When the consumer likes both goods (i.e., when MUx and MUy are both positive), all the indifference curves have a negative slope.
2. Indifference curves cannot intersect.
3. Every consumption basket lies on one and only one indifference curve.
4. Indifference curves are not “thick.’’
We will now explore these properties in further detail.
1. When the consumer likes both goods (i.e., when MUx and MUy are both positive), all the
indifference curves will have a negative slope. Consider the graph in Figure 3.6. Suppose
the consumer currently has basket A. Since the consumer has positive marginal utility
for both goods, she will prefer any baskets to the north, east, or northeast of A. We
indicate this in the graph by drawing arrows to indicate preference directions. The
arrow pointing to the east reflects the fact that MUx 7 0. The arrow pointing to the
north reflects the fact that MUy 7 0.
Points to the northeast or southwest of A cannot be on the same indifference
curve as A because they will be preferred to A or less preferred than A, respectively.
Thus, points on the same indifference curve as A must lie either to the northwest or
southeast of A. This shows that indifference curves will have a negative slope when
both goods have positive marginal utilities.
3.2 UTILITY FUNCTIONS
Preference
directions
y
A
Indifference
curve
x
FIGURE 3.6 Slope of Indifference Curves
Suppose that goods x and y are both liked by the consumer
(MUx ⬎ 0 and MUy ⬎ 0, indicating that the consumer
prefers more of y and more of x). Points in the shaded
region to the northeast of A cannot be on the same indifference curve as A since they will be preferred to A. Points
in the shaded region to the southwest of A also cannot be
on the same indifference curve as A since they will be less
preferred than A. Thus, points on the same indifference
curve as basket A must lie to the northwest or southeast of
A, and the slope of the indifference curve running through
A must be negative.
2. Indifference curves cannot intersect. To understand why, consider Figure 3.7,
which shows two hypothetical indifference curves (with levels of utility U1 and U2)
that cross. The basket represented by point S on U1 is preferred to the basket represented by point T on U2, as shown by the fact that S lies to the northeast of T;
thus, U1 ⬎ U2. Similarly, the basket represented by point R on U2 is preferred
to the basket represented by point Q on U1 (R lies to the northeast of Q); thus,
U2 ⬎ U1. Obviously, it cannot be true that U1 ⬎ U2 and that U2 ⬎ U1. This
logical inconsistency arises because U1 and U2 cross; therefore, indifference curves
cannot intersect.
3. Every consumption basket lies on one and only one indifference curve. This follows
from the property that indifference curves cannot intersect. In Figure 3.7, the basket
represented by point A lies on the two intersecting indifference curves (U1 and U2);
a point can lie on two curves only at a place where the two curves intersect. Since
indifference curves cannot intersect, every consumption basket must lie on a single
indifference curve.
4. Indifference curves are not “thick.” To see why, consider Figure 3.8, which shows a
thick indifference curve passing through distinct baskets A and B. Since B lies to the
northeast of A, the utility at B must be higher than the utility at A. Therefore, A
and B cannot be on the same indifference curve.
Preference
directions
y
S
T
A
FIGURE 3.7
R
Q
x
U1
U2
Indifference Curves Cannot Intersect
If we draw two indifference curves (with different levels of
utility U1 and U2) that intersect each other, then we create
a logical inconsistency in the graph. Since S lies to the
northeast of T, then U1 ⬎ U2. But since R lies to the northeast of Q, then U2 ⬎ U1. This logical inconsistency (that
U1 ⬎ U2 and U2 ⬎ U1) arises because the indifference curves
intersect one another.
85
86
CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
Preference
directions
y
B
A
FIGURE 3.8
Indifference Curves Are Not “Thick”
A thick indifference curve U0 contains baskets A and B. But
B lies to the northeast of A, so the utility at B must be
higher than the utility at A. Therefore, A and B cannot be
on the same indifference curve.
U0
x
The Marginal Rate of Substitution
which the consumer will
give up one good to get
more of another, holding
the level of utility constant.
A consumer’s willingness to substitute one good for another while maintaining the
same level of satisfaction is called the marginal rate of substitution. For example, a
consumer’s marginal rate of substitution of hamburgers for lemonade is the rate at
which the consumer would be willing to give up glasses of lemonade to get more hamburgers, with the same overall satisfaction.
When two goods have positive marginal utilities, the trade-off that the consumer
is willing to make between the two goods is illustrated by the slope of the indifference curve. To see why, consider the indifference curve U0 in Figure 3.9, which
shows the weekly consumption of hamburgers and glasses of lemonade by a particular consumer, Eric. When Eric moves from any given basket, such as basket A, to an
equally preferred basket farther to the right on the curve, such as basket B, he must
give up some of one good (glasses of lemonade) to get more of the other good (hamburgers). The slope of the indifference curve at any point (i.e., the slope of the line
tangent to the curve at that point) is ¢y Ⲑ ¢ x—the rate of change of y relative to the
change of x. But this is exactly Eric’s marginal rate of substitution of hamburgers for
lemonade––the amount of lemonade he would give up (⌬y) to gain additional hamburgers (⌬x).
FIGURE 3.9 The Marginal Rate of Substitution of x for y
(MRSx,y)
The marginal rate of substitution of x for y (MRSx,y) is the rate
at which the consumer is willing to give up y in order to get
more of x, holding utility constant. On a graph with x on the
horizontal axis and y on the vertical axis, MRSx,y at any basket
is the negative of the slope of the indifference curve through
that basket. At basket A the slope of the indifference curve is
⫺5, so MRSx,y ⫽ 5. At basket D the slope of the indifference
curve is ⫺2, so MRSx,y ⫽ 2.
y, glasses of lemonade per week
marginal rate of substitution The rate at
Preference
directions
A
B
C
D
U0
x, hamburgers per week
87
3.2 UTILITY FUNCTIONS
A P P L I C A T I O N
3.1
Influencing Your Preferences
The theory of consumer behavior assumes that the indifference map for a consumer is given exogenously
and remains fixed. In reality, a consumer’s preferences
can change over time, and with age, education, or experience. Preferences may also change as a result of
actions designed to influence consumer attitudes
about goods and services.
Firms often pay great sums of money for the opportunity to influence your preferences by advertising. For example, for the telecast of the 2009 Super
Bowl, NBC was able to charge an average of $3 million for each 30-second commercial. Why would an
advertiser pay so much? Super Bowl ratings are always high, regardless of how interesting the game
is. When ratings are high, advertisers know their
messages will reach millions of households. In addition, while TV viewers often find commercials to be
an annoyance, that changes during the Super Bowl.
Many viewers look forward to the humorous and
creative ads that companies run during the game.
Furthermore, advertisers get extra publicity from
good ads, since the media discusses Super Bowl ads
at great length.
As can be seen in Figure 3.10, Super Bowl ad
prices gradually rose over time to the record price in
2009. Ad prices tend to be higher when a more exciting game is anticipated. For example, prices rose dramatically for the 1998 Super Bowl, when the Denver
Broncos upset the Green Bay Packers for the championship of the National Football League in a very close
game. Prices sometimes decline during a recession, as
they did in 2001 and 2002. Despite the severe recession in 2009, prices rose. As a result, NBC was reported to have more difficulty selling all of the commercial slots than in prior years (both FedEx and
General Motors, regular Super Bowl advertisers, did
not buy ads that year). Average prices may have been
higher because many ads were sold prior to
September 2008, when the recession began to be felt
most strongly.
The government and interest groups can also
influence consumer preferences. For example, in
1953 the American Cancer Society issued its own
warning about smoking, when it published a report
linking cigarette smoking with cancer. Some governments require cigarette producers to place graphic
pictures (e.g., of oral cancer) on packages as a warning to consumers about the dangers of smoking.
Super Bowl Ad Prices
(30-second ad)
$3,000,000
Price (2009 dollars)
$2,500,000
$2,000,000
$1,500,000
$1,000,000
$500,000
$0
1969
1974
1979
1984
1989
Year
FIGURE 3.10 Prices of Super Bowl Television Ads
The prices of 30-second ads are expressed in 2009 dollars.
Sources: Advertising Age for 1969–2007; Reuters for 2008–2009.
1994
1999
2004
2009
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CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
In June 2009 the Family Smoking Prevention and
Tobacco Control Act was enacted in the United States.
It bans promotions and advertising believed to be
focused on youth. It also requires that the top half
of cigarette packs, front and back, have stern health
warnings. Within two years the law requires the
Food and Drug Administration to add graphic
warning labels similar to those used in other countries. Studies by economists have found that such
warnings and advertising restrictions can have significant negative impacts on consumer demand for
cigarettes.
For instance, the slope of Eric’s indifference curve at point A is ⫺5, which means that
at the level of consumption represented by basket A, Eric would be willing to trade 5
glasses of lemonade for 1 additional hamburger: his marginal rate of substitution of
hamburgers for lemonade at point A is therefore 5. At point D, the slope of the indifference curve is ⫺2: at this level of consumption, Eric’s marginal rate of substitution
is 2—he would be willing to give up only two glasses of lemonade for an additional
hamburger.
This discussion suggests a clear relationship between the marginal rate of substitution of x for y (denoted by MRSx, y) and the slope of the indifference curve. On a
graph with x on the horizontal axis and y on the vertical axis, MRSx, y at any point is
the negative of the slope of the indifference curve at that point.
We can also express the marginal rate of substitution for any basket as a ratio of
the marginal utilities of the goods in that basket. To see how, consider any specific basket on the indifference curve U0. Suppose the consumer changes the level of consumption of x and y by ⌬x and ⌬y, respectively. The corresponding impact on utility
⌬U will be4
¢U ⫽ MUx (¢x) ⫹ MUy (¢y)
(3.4)
But it must be that ⌬U ⫽ 0, because changes in x and y that move us along the indifference curve U0 must keep utility unchanged. So 0 ⫽ MUx (⌬x) ⫹ MUy (⌬y), which
can be rewritten as MUy (⌬y) ⫽ ⫺MUx(⌬x). We can now solve for the slope of the
indifference curve ¢yⲐ ¢x:
¢y
¢x
2
⫽⫺
holding utility constant
MUx
MUy
Finally, since MRSx, y is the negative of the slope of the indifference curve, we
observe that
¢y
⫺
¢x
2
⫽
holding utility constant
MUx
⫽ MRSx, y
MUy
(3.5)
4
You may recognize that this equation is an approximation of the change in utility that results from
changing x and y by ⌬x and ⌬y, respectively. The approximation becomes more accurate when ⌬x
and ⌬y are small because the marginal utilities will be approximately constant for small changes in
x and y.
3.2 UTILITY FUNCTIONS
89
Diminishing Marginal Rate of Substitution
For many (but not all) goods, MRSx, y diminishes as the amount of x increases along an
indifference curve. To see why, refer to Figure 3.9. At basket A, to get 1 more hamburger,
Eric would be willing to forgo as many as 5 glasses of lemonade. And this makes sense
because at basket A Eric is drinking much lemonade and eating only a few hamburgers.
So we might expect MRSx,y to be large. However, if Eric were to move to basket D, where
he is consuming more hamburgers and less lemonade, he might not be willing to give up
as many glasses of lemonade to get still another hamburger. Thus, his MRSx,y will be
lower at D than at A. We have already shown that Eric’s MRSx,y at basket D is 2, which
is lower than his MRSx,y at basket A. In this case Eric’s preferences exhibit a diminishing
marginal rate of substitution of x for y. In other words, the marginal rate of substitution of x for y declines as Eric increases his consumption of x along an indifference curve.
What does a diminishing marginal rate of substitution of x for y imply about the
shape of the indifference curves? Remember that the marginal rate of substitution of
x for y is just the negative of the slope of the indifference curve on a graph with x on
the horizontal axis and y on the vertical axis. If MRSx, y diminishes as the consumer increases x along an indifference curve, then the slope of the indifference curve must be
getting flatter (less negative) as x increases. Therefore, indifference curves with diminishing MRSx, y must be bowed in toward the origin, as in Figure 3.9.
A P P L I C A T I O N
A feature of consumer preferences for which the marginal rate of substitution of
one good for another good
diminishes as the consumption of the first good increases along an indifference curve.
3.2
How People Buy Cars: The
Importance of Attributes
We began this chapter by discussing one of the choices
you would face as you decide whether to buy an automobile, the level of fuel efficiency. But you will probably also care about other attributes of the car you
might buy. Should it be big or small? Should it have a
big engine and lots of horsepower, or should it have a
smaller engine and thus get better gas mileage?
In other words, when you buy a car you are really
buying a bundle of attributes. Just as we can build a
theory of consumer choice among different goods by
means of a utility function defined over those goods,
we can also build a model of consumer choice among
different varieties of the same good (such as automobiles) by means of a utility function defined over the
attributes of this good. For example, the satisfaction
that consumers would derive from different brands of
cars could be described by a utility function over horsepower, gas mileage, luggage space, and so forth. Market
researchers often use this attribute-based approach
5
diminishing marginal
rate of substitution
when companies attempt to forecast the potential
market for a new product.
Nestor Arguea, Cheng Hsiao, and Grant Taylor
(AHT) used data on prices in the U.S. automobile market to estimate what are known as hedonic prices for
automobile attributes.5 A discussion of hedonic prices
is the stuff of an advanced econometrics course, so
we won’t go into the details of AHT’s methods here.
Roughly speaking, a hedonic price is a measure of the
marginal utility of a particular attribute. Given this,
the ratio of hedonic prices for two different automobile attributes, such as horsepower and gas mileage,
represents the marginal rate of substitution between
these attributes for the typical automobile consumer.
Based on AHT’s estimates, the marginal rate of
substitution of gas mileage for horsepower for a typical U.S. auto consumer in 1969 was 3.79. This means
that the typical consumer would be willing to forgo
3.79 horsepower to get an additional one mile per
gallon in gas mileage. Between 1969 and 1986 the
marginal rate of substitution of gas mileage for horsepower gradually fell, reaching 0.71 by 1986.
N. M. Arguea, C. Hsiao, and G. A. Taylor, “Estimating Consumer Preferences Using Market Data—An
Application to U.S. Automobile Demand,” Journal of Applied Econometrics 9 (1994): 1–18.
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CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
This decline in the marginal rate of substitution
of gas mileage for horsepower could reflect changes
in consumer tastes, or it could also reflect simultaneous changes in automobile prices, gasoline prices, and
consumer incomes. As we will see in the next chapter,
when changes in prices and income occur, consumers
move from one consumption bundle (and correspon-
S
E
ding indifference curve) to another, and at these bundles the marginal rates of substitution may differ.
The key point of this example is that marginal
rate of substitution is more than just a theoretical
concept. It can be estimated and used to help us understand the trade-offs that consumers are willing to
make between products and product attributes.
L E A R N I N G - B Y- D O I N G E X E R C I S E 3 . 3
D
Indifference Curves with Diminishing MRSx,y
Suppose a consumer has preferences
between two goods that can be represented by the utility function U xy. For this utility function, MUx y
and MUy x.6
(b) On the same graph draw a second indifference curve,
U2 200. Show how MRSx, y depends on x and y, and
use this information to determine if MRSx, y is diminishing for this utility function.
Problem
Solution
(a) On a graph, draw the indifference curve associated
with the utility level U1 128. Then answer the following questions:
1. Does the indifference curve intersect either axis?
2. Does the shape of the indifference curve indicate
that MRSx, y is diminishing?
(a) To draw the indifference curve U1 128 for the utility
function U xy, we plot points where xy 128—for
example, point G (x 8, y 16), point H (x 16, y
8), and point I (x 32, y 4)—and then connect these
points with a smooth line. Figure 3.11 shows this indifference curve.
20
y
Preference
directions
G
15
10
H
5
I
U2 = 200
U1 = 128
0
10
20
30
40
x
FIGURE 3.11 Indifference Curves with Diminishing MRSx,y
The indifference curves on this graph are for the utility function U xy, for which
MRSx,y yⲐx . On curve U1, the MRSx,y at basket G is 16Ⲑ8 ⫽ 2; therefore, the slope of the
indifference curve at G is ⫺2. The MRSx,y at basket I is 4Ⲑ32 ⫽ ⫺1Ⲑ8; therefore, the slope
of the indifference curve at I is ⫺1Ⲑ8. Thus, for U1 (and for U2) MRSx,y diminishes as x
increases, and the indifference curves are bowed in toward the origin.
6
To see how these marginal utilities can be derived from the utility function, you would use the calculus
techniques illustrated in Learning-By-Doing Exercise A.7 in the Mathematical Appendix.
3.2 UTILITY FUNCTIONS
Can the indifference curve U1 intersect either axis?
Since U1 is positive, x and y must both be positive (assuming the consumer is buying positive amounts of both
goods). If U1 intersected the x axis, the value of y at that
point would be zero; similarly, if U1 intersected the y axis,
the value of x at that point would be zero. If either x or y
were zero, the value of U1 would also be zero, not 128.
Therefore, the indifference curve U1 cannot intersect
either axis.
Is MRSx, y diminishing for U1? Figure 3.11 shows
that U1 is bowed in toward the origin; therefore, MRSx, y
is diminishing for U1.
91
Note that both MUx and MUy are positive whenever the consumer has positive amounts of x and y.
Therefore, indifference curves will be negatively sloped.
This means that as the consumer increases x along an indifference curve, y must decrease. Since MRSx,y ⫽
MUx ⲐMUy ⫽ yⲐx, as we move along the indifference
curve by increasing x and decreasing y, MRSx, y yⲐx
will decrease. So MRSx, y depends on x and y, and we
have diminishing marginal rate of substitution of x for y.
Similar Problems: 3.10, 3.11
(b) Figure 3.11 also shows the indifference curve
U2 200, which lies up and to the right of U1 128.
Learning-By-Doing Exercise 3.4 involves indifference curves with an increasing
marginal rate of substitution. Such curves are theoretically possible but not usually
encountered.
S
L E A R N I N G - B Y- D O I N G E X E R C I S E 3 . 4
D
E
Indifference Curves with Increasing MRSx,y
Consider what happens when a utility
function has an increasing marginal rate of substitution.
Problem Suppose a consumer’s preferences between
two goods (x and y) can be represented by the utility function U ⫽ Ax 2 ⫹ By2, where A and B are positive constants.
For this utility function MUx ⫽ 2Ax and MUy ⫽ 2By.
Show that MRSx, y is increasing.
Solution Since both MUx and MUy are positive,
indifference curves will be negatively sloped. This
means that as x increases along an indifference curve, y
must decrease. We know that MRSx,y ⫽ MUx ⲐMUy ⫽
2Ax Ⲑ(2By) ⫽ Ax Ⲑ(By). This means that as we move
along the indifference curve by increasing x and decreasing y, MRSx, y will increase. So we have an increasing marginal rate of substitution of x for y. Figure 3.12
illustrates the indifference curves for this utility function. With increasing MRSx, y they are bowed away
from the origin.
Similar Problems:
3.10, 3.11
Preference
directions
G
y
U2
U1
H
x
FIGURE 3.12 Indifference Curves
with Increasing MRSx,y
If the MRSx,y is higher at basket H than at
basket G, then the slope of indifference
curve U1 will be more negative (steeper)
at H than at G. Thus, with increasing MRSx,y,
the indifference curves will be bowed away
from the origin.
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CHAPTER 3
3.3
A consumer’s willingness to substitute one good for another will depend on the com-
SPECIAL
PREFERENCES
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
modities in question. For example, one consumer may view Coke and Pepsi as perfect
substitutes and always be willing to substitute a glass of one for a glass of the other. If
so, the marginal rate of substitution of Coke for Pepsi will be constant and equal to 1,
rather than diminishing. Sometimes a consumer may simply be unwilling to substitute
one commodity for another. For example, a consumer might always want exactly 1
ounce of peanut butter for each ounce of jelly on his sandwiches and be unwilling to
consume peanut butter and jelly in any other proportions. To cover cases such as these
and others, there are several special utility functions. Here we discuss four: utility functions in the case of perfect substitutes and the case of perfect complements, the
Cobb–Douglas utility function, and quasilinear utility functions.
PERFECT SUBSTITUTES
perfect substitutes
(in consumption) Two
goods such that the marginal rate of substitution
of one good for the other
is constant; therefore, the
indifference curves are
straight lines.
In some cases, a consumer might view two commodities as perfect substitutes for one
another. Two goods are perfect substitutes when the marginal rate of substitution of one
for the other is a constant. For example, suppose David likes both butter (B) and margarine (M ) and that he is always willing to substitute a pound of either commodity for a
pound of the other. Then MRSB,M MRSM,B 1. We can use a utility function such as
U aB ⫹ aM, where a is any positive constant, to describe these preferences. (With this
utility function, MUB ⫽ a and MUM ⫽ a. It also follows that MRSB, M ⫽ MUBⲐMUM ⫽
a Ⲑa ⫽ 1, and the slope of the indifference curves will be constant and equal to ⫺1.)
More generally, indifference curves for perfect substitutes are straight lines, and
the marginal rate of substitution is constant, though not necessarily equal to 1. For
example, suppose a consumer likes both pancakes and waffles and is always willing
to substitute two pancakes for one waffle. A utility function that would describe
his preferences is U ⫽ P ⫹ 2W, where P is the number of pancakes and W the
number of waffles. With these preferences, MUP ⫽ 1 and MUW ⫽ 2, so each waffle yields twice the marginal utility of a single pancake. We also observe that
MRSP,W ⫽ MUPⲐMUW ⫽ 1Ⲑ2. Two indifference curves for this utility function are
A P P L I C A T I O N
3.3
Taste Tests
If you listen to advertisements on television, you
might believe that most goods are highly differentiated products and that most consumers have strong
preferences for one brand over another. To be sure,
there are differences among brands, and brands vary
in price. But are brands so different that one producer
could raise the price of its product without losing a
significant portion of its sales?
In looking at the U.S. beer industry, Kenneth
Elzinga observed, “Several studies indicate that, at
7
least under blindfold test conditions, most beer
drinkers cannot distinguish between brands of
beer.” He also noted that brewers have devoted
“considerable talent and resources . . . to publicizing real or imagined differences in beers, with the
hope of producing product differentiation.” In the
end, Elzinga suggested, despite brewers’ efforts to
differentiate their products from those of their
competitors, most consumers would be quite willing to substitute one brand of beer for another,
especially if one brand were to raise its price significantly.7
K. Elzinga, “The Beer Industry,” in W. Adams, The Structure of American Industry, 8th ed. pp.142–143
(New York: Macmillan Publishing Company, 1990).
3.3 SPECIAL PREFERENCES
A recent study of wine drinkers came to a similar
conclusion.8 The food and wine publishing firm
Fearless Critic Media organized 17 blind tastings of
wine by 506 participants in 2007–2008. Wines ranged
from $1.65 to $150 per bottle. Tasters were asked to assign a rating to each wine. The data were then statistically analyzed by economists. They found a small and
negative correlation between price and rated quality.
They did find a positive correlation between price and
quality among tasters with wine training, but the correlation was small and had low statistical significance.
Two members of that research team recently collaborated on a similar study that is perhaps a bit more
troubling than the wine research.9 Noting that canned
dog food and paté are both made at least partially from
small pieces of ground meat, they studied whether
(human) tasters could distinguish the two products in a
blind taste test. The team blended a high-end organic
dog food made exclusively from “human grade” agricultural products until it had consistency similar to
paté. This was compared to Spam,10 supermarket liverwurst, and two types of gourmet paté. The good news
is that 72 percent of tasters ranked dog food as the
worst tasting of the five products. The bad news is that
this result was not statistically significant!
These kinds of studies do not suggest that all
consumers regard all beer, wine, or paté style products to be perfect substitutes. However, when a consumer does not have a strong preference for one
brand over another, the marginal rate of substitution
of brand A for brand B might be nearly constant, and
probably near 1, since a consumer would probably be
willing to give up one unit of one brand for one unit
of another.
shown in Figure 3.13. Since MRSP,W ⫽ 1Ⲑ2, on a graph with P on the horizontal
axis and W on the vertical axis, the slope of the indifference curves is 1 Ⲑ2.
Preference
directions
W, waffles
4
U=8
2
U=4
4
P, pancakes
8
FIGURE 3.13 Indifference Curves with Perfect
Substitutes
A consumer with the utility function U ⫽ P ⫹ 2W
always views two pancakes as a perfect substitute for
one waffle. MRSP,W ⫽ 1Ⲑ2, and so indifference curves
are straight lines with a slope of 1Ⲑ2.
PERFECT COMPLEMENTS
In some cases, consumers might be completely unwilling to substitute one good for another. Consider a typical consumer’s preferences for left shoes and right shoes, depicted
in Figure 3.14. The consumer wants shoes to come in pairs, with exactly one left shoe for
every right shoe. The consumer derives satisfaction from complete pairs of shoes, but
gets no added utility from extra right shoes or extra left shoes. The indifference curves in
this case comprise straight-line segments at right angles, as shown in Figure 3.14.
8
93
R. Goldstein et al., “Do More Expensive Wines Taste Better? Evidence from a Large Sample of Blind
Tastings,” Journal of Wine Economics (Spring 2008).
9
J. Bohannon et al., “Can People Distinguish Paté from Dog Food?” American Association of Wine
Economists’ Working Paper #36, April 2009.
10
Spam is an inexpensive canned food made out of precooked chopped pork and gelatin.
94
CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
U3, the utility from
three pairs of shoes
FIGURE 3.14 Indifference Curves with Perfect
Complements
The consumer wants exactly one left shoe for every
right shoe. For example, his utility at basket G, with
2 left shoes and 2 right shoes, is not increased by
moving to basket H, containing 2 left shoes and 3
right shoes.
perfect complements
(in consumption) Two
goods that the consumer
always wants to consume
in fixed proportion to each
other.
L, Left shoes
3
H
G
2
U1, the utility from
one pair of shoes
1
0
U2, the utility from
two pairs of shoes
1
2
3
R, Right shoes
The consumer with the preferences illustrated in Figure 3.14 regards left shoes
and right shoes as perfect complements in consumption. Perfect complements are
goods the consumer always wants in fixed proportion to each other; in this case, the
desired proportion of left shoes to right shoes is 1:1.11
A utility function for perfect complements—in this case, left shoes (L) and right
shoes (R)—is U(R, L) 10min(R, L), where the notation “min” means “take the minimum value of the two numbers in parentheses.” For example, at basket G, R 2 and
L 2; so the minimum of R and L is 2, and U 10(2) 20. At basket H, R 3 and
L 2; so the minimum of R and L is still 2, and U 10(2) 20. This shows that baskets G and H are on the same indifference curve, U2 (where U2 20).
THE COBB–DOUGLAS UTILITY FUNCTION
Cobb–Douglas utility
function A function of
the form U ⫽ Ax ␣y ,
where U measures the consumer’s utility from x units
of one good and y units of
another good and where
A, ␣, and  are positive
constants.
The utility functions U ⫽ 1xy and U xy are examples of the Cobb–Douglas utility
function. For two goods, the Cobb–Douglas utility function is more generally represented as U Ax␣ y, where A, ␣, and  are positive constants.12
The Cobb–Douglas utility function has three properties that make it of interest
in the study of consumer choice.
• The marginal utilities are positive for both goods. The marginal utilities are
MUx ␣Ax␣⫺1 y  and MUy ⫽ Ax␣ y ⫺1; thus, both MUx and MUy are positive
when A, ␣, and  are positive constants. This means that “the more is better”
assumption is satisfied.
• Since the marginal utilities are both positive, the indifference curves will be
downward sloping.
• The Cobb–Douglas utility function also exhibits a diminishing marginal rate
of substitution. The indifference curves will therefore be bowed in toward the
11
The fixed-proportions utility function is sometimes called a Leontief utility function, after the economist
Wassily Leontief, who employed fixed-proportion production functions to model relationships between
sectors in a national economy. We shall examine Leontief production functions in Chapter 6.
12
This type of function is named for Charles Cobb, a mathematician at Amherst College, and Paul
Douglas, a professor of economics at the University of Chicago (and later a U.S. senator from Illinois).
It has often been used to characterize production functions, as we shall see in Chapter 6 when we study
the theory of production. The Cobb–Douglas utility function can easily be extended to cover more than
two goods. For example, with three goods the utility function might be represented as U⫽Ax ␣ y z␥,
where z measures the quantity of the third commodity, and A, ␣, , and ␥ are all positive constants.
95
3.3 SPECIAL PREFERENCES
y
A
B
U3
C
U2
U1
x1
x
FIGURE 3.15 Indifference Curves for a Quasilinear
Utility Function
A quasilinear utility function has the form U(x, y) ⫽ v (x) ⫹
by, where v(x) is a function that increases in x and b is a
positive constant. The indifference curves are parallel, so
for any value of x (such as x1), the slopes of the indifference curves will be the same (e.g., the slopes of the indifference curves are identical at baskets A, B, and C ).
origin, as in Figure 3.11. Problem 3.21 at the end of the chapter asks you to
verify that the marginal rate of substitution is diminishing.
Q UA S I L I N E A R U T I L I T Y F U N C T I O N S
The properties of a quasilinear utility function often simplify analysis. Further, economic studies suggest that such functions may reasonably approximate consumer
preferences in many settings. For example, as we shall see in Chapter 5, a quasilinear
utility function can describe preferences for a consumer who purchases the same
amount of a commodity (such as toothpaste or coffee) regardless of his income.
Figure 3.15 shows the indifference curves for a quasilinear utility function. The
distinguishing characteristic of a quasilinear utility function is that, as we move due
north on the indifference map, the marginal rate of substitution of x for y remains
the same. That is, at any value of x, the slopes of all of the indifference curves will be
the same, so the indifference curves are parallel to each other.
The equation for a quasilinear utility function is U(x, y) v(x) ⫹ by, where b is a
positive constant and v(x) is a function that increases in x—that is, the value of v(x) increases as x increases [e.g., v(x) ⫽ x2or v(x) ⫽ 1x]. This utility function is linear in y,
but generally not linear in x. That is why it is called quasilinear.
A P P L I C A T I O N
quasilinear utility
function A utility function that is linear in at
least one of the goods
consumed, but may be a
nonlinear function of the
other good(s).
3.4
Hula Hoops and Beanie Babies
The preferences of individual consumers are often influenced by fads, typically short-lived episodes during
which the consumption of a good or service enjoys
widespread popularity. One of the greatest fads of
the past century was the Hula Hoop, a light plastic
circular tube developed in 1957 by Wham-O. The Hula
Hoop was patterned after bamboo hoops that children
in Australia twirled around their waists in physical
education classes, and was named after the Hawaiian
dance involving similar movements.
Although children have long played with
wooden or metal hoops by rolling, tossing, or spinning them, Wham-O found the durable, light, plastic
version of the hoop to be especially popular. When
Wham-O test-marketed a prototype of the Hula
Hoop in California, interest in the new toy spread
quickly. Wham-O sold 25 million units in the early part
of 1958, and orders for many more units followed as
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CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
R, number of Beanie Babies
the fad spread to Europe and Japan. By the end of
1958, the fad had subsided, and Wham-O moved on
to its next major product, the Frisbee.
Of course, there have been many fads over time.
In 1993 Ty Incorporated introduced Beanie Babies,
small stuffed animals. The line of toys became perhaps the biggest fad of all time, with Ty’s revenue
topping $6 billion. While Beanie Babies sold for about
$6 at stores, their resale value on the secondary market
was often $100 or more, especially for rare varieties.
People often waited in line to purchase new designs.
However, by 1999 the craze for Beanie Babies was
subsiding, and Ty announced the end of the product
line by releasing a bear named “The End.” Ty later
brought back Beanie Babies, and in 2008 released a
new line called Beanie Babies 2.0, but the product
never again became the fad it had been in the 1990s.
Fads change consumer preferences. For example,
suppose a consumer purchases only two goods,
Beanie Babies and food. During the fad, as shown in
panel (a) of Figure 3.16, when the consumer increases
his utility significantly by purchasing more Beanie
Babies (e.g., by changing his consumption from basket A to basket B), indifference curves are relatively
flat. After the fad, as shown in panel (b) of Figure 3.16,
when the consumer gains little extra utility by purchasing more Beanie Babies, indifference curves are much
steeper (i.e., the marginal rate of substitution of food
for Beanie Babies has increased). Note that in panel
(b) the consumer still has some interest in Beanie
Babies; if he entirely stopped caring about them, the
indifference curves would become vertical, with
higher indifference curves located farther to the right.
B
R2
R1
U4 = 50
U3 = 40
U2 = 30
U1 = 20
A
F1
F, units of food
FIGURE 3.16 Fads and Preferences
During the Beanie Baby fad, as shown in panel
(a), the consumer can achieve much added satisfaction (moving from indifference curve U1 to
U4) by purchasing more Beanie Babies (moving
from basket A to basket B). When the fad is
over, as shown in panel (b), the move from basket A to basket B generates much less additional satisfaction (the utility increases from U1
to only U2); the consumer now has less interest
in Beanie Babies. The indifference curves become steeper as his interest in Beanie Babies
fades.
R, number of Beanie Babies
(a)
B
R2
R1
U4 = 40
U3 = 30
A
U2 = 20
U1 = 10
F1
F, units of food
(b)
3.3 SPECIAL PREFERENCES
97
In this chapter we have kept the discussion of preferences (including the graphs)
simple by analyzing cases in which the consumer buys two goods. But the principles
presented here also apply to much more complicated consumer choice problems,
including choices among many different goods. For example, as observed in
Application 3.2, a consumer typically considers many factors when buying an
automobile, including the dimensions of the car, the size of the engine, the fuel
used, fuel efficiency, reliability, the availability of options, and safety features. Using
the framework developed in this chapter, we would say that the utility a consumer derives from an automobile depends on the characteristics of that vehicle. As the research
described in Application 3.2 shows, consumers are often willing to trade off one
attribute for another.
A P P L I C A T I O N
3.5
Does More Make You Happier?
Reference-Dependent Preferences13
As you consume more and more of the goods you
typically purchase, do you become ever and ever happier? In other words, does your utility increase? An
assumption that we have maintained throughout
this chapter—”more is better”—would imply that
your answer would be yes.14 As we will see in the
next two chapters, increases in your income will enable you to purchase bigger bundles of goods and
services, which in turn will move you to higher and
higher levels of utility.
If you are like most people, however, it is likely
that increased consumption does not always bring
with it feelings of greater happiness. Research on the
determinants of happiness suggests that more is
often not better. One of the most influential researchers in this field is Richard Easterlin, who in 2009
received the prestigious IZA Prize in Labor Economics
for his pioneering research on the economics of happiness. (IZA is the Institut zur Zukunft der Arbeit, or
Institute for the Study of Labor.) To quote from the
press release announcing the prize:
Richard Easterlin first showed in the 1970s that rising wealth does not necessarily improve individual well-being. It is true that wealthier societies
are more satisfied on average than poorer ones.
13
However, once labor income ensures a certain
level of material wealth guaranteeing basic
needs, individual and societal well-being no
longer increases with growing economic wealth.
Social comparisons and changes in expected living standard strongly influence individual wellbeing. . . . Overall, Easterlin’s research shows that
people in wealthy nations show no higher life
satisfaction than people in poorer nations once
the level of income is high enough to provide for
food, shelter and other fundamental needs. This
apparently contradictory finding became known
as the “Easterlin Paradox.”15
Is there a way to adapt the traditional theory of
consumer choice from microeconomics, so that its
implications are consistent with the empirical findings from the literature on psychological well-being?
Bridging psychology and economic theory is the central purpose of an important area within economics
known as behavioral economics. Research in behavioral economics seeks to strengthen the psychological foundations of economic models so that they can
make better predictions about individual decision
making.
Behavioral economists Botond Koszegi and
Matthew Rabin have proposed a theory of referencebased preferences that yields implications consistent
with psychological research on happiness.16 Koszegi
and Rabin posit that an individual’s utility depends not
We would like to thank Eric Schultz for his comments and suggestions on this application.
We are, of course, talking about “goods” rather than “bads,” which would include phenomena such as pollution or traffic congestion.
15
IZA press release, May 4, 2009.
16
B. Koszegi and M. Rabin, “A Model of Reference-Dependent Preferences,” Quarterly Journal of Economics 12, no. 4 (2006): 1133–1165.
14
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CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
on the individual’s absolute consumption of goods and
services but on the consumption of goods and services
relative to some sort of reference level. In Koszegi and
Rabin’s theory, reference levels represent a consumer’s
expectation (prior to making consumption decisions)
about how much of each good the consumer is likely to
end up consuming. If the consumer ends up consuming
less than the expected amount, the consumer experiences a loss; if the consumer ends up consuming more
than the expected amount, the consumer experiences
a gain (which typically would be expected to be smaller
than the loss). These assumptions imply that it could
easily be the case that an increase in consumption
could leave a consumer no happier than he or she
was before. This would be the case if the consumer
ends up consuming exactly what he or she expected to
consume.
Utility functions that include reference levels of
consumption are a special case of a broader phenomenon in which individuals tend to adapt to the situations in which they find themselves. Psychologists
define hedonic adaptation as the tendency of our
moods to settle back to some set range after a temporary burst of emotion in response to certain events.
This would explain why individuals predict that they
would be miserable if they were to suffer a physical
handicap, while at the same time, people who do
suffer from such handicaps adapt and tend to find life
satisfying (or, at least, less miserable than those who
were merely projecting themselves into that situation
would predict). It would also explain why people who
marry tend to experience a large positive deviation in
happiness in the short to medium term but eventually
experience a significant drop in this happiness, In addition, people who divorce tend eventually to be no
happier than before (in part because they tend to
also overlook the additional problems that the divorce will create).
Behavioral economics is an important field of
economics because it highlights, through empirical
and experimental analysis, anomalies in behavior
that cannot be explained using the tools of traditional microeconomic theory. In addition, it points
out how traditional theory tools need to be modified
in order for predicted decisions to be consistent with
real-world evidence. For example, behavioral economists have formulated theories that explain procrastination, lack of self-control, and a willingness to go
against self-interest (e.g., the willingness of a household to heed the call for voluntary reductions in the
use of water during a drought). These contributions
enhance the richness of economic theory and (to paraphrase one account in the business press), “put a
human face on economics.”17
CHAPTER SUMMARY
• Consumer preferences tell us how a consumer ranks
(compares the desirability of ) any two baskets, assuming
the baskets are available at no cost. In most situations, it
is reasonable to make three assumptions about consumer
preferences:
1. They are complete, so that the consumer is able to
rank all baskets.
2. They are transitive, meaning that if the consumer
prefers basket A to basket B and he prefers basket B to
basket E, then he prefers basket A to basket E.
3. They satisfy the property that more is better, so that
having more of either good increases the consumer’s
satisfaction.
17
• A utility function measures the level of satisfaction
that a consumer receives from any basket of goods.
The assumptions that preferences are complete, that
preferences are transitive, and that more is better
imply that preferences can be represented by a utility
function.
• The marginal utility of good x (MUx) is the rate at
which total utility changes as the consumption of x rises.
(LBD Exercises 3.1 and 3.2)
• An indifference curve shows a set of consumption
baskets that yield the same level of satisfaction to the
consumer. Indifference curves cannot intersect. If the
consumer likes both goods x and y (i.e., if MUx and MUy
“Putting a Human Face on Economics,” Business Week ( July 31, 2000), pp. 76–77.
REVIEW QUESTIONS
99
are both positive), then indifference curves will have a
negative slope.
other will be constant, and the indifference curves will
be straight lines.
• The marginal rate of substitution of x for y (MRSx, y)
at any basket is the rate at which the consumer will give
up y to get more x, holding the level of utility constant.
On a graph with x on the horizontal axis and y on the
vertical axis, the MRSx, y at any basket is the negative of
the slope of the indifference curve at that basket. (LBD
• If two goods are perfect complements in consumption, the consumer wants to purchase the two goods in a
fixed proportion. The indifference curves in this case
will be L-shaped.
Exercises 3.3 and 3.4)
• For most goods we would expect to observe a diminishing MRSx, y. In this case the indifference curves will be
bowed in toward the origin.
• If a consumer’s utility function is quasilinear (e.g.,
linear in y, but generally not linear in x), the indifference
curves will be parallel. At any value of x, the slopes of all
of the indifference curves (and thus the MRSx, y) will be
the same.
• If two goods are perfect substitutes in consumption,
the marginal rate of substitution of one good for the
REVIEW QUESTIONS
1.
What is a basket (or a bundle) of goods?
2. What does the assumption that preferences are complete mean about the consumer’s ability to rank any two
baskets?
3. Consider Figure 3.1. If the more is better assumption
is satisfied, is it possible to say which of the seven baskets
is least preferred by the consumer?
4. Give an example of preferences (i.e., a ranking of
baskets) that do not satisfy the assumption that preferences are transitive.
5. What does the assumption that more is better imply
about the marginal utility of a good?
6. What is the difference between an ordinal ranking
and a cardinal ranking?
7. Suppose Debbie purchases only hamburgers. Assume
that her marginal utility is always positive and diminishing. Draw a graph with total utility on the vertical axis
and the number of hamburgers on the horizontal axis.
Explain how you would determine marginal utility at any
given point on your graph.
8. Why can’t you plot the total utility and marginal
utility curves on the same graph?
9. Adam consumes two goods: housing and food.
a) Suppose we are given Adam’s marginal utility of housing and his marginal utility of food at the basket he currently consumes. Can we determine his marginal rate of
substitution of housing for food at that basket?
b) Suppose we are given Adam’s marginal rate of substitution of housing for food at the basket he currently consumes. Can we determine his marginal utility of housing
and his marginal utility of food at that basket?
10. Suppose Michael purchases only two goods,
hamburgers (H ) and Cokes (C).
a) What is the relationship between MRSH,C and the
marginal utilities MUH and MUC ?
b) Draw a typical indifference curve for the case in which
the marginal utilities of both goods are positive and the
marginal rate of substitution of hamburgers for Cokes is
diminishing. Using your graph, explain the relationship
between the indifference curve and the marginal rate of
substitution of hamburgers for Cokes.
c) Suppose the marginal rate of substitution of hamburgers for Cokes is constant. In this case, are hamburgers and Cokes perfect substitutes or perfect
complements?
d) Suppose that Michael always wants two hamburgers
along with every Coke. Draw a typical indifference curve.
In this case, are hamburgers and Cokes perfect substitutes or perfect complements?
11. Suppose a consumer is currently purchasing 47
different goods, one of which is housing. The quantity of
housing is measured by H. Explain why, if you wanted to
measure the consumer’s marginal utility of housing
(MUH) at the current basket, the levels of the other 46
goods consumed would be held fixed.
100
CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
PROBLEMS
3.1. Bill has a utility function over food and gasoline
with the equation U ⫽ x2y, where x measures the quantity of food consumed and y measures the quantity of
gasoline. Show that a consumer with this utility function
believes that more is better for each good.
3.2. Consider the single-good utility function
U(x) ⫽ 3x2, with a marginal utility given by MUx ⫽ 6x.
Plot the utility and marginal utility functions on two
separate graphs. Does this utility function satisfy the
principle of diminishing marginal utility? Explain.
3.3. Jimmy has the following utility function for hot
dogs: U(H) ⫽ 10H ⫺ H2, with MUH ⫽ 10 ⫺ 2H.
a) Plot the utility and marginal utility functions on two
separate graphs.
b) Suppose that Jimmy is allowed to consume as many
hot dogs as he likes and that hot dogs cost him nothing.
Show, both algebraically and graphically, the value of H
at which he would stop consuming hot dogs.
3.4. Consider the utility function U(x, y) y1x with
the marginal utilities MUx y(2 1x) and MUy 1x.
a) Does the consumer believe that more is better for
each good?
b) Do the consumer’s preferences exhibit a diminishing marginal utility of x? Is the marginal utility of y
diminishing?
3.5. Carlos has a utility function that depends on the
number of musicals and the number of operas seen each
month. His utility function is given by U ⫽ xy2, where x
is the number of movies seen per month and y is the
number of operas seen per month. The corresponding
marginal utilities are given by: MUx ⫽ y2 and MUy ⫽ 2xy.
a) Does Carlos believe that more is better for each good?
b) Does Carlos have a diminishing marginal utility for
each good?
3.6. For the following sets of goods draw two indifference curves, U1 and U2, with U2 U1. Draw each graph
placing the amount of the first good on the horizontal
axis.
a) Hot dogs and chili (the consumer likes both and has a
diminishing marginal rate of substitution of hot dogs for
chili)
b) Sugar and Sweet’N Low (the consumer likes both and
will accept an ounce of Sweet’N Low or an ounce of
sugar with equal satisfaction)
c) Peanut butter and jelly (the consumer likes exactly
2 ounces of peanut butter for every ounce of jelly)
d) Nuts (which the consumer neither likes nor dislikes)
and ice cream (which the consumer likes)
e) Apples (which the consumer likes) and liver (which
the consumer dislikes)
3.7. Alexa likes ice cream, but dislikes yogurt. If you
make her eat another gram of yogurt, she always requires
two extra grams of ice cream to maintain a constant level
of satisfaction. On a graph with grams of yogurt on the
vertical axis and grams of ice cream on the horizontal
axis, graph some typical indifference curves and show the
directions of increasing utility.
3.8. Joe has a utility function over hamburgers and hot
dogs given by U x 1y, where x is the quantity of
hamburgers and y is the quantity of hot dogs. The marginal utilities for this utility function are MUx 1 and
MUy 1(2 1y ).
Does this utility function have the property that MRSx,y
is diminishing?
3.9. Julie and Toni consume two goods with the following utility functions:
U Julie (x y)2,
U Toni x y,
MU Julie
2(x y),
x
MU Toni
1,
x
2(x y)
MU Julie
y
MU Toni
1
y
a) Graph an indifference curve for each of these utility
functions.
b) Julie and Toni will have the same ordinal ranking of
different baskets if, when basket A is preferred to basket
B by one of the functions, it is also preferred by the other.
Do Julie and Toni have the same ordinal ranking of different baskets of x and y? Explain.
3.10. The utility that Julie receives by consuming food
F and clothing C is given by U(F, C) FC. For this
utility function, the marginal utilities are MUF C and
MUC F.
a) On a graph with F on the horizontal axis and C on the
vertical axis, draw indifference curves for U 12, U 18,
and U 24.
b) Do the shapes of these indifference curves suggest
that Julie has a diminishing marginal rate of substitution
of food for clothing? Explain.
c) Using the marginal utilities, show that the MRSF,C
C F. What is the slope of the indifference curve U 12
at the basket with 2 units of food and 6 units of clothing?
What is the slope at the basket with 4 units of food and
3 units of clothing? Do the slopes of the indifference
PROBLEMS
curves indicate that Julie has a diminishing marginal rate
of substitution of food for clothing? (Make sure your answers to parts (b) and (c) are consistent!)
3.11. Sandy consumes only hamburgers (H ) and milkshakes (M ). At basket A, containing 2 hamburgers and
10 milkshakes, his MRSH,M is 8. At basket B, containing
6 hamburgers and 4 milkshakes, his MRSH,M is 1 Ⲑ2. Both
baskets A and B are on the same indifference curve. Draw
the indifference curve, using information about the
MRSH,M to make sure that the curvature of the indifference curve is accurately depicted.
3.12. Adam likes his caffé latte prepared to contain
exactly 1Ⲑ4 espresso and 3 Ⲑ4 steamed milk by volume. On
a graph with the volume of steamed milk on the horizontal axis and the volume of espresso on the vertical axis,
draw two of his indifference curves, U1 and U2, with
U1 ⬎ U2.
3.13. Draw indifference curves to represent the following types of consumer preferences.
a) I like both peanut butter and jelly, and always get the
same additional satisfaction from an ounce of peanut
butter as I do from 2 ounces of jelly.
b) I like peanut butter, but neither like nor dislike jelly.
c) I like peanut butter, but dislike jelly.
d) I like peanut butter and jelly, but I only want 2 ounces
of peanut butter for every ounce of jelly.
3.14. Dr. Strangetaste buys only food (F) and clothing
(C) out of his income. He has positive marginal utilities
for both goods, and his MRSF,C is increasing. Draw two of
Dr. Strangetaste’s indifference curves, U1 and U2, with
U2 ⬎ U1.
The following exercises will give you practice in working with
a variety of utility functions and marginal utilities and will
help you understand how to graph indifference curves.
3.15. Consider the utility function U(x, y) ⫽ 3x ⫹ y,
with MUx ⫽ 3 and MUy ⫽ 1.
a) Is the assumption that more is better satisfied for both
goods?
b) Does the marginal utility of x diminish, remain constant, or increase as the consumer buys more x? Explain.
c) What is MRSx,y?
d) Is MRSx,y diminishing, constant, or increasing as the
consumer substitutes x for y along an indifference curve?
e) On a graph with x on the horizontal axis and y on the
vertical axis, draw a typical indifference curve (it need not
be exactly to scale, but it needs to reflect accurately
whether there is a diminishing MRSx,y). Also indicate on
your graph whether the indifference curve will intersect
either or both axes. Label the curve U1.
101
f ) On the same graph draw a second indifference curve
U2, with U2 ⬎ U1.
3.16. Answer all parts of Problem 3.15 for the utility
function U(x, y) ⫽ 1xy. The marginal utilities are
MUx ⫽ 1yⲐ(2 1x) and MUy ⫽ 1xⲐ(2 1y).
3.17. Answer all parts of Problem 3.15 for the utility
function U(x, y) ⫽ x y ⫹ x. The marginal utilities are
MUx ⫽ y ⫹1 and MUy ⫽ x.
3.18. Answer all parts of Problem 3.15 for the utility
function U(x, y) ⫽ x0.4y0.6. The marginal utilities are
MUx ⫽ 0.4 ( y0.6 Ⲑx0.6 ) and MUy ⫽ 0.6 (x0.4Ⲑy0.4 ) .
3.19. Answer all parts of Problem 3.15 for the utility
function U ⫽ 1x ⫹ 21y. The marginal utilities for x
and y are, respectively, MUx ⫽ 1Ⲑ(2 1x) and
MUy ⫽ 1Ⲑ 1y.
3.20. Answer all parts of Problem 3.15 for the utility
function U(x, y) ⫽ x 2 ⫹ y 2. The marginal utilities are
MUx ⫽ 2x and MUy ⫽ 2y.
3.21. Suppose a consumer’s preferences for two goods
can be represented by the Cobb–Douglas utility function
U ⫽ Ax ␣y, where A, ␣, and  are positive constants.
The marginal utilities are MUx ⫽ ␣Ax ␣⫺1y  and MUy ⫽
Ax ␣y ⫺1. Answer all parts of Problem 3.15 for this
utility function.
3.22. Suppose a consumer has preferences over two
goods that can be represented by the quasilinear utility
function U(x, y) ⫽ 21x ⫹ y The marginal utilities are
MUx ⫽ 1Ⲑ 1x and MUy ⫽ 1.
a) Is the assumption that more is better satisfied for both
goods?
b) Does the marginal utility of x diminish, remain constant, or increase as the consumer buys more x? Explain.
c) What is the expression for MRSx,y?
d) Is the MRSx,y diminishing, constant, or increasing as
the consumer substitutes more x for y along an indifference curve?
e) On a graph with x on the horizontal axis and y on the
vertical axis, draw a typical indifference curve (it need not
be exactly to scale, but it should accurately reflect
whether there is a diminishing MRSx,y ). Indicate on your
graph whether the indifference curve will intersect either
or both axes.
f ) Show that the slope of every indifference curve will be
the same when x ⫽ 4. What is the value of that slope?
3.23. Daniel and Will each consume two goods. When
they consume the same basket, Daniel’s marginal utility
of each good is higher than Will’s. But at any basket they
both have the same marginal rate of substitution of one
102
CHAPTER 3
CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY
good for the other. Do they have the same ordinal ranking of different baskets?
Does this utility function exhibit the property of diminishing MRSx,y?
3.24. Claire consumes three goods out of her income:
food (F), shelter (S), and clothing (C). At her current
levels of consumption, her marginal utility of food is
3 and her marginal utility of shelter is 6. Her marginal
rate of substitution of shelter for clothing is 4. Do you
have enough information to determine her marginal rate
of substitution of food for clothing? If so, what is it? If
not, why not?
3.26. Annie consumes three goods out of her income:
food (F) shelter (S), and clothing (C). At her current
levels of consumption, her marginal rate of substitution
of food for clothing is 2 and her marginal rate of substitution of clothing for shelter is 3.
a) Do you have enough information to determine her
marginal rate of substitution of food for shelter? If so,
what is it? If not, why not?
b) Do you have enough information to determine her
marginal utility of shelter? If so, what is it? If not, why
not?
3.25. Suppose a person has a utility function given by
U ⫽ [x y ]1/ where is a number between ⫺q and
1. This is called a constant elasticity of substitution
(CES) utility function. You will encounter CES functions
in Chapter 6, where the concept of elasticity of substitution will be explained. The marginal utilities for this utility function are given by
1
MUx ⫽ 冤x r ⫹ y r冥 r ⫺1x
1
r⫺1
MUy ⫽ 冤x r ⫹ y r冥 r ⫺1y r⫺1
4
CONSUMER CHOICE
4.1
T H E BU D G E T C O N S T R A I N T
APPLICATION 4.1
The Rising Price of Gasoline
4.2
OPTIMAL CHOICE
The Marginal Utility of “Home
Cooking” versus “Eating Out”: Exploring the
Implications of the “Equal Bang for the Buck
Condition”
APPLICATION 4.2
4.3
CONSUMER CHOICE WITH
COMPOSITE GOODS
APPLICATION 4.6
Coupons versus Cash: SNAP
Pricing a Calling Plan
To Lend or Not to Lend?
Flying Is Its Own Reward
APPLICATION 4.7
Is Altruism Rational?
APPLICATION 4.3
APPLICATION 4.4
APPLICATION 4.5
4.4
REVEALED PREFERENCE
APPENDICES
T H E M AT H E M AT I C S O F
CONSUMER CHOICE
T H E T I M E VA L U E O F M O N E Y
How Much of What You Like Should You Buy?
According to the United States Bureau of Labor Statistics, in 2007 there were about 120 million households
in the United States. The average household had before-tax annual income of about $63,100. Consumers
in these households faced many decisions. How much should they spend out of their income, and how
much should they save? On average, they spent about $49,600. They also had to decide how to divide their
expenditures among various types of goods and services, including food, housing, clothing, transportation,
health care, entertainment, and other items.
103
Of course, the average values of statistics reported for all households mask the great variations in
consumption patterns by age, location, income level, marital status, and family composition. Table 4.1
compares expenditure patterns for all households and for selected levels of income.
A casual examination of the table reveals some interesting patterns in consumption. Consumers
with lower income tend to spend more than their current after-tax income, electing to borrow today
and repay their loans in the future. For example, households with incomes in the $20,000–$30,000 range
spend about $5,000 per year more than their after-tax income. By contrast, households with incomes
in excess of $70,000 save more than 30 percent their after-tax income. The table also indicates that
consumers who attend college can expect to earn substantially higher incomes, a fact that influences
the choice to attend college.
Consumer decisions have a profound impact on the economy as a whole and on the fortunes of individual
firms and institutions. For example, consumer expenditures on transportation affect the financial viability of
the airline and automobile sectors of the economy, as well as the demand for related items such as fuel and
insurance. The level of spending on health care will affect not only providers of health care services in the
private sector, but also the need for public sector programs such as Medicare and Medicaid.
This chapter develops the theory of consumer choice, explaining how consumers allocate their
limited incomes among available goods and services. It begins where Chapter 3 left off. In that chapter,
we developed the first building block in the study of consumer choice: consumer preferences. However,
preferences alone do not explain why consumers make the choices they do. Consumer preferences tell
us whether a consumer likes one particular basket of goods and services better than another, assuming
that all baskets could be “purchased” at no cost. But it does cost the consumer something to purchase
baskets of goods and services, and a consumer has limited resources with which to make these
purchases.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Write the equation of the budget constraint and graph the budget line.
• Illustrate graphically how a change in income or a change in a price affects the budget line.
• Describe the conditions for optimal consumer choice.
• Illustrate graphically the tangency condition for optimal
consumer choice.
• Solve for an optimal consumption basket, given
information about income, prices, and marginal utilities.
• Explain why the optimal consumption basket solves
both a utility maximization problem and an expenditure minimization problem.
• Explain why the optimal consumption basket could
occur at a corner point.
104
105
4 . 1 T H E BU D G E T C O N S T R A I N T
TABLE 4.1
U.S. Average Expenditures by Household, 2007
All
Households
Households
with Income
$20,000–
$29,999
Households
with Income
$40,000–
$49,999
Households
with Income
over
$70,000
121,171,000
2.5
14,720,000
2.2
11,824,000
2.4
37,322,000
3.1
48.8
60
52.3
44
46.8
59
47.0
79
$ 63,091
$60,858
$49,638
$ 24,893
$24,709
$29,704
$44,555
$43,628
$ 41,083
$130,455
$ 124,613
$ 84,072
$ 6,133
$ 16,920
$ 4,071
$ 10,994
$ 5,689
$ 13,997
$ 9,464
$ 27,408
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
Number of households
Average number
of people in household
Age of reference person*
Percent (reference person)
having attended college
Income before taxes
Income after taxes
Average annual expenditures
Expenditure on
selected categories
Food
Housing (including
shelter, utilities, supplies,
furnishings, and equipment)
Apparel and services
Transportation
Health care
Entertainment
1,881
8,758
2,853
2,698
1,016
5,434
2,841
1,375
1,517
7,346
2,800
2,029
3,275
14,362
3,928
4,927
*Reference person: The first member mentioned by the respondent when asked to “Start with the name of the person
or one of the persons who owns or rents the home.”
Source: Bureau of Labor Statistics. All data come from Table 2, Income Before Taxes: Average Annual Expenditures and
Characteristics, Consumer Expenditure Survey, 2007. The Consumer Expenditure Survey tables are available online at
www.bls.gov/cex/tables.htm, August 19, 2009.
• Illustrate the budget line and optimal consumer choice graphically when one of the goods a consumer
can choose is a composite good.
• Describe the concept of revealed preference.
• Employ the concept of revealed preference to determine whether observed choices are consistent
with utility maximization.
T
he budget constraint defines the set of baskets that a consumer can purchase with
a limited amount of income. Suppose a consumer, Eric, purchases only two types of
goods, food and clothing. Let x be the number of units of food he purchases each
month and y the number of units of clothing. The price of a unit of food is Px, and the
price of a unit of clothing is P y. Finally, to keep matters simple, let’s assume that Eric
has a fixed income of I dollars per month.
4.1
THE BUDGET
CONSTRAINT
106
CHAPTER 4
budget constraint
Eric’s total monthly expenditure on food will be Px x (the price of a unit of food times
the amount of food purchased). Similarly, his total monthly expenditure on clothing will
be Py y (the price of a unit of clothing times the number of units of clothing purchased).
The budget line indicates all of the combinations of food (x) and clothing ( y) that
Eric can purchase if he spends all of his available income on the two goods. It can be
expressed as
The set of baskets that a
consumer can purchase
with a limited amount of
income.
budget line
The set of
baskets that a consumer can
purchase when spending
all of his or her available
income.
(4.1)
Px x ⫹ Py y ⫽ I
Figure 4.1 shows the graph of a budget line for Eric based on the following assumptions: Eric has an income of I ⫽ $800 per month, the price of food is Px ⫽ $20
per unit, and the price of clothing is Py ⫽ $40 per unit. If he spends all $800 on food,
he will be able to buy, at most, IⲐPx ⫽ 800 Ⲑ20 ⫽ 40 units of food. So the horizontal
intercept of the budget line is at x ⫽ 40. Similarly, if Eric buys only clothing, he will
be able to buy at most I ⲐPy ⫽ 800 Ⲑ40 ⫽ 20 units of clothing. So the vertical intercept
of the budget line is at y ⫽ 20.
As explained in Figure 4.1, Eric’s income permits him to buy any basket on or inside
the budget line (baskets A–F ), but he cannot buy a basket outside the budget line, such
as G. To buy G he would need to spend $1,000, which is more than his monthly income.
These two sets of baskets—those Eric can buy and those he cannot buy—exemplify what
is meant by the budget constraint.
Since the budget constraint permits a consumer to buy baskets both on and inside
the budget line, the equation for the budget constraint is somewhat different from
I = 20
Py
20
y, units of clothing
CONSUMER CHOICE
A
B
15
10
F
G
Budget line:
Income = $800 per month
C
D
5
Slope = –
Px
1
=–
Py
2
E
0
10
20
x, units of food
FIGURE 4.1
30
40
I = 40
Px
Budget Line
The line connecting baskets A and E is Eric’s budget line when he has an income of I ⫽ $800
per month, the price of food is Px ⫽ $20 per unit, and the price of clothing is Py ⫽ $40 per unit.
The equation of the budget line is Px x ⫹ Py y ⫽ I (i.e., 20x ⫹ 40y ⫽ 800). Eric can buy any basket
on or inside the budget line—baskets A–F (note that basket F would cost him only $600).
However, he cannot buy a basket outside the budget line, such as basket G, which would cost
him $1,000, more than his monthly income.
4 . 1 T H E BU D G E T C O N S T R A I N T
equation (4.1) for the budget line. The budget constraint can be expressed as:
Px x ⫹ Py y ⱕ I
(4.1a)
What does the slope of the budget line tell us? The slope of the budget line is
⌬y/⌬x. If Eric is currently spending his entire income on basket B in Figure 4.1—that
is, consuming 10 units of food (x) and 15 units of clothing ( y)—and he wants to move
to basket C, he must give up 5 units of clothing (⌬y ⫽ ⫺5) in order to gain 10 units
of food (⌬x ⫽ 10). We can see that, in general, since food is half as expensive as clothing, Eric must give up 1Ⲑ2 unit of clothing for each additional unit of food, and the
slope of the budget line reflects this (⌬ y Ⲑ⌬x ⫽ ⫺5Ⲑ10 ⫽ ⫺1Ⲑ2). Thus, the slope of the
budget line tells us how many units of the good on the vertical axis a consumer must give up
to obtain an additional unit of the good on the horizontal axis.
Note that the slope of the budget line is ⫺Px ⲐPy.1 If the price of good x is three
times the price of good y, the consumer must give up 3 units of y to get 1 more unit of
x, and the slope is ⫺3. If the prices are equal, the slope of the budget line is ⫺1—the
consumer can always get 1 more unit of x by giving up 1 unit of y.
HOW DOES A CHANGE IN INCOME
AFFECT THE BUDGET LINE?
As we have shown, the location of the budget line depends on the level of income and
on the prices of the goods the consumer purchases. As you might expect, when income
rises, the set of choices available to the consumer will increase. Let’s see how the
budget line changes as income varies.
In the example just discussed, suppose Eric’s income rises from I1 ⫽ $800 per
month to I2 ⫽ $1,000 per month, with the prices Px ⫽ $20 and Py ⫽ $40 unchanged. As
shown in Figure 4.2, if Eric buys only clothing, he can now purchase I2ⲐPy ⫽ 1000Ⲑ40 ⫽
25 units of clothing, corresponding to the vertical intercept of the new budget line. The
extra $200 of income allows him to buy an extra 5 units of y, since Py ⫽ $40.
If he buys only food, he could purchase I2 ⲐPx ⫽ 1000Ⲑ20 ⫽ 50 units, corresponding to the horizontal intercept on the new budget line. With the extra $200 of income he can buy an extra 10 units of x, since Px ⫽ $20. With his increased income of
$1,000, he can now buy basket G, which had formerly been outside his budget line.
The slopes of the two budget lines are the same because the prices of food and
clothing are unchanged (⌬yⲐ⌬x ⫽ ⫺Px ⲐPy ⫽ ⫺1Ⲑ2).
Thus, an increase in income shifts the budget line outward in a parallel fashion.
It expands the set of possible baskets from which the consumer may choose.
Conversely, a decrease in income would shift the budget line inward, reducing the set
of choices available to the consumer.
HOW DOES A CHANGE IN PRICE
AFFECT THE BUDGET LINE?
How does Eric’s budget line change if the price of food rises from Px1 ⫽ $20 to
Px2 ⫽ $25 per unit, while income and the price of clothing are unchanged? As shown in
1
To see why this is so, first solve equation (4.1) for y, which gives y ⫽ (I ⲐPy) ⫺ (Px ⲐPy)x. Then, recall from
algebra that the general equation for a straight line is y ⫽ mx ⫹ b, where m is the slope of the graph and
b is the intercept on the y axis. This matches up with the budget line equation solved for y: the y intercept
is IⲐPy, and the slope is ⫺Px ⲐPy.
107
108
CHAPTER 4
I2
Py
I1
= 25
25
= 20
A
20
y, units of clothing
Py
CONSUMER CHOICE
B
15
F
10
G
Slope of BL2 =
∆y
Px
=–
Py
∆x
=–
1
2
C
D
5
BL1
BL2
E
0
10
20
30
x, units of food
40
I1
Px
= 40
50
I2
Px
= 50
FIGURE 4.2
Effect of a Change in Income on the Budget Line
The price of food is Px ⫽ $20 per unit, and the price of clothing is Py ⫽ $40 per unit. If the consumer has an income of I1 ⫽ $800 per month, the budget line is BL1, with a vertical intercept of
y ⫽ 20, a horizontal intercept of x ⫽ 40, and a slope of ⫺1/2. If income grows to I2 ⫽ $1,000
per month, the budget line is BL2, with a vertical intercept of y ⫽ 25, a horizontal intercept of
x ⫽ 50, and the same slope of ⫺1/2. The consumer cannot buy basket G with an income of
$800, but he can afford it if income rises to $1,000.
Figure 4.3, the vertical intercept of the budget line remains unchanged since I and
Py do not change. However, the horizontal intercept decreases from I/Px1 ⫽ 800/20 ⫽
40 units to I/Px2 ⫽ 800/25 ⫽ 32 units. The higher price of food means that if Eric
spends all $800 on food, he can purchase only 32 units of food instead of 40. The slope
of the budget line changes from ⫺(Px1/Py ) ⫽ ⫺(20/40) ⫽ ⫺1/2 to ⫺(Px2 /Py ) ⫽
⫺(25/40) ⫽ ⫺5/8. The new budget line BL2 has a steeper slope than BL1, which
means that Eric must give up more units of clothing than before to purchase one more
unit of food. When the price of food was $20, Eric needed to give up only 1/2 unit of
clothing; at the higher price of food ($25), he must give up 5/8 of a unit of clothing.
Thus, an increase in the price of one good moves the intercept on that good’s axis
toward the origin. Conversely, a decrease in the price of one good would move the intercept on that good’s axis away from the origin. In either case, the slope of the budget
line would change, reflecting the new trade-off between the two goods.
When the budget line rotates in, the consumer’s purchasing power declines because the set of baskets from which he can choose is reduced. When the budget line
rotates out, the consumer is able to buy more baskets than before, and we say that the
consumer’s purchasing power has increased. As we have seen, an increase in income
or a decrease in price increases purchasing power, whereas an increase in price or a
decrease in income decreases purchasing power.
109
4 . 1 T H E BU D G E T C O N S T R A I N T
I = 800 = 20
40
Py
20
y, units of clothing
Slope of BL1 = –
Slope of BL2 = –
Px
2
Py
= –
Px
1
Py
= –
5
8
1
2
BL2
x, units of food
0
BL1
32
I = 800 = 32
25
Px
2
40
I = 800 = 40
20
Px
1
FIGURE 4.3
Effect of a Price Increase on the Budget Line
When the price of food rises from $20 to $25 per unit, the budget line rotates in toward the
origin, from BL1 to BL2, and the horizontal intercept shifts from 40 to 32 units. The vertical
intercept does not change because income and the price of clothing are unchanged. The new
budget line BL2 has a steeper slope than BL1.
A P P L I C A T I O N
4.1
The Rising Price of Gasoline
The average retail price for a gallon of gasoline in the
United States has varied greatly in recent years. As
the table shows, the retail price of regular gasoline
increased dramatically in Summer 2008. From the end
of January to the end of June, prices rose from about
$2.95 per gallon to $4.03 per gallon—the first time
that gasoline prices in the United States had topped
$4.00.
Average Retail Price of Regular Gasoline in the United States in 2008
Date
January
February
March
April
May
June
July
Price
(per $ gallon)
$2.95
$3.12
$3.26
$3.57
$3.91
$4.03
$3.90
Source: U.S. Energy Information Administration, http://tonto.eia.doe.gov/oog/info/gdu/gasdiesel.asp (accessed September 25,
2009).
How would an increase in the price of gasoline
affect a consumer’s budget line? To keep matters
simple, suppose the consumer buys only two goods,
gasoline and clothing, and suppose further that the
consumer’s income and the price of clothing do not
change. We could draw budget lines on a graph like
that in Figure 4.3, with a horizontal axis measuring
gallons of gasoline (instead of units of food). An increase in the price of gasoline would rotate the
budget line in toward the origin from BL1 to BL2.
Consumers responded to the rise in gasoline
prices in 2008 in several ways. As prices rose from
110
CHAPTER 4
CONSUMER CHOICE
January through June, total highway miles driven
declined every month. The U.S. Department of
Transportation estimated a total decline of 20 billion
miles traveled during the first half of 2008. At the same
time, commuter rail usage increased. Gasoline prices also
affected car sales. Purchases of gas-guzzling vehicles
such as SUVs (Sport Utility Vehicles) and pickup trucks
fell approximately 40 percent in May, and again in
S
E
June. Relative sales of smaller cars rose. In addition,
sales of diesel cars increased (the price of diesel gasoline did not rise as sharply).
In the next section we will combine budget lines
with the utility theory from Chapter 3. After studying
that section, you will be able to explain why consumers changed their spending habits in response to
the rise in gasoline prices as described here.
L E A R N I N G - B Y- D O I N G E X E R C I S E 4 . 1
D
Good News/Bad News and the Budget Line
Suppose that a consumer’s income (I )
doubles and that the prices (Px and P y) of both goods in
his basket also double. He views the doubling of income
as good news because it increases his purchasing power.
However, the doubling of prices is bad news because it
decreases his purchasing power.
Problem What is the net effect of the good and bad
news?
Solution The location of the budget line is deter-
of income and prices, the y intercept was IⲐPy; afterward, the y intercept is 2IⲐ2Py ⫽ IⲐPy, so the y intercept
is unchanged. Similarly, the x intercept is unchanged.
Thus, the location of the budget line is unchanged,
as is its slope, since ⫺(2Px Ⲑ2Py) ⫽ ⫺(Px ⲐPy). The doubling
of income and prices has no net effect on the budget
line, on the trade-off between the two goods, or on the
consumer’s purchasing power.
Similar Problems: 4.1, 4.2.
mined by the x and y intercepts. Before the doubling
We have learned that the consumer can choose any basket on or inside the budget
line. But which basket will he choose? We are now ready to answer this question.
4.2
OPTIMAL
CHOICE
optimal choice
Consumer choice of a basket
of goods that (1) maximizes
satisfaction (utility) while
(2) allowing him to live
within his budget constraint.
I
f we assume that a consumer makes purchasing decisions rationally and we know
the consumer’s preferences and budget constraint, we can determine the consumer’s
optimal choice—that is, the optimal amount of each good to purchase. More precisely, optimal choice means that the consumer chooses a basket of goods that (1) maximizes his satisfaction (utility) and (2) allows him to live within his budget constraint.
Note that an optimal consumption basket must be located on the budget line. To
see why, refer back to Figure 4.1. Assuming that Eric likes more of both goods (food
and clothing), it’s clear that a basket such as F cannot be optimal because basket F
doesn’t require Eric to spend all his income. The unspent income could be used to increase satisfaction with the purchase of additional food or clothing.2 For this reason,
no point inside the budget line can be optimal.
Of course, consumers do not always spend all of their available income at any given
time. They often save part of their income for future consumption. The introduction
of time into the analysis of consumer choice really means that the consumer is making
choices over more than just two goods, including for instance the consumption of food
2
This observation can be generalized to the case in which the consumer is considering purchases of more
than two goods, say N goods, all of which yield positive marginal utility to the consumer. At an optimal
consumption basket, all income must be exhausted.
4.2 OPTIMAL CHOICE
111
today, clothing today, food tomorrow, and clothing tomorrow. For now, however, let
us keep matters simple and assume that there is no tomorrow. Later, we will introduce
time (with the possibility of borrowing and saving) into the discussion.
To state the problem of optimal consumer choice, let U(x, y) represent the consumer’s utility from purchasing x units of food and y units of clothing. The consumer
chooses x and y, but must do so while satisfying the budget constraint Px x ⫹ Py y ⱕ I.
The optimal choice problem for the consumer is expressed like this:
max U(x, y)
(4.2)
(x, y)
subject to: Px x ⫹ Py y ⱕ I
where the notation ‘‘max U(x, y)” means “choose x and y to maximize utility,” and the
(x, y)
notation “subject to: Px x ⫹ Py y ⱕ I ” means “the expenditures on x and y must not
exceed the consumer’s income.” If the consumer likes more of both goods, the marginal
utilities of food and clothing are both positive. At an optimal basket all income will be
spent (i.e., the consumer will choose a basket on the budget line Px x ⫹ Py y ⫽ I).
Figure 4.4 represents Eric’s optimal choice problem graphically. He has an income of I ⫽ $800 per month, the price of food is Px ⫽ $20 per unit, and the price of
clothing is Py ⫽ $40 per unit. The budget line has a vertical intercept at y ⫽ 20, indicating that if he were to spend all his income on clothing, he could buy 20 units of
clothing each month. Similarly, the horizontal intercept at x ⫽ 40 shows that Eric
could buy 40 units of food each month if he were to spend all his income on food. The
slope of the budget line is ⫺Px Ⲑ Py ⫽ ⫺1Ⲑ 2. Three of Eric’s indifference curves are
shown as U1, U2, and U3.
I
800
= 20
=
40
Py
Preference
directions
FIGURE 4.4
20
y, units of clothing
D
B
16
15
Budget line BL slope = –
A
10
E
U3
C
5
U2
BL
0
8
11
20
x, units of food
30
U1
40
I
800 = 40
=
20
Px
1
2
Optimal
Choice: Maximizing Utility
with a Given Budget
Which basket should the
consumer choose if he wants
to maximize utility while living
within a budget constraint
limiting his expenditures to
$800 per month? He should
select basket A, achieving a
level of utility U2. Any other
basket on or inside the budget
line BL (such as B, E, or C ) is
affordable, but leads to less
satisfaction. A basket outside
the budget line (such as D) is
not affordable.
At the optimal basket A the
budget line is tangent to an
indifference curve. The slope
of the indifference curve U2 at
point A and the slope of the
budget line are both ⫺1Ⲑ2.
112
CHAPTER 4
CONSUMER CHOICE
To maximize utility while satisfying the budget constraint, Eric will choose the
basket that allows him to reach the highest indifference curve while being on or inside the budget line. In Figure 4.4 that optimal basket is A, where Eric achieves a level
of utility U2. Any other point on or inside the budget line will leave him with a lower
level of utility.
To further understand why basket A is the optimal choice, let’s explore why other
baskets are not optimal. First, baskets outside the budget line, such as D, cannot be
optimal because Eric cannot afford them. We can therefore restrict our attention to
baskets on or inside the budget line. Any basket inside the budget line, such as E or
C, is also not optimal, since, as we have shown, an optimal basket must lie on the
budget line.
If Eric were to move along the budget line away from A, even by a small amount,
his utility would fall because the indifference curves are bowed in toward the origin
(in economic terms, because there is diminishing marginal rate of substitution of x
for y). At the optimal basket A, the budget line is just tangent to the indifference curve
U2. This means that the slope of the budget line (⫺Px ⲐPy) and the slope of the indifference curve are equal. Recall from equation (3.5) that the slope of the indifference
curve is ⫺MUx ⲐMUy ⫽ ⫺MRSx,y. Thus, at the optimal basket A, this tangency condition requires that
MUx
Px
⫽
MUy
Py
interior optimum An
optimal basket at which a
consumer will be purchasing positive amounts of all
commodities.
(4.3)
or MRSx, y ⫽ Px ⲐPy. In Appendix 1, we show how this condition can be derived using
formal mathematical tools.
In Figure 4.4 the optimal basket A is said to be an interior optimum, that is, an
optimum at which the consumer will be purchasing both commodities (x ⬎ 0 and y ⬎
0). The optimum occurs at a point where the budget line is tangent to the indifference
curve. In other words, at an interior optimal basket, the consumer chooses commodities so that the ratio of the marginal utilities (i.e., the marginal rate of substitution)
equals the ratio of the prices of the goods.
We can also express the tangency condition by rewriting equation (4.3) as follows:
MUy
MUx
⫽
Px
Py
(4.4)
This form of the tangency condition states that, at an interior optimal basket, the
consumer chooses commodities so that the marginal utility per dollar spent on each
commodity is the same. Put another way, at an interior optimum, the extra utility per
dollar spent on good x is equal to the extra utility per dollar spent on good y. Thus, at
the optimal basket, each good gives the consumer equal “bang for the buck.”
Although we have focused on the case in which the consumer purchases only two
goods, such as food and clothing, the consumer’s optimal choice problem can also be
analyzed when the consumer buys more than two goods. For example, suppose the
consumer chooses among baskets of three commodities. If all of the goods have positive marginal utilities, then at the optimal basket the consumer will spend all of his
income. If the optimal basket is an interior optimum, the consumer will choose the
goods so that the marginal utility per dollar spent on all three goods will be the same.
The same principles apply to the case in which the consumer buys any given number
of goods.
4.2 OPTIMAL CHOICE
A P P L I C A T I O N
4.2
The Marginal Utility of “Home
Cooking” versus “Eating Out”:
Exploring the Implications of the
“Equal Bang for the Buck” Condition
Economic theory implies that at an optimal consumption basket, each good that is purchased in positive
quantities gives the consumer equal “bang for the
buck.” We can use this condition to derive some interesting implications about the marginal value that the
typical U.S. household enjoys from dining out versus
eating at home. You may recall from Table 4.1 in the
introduction to this chapter that in 2007 the average
U.S. household spent $6,133 per year on food. Of this
amount, $3,465 (or 56.5 percent) was spent on food
consumed at home, and $2,668 (or 43.5 percent) on
food consumed away from home (e.g., food purchased at restaurants and fast-food outlets).3 The U.S.
Department of Agriculture has estimated that in
1995, nearly two-thirds (66 percent) of the total calorie intake of the typical U.S. household came from
food consumed at home, while slightly more than
one-third (34 percent) of total calorie intake came
from food consumed away from home. This latter
percentage has been increasing steadily over time: In
the late 1970s, only 18 percent of total calories came
from food consumed away from home.4
We can use these data, along with the “equal
bang for the buck” condition, to draw inferences
about the marginal utility of a calorie from food consumed at home and the marginal utility of a calorie
from food consumed away from home. Letting X denote the quantity of food consumed at home (measured in calories) and Y denote the quantity of food
consumed away from home (also measured in calories), we can rewrite the “equal bang for the buck”
condition in equation (4.4) as
3
113
MUyY
MUxX
⫽
PxX
PyY
(To derive this expression, we multiplied the top
and bottom of the left-hand side of equation (4.4)
by X and the top and bottom of the right-hand
side of equation (4.4) by Y.) In the expression
above, Px and Py are the prices of a calorie of food
consumed at home and away, respectively, and Px X
and PyY are total expenditures on food consumed
at home and away, respectively. As has been noted,
for the typical U.S. household in 2007, Px X ⫽
$3,465 and PyY ⫽ $2,668. Thus, for the typical U.S.
household, the “equal bang for the buck” condition implies:
MUyY
MUxX
⫽
$3,465
$2,668
which, by rearranging terms, can be rewritten as
MUy
$2,668 X
⫽
$3,465 Y
MUx
Now, as noted earlier, in the mid-1990s, the typical
U.S. household consumed 66 percent of its calories
from food at home and 34 percent of its calories from
food away from home. If this ratio held for U.S.
household in 2007, this implies that XY —the ratio of
total calories from food consumed at home to total
calories of food consumed away from home—would
66
equal 34
or about 1.94. The “equal bang for the buck”
condition would then imply that
MUy
MUx
⫽ 1.49
This tells us that for the typical U.S. household, the
marginal utility of calories from eating out is 1.49 times
as large as the marginal utility of calories from eating
at home. That is, the marginal calorie consumed away
These data (which are not presented in Table 4.1) come from Bureau of Labor Statistics, Table 2, Income
Before Taxes: Average Annual Expenditures and Characteristics, Consumer Expenditure Survey, 2007,
http://www.bls.gov/cex/2007/Standard/income.pdf (accessed September 25, 2009).
4
U.S. Department of Agriculture, Economic Research Service, Agriculture Information Bulletin No.
(AIB750), 484 pp, May 1999, America’s Eating Habits: Changes and Consequences, Chapter 12, “Nutrients
away from Home,” Table 2, p. 219, http://www.ers.usda.gov/Publications/AIB750.
114
CHAPTER 4
CONSUMER CHOICE
from home provides 49 percent more utility than the
marginal calorie consumed at home.
This calculation seems plausible. Food consumption away from home often occurs on special occasions
(e.g., dining out to celebrate a wedding anniversary)
or is bound up in enjoyable moments (e.g., dining out
on a date or at the end of a long week of work).
Sometimes households eat out because it provides a
welcome break from the “same-old-same-old” menus
and routines of home cooking. For all these reasons, it
seems reasonable that the marginal calorie taken in
away from home generates more utility than the marginal calorie taken in from food eaten at home.
Back-of-the envelope calculations like this one
typically rely on a number of simplifying assumptions.
For example, in the calculation, the prices of food
reflect what is spent to purchase the food at home or
in a restaurant. But they do not reflect the prices of
other activities related to eating at home (like traveling to and from the grocery store or preparing the
food) or away from home (like traveling to and from
a restaurant). Also, our calculation assumes that for
the typical U.S. household in the late 2000s, the mix
of calories consumed at home and away from home
has remained the same as it was in the mid-1990s.
Since the fraction of calories consumed at home
steadily decreased in the 1980s and early 1990s, this
assumption might not be valid. Indeed, it seems plausible that this fraction would have fallen somewhat,
perhaps to close to 60 percent, or maybe even slightly
below. If, for example, the percentage of calories
from home consumption was actually 60 percent in
2007, then the marginal utility of calories consumed
outside the home would only have been 15 percent
greater than the marginal utility of calories consumed at home.
This example illustrates how the “equal bang for
the buck condition,” combined with data on expenditures and ratios of consumption levels, can provide
interesting and fun insights into the preferences of
groups of consumers.
U S I N G T H E TA N G E N C Y C O N D I T I O N TO U N D E R S TA N D
W H E N A BA S K E T I S N O T O P T I M A L
Let’s use the tangency condition represented in equations (4.3) and (4.4) to explore why
an interior basket such as B in Figure 4.4 is not optimal. In the figure we are given an
indifference map, which comes from the utility function U(x, y) ⫽ xy. As we noted in
Learning-By-Doing Exercise 3.3, the marginal utilities for this utility function are
MUx ⫽ y and MUy ⫽ x. For example, at basket B (where y ⫽ 16 and x ⫽ 8), the marginal
utilities are MUx ⫽ 16 and MUy ⫽ 8. We also are given that Px ⫽ $20 and Py ⫽ $40.
How does the tangency condition indicate that B is not an optimal choice?
Consider equation (4.3). The left-hand side of that equation tells us that MUx ⲐMUy ⫽
16Ⲑ8 ⫽ 2 at B; that is, at B, Eric’s marginal rate of substitution of x for y is 2. At B he
would be willing to give up two units of clothing ( y) to get one more unit of food (x).5
But given the prices of the goods, will Eric have to give up two units of clothing to get
one more unit of food? The right-hand side of equation (4.3) tells us that Px ⲐPy ⫽
20Ⲑ40 ⫽ 1Ⲑ2 because clothing is twice as expensive as food. So, to buy one more unit
of food, he needs to give up only 1 Ⲑ2 unit of clothing. Thus, at B, to get one more unit
of food, he is willing to give up two units of clothing, but he is only required to give up
1Ⲑ2 unit of clothing. Since basket B leaves him willing to give up more clothing than
he needs to give up to get additional food, basket B cannot be his optimal choice.
Now let’s examine the other form of the tangency condition in equation (4.4) to
see why the marginal utility per dollar spent must be equal for all goods at an interior
optimum, which is another reason basket B cannot be optimal.
5
Remember, MRSx,y ⫽ MUx ⲐMUy ⫽ ⫺(slope of the indifference curve). In Figure 4.4, the slope of the
indifference curve at B is ⫺2 (the same as the slope of the line tangent to the indifference curve at B).
4.2 OPTIMAL CHOICE
115
If we compare the marginal utility per dollar spent on the two commodities at B,
we find that MUx Px ⫽ 16Ⲑ20 ⫽ 0.8 and that MUy ⲐPy ⫽ 8/40 ⫽ 0.2. Eric’s marginal
utility per dollar spent on food (MUx ⲐPx) is higher than his marginal utility per dollar
spent on clothing (MUy ⲐPy). He should therefore take the last dollar he spent on
clothing and instead spend it on food. How would this reallocation of income affect
his utility? Decreasing clothing expenditures by a dollar would decrease utility by
about 0.2, but increasing food expenditures by that dollar would increase utility by
about 0.8; the net effect on utility is the difference, a gain of about 0.6.6 So if Eric is
currently purchasing basket B, he is not choosing his optimal basket.
F I N D I N G A N O P T I M A L C O N S U M P T I O N BA S K E T
As we have seen, when both marginal utilities are positive, an optimal consumption
basket will be on the budget line. Furthermore, when there is a diminishing marginal
rate of substitution, then an interior optimal consumption basket will occur at the tangency between an indifference curve and the budget line. This is the case illustrated
at basket A in Figure 4.4.
Learning-By-Doing Exercise 4.2 illustrates how to use information about the
consumer’s budget line and preferences to find his optimal consumption basket.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 4 . 2
D
Finding an Interior Optimum
Eric purchases food (measured by x) and
clothing (measured by y) and has the utility function
U(x, y) ⫽ xy. His marginal utilities are MUx ⫽ y and
MUy ⫽ x. He has a monthly income of $800. The price
of food is Px ⫽ $20, and the price of clothing is Py ⫽ $40.
•
Problem
So we have two equations with two unknowns. If we
substitute x ⫽ 2y into the equation for the budget line,
we get 20(2y) ⫹ 40y ⫽ 800. So y ⫽ 10 and x ⫽ 20. Eric’s
optimal basket involves the purchase of 20 units of food
and 10 units of clothing each month, as is indicated at
basket A in Figure 4.4.
Find Eric’s optimal consumption bundle.
Solution In Learning-By-Doing Exercise 3.3, we
learned that the indifference curves for this utility function are bowed in toward the origin and do not intersect
the axes. So the optimal basket must be interior, with
positive amounts of food and clothing being consumed.
How do we find an optimal basket? We know two
conditions that must be satisfied at an optimum:
•
Since the optimum is interior, the indifference curve
must be tangent to the budget line. From equation
(4.3), we know that a tangency requires that MUx Ⲑ
MUy ⫽ Px ⲐPy, or, with the given information, y/x ⫽
20/40, or x ⫽ 2y.
Similar Problems:
4.3, 4.4
An optimal basket will be on the budget line. This
means that Px x ⫹ Py y ⫽ I, or, with the given information, 20x ⫹ 40y ⫽ 800.
6
Since Px ⫽ $20, the increased spending of a dollar on food means that the consumer will buy an
additional 1/20 unit of food, so that ⌬x ⫽ ⫹1/20. Similarly, since Py ⫽ $40, a decreased expenditure of
one dollar on clothing will mean that the consumer reduces consumption of clothing by 1/40, so that
⌬y ⫽ ⫺1/40. Recall from equation (3.4) that the effect of changes in consumption on total utility can be
approximated by ⌬U ⫽ (MUx ⫻ ⌬x) ⫹ (MUy ⫻ ⌬y). Thus, the reallocation of one dollar of expenditures
from clothing to food will affect utility by approximately ⌬U ⫽ (16 ⫻ 1/20) ⫹ [8 ⫻ (⫺1/40)] ⫽ 0.6.
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CHAPTER 4
CONSUMER CHOICE
T W O WAYS O F T H I N K I N G A B O U T O P T I M A L I T Y
We have shown that basket A in Figure 4.4 is optimal for the consumer because it answers
this question: What basket should the consumer choose to maximize utility, given a budget
constraint limiting expenditures to $800 per month? In this case, since the consumer
chooses the basket of x and y to maximize utility while spending no more than $800
on the two goods, optimality can be described as follows:
(4.5)
max Utility ⫽ U(x, y)
(x, y)
subject to: Px x ⫹ Py y ⱕ I ⫽ 800
In this example, the endogenous variables are x and y (the consumer chooses the
basket). The level of utility is also endogenous. The exogenous variables are the prices
Px and Py and income I (i.e., the level of expenditures). The graphical approach solves
the consumer choice problem by locating the basket on the budget line that allows the
consumer to reach the highest indifference curve. That indifference curve is U2 in
Figure 4.4.
There is another way to look at optimality, by asking a different question: What
basket should the consumer choose to minimize his expenditure (Px x ⫹ Py y) and also achieve
a given level of utility U2? Equation (4.6) expresses this algebraically:
(4.6)
min expenditure ⫽ Px x ⫹ Py y
(x, y)
subject to: U(x, y) ⫽ U2
expenditure
minimization problem
Consumer choice between
goods that will minimize
total spending while achieving a given level of utility.
This is called the expenditure minimization problem. In this problem the endogenous variables are still x and y, but the exogenous variables are the prices Px, Py, and
the required level of utility U2. The level of expenditure is also endogenous. Basket A
in Figure 4.5 is optimal because it solves the expenditure minimization problem. Let’s
see why.
25
BL1: spending =
FIGURE 4.5
$640 per month
20
R
BL2: spending =
$800 per month
y, units of clothing
Optimal Choice:
Minimizing Expenditure to Achieve a
Given Utility
Which basket should the consumer
choose if he wants to minimize the
expenditure necessary to achieve a
level of utility U2? He should select
basket A, which can be purchased at
a monthly expenditure of $800.
Other baskets on U2 will cost the
consumer more than $800. For example, to purchase R or S (also on U2),
the consumer would need to spend
$1,000 per month (since R and S are
on BL3). Any total expenditure less
than $800 (e.g., $640, represented by
BL1) will not enable the consumer to
reach the indifference curve U2.
16
15
BL3: spending =
$1,000 per month
A
10
S
5
U2 = 200
BL1
0
10
20
30 32
x, units of food
BL2
40
BL3
50
4.2 OPTIMAL CHOICE
117
Using Figure 4.5, let’s look for a basket that would require the lowest expenditure
to reach indifference curve U2. (In this figure, U2 corresponds to a utility level of 200.)
In the figure, we have drawn three different budget lines. All baskets on the
budget line BL1 can be purchased if the consumer spends $640 per month.
Unfortunately, none of the baskets on BL1 allows him to reach the indifference curve
U2, so he will need to spend more than $640 to achieve the required utility. Could he
reach the indifference curve U2 with a monthly expenditure of $1,000? All baskets on
budget line BL3, such as baskets R and S, can be purchased by spending $1,000 a
month. But there are other baskets on U2 that would cost the consumer less than
$1,000. To find the basket that minimizes expenditure, we have to find the budget line
that is tangent to the indifference curve U2. That budget line is BL2, which is tangent
to BL2 at point A. Thus, the consumer can reach U2 by purchasing basket A, which
costs only $800. Any expenditure less than $800 will not be enough to purchase a basket
on indifference curve U2.
The utility maximization problem of equation (4.5) and the expenditure minimizing problem of equation (4.6) are said to be dual to one another. The basket that maximizes utility with a given level of income leads the consumer to a level of utility U2.
That same basket minimizes the level of expenditure necessary for the consumer to
achieve a level of utility U2.
We have already seen that a basket such as B in Figure 4.6 is not optimal because
the budget line is not tangent to the indifference curve at that basket. How might the
consumer improve his choice if he is at basket B, where he is spending $800 per month
and realizing a level of utility U1 ⫽ 128? We can answer this question from either of
our dual perspectives: utility maximization or expenditure minimization. Thus, the
consumer could ask, “If I spend $800 per month, what basket will maximize my satisfaction?” He will choose basket A and realize a higher level of utility U2. Alternatively,
the consumer might say, “If I am content with a level of utility U1, what is the least
amount of money I will need to spend?” As the graph shows, the answer to this question is basket C, where he needs to spend only $640 per month.
20
y, units of clothing
16
15
E
Preference
directions
B
BL1: spending = $640
BL2: spending = $800
FIGURE 4.6
A
10
C
8
5
U2 = 200
U1 = 128
BL1
0
8 10
20
16
x, units of food
30 32
BL2
40
Nonoptimal Choice
At basket B the consumer spends $800
monthly and realizes a level of utility
U1. There are two ways to see that
basket B is not an optimal choice. The
consumer could continue to spend $800
per month but realize greater utility by
choosing basket A, reaching indifference curve U2. Or the consumer could
continue to achieve U1 but spend less
than $800 per month by choosing
basket C.
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CHAPTER 4
CONSUMER CHOICE
So we have demonstrated the nonoptimality of B in two ways: The consumer can
increase utility if he continues to spend $800 monthly, or he can spend less money to
stay at the same level of utility he is currently realizing at B.
CORNER POINTS
to the consumer’s optimal
choice problem at which
some good is not being
consumed at all, in which
case the optimal basket lies
on an axis.
FIGURE 4.7
Corner Point
At basket S the slope of the
indifference curve U1 is steeper
(more negative) than the
budget line. This means that
the marginal utility per dollar
spent on food is higher than
on clothing, so the consumer
would like to purchase less
clothing and more food. He
would move along the budget
line until he reaches the corner
point basket R, where no
further substitution is possible
because he purchases no
clothing at R.
Preference
directions
Budget
line BL P
slope = – x
y, units of clothing
corner point A solution
In all the examples considered so far, the optimal consumer basket has been interior, meaning that the consumer purchases positive amounts of both goods. In
reality, though, a consumer might not purchase positive amounts of all available
goods. For example, not all consumers own an automobile or a house. Some consumers may not spend money on tobacco or alcohol. If the consumer cannot find
an interior basket at which the budget line is tangent to an indifference curve, then
the consumer might find an optimal basket at a corner point, that is, at a basket
along an axis, where one or more of the goods is not purchased at all. If an optimum occurs at a corner point, the budget line may not be tangent to an indifference curve at the optimal basket.
To see why, let’s consider again our consumer who chooses between just two
goods, food and clothing. If his indifference map is like the one shown in Figure 4.7,
no indifference curve is tangent to his budget line. At any interior basket on the
budget line, such as basket S, the slope of the indifference curve is steeper (more
negative) than the slope of the budget line. This means ⫺MUx ⲐMUy ⬍ ⫺Px ⲐPy, or
(reversing the inequality) MUx ⲐMUy ⬎ Px ⲐPy. Then, by cross multiplying, MUx ⲐPx ⬎
MUy ⲐPy, which tells us the marginal utility per dollar spent is higher for food than
for clothing, so the consumer would like to purchase more food and less clothing.
This is true not only at basket S, but at all baskets on the budget line. The consumer
would continue to substitute food for clothing, moving along the budget line until
he reaches the corner point basket R. At basket R the slope of the indifference curve
U2 is still steeper than the slope of the budget line. He would like to continue substituting food for clothing, but no further substitution is possible because no clothing is purchased at basket R. Therefore, the optimal choice for this consumer is
basket R because that basket gives the consumer the highest utility possible (U2) on
the budget line.
Py
U1
U2
U3
BL
Slope of
indifference curve
at any
MUx
basket = –
S
MUy
R
x, units of food
4.2 OPTIMAL CHOICE
L E A R N I N G - B Y- D O I N G E X E R C I S E 4 . 3
S
E
119
D
Finding a Corner Point Solution
x ⫹ 2y ⫽ 10
David is considering his purchases of food
(x) and clothing ( y). He has the utility function U(x, y) ⫽
xy ⫹ 10x, with marginal utilities MUx ⫽ y ⫹ 10 and
MUy ⫽ x. His income is I ⫽ 10. He faces a price of food
Px ⫽ $1 and a price of clothing Py ⫽ $2.
Problem
If the basket is at a point of tangency, then
MUx ⲐMUy ⫽ Px ⲐPy , or ( y ⫹ 10)Ⲑx = 1Ⲑ2, which simplifies to
x ⫽ 2y ⫹ 20
What is David’s optimal basket?
These two equations with two unknowns are solved
by x ⫽ 15 and y ⫽ ⫺2.5. But this algebraic “solution,”
which suggests that David would buy a negative amount
of clothing, does not make sense because neither x nor y
can be negative. This tells us that there is no basket on
the budget line where the budget line is tangent to an
indifference curve. The optimal basket is therefore not
interior, and the optimum will be at a corner point.
Where is the optimal basket? As we can see in the
figure, the optimum will be at basket R (a corner
point), where David spends all his income on food, so
that x ⫽ 10 and y ⫽ 0. At this basket MUx ⫽ y ⫹ 10 ⫽
10 and MUy ⫽ x ⫽ 10. So at R the marginal utility per
dollar spent on x is MUx ⲐPx ⫽ 10/1 ⫽ 10, while the marginal utility per dollar spent on y is MUy ⲐPy ⫽ 10 Ⲑ2 ⫽ 5.
At R, David would like to purchase more food and less
clothing, but he cannot because basket R is at a corner
point on the x axis. At R, David reaches the highest indifference curve possible while choosing a basket on
the budget line.
Solution The budget line, shown in Figure 4.8, has
a slope of ⫺(Px ⲐPy) ⫽ ⫺1Ⲑ2. The equation of the budget
line is Px x ⫹ Py y ⫽ I, or x ⫹ 2y ⫽ 10. To find an optimum, we must make sure that we understand what the
indifference curves look like. Both marginal utilities are
positive, so the indifference curves are negatively sloped.
The marginal rate of substitution of x for y [ MRSx,y ⫽
MUx ⲐMUy ⫽ ( y ⫹ 10)Ⲑx] diminishes as we increase x and
decrease y along an indifference curve. The indifference
curves are therefore bowed in toward the origin. Finally,
the indifference curves do intersect the x axis because it is
possible to achieve a positive level of utility with purchases
of food (x ⬎ 0) but no purchases of clothing ( y ⫽ 0). This
means that the consumer’s optimal basket may be at a
corner point along the x axis. We have plotted three of
David’s indifference curves in the figure.
Suppose we (mistakenly) assume that David’s optimal basket is interior, on the budget line at a tangency
between the budget line and an indifference curve. If the
optimal basket is on the budget line, then it must satisfy
the equation for the budget line:
U = 100
10
U = 80
Similar Problems: 4.9, 4.10
Preference
directions
U = 120
y, units of clothing
8
Slope of
indifference curve
at basket R =
6
5
–
4
2
Slope of BL = –
0
2
4
Px
1
=–
2
Py
6
BL
8
x, units of food
MUx
MUy
= –1
FIGURE 4.8
R
10
12
14
Corner Point Solution
(for Learning-By-Doing Exercise 4.3)
The budget line: The consumer has an income of
10, with prices Px ⫽ 1 and Py ⫽ 2. The budget line
has a slope of ⫺1Ⲑ2.
The indifference map: Indifference curves are
drawn for three levels of utility, U ⫽ 80, U ⫽ 100,
and U ⫽ 120.
The optimal consumption basket: The optimal basket
is R, where the slope of the indifference curve is ⫺1.
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CHAPTER 4
CONSUMER CHOICE
Learning-By-Doing Exercise 4.3 illustrates that a corner point may exist when
the consumer has a diminishing marginal rate of substitution (the indifference curves
are bowed in toward the origin). Learning-By-Doing Exercise 4.4 shows that a corner point is often optimal when a consumer is quite willing to substitute one commodity for another. (For example, if you view butter and margarine as perfect substitutes
and are always willing to substitute an ounce of one for an ounce of the other, you
would buy only the product that has a lower price per ounce.)
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 4 . 4
D
Corner Point Solution with Perfect Substitutes
Sara views chocolate and vanilla ice cream
as perfect substitutes. She likes both and is always willing to trade one scoop of chocolate for two scoops of
vanilla ice cream. In other words, her marginal utility for
chocolate is twice as large as her marginal utility for
vanilla. Thus, MRSC,V ⫽ MUC ⲐMUV ⫽ 2.
Problem If the price of a scoop of chocolate ice cream
(PC) is three times the price of vanilla (PV), will Sara buy
both types of ice cream? If not, which will she buy?
Solution If Sara buys both types of ice cream, then
there is an interior optimum and the tangency condition
must be satisfied. But the slopes of the indifference
curves are all ⫺2, and the slope of the budget line is
⫺3 (PC ⲐPV ⫽ 3), so the budget line can never be tangent
to an indifference curve. This is shown in Figure 4.9: the
indifference curves are straight lines and less steeply
sloped (flatter) than the budget line. Thus, the optimal
basket will be at a corner point (basket A), at which Sara
buys only vanilla ice cream.
Another way of seeing this is to observe that Sara’s
marginal utility per dollar spent on chocolate ice cream
is less than her marginal utility per dollar spent on
vanilla ice cream: (MUC ⲐMUV ⫽ 2) ⬍ (PC ⲐPV ⫽ 3), so
MUC ⲐMUV ⬍ PC ⲐPV, or MUC ⲐPC ⬍ MUV ⲐPV. Sara will
always try to substitute more vanilla for chocolate, and
this will lead her to a corner point such as basket A.
Similar Problem: 4.18
Preference
directions
Budget
line BL P
C
= –3
slope = –
V, units of vanilla ice cream
A
PV
Slopes of all
indifference curves =
–MRSC,V = –2
BL
FIGURE 4.9
Perfect Substitutes
The marginal utility per dollar spent on
vanilla ice cream is always larger than
the marginal utility per dollar spent on
chocolate ice cream. Thus, the optimal
basket A is at a corner point.
U1
U3 U4
U2
C, units of chocolate ice cream
121
4.3 CONSUMER CHOICE WITH COMPOSITE GOODS
A
lthough consumers typically purchase many goods and services, economists often
want to focus on the consumer’s selection of a particular good or service, such as the
consumer’s choice of housing or level of education. In that case, it is useful to present
the consumer choice problem using a two-dimensional graph with the amount of the
commodity of interest (say, housing) on the horizontal axis, and the amount of all
other goods combined on the vertical axis. The good on the vertical axis is called a
composite good because it is the composite of all other goods. By convention, the
price of a unit of the composite good is Py ⫽ 1. Thus, the vertical axis represents not
only the number of units y of the composite good, but also the total expenditure on
the composite good (Py y).
In this section we will use composite goods to illustrate four applications of the
theory of consumer choice. Let’s begin by considering Figure 4.10. Here we are interested in the consumer’s choice of housing. On the horizontal axis are the units of
housing h (measured, e.g., in square feet). The price of housing is Ph. On the vertical
axis is the composite good, measured in units by y and with a price Py ⫽ 1. If the consumer spends all his income I on housing, he could purchase at most IⲐPh units of
housing, the intercept of the budget line on the horizontal axis. If he spends all of his
income on other goods, he could purchase at most I units of the composite good, the
intercept of the budget line on the vertical axis. With the indifference curve pictured,
the optimal basket will be at point A.
4.3
CONSUMER
CHOICE WITH
COMPOSITE
GOODS
composite good A
good that represents the
collective expenditures on
every other good except
the commodity being
considered.
A P P L I C AT I O N : C O U P O N S A N D C A S H S U B S I D I E S
FIGURE 4.10 Optimal Choice of
Housing (with Composite Good)
The horizontal axis measures the number of
units of housing h. The price of housing is
Ph. If the consumer has an income of I, he
could purchase at most I ⲐPh units of housing (the intercept of the budget line on the
horizontal axis). The vertical axis measures
the number of units of the composite good
y (all other goods). The price of the composite good is Py ⫽ 1. If the consumer were
to spend all his income on the composite
good, he could purchase I units of the composite good. Thus, the intercept of the
budget line on the vertical axis is I, the level
of income. The budget line BL has a slope
equal to ⫺Ph ⲐPy ⫽ ⫺Ph. Given the consumer’s preferences, the optimal basket is
A, where the consumer purchases hA units
of housing and spends yA dollars on other
goods.
y, units of the composite good
(= amount of expenditure on all other goods)
Governments often have programs aimed at helping low-income consumers purchase
more of an essential good, such as food, housing, or education. For example, the U.S.
I
Preference
directions
A
yA
U1
BL
Budget
line BL
Ph
= –Ph
slope = –
Py
hA
I
Ph
h, units of housing
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CHAPTER 4
CONSUMER CHOICE
FIGURE 4.11 Optimal Choice of
Housing: Subsidy and Voucher
Consider two types of programs that might
be implemented to increase the consumer’s
purchases of housing.
Income subsidy: If the consumer receives an
income subsidy of S dollars from the government, the budget line moves from KJ
to EG.
Housing voucher: If the government gives
the consumer a voucher of S dollars that
can only be spent on housing, the budget
line moves from KJ to KFG.
If the consumer has the indifference
map shown in the graph, he is indifferent
between receiving an income subsidy of
S dollars and a housing voucher worth S dollars. In either case, he will select basket B.
y, units of the composite good
(= amount of expenditure on all other goods)
I+S
I
yA
Preference
directions
E
Slope of budget
lines KJ and EG
K
F
=–
B
A
U1
hA
hB
Ph
Py
= –P h
U2
J
G
I
Ph
(I + S)
Ph
h, units of housing
government administers a food stamp program that subsidizes purchases of food and
beverages (see Application 4.3). The U.S. government also provides assistance to help
low-income consumers purchase housing. Let’s use the theory of consumer choice to
examine how a government program might increase the amount of housing chosen by
a consumer.
Suppose the consumer has preferences for housing and other goods as shown by
the indifference curves in Figure 4.11. The consumer has an income I and must pay
a price Ph for each “unit” (e.g., square foot) of housing he rents and Py ⫽ 1 for each
unit of the composite “other goods” he buys. The budget line is K J. If he spends all
his income on housing, he could rent IⲐPh units of housing. If he spends all his
income on other goods, he could buy IⲐPy ⫽ I units of the composite good. With his
preferences and the budget line K J, he chooses bundle A, with hA units of housing
and utility U1.
Now suppose that the government concludes that an amount of housing such as
hA does not provide an adequate standard of living and mandates that every consumer should have at least hB units of housing, where hB ⬎ hA. How might the
government induce the consumer to increase his consumption of housing from hA
to hB?
One way is to give the consumer an income subsidy of S dollars in cash. This increase in income shifts the budget line out from K J to EG in Figure 4.11. If the consumer spent all his income of I and the S cash subsidy on the composite good, he
would be able to purchase basket E, which contains I ⫹ S units of the composite good
and no housing. If he were to spend all of his income and the cash subsidy on
4.3 CONSUMER CHOICE WITH COMPOSITE GOODS
housing, he would be able to buy basket G, which contains (I ⫹ S )ⲐPh units of housing. With the budget line EG and the indifference curves in the figure, his optimal
choice will be basket B, with hB units of housing and utility U2. Note that the cash subsidy S is just large enough to induce the consumer to satisfy the government standard
for housing hB.
Another way to stimulate housing consumption would be to give the consumer
a housing coupon (sometimes called a voucher) worth some amount of money
that can be redeemed only for housing. Suppose the housing voucher is also
worth S dollars. With the voucher the budget line for the consumer would become
KFG, because the consumer cannot apply the voucher to purchase other goods.
The maximum amount he could spend on other goods is his cash income I, so
he could not purchase baskets to the north of the segment KF under the voucher
program.
If he spends all his cash income I on other goods, using only the voucher to purchase housing, he will be able to consume basket F, with I units of the composite good
and S ⲐPh units of housing. If he were to spend all his cash income and the voucher on
housing, he would be able to acquire basket G, with (I ⫹ S) ⲐPh units of housing and
none of the composite good.
Would it matter to the consumer or to the government whether the consumer
receives an income subsidy of S dollars or a housing voucher that can be redeemed for
S dollars worth of housing? If the indifference map is as depicted in Figure 4.11, the
consumer will be equally happy under either program, choosing basket B and reaching the indifference curve U2.
But suppose the indifference map is as depicted in Figure 4.12. Then the type
of program does matter. With no government program, the budget line is again K J,
and the consumer chooses basket A, with a level of housing hA. To induce the consumer to rent hB units of housing with a cash subsidy, the size of the subsidy must
be S. With that subsidy the consumer will choose basket T, with utility U4.
However, the government can also induce the consumer to rent hB units of housing
with a voucher that can be redeemed for V dollars (note that V ⬍ S ). With such a
voucher the budget line will be KRG. The consumer will purchase basket R with
utility U2.7
With the indifference map illustrated in Figure 4.12, the consumer is worse off
with the voucher worth V dollars than with an income subsidy of S dollars. But if the
government’s primary goal is to increase the consumption of housing to hB, the government can save (S ⫺ V ) dollars if it uses the voucher program instead of an income
subsidy.
We could also ask how the consumer would act if given a cash subsidy of V dollars. Then the budget line would be EG, and the consumer would choose basket F
with utility U3. The consumer would prefer this to the voucher worth V dollars,
when he would choose basket R and only reach utility U2. However, with a cash subsidy of V dollars, the consumer’s choice of housing (hF) is below the government’s
target level (hB).
7
While the slope of the indifference curve U2 is defined at basket R, the slope of the budget line is not
defined at that point because the budget “line” has a corner at R. Thus, one cannot apply a tangency
condition to find an optimum such as R.
123
124
CHAPTER 4
CONSUMER CHOICE
y, units of the composite good
(= amount of expenditure on all other goods)
I+S
FIGURE 4.12
Optimal Choice of
Housing: Subsidy and
Voucher
If a consumer has an
income I, he will choose
hA units of housing. The
government could induce
him to choose hB units of
housing with either of the
following two programs:
I+V
M
Preference
directions
E
T
F
I
K
U4
R
U3
U2
A
Slope of budget
lines KJ, EG, and MN
U1
=–
J
hA hF
hB
I
Ph
G
I+V
Ph
Ph
Py
= –Ph
N
I+S
Ph
h, units of housing
•
•
Give him an income subsidy of S dollars, moving the budget line to MN. The consumer
chooses basket T.
Give him a housing voucher worth V dollars that can be spent only on housing, moving
the budget line to KRG. The consumer chooses basket R.
Since basket T lies on a higher indifference curve than basket R, a consumer with
the preferences in the graph would prefer an income subsidy of S dollars over a housing
voucher worth V dollars. However, the government might choose the voucher program
because it would cost less. To induce the consumer to choose hB units of housing, the
government must spend (S ⫺ V ) dollars more if it chooses the cash subsidy program
instead of the voucher program.
A P P L I C A T I O N
4.3
Coupons versus Cash: SNAP
The Supplemental Nutrition Assistance Program (SNAP;
known as the Food Stamp Program prior to 2008) is the
largest food assistance program in the United States. It
began in 1964, though earlier programs date back to
1939. The program is designed to improve the nutrition
and food purchasing power of people with low incomes.
Food stamps were paper coupons issued by the
government. In June 2009, all food stamps were elim-
inated as the program completed its transition over to
electronic cards that recipients now use at authorized
stores to buy food, beverages, and food-producing
seeds or plants. The cards cannot be used to buy nonfood items such as alcohol, tobacco, pet food, and
nonprescription drugs.
Federal expenditures under the program were
nearly $34.6 billion in 2008, when the program
provided an average monthly benefit of $227 per
4.3 CONSUMER CHOICE WITH COMPOSITE GOODS
household to about 12.7 million households.8 The
federal government provides the funds used to pay
for the cards. The administrative costs of the program
are shared by federal, state, and local governments.
To be eligible for SNAP assistance, a household
must have assets and income below governmentspecified levels. Since 1979, recipients have not had to
pay for SNAP assistance. However, the amount of assistance that an individual or household receives
depends on the household size, composition, and
location. In 2007 the average monthly gross income of
households receiving SNAP was $691. The maximum
benefit for a family of four was $506 per month.9
The effect of the SNAP program on the consumer
can be illustrated on graphs like the ones in Figures 4.11
and 4.12, with the composite good on the vertical axis
and the amount of food consumed on the horizontal
axis. As the analysis in Figure 4.11 suggests, some consumers will be equally happy with SNAP assistance or
cash. However, other consumers will prefer to have
cash instead of SNAP, as suggested in Figure 4.12.
Many people believe that the government should
help low-income households with cash supplements instead of in-kind supplements such as SNAP. Proponents
of cash supplements argue that coupon programs are
very expensive to administer and that it is inappropriate
for the government to place requirements on individuals’ consumption decisions. Proponents of in-kind supplements argue that in-kind programs are often significantly less costly to taxpayers than cash supplements.
A P P L I C AT I O N : J O I N I N G A C L U B
Consumers can join clubs that let them purchase goods and services at a discount.
Suppose a music-loving college student spends his income of $300 per month on music
CDs and other goods. He has positive marginal utilities of CDs and other goods, and
his marginal rate of substitution is diminishing. He currently must pay $20 per CD,
and given this price, he buys 10 CDs per month and spends $100 on other goods.
He has just received an advertisement announcing that he can join a CD club. He
would have to pay a membership fee of $100 per month, but then he would be able to
buy as many CDs as he wishes at $10 each. The theory of consumer choice explains
why he might want to join the club and how joining the club would affect the basket
he would choose.
This consumer’s choice problem is illustrated in Figure 4.13. The number of CDs
consumed per month is measured on the horizontal axis, and the number of units of
the composite other good ( y) appears on the vertical axis. The price of a CD is PCD,
and the price of the composite good is Py ⫽ 1. Before the consumer joins the club, the
budget line is BL1. He could spend all his money to buy 300 units of other goods.
Or he could spend all of it to buy 15 CDs. The slope of BL1 is ⫺PCD ⲐPy ⫽ ⫺20.
With BL1 the consumer chooses basket A, where BL1 is tangent to the indifference
curve U1. The tangency at basket A tells us that MRSCD, y ⫽ 20 ⫽ PCD ⲐPy.
If he were to join the music club, the budget line would be BL2. If he joins the club,
he must pay the fee of $100 per month. That means he has only $200 remaining for
other goods and CDs. He could buy as many as 20 CDs (the horizontal intercept
of BL2). Or, he could spend the remaining $200 to buy only the composite good (at the
vertical intercept of BL2). The slope of BL2 is ⫺PCD ⲐPy ⫽ ⫺10.
As the figure indicates, the budget lines BL1 and BL2 happen to intersect at basket A. This means that the consumer could continue to choose basket A after joining
the club, spending $100 for the membership, $100 on CDs (buying 10 CDs at the club
8
125
Data are from a summary of the SNAP program available at SNAP’s website, http://www.fns.usda.gov/
pd/SNAPsummary.htm (accessed September 25, 2009).
9
“Characteristics of Food Stamp Households: Fiscal Year 2007,” U.S. Department of Agriculture, Food &
Nutrition Service, http://www.fns.usda.gov/ora/menu/Published/SNAP/FILES/ Participation/
2007CharacteristicsSummary.pdf (accessed on September 25, 2009).
126
CHAPTER 4
CONSUMER CHOICE
Preference
directions
E
y, units of the composite good
300
Slope of BL1 (budget line if not
a member of the club) = –20
Slope of BL2 (budget line if
a member of the club) = –10
200
BL2
A
FIGURE 4.13
Joining a Club
If the consumer does not belong to the CD club,
his budget line is BL1 and his optimal basket is A,
with utility U1. If he joins the club, his budget line
is BL2 and his optimal basket is B, with utility U2.
The consumer will be better off joining the club
(i.e., will achieve a higher level of utility) and will
buy more CDs.
BL1
B
U1
0
10
15
U2
20
CD, number of CDs
price of $10 each), and $100 on other goods. This tells us that the consumer can be
no worse off after joining the club because he can still purchase the basket he chose
when he was not in the club.
However, basket A will not be optimal for the consumer if he joins the club. We
already know that at A, MRSCD, y ⫽ 20; with the new price of CDs, PCD ⲐPy ⫽10. So
the budget line BL2 is not tangent to the indifference curve passing through basket
A. The consumer will seek a new basket, B, at which the budget line BL2 will be tangent to the indifference curve (and MRSCD, y ⫽ 10 ⫽ PCD ⲐPy). The consumer will be
better off in the club at basket B (achieving a level of utility U2) and will purchase
more CDs (15).
Consumers make similar decisions when deciding on many other types of purchases. For example, when customers subscribe to cellular telephone service, they
can pay a smaller monthly subscription charge and a higher price per minute of telephone usage or vice versa. Similarly, a consumer who joins a country club pays a
membership fee, but also pays less for each round of golf than someone who does not
join the club.
A P P L I C AT I O N : B O R R O W I N G A N D L E N D I N G
Up to this point, we have simplified the discussion by assuming that the consumer has
a given amount of income and neither borrows nor lends. Using composite goods, we
can modify the model of consumer choice to allow for borrowing and lending. (In the
following analysis, note that saving—putting money in the bank—is, in effect, lending money to the bank at the interest rate offered by the bank.)
Suppose that a consumer’s income this year is I1 and that next year he will have
an income of I2. If the consumer cannot borrow or lend, he will spend I1 this year and
I2 next year on goods and services.
4.3 CONSUMER CHOICE WITH COMPOSITE GOODS
A P P L I C A T I O N
4.4
sures the dollars spent per month on a composite good
whose price is $1. The consumer has a monthly income
of $500. If he spends all of his income on the composite good, he will be able to buy 500 units (basket E).
Suppose the consumer subscribes to Plan A.
After paying the $40 subscription fee, he will be
able to buy 460 units of the composite good as long
as he uses cellular service for less than 450 minutes
during the month. Until he reaches 450 minutes, his
budget line is flat. This means that once the monthly
fee is paid, the consumer, in effect, gets the first 450
minutes at a price of zero dollars. Indeed, this is how
these plans are often advertised: “Pay $40 and your
first 450 minutes are free.” Since he must pay an
extra $0.40 for calls exceeding the 450-minute limit
on Plan A, the slope of the budget line to the right
of basket R is –0.40. If the consumer were to use the
network for 500 minutes under Plan A, his total bill
would be $60 [i.e., $40 ⫹ $0.40(500–450)]. The
budget line under Plan A is MRT. If he spends his entire budget on cell phone calls, he will be able to
consume 1,600 minutes per month (basket T ).
Figure 4.14 shows the budget line for Plan B,
labeled NSV.
Pricing a Calling Plan
Companies that provide cellular phone and wireless
communications services often offer customers a menu
of pricing and service options. Customers choose a plan
from a menu and are billed accordingly. For example,
AT&T offers several options for its Apple iPhone 3GS
service for the Chicago area as of September 2009. The
following two calling plans are similar to actual options
offered by AT&T, although they have been somewhat
simplified for illustrative purposes.
• For $40 per month, you can call up to 450 minutes
per month. Each additional minute beyond 450
costs you $0.40. Let’s call this Plan A.
• For $60 per month, you can call up to 900 minutes
per month. Each additional minute beyond 900
costs you $0.40. Let’s call this Plan B.
Units of composite goods
(dollars per month)
Which plan would a utility-maximizing consumer
choose? A first step in answering this question is to
draw the budget line that corresponds to each plan. In
Figure 4.14, the horizontal axis measures the number
of minutes of telephone calls. The vertical axis mea-
500
460
440
E
M
R
U2
S
N
U1
Budget line
for plan B
Budget line
for plan A
T
0
450
127
900
1,600
Minutes of service per month
V
2,000
FIGURE 4.14 Choosing among Cellular Telephone Plans
Under Plan A, the consumer pays $40 and can use the phone up to 450 minutes at no extra
charge. If he makes more calls, he must pay $0.40 for each extra minute. His budget line is
therefore MRT. With Plan B, he pays $60 and can use the phone up to 900 minutes at no
extra charge. If he makes more calls, he must pay $0.40 for each extra minute. His budget
line is therefore NSV. The optimal choice will depend on the indifference map. With the
indifference map in the figure, he chooses Plan B and uses the telephone 900 minutes.
128
CHAPTER 4
CONSUMER CHOICE
The figure helps us understand why some consumers might choose one plan, while others choose
another plan. If a consumer needs 450 minutes per
month, he will choose Plan A. His cellular phone bill
will be $40. He could choose Plan B, but it would be
more costly for the level of service he needs. (If he
chooses only 450 minutes under Plan B, it will cost
him $60.)
Similarly, if the consumer needs 900 minutes of
service per month, he will choose Plan B and consume
basket S. His bill under Plan B will be $60. He could
choose Plan A, but it would be more expensive ($220)
given the level of service he needs.
If the consumer has an indifference map like the
one in Figure 4.14, he will choose Plan B and consume
basket S, consuming 900 minutes of service each month.
We can now use the composite good to help us represent the consumer’s choice of
consumption in each of the two years, both with and without borrowing and lending.
In Figure 4.15, the horizontal axis shows the consumer’s spending on the composite
good this year (C1); since the price of the composite good is $1, the horizontal axis also
shows the amount of the composite good purchased this year. Similarly, the vertical axis
shows the consumer’s spending on the composite good next year (C2), likewise equivalent to the amount of the composite good purchased that year. With no borrowing or
lending, the consumer can purchase basket A over the two-year period.
Now suppose the consumer can put money in the bank and earn an interest rate
r of 10 percent this year (r ⫽ 0.1). If he saves $100 this year, he will receive $100 plus
interest of $10 (0.1 ⫻ $100) next year, a total of $110. So, if he starts at A, every time
he decreases consumption this year (moves to the left on the budget line) by $1, he increases consumption next year (moves up on the budget line) by (1 ⫹ r) dollars. The
slope of the budget line is ⌬C2Ⲑ⌬C1 ⫽ (1 ⫹ r) Ⲑ(⫺1) ⫽ ⫺(1 ⫹ r).
Suppose, also, that the consumer can borrow money at the same annual interest
rate r of 10 percent this year (r ⫽ 0.1). If he borrows $100 in this year, he will have to
pay back $110 next year. If he starts at A, every time he increases consumption this
year (moves to the right on the budget line) by $1, he needs to decrease consumption
next year (move down on the budget line) by (1 ⫹ r) dollars. Again, the slope of the
budget line is ⫺(1 ⫹ r).
To determine the location of the budget line, we need to find its horizontal and
vertical intercepts. If the consumer spends nothing this year, and instead puts I1 in the
bank, next year he will be able to spend I2 ⫹ I1(1 ⫹ r); this is the vertical intercept of
the budget line. Similarly, if he borrows the maximum amount possible this year and
saves nothing, he will be able to spend up to I1 ⫹ I2Ⲑ(1 ⫹ r) this year; this is the horizontal intercept of the budget line.10
A consumer with the indifference map shown in Figure 4.15 would choose basket
B, borrowing some money (C1B – I1) from the bank this year and repaying the loan
next year, when he will be able to consume only C2B. Borrowing has increased his utility from U1 to U2.
The analysis shows how consumer preferences and interest rates determine why
some people are borrowers and others are savers. Can you draw an indifference map
for a consumer who would want to save money in the first period?
10
I2 ⫹ I1(1 ⫹ r) is what economists refer to as the future value of the consumer’s stream of income, and
I1 ⫹ I2 Ⲑ(1 ⫹ r) is what economists call the present value of the consumer’s stream of income. In Appendix 2
to this chapter, we discuss the concepts of future value and present value, as well as a number of other
concepts relating to the time value of money.
C2, amount of spending next year
(= amount of composite good purchased next year)
4.3 CONSUMER CHOICE WITH COMPOSITE GOODS
I2 + I1(1 + r)
E
129
Slope of budget line EG
= –(1 + r )
Preference
directions
I2
A
B
C2B
G
I1
C1B
I1 +
U2
U1
I2
1+r
C1, amount of spending this year
(= amount of composite good purchased this year)
A P P L I C A T I O N
FIGURE 4.15 Borrowing and
Lending
A consumer receives income I1 this
year and I2 next year. If he neither
borrows nor lends, he will be at
basket A. Suppose he can borrow
or lend at an interest rate r. If
his indifference map is as shown
in the graph, he would choose
basket B, borrowing (C1B ⫺ I1) from
the bank this year and repaying the
loan next year. Borrowing has increased his utility from U1 to U2.
4.5
To Lend or Not to Lend
Thus far in our discussion of borrowing and lending,
we have assumed that the interest rate the consumer
receives if he saves money (which means, in effect, he
lends it to the bank) is the same as the rate that the
consumer must pay if he borrows money. In reality,
however, the interest rate you pay when you borrow
is generally higher than the rate you earn when you
save, and financial institutions rely on this difference
to make money.
Let’s consider how different interest rates for borrowing and lending affect the shape of a consumer’s
budget line. In Summer 2009, the U.S. economy was in
a deep recession. During this recession many consumers had high levels of personal debt, in many
cases including mortgage debt that was higher than
the market value of their home. In addition, banks
were under strong financial pressure (and many
closed). For these reasons, interest rates on credit card
debt were higher than in more normal economic
times, while rates of return on investments in certificates of deposit (CDs) were quite low. In August 2009,
the average interest rate on a new credit card was
approximately 15%, while a typical 1-year CD offered
an interest rate of 1.5%.
Suppose that Mark receives an income of
$20,000 in year 1 and $24,150 in year 2. If he neither
borrows nor lends, he can purchase basket A in
Figure 4.16.
Let’s find the corner point of the budget line
along the vertical axis, representing the basket Mark
can choose if he consumes nothing in the first year
and saves all his income to spend in the second year.
If he can save at an interest rate of 1.5 percent (rL ⫽
0.015), he will have $44,450 available next year (the
$20,000 income in year 1, plus the interest payment of
$300, plus the income of $24,150 in year 2) and can
purchase basket E. The slope of the budget line between baskets A and E is ⫺(1 ⫹ rL) ⫽ –1.015, reflecting
FIGURE 4.16
CHAPTER 4
Consumer Choice
with Different Interest Rates for
Borrowing and Lending
A consumer receives an income of
$20,000 this year and $24,150 next year.
If he neither borrows nor lends, he will
be at basket A. Suppose he can save
(lend money to the bank) at an interest
rate of 1.5 percent. Every dollar he
saves this year will give him an additional $1.015 to spend next year. The
slope of the budget line between E and
A is therefore ⫺1.015. Similarly, if he
elects to borrow a dollar from the bank
this year, he will have to pay back $1.15
next year. The slope of the budget line
between A and G is therefore ⫺1.15.
CONSUMER CHOICE
C2, amount of spending next year
130
Preference
directions
$44,450
E
Slope of budget line EA = –1.015
Slope of budget line AG = –1.15
$24,150
the fact that for each dollar Mark saves this year, he
will have an extra $1.015 to spend next year.
Now let’s find the corner point of the budget line
along the horizontal axis, representing the budget
Mark could choose if he buys as much as possible in
year 1 and nothing in year 2. In order to buy as much
as possible in year 1, he would borrow as much as possible in that year by running up debt on his credit
card, and pay it back in year 2. The most that Mark
can borrow in year 1 is $21,000, since that credit card
debt would require a repayment equal to his entire
income in year 2 ($21,000 plus $3,150 in interest payments equals $24,150). Thus, his maximum spending
in year 1 is $41,000 ($20,000 income plus $21,000 borrowed), which would allow him to purchase basket G.
If he starts at A, every time he increases consumption
this year (moves to the right on the budget line) by
$1, he will need to decrease consumption next year
(move down on the budget line) by (1 ⫹ rB) dollars.
The slope of the budget line between baskets A and
G is ⫺1.15.
The borrowing and saving interest rates determine the slopes of the two parts of the budget line
A
G
$20,000
$41,000
C1, amount of spending this year
(EA and AG in Figure 4.16). The difference between
the two slopes on the budget line is quite dramatic
in this example, because the difference between interest rates for borrowing and saving was so great in
Summer 2009. This puts a prominent “kink” in the
budget line at basket A, and we would expect that
many consumers would choose this as their optimum.
At A Mark would neither save nor borrow. If the
two interest rates became closer to each other (as
would occur in a more typical year without deep recession), the budget line would still have a kink at A
but the slopes would be more similar.
To determine whether the consumer is a borrower or a lender, we would need to draw the consumer’s indifference map. Can you draw an indifference map for a consumer who would want to save
money in year 1? For such preferences, the highest indifference curve he can reach must be tangent to the
budget line between baskets A and E. Can you draw
an indifference map for a consumer who would want
to borrow in year 1? For such preferences, the highest
indifference curve he can reach must be tangent to
the budget line between baskets A and G.
4.3 CONSUMER CHOICE WITH COMPOSITE GOODS
A P P L I C AT I O N : Q UA N T I T Y D I S C O U N T S
In many product markets, sellers offer consumers quantity discounts. We can use the
theory of consumer choice to understand how such discounts affect consumer behavior.
Firms offer many kinds of quantity discounts. Here we consider an example that
is commonly observed in the electric power industry. In Figure 4.17 the horizontal
axis measures the number of units of electricity a consumer buys each month. The
vertical axis measures the number of units of a composite good, whose price is $1. The
consumer has a monthly income of $440.
Suppose the power company sells electricity at a price of $11 per unit, with no
quantity discount. The budget line facing the consumer would be MN, and the slope
of the budget line would be ⫺11. With the indifference map shown in Figure 4.17,
she would choose basket A, with 9 units of electricity.
Now suppose the supplier offers the following quantity discount: $11 per unit for
the first 9 but only $5.50 per unit for additional units. The budget line is now composed of two segments. The first segment is MA. The second segment is AR, having
a slope of ⫺5.5 because the consumer pays a price of $5.50 for units of electricity purchased beyond 9 units. Given the indifference map in the figure, the consumer will
buy a total of 16 units (at basket B) when she is offered the quantity discount. The discount has induced her to buy 7 extra units of electricity.
Quantity discounts expand the set of baskets a consumer can purchase. In Figure 4.17,
the additional baskets are the ones in the area bounded by RAN. As the figure
illustrates, a discount may enable the consumer to purchase a basket that gives her a higher
level of satisfaction than would otherwise be possible.
Units of composite good
comsumed per month
440
M
Slope of budget line MN = –11
Slope of budget line AR = – 5.5
B
A
U2
U1
BL2
BL1
N
9
R
16
Units of electricity consumed per month
FIGURE 4.17 Quantity Discount
If the electric power company sells electricity at a price of $11 per unit, the budget line
facing the consumer is MN. Given the indifference map shown in the graph, the consumer would choose basket A, with 9 units of electricity. If the supplier offers a quantity discount, charging $11 for each of the first 9 units, but only $5.50 per additional
units, the budget line is now composed of two segments, MA and AR. The consumer
will buy a total of 16 units of electricity (at basket B). Thus, the quantity discount has
induced her to buy 7 extra units of electricity. The figure shows that a quantity discount
may enable the consumer to achieve a higher level of satisfaction.
131
132
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A P P L I C A T I O N
CONSUMER CHOICE
4.6
Flying Is Its Own Reward
In 1981 American Airlines launched the industry’s first
frequent flyer program, AAdvantage Travel Awards.
Later the same year, United Airlines created its own
frequent flyer program, United Airlines Mileage Plus.
Many other airlines around the world now offer such
programs. These programs provide a number of rewards to travelers who repeatedly give their business
to a particular airline. Members may accumulate credit
for miles they have flown and redeem these miles for
upgrades and free tickets. They also receive other benefits, including priority for upgrades to a higher class
of service, preferred seating, and special treatment at
ticket counters and in airport lounges.
Frequent flyer programs typically have different
levels of membership, depending on the number of
miles a consumer flies with the airline during the year.
For example, under the AAdvantage program, a consumer traveling less than 25,000 miles per year receives credit in a mileage account for each mile flown.
4.4
REVEALED
PREFERENCE
revealed preference
Analysis that enables us to
learn about a consumer’s
ordinal ranking of baskets
by observing how his or her
choices of baskets change
as prices and income vary.
Y
A member traveling between 25,000 and 50,000
miles in a year attains AAdvantage Gold status for
the next year and receives credit for miles flown plus
a 25 percent mileage bonus. A consumer flying between 50,000 and 100,000 miles in a year attains
AAdvantage Platinum status for the next year.
Among other benefits, a Platinum member receives
credit for miles flown plus a 100 percent mileage
bonus. There is also a higher level of membership
(Executive Platinum) with additional benefits for
members who travel more than 100,000 miles per
year.
The provisions of frequent flyer programs are
often quite complicated, with a number of special
rules and rewards not discussed here. The important
idea is this: The more you travel, the less expensive
additional travel becomes—that is, you receive a
quantity discount. That is why frequent flyer programs are so popular today. As of 2009, American’s
AAdvantage program had enrolled more than 63 million members worldwide.
ou have now learned how to find a consumer’s optimal basket given preferences (an
indifference map) and given a budget line. In other words, if you know how the consumer ranks baskets, you can determine the optimal basket for any budget constraint
the consumer faces.
But suppose you do not know the consumer’s indifference map. Can you infer how
he ranks baskets by observing his behavior as his budget line changes? In other words,
do the consumer’s choices of baskets reveal information about his preferences?
The main idea behind revealed preference is simple: If the consumer chooses
basket A when basket B costs just as much, then we know that A is weakly preferred
to (i.e., at least as preferred as) B. (We write this as A Ɒ B, meaning that either A Ɑ B
or A L B.) When he chooses basket C, which is more expensive than basket D, then
we know that he must strongly prefer C to D (C Ɑ D). Given enough observations
about his choices as prices and income vary, we can learn much about how he ranks
baskets, even though we may not be able to determine the exact shape of his indifference map. Revealed preference analysis assumes that the consumer always chooses an
optimal basket and that, although prices and income may vary, his underlying preferences do not change.
Figure 4.18 illustrates how consumer behavior can reveal information about preferences. Given an initial level of income and prices for two goods (housing and clothing) the consumer faces budget line BL1 and chooses basket A. Suppose prices and income change so that the budget line becomes BL2, and he chooses basket B. What do
the consumer’s choices reveal about his preferences?
4.4 REVEALED PREFERENCE
133
F
Units of clothing
Preference
directions
C
B
A
E
BL2
BL1
H
Units of housing
FIGURE 4.18 Revealed
Preference
Suppose we do not know the consumer’s indifference map, but we
do have observations about consumer choice with two different
budget lines. When the budget
line is BL1, the consumer chooses
basket A. When the budget line is
BL2, the consumer chooses basket B.
What does the consumer’s behavior
reveal about his preferences? As
shown by the analysis in the text,
the consumer’s indifference curve
through A must pass somewhere
through the yellow area, perhaps
including other baskets on EF.
First, the consumer chooses basket A when he could afford any other basket on
or inside BL1, such as basket B. Therefore, A is at least as preferred as B (A Ɒ B). But
he has revealed even more about how he ranks A and B. Consider basket C. Since the
consumer chooses A when he can afford C, we know that A Ɒ C. And since C lies to
the northeast of B, C must be strongly preferred to B (C Ɑ B). Then, by transitivity, A
must be strongly preferred to B (if A Ɒ C and C Ɑ B, then A Ɑ B).
The consumer’s behavior also helps us learn about the shape of the indifference
curve through A. All baskets to the north, east, or northeast of A are strongly preferred
to A (including baskets in the darkly shaded area). A is strongly preferred to all baskets
in the region shaded light green, and at least as preferred as any other basket between
F and E. We also know that A is strongly preferred to any basket on the segment EH
because A is strongly preferred to B, and B is at least as preferred as any other basket
on BL2. Therefore, although we do not know exactly where the indifference curve
through A lies, it must pass somewhere through the yellow area, perhaps including baskets on EF other than A, but not including basket E.
A R E O B S E RV E D C H O I C E S C O N S I S T E N T
W I T H U T I L I T Y M A X I M I Z AT I O N ?
In our discussion of revealed preference, we have assumed that the consumer always
maximizes his utility by choosing the best basket given his budget constraint. Yet the
consumer could be choosing his basket in some other way. Can revealed preference
analysis tell us if a consumer is choosing baskets in a manner consistent with utility
maximization? Or, to pose the question differently, what observations about consumer choice would lead us to conclude that the consumer is not always maximizing
utility?
134
CHAPTER 4
CONSUMER CHOICE
Consider a case in which a utility-maximizing consumer buys only two goods.
Suppose that when the prices of the goods are initially (Px , Py), the consumer chooses
苲 苲
basket 1, containing (x1, y1). At a second set of prices (Px , Py ), he chooses basket 2, containing (x2, y2).
At the initial prices, basket 1 will cost the consumer Px x1 ⫹ Py y1. Let’s suppose
that basket 2 is also affordable at the initial prices, so that
Px x1 ⫹ Py y1 ⱖ Px x2 ⫹ Py y2
(4.7)
The left-hand side of equation (4.7) tells us how much the consumer would need
to spend to buy basket 1 at the initial prices. The right-hand side measures the expenditure necessary to buy basket 2 at the initial prices.
Since at the initial prices he chose basket 1 (and basket 2 was also affordable), he
has revealed that he likes basket 1 at least as much as basket 2.
We also know that at the second set of prices, he chose basket 2 instead of basket
1. Since he has already revealed that he prefers basket 1 at least as much as basket 2,
it must also be true that at the new prices basket 2 is no more expensive than basket 1.
Otherwise, he would have chosen basket 1 at the new prices. Equation (4.8) states that
basket 2 costs no more than basket 1 at the new prices.
苲
苲
苲
苲
Px x2 ⫹ Py y2 ⱕ Px x1 ⫹ Py y1
(4.8)
Why must equation (4.8) be satisfied if the consumer’s choices are consistent with
utility maximization? If it is not satisfied, then
苲
苲
苲
苲
Px x2 ⫹ Py y2 7 Px x1 ⫹ Py y1
(4.9)
If equation (4.9) were true, it would tell us that basket 2 is more expensive than
basket 1 at the second set of prices. Since the consumer chooses basket 2 at the second set of prices (when basket 1 is also affordable), he would then have to strongly
prefer basket 2 to basket 1. But this would be inconsistent with the earlier conclusion that he likes basket 1 at least as much as basket 2. To eliminate this inconsistency, equation (4.8) must be satisfied (and, equivalently, equation (4.9) must not be
satisfied).
Thus, if equation (4.8) is not satisfied, the consumer must be making choices that
fail to maximize utility. Learning-By-Doing Exercise 4.5 illustrates the use of revealed
preference analysis to detect such behavior.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 4 . 5
D
Consumer Choice That Fails to Maximize Utility
Problem
A consumer has an income of
$24 per week and buys two goods in quantities measured
by x and y. Initially he faces prices (Px , Py) ⫽ ($4, $2) and
chooses basket A containing (x1, y1) ⫽ (5, 2). Later the
苲 苲
prices change to (Px , Py) ⫽ ($3, $3). He then chooses
basket B, containing (x2, y2) ⫽ (2, 6). These choices and
his budget lines are illustrated in Figure 4.19. Show that
he cannot be choosing baskets that maximize his utility
in both periods.
4.4 REVEALED PREFERENCE
135
12
10
FIGURE 4.19
8
C
B
y
6
4
D
A
2
BL1
0
2
5
4
x
6
BL2
8
Consumer Choice
That Fails to Maximize Utility
When the budget line is BL1, the
consumer chooses basket A when
he can afford basket C; thus, A Ɒ C.
Since basket C lies northeast of basket B, it must be that C Ɑ B. This
implies A Ɑ B (if A Ɒ C and C Ɑ B,
then A Ɑ B).
When the budget line is BL2, the
consumer chooses basket B when he
can afford basket D; thus, B Ɒ D.
Since basket D lies northeast of basket A, it must be that D Ɑ A. This
implies B Ɑ A (if B Ɒ D and D Ɑ A,
then B Ɑ A).
Since it can’t be true that A Ɑ B
and B Ɑ A, the consumer must not
always be choosing the optimal
basket.
Solution There are two ways to demonstrate that
the consumer is failing to maximize utility. First, let’s use
a graphical approach. Observe that with BL1, he chose
basket A when he could afford basket C. Thus basket A
is at least as preferred as basket C ( A Ɒ C ). Further,
since basket C lies to the northeast of basket B, he
must strongly prefer basket C to basket B (C Ɑ B).
Using transitivity, we can conclude that basket A is
strongly preferred to basket B (if A Ɒ C and C Ɑ B,
then A Ɑ B).
Let’s apply similar reasoning to the consumer’s
choice of basket B when given BL2. Here the consumer
chose basket B when he could afford basket D. Thus basket B is at least as preferred as basket D. Further, since
basket D lies to the northeast of basket A, he must
strongly prefer basket D to basket A. By transitivity we
conclude that basket B is strongly preferred to basket A
(if B Ɒ D and D Ɑ A, then B Ɑ A).
It cannot simultaneously be true that basket A is
strongly preferred to basket B and that basket B is
strongly preferred to basket A. Therefore, the consumer must not be choosing the best basket with each
budget line.
We can reach the same conclusion using an algebraic approach. At the initial prices (Px , Py) ⫽ ($4,
$2), the consumer chose basket A when he could afford basket B. He paid Px x1 ⫹ Py y1 ⫽ $4(5) ⫹ $2(2) ⫽
$24 for basket A when he could have paid Px x2 ⫹
Py y2 ⫽ $4(2) ⫹ $2(6) ⫽ $20 for basket B. This implies that he strongly prefers basket A to basket B.
(Note that equation (4.7) is satisfied: Px x1 ⫹ Py y1 ⱖ
Px x2 ⫹ Py y2.)
苲 苲
However, at the new prices (Px, Py) ⫽ ($3, $3), he
chose basket B when he could afford basket A. He paid
苲
苲
Px x2 ⫹ Py y2 ⫽ $3(2) ⫹ $3(6) ⫽ $24 for basket B when
苲
苲
he could have paid Px x1 ⫹ Py y1 ⫽ $3(5) ⫹ $3(2) ⫽ $21
for basket A. This implies that he strongly prefers basket
B to basket A.
Thus, his behavior at the two price levels is inconsistent, which means that he is not always choosing the
best basket. (Note that equation (4.8) is not satisfied:
苲
苲
苲
苲
Px x2 ⫹ Py y2 7 Px x1 ⫹ Py y1.)
Similar Problems: 4.25, 4.27, 4.30
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CHAPTER 4
A P P L I C A T I O N
CONSUMER CHOICE
4.7
Is Altruism Rational?
Is altruism consistent with utility-maximizing behavior? After reading Chapters 3 and 4, you might be
tempted to conclude that the answer is no. After all,
in the theory of consumer choice that we have developed so far, individuals seek to maximize their own
utility. This behavior seems selfish and therefore inconsistent with the idea that individuals might act
benevolently toward others. Yet, in the real world,
individuals do exhibit altruistic behavior. And in laboratory experiments in which individuals have the
opportunity to behave selfishly or altruistically, they
often (voluntarily!) choose to be altruistic.
One explanation for altruistic behavior that is
consistent with the theory of consumer choice is that
an individual’s utility function could be an increasing
function of both the individual’s own consumption
and that of fellow individuals. If so, some degree of
altruism could be consistent with individual optimizing behavior. Using experimental methods and the
theory of revealed preference, James Andreoni and
John Miller sought to test whether altruism can indeed be the result of utility-maximizing behavior.11 In
their experiments, a subject was faced with the task of
allocating tokens (each worth a certain amount of
money) to him- or herself and to another subject. By
varying the number of tokens the subject was allocated,
as well as the relative price of donating tokens to the
other person versus keeping them, Andreoni and
Miller were able to shift a subject’s budget line in
such a way that revealed preference could be used to
test whether the subject’s choices were consistent
with utility maximization.
Andreoni and Miller found that the choices of
nearly all subjects—whether or not they exhibited altruism—were consistent with utility maximization.
About 22 percent of subjects were completely selfish.
Their behavior was consistent with a utility function
that depended only on their own allocation of tokens. The vast majority of the remaining subjects exhibited altruistic behavior that was consistent with
the maximization of a utility function subject to a
budget constraint. For example, 16 percent of the
subjects always split the tokens evenly. The utility
function that rationalizes this behavior reflects a perfect complementarity between one’s own consumption and that of other subjects: U ⫽ min (xS, xO),
where xS is the allocation of tokens to oneself and xO
is the allocation of tokens to others.
The lesson? While not everyone is altruistic—the
world does contain some selfish maximizers—one
should not assume that altruistic behavior is inconsistent with utility maximization. The impulse to be
generous could go hand in hand with the desire to
maximize one’s own utility.
Learning-By-Doing Exercise 4.5 demonstrated one of the potentially powerful
applications of revealed preference analysis. Even though we did not know the consumer’s indifference map, we used evidence from the consumer’s choices to infer that
he was not always maximizing utility. We conclude this section with an exercise that
will help you see some of the other types of inferences that can be drawn from revealed preference analysis.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 4 . 6
D
Other Uses of Revealed Preference
Each of the graphs in Figure 4.20 depicts
choices by an individual consuming two commodities, x
and y. The consumer likes x and y (more of x is better
11
and more of y is better). In each case, when the budget
line is BL1, the consumer selects basket A, and when the
budget line is BL2, the consumer selects basket B.
J. Andreoni and J. H. Miller, “Analyzing Choice with Revealed Preference: Is Altruism Rational?,” in C.
Plott and V. Smith, eds. Handbook of Experimental Economics Results (Amsterdam: Elsevier, 2004).
4.4 REVEALED PREFERENCE
Problem What can be said about the way the consumer ranks the two baskets in each case?
This contradiction (it can’t be that B Ɑ A and A Ɒ B)
indicates that the consumer isn’t always maximizing utility by purchasing the best basket.
Case 3: With BL1, the consumer chose basket A
when he could afford B (both are on BL1). Therefore,
A Ɒ B.
With BL2, the consumer chose basket B but couldn’t
afford basket A, which doesn’t tell us anything new. The
ranking A Ɒ B is all we can determine.
Case 4: With BL1, the consumer chose basket A but
couldn’t afford basket B; with BL2, the consumer chose
basket B but couldn’t afford basket A. Neither choice
tells us anything about how the consumer ranks baskets
A and B. (To learn anything about how a consumer
ranks two baskets, we must observe at least one instance
where he chooses between them when he can afford
both.)
Solution Case 1: With BL2, the consumer chose bas-
B
A
BL1 BL2
x
(a) Case 1
y
B
Similar Problems: 4.21, 4.23, 4.24, 4.28
A
y
y
y
ket B when he could afford A (we know this because A is
inside BL2); thus B Ɒ A.
But consider basket C, which is also on BL2. Since
the consumer chose basket B over basket C, it must be
that B Ɒ C. And, since C is northeast of A, it must be
that C Ɑ A. Therefore, B Ɑ A (if B Ɒ C and, C Ɑ A, then
B Ɑ A).
This case shows that when a consumer chooses a
basket on a budget line, it is strongly preferred to any
basket inside that budget line.
Case 2: With BL2, the consumer chose basket B
when he could afford A (we know this because A is inside BL2). By the reasoning in Case 1, since A is inside
BL2, we know that B Ɑ A.
Now consider BL1. Both baskets A and B are on
BL1, and the consumer chose A. Therefore, A Ɒ B.
C
A
B
B
A
BL1 BL2
x
(b) Case 2
137
BL1 BL2
BL1 BL2
x
(c) Case 3
x
(d) Case 4
FIGURE 4.20 Revealed Preference
In each case, when the budget line is BL1, the consumer selects basket A, and when the
budget line is BL2, the consumer selects basket B. What can be said about the way the
consumer ranks the two baskets in each case? In Case 1 we conclude that B is strongly
preferred to A. In Case 2 the consumer’s choices are inconsistent with utility-maximizing
behavior. In Case 3 we infer that A is weakly preferred to B. In Case 4 we cannot infer any
ranking.
The theory of revealed preference is surprisingly powerful. It allows us to use
information about consumer choices to infer how the consumer must rank baskets
if he is maximizing utility with a budget constraint. It also allows us to discover
when a consumer is failing to choose his optimal basket given a budget constraint.
We can draw these inferences without knowing the consumer’s utility function or
indifference map.
138
CHAPTER 4
CONSUMER CHOICE
CHAPTER SUMMARY
• A budget line represents the set of all baskets that a
consumer can buy if she spends all of her income. A
budget line shifts out in a parallel fashion if the consumer receives more income. A budget line will rotate
about its intercept on the vertical axis if the price of the
good on the horizontal axis changes (holding constant
the consumer’s income and the price of the good on the
vertical axis). (LBD Exercise 4.1)
• If the consumer maximizes utility while living within
her budget constraint (i.e., choosing a basket on or inside the budget line), and if there are positive marginal
utilities for all goods, the optimal basket will be on the
budget line. (LBD Exercise 4.2)
• When a utility-maximizing consumer buys positive
amounts of two goods, she will choose the amounts of
those goods so that the ratio of the marginal utilities of
the two goods (which is the marginal rate of substitution) is equal to the ratio of the prices of the goods.
(LBD Exercise 4.2)
• When a utility-maximizing consumer buys positive
amounts of two goods, she will choose the amounts of
those goods so that the marginal utility per dollar spent
will be equal for the two goods. (LBD Exercises 4.3
and 4.4)
• It may not be possible for a utility-maximizing consumer to buy two goods so that the marginal utility per
dollar spent is equal for the two goods. An optimal basket would then be at a corner point. (LBD Exercises 4.3
and 4.4)
• The analysis of revealed preference may help us to
infer how an individual ranks baskets without knowing
the individual’s indifference map. We learn about preferences by observing which baskets the consumer chooses
as prices and income vary. When the consumer chooses
basket A over an equally costly basket B, then we know
that A is at least as preferred as B. When she chooses
basket C over a less costly basket D, then we know that
C is strongly preferred to D. Revealed preference analysis may also help us identify cases in which observed
consumer behavior is inconsistent with the assumption
that the consumer is maximizing her utility. (LBD
Exercises 4.5 and 4.6)
REVIEW QUESTIONS
1. If the consumer has a positive marginal utility for
each of two goods, why will the consumer always choose
a basket on the budget line?
2. How will a change in income affect the location of
the budget line?
3. How will an increase in the price of one of the goods
purchased by a consumer affect the location of the
budget line?
4. What is the difference between an interior optimum
and a corner point optimum in the theory of consumer
choice?
5. At an optimal interior basket, why must the slope of
the budget line be equal to the slope of the indifference
curve?
6. At an optimal interior basket, why must the marginal
utility per dollar spent on all goods be the same?
7. Why will the marginal utility per dollar spent not
necessarily be equal for all goods at a corner point?
8. Suppose that a consumer with an income of $1,000
finds that basket A maximizes utility subject to his budget
constraint and realizes a level of utility U1. Why will this
basket also minimize the consumer’s expenditures necessary to realize a level of utility U1?
9.
What is a composite good?
10. How can revealed preference analysis help us learn
about a consumer’s preferences without knowing the
consumer’s utility function?
PROBLEMS
4.1. Pedro is a college student who receives a monthly
stipend from his parents of $1,000. He uses this stipend
to pay rent for housing and to go to the movies (assume
that all of Pedro’s other expenses, such as food and clothing have already been paid for). In the town where Pedro
goes to college, each square foot of rental housing costs
PROBLEMS
$1.50 per month. Each movie he attends costs $10. Let x
denote the square feet of housing, and let y denote the
number of movies he attends per month.
a) What is the expression for Pedro’s budget constraint?
b) Draw a graph of Pedro’s budget line.
c) What is the maximum number of square feet of housing he can purchase given his monthly stipend?
d) What is the maximum number of movies he could
attend given his monthly stipend?
e) Suppose Pedro’s parents increase his stipend by 10
percent. At the same time, suppose that in the college
town where he lives, all prices, including housing rental
rates and movie ticket prices, increase by 10 percent.
What happens to the graph of Pedro’s budget line?
4.2. Sarah consumes apples and oranges (these are the
only fruits she eats). She has decided that her monthly
budget for fruit will be $50. Suppose that one apple costs
$0.25, while one orange costs $0.50.
a) What is the expression for Sarah’s budget constraint?
b) Draw a graph of Sarah’s budget line.
c) Show graphically how Sarah’s budget line changes if
the price of apples increases to $0.50.
d) Show graphically how Sarah’s budget line changes if
the price of oranges decreases to $0.25.
e) Suppose Sarah decides to cut her monthly budget for
fruit in half. Coincidentally, the next time she goes to the
grocery store, she learns that oranges and apples are on
sale for half price, and will remain so for the next month;
that is, the price of apples falls from $0.25 per apple to to
$0.125 per apple, and the price of oranges falls from
$0.50 per orange to $0.25 per orange. What happens to
the graph of Sarah’s budget line?
4.3. Julie has preferences for food F and clothing C described by a utility function U(F, C ) ⫽ FC. Her marginal
utilities are MUF ⫽ C and MUC ⫽ F. Suppose that food
costs $1 a unit and that clothing costs $2 a unit. Julie has
$12 to spend on food and clothing.
a) On a graph draw indifference curves corresponding
to u ⫽ 12, u ⫽ 18, and u ⫽ 24. Using the graph (and no
algebra), find the optimal (utility-maximizing) choice of
food and clothing. Let the amount of food be on the
horizontal axis and the amount of clothing be on the
vertical axis.
b) Using algebra (the tangency condition and the budget
line), find the optimal choice of food and clothing.
c) What is the marginal rate of substitution of food for
clothing at her optimal basket? Show this graphically and
algebraically.
d) Suppose Julie decides to buy 4 units of food and
4 units of clothing with her $12 budget (instead of the optimal basket). Would her marginal utility per dollar spent
on food be greater than or less than her marginal utility
139
per dollar spent on clothing? What does this tell you about
how she should substitute food for clothing if she wanted
to increase her utility without spending any more money?
4.4. The utility that Ann receives by consuming food F
and clothing C is given by U(F, C ) ⫽ FC ⫹ F. The marginal utilities of food and clothing are MUF ⫽ C ⫹ 1 and
MUC ⫽ F. Food costs $1 a unit, and clothing costs $2 a
unit. Ann’s income is $22.
a) Ann is currently spending all of her income. She is
buying 8 units of food. How many units of clothing is she
consuming?
b) Graph her budget line. Place the number of units of
clothing on the vertical axis and the number of units of
food on the horizontal axis. Plot her current consumption basket.
c) Draw the indifference curve associated with a utility
level of 36 and the indifference curve associated with a
utility level of 72. Are the indifference curves bowed in
toward the origin?
d) Using a graph (and no algebra), find the utilitymaximizing choice of food and clothing.
e) Using algebra, find the utility-maximizing choice of
food and clothing.
f ) What is the marginal rate of substitution of food for
clothing when utility is maximized? Show this graphically and algebraically.
g) Does Ann have a diminishing marginal rate of substitution of food for clothing? Show this graphically and
algebraically.
4.5. Consider a consumer with the utility function
U(x, y) ⫽ min(3x, 5y); that is, the two goods are perfect
complements in the ratio 3:5. The prices of the two goods
are Px ⫽ $5 and Py ⫽ $10, and the consumer’s income is
$220. Determine the optimum consumption basket.
4.6. Jane likes hamburgers (H) and milkshakes (M). Her
indifference curves are bowed in toward the origin and
do not intersect the axes. The price of a milkshake is $1
and the price of a hamburger is $3. She is spending all
her income at the basket she is currently consuming, and
her marginal rate of substitution of hamburgers for milkshakes is 2. Is she at an optimum? If so, show why. If not,
should she buy fewer hamburgers and more milkshakes,
or the reverse?
4.7. Ray buys only hamburgers and bottles of root beer
out of a weekly income of $100. He currently consumes
20 bottles of root beer per week, and his marginal utility
of root beer is 6. The price of root beer is $2 per bottle.
Currently, he also consumes 15 hamburgers per week,
and his marginal utility of a hamburger is 8. Is Ray maximizing utility at his current consumption basket? If not,
should he buy more hamburgers each week, or fewer?
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CHAPTER 4
CONSUMER CHOICE
4.8. Dave currently consumes 10 hot dogs and 6 sodas
each week. At his current consumption basket, his marginal utility for hot dogs is 5 and his marginal utility for
sodas is 3. If the price of one hot dog is $1 and the price
of one soda is $0.50, is Dave currently maximizing his
utility? If not, how should he reallocate his spending in
order to increase his utility?
4.9. Helen’s preferences over CDs (C ) and sandwiches
(S) are given by U(S, C) ⫽ SC ⫹ 10(S ⫹ C), with MUC ⫽
S ⫹ 10 and MUS ⫽ C ⫹ 10. If the price of a CD is $9 and
the price of a sandwich is $3, and Helen can spend a combined total of $30 each day on these goods, find Helen’s
optimal consumption basket.
4.10. The utility that Corey obtains by consuming
hamburgers (H) and hot dogs (S) is given by U(H, S) ⫽
1H ⫹ 1S ⫹ 4. The marginal utility of hamburgers is
0.5
0.5 #
and the marginal utility of steaks is equal to
1H
1S ⫹ 4
a) Sketch the indifference curve corresponding to the
utility level U ⫽ 12.
b) Suppose that the price of hamburgers is $1 per hamburger and the price of steak is $8 per steak. Moreover,
suppose that Corey can spend $100 per month on these
two foods. Sketch Corey’s budget line for hamburgers
and steak given this budget.
c) Based on your answer to parts (a) and (b), what is
Corey’s optimal consumption basket given his budget?
4.11. This problem will help you understand what
happens if the marginal rate of substitution is not diminishing. Dr. Strangetaste buys only french fries (F ) and
hot dogs (H ) out of his income. He has positive marginal
utilities for both goods, and his MRSH,F is increasing. The
price of hot dogs is PH, and the price of french fries is PF.
a) Draw several of Dr. Strangetaste’s indifference curves,
including one that is tangent to his budget line.
b) Show that the point of tangency does not represent
a basket at which utility is maximized, given the budget
constraint. Using the indifference curves you have
drawn, indicate on your graph where the optimal basket
is located.
4.12. Julie consumes two goods, food and clothing, and
always has a positive marginal utility for each good. Her
income is 24. Initially, the price of food is 2 and the price
of clothing is 2. After new government policies are implemented, the price of food falls to 1 and the price of
clothing rises to 4. Suppose, under the initial budget constraint, her optimal choice is 10 units of food and 2 units
of clothing.
a) After the prices change, can you predict whether her
utility will be higher, lower, or the same as under the
initial prices?
b) Does your answer require that there be a diminishing
marginal rate of substitution of food for clothing?
Explain.
4.13. Toni likes to purchase round trips between the
cities of Pulmonia and Castoria and other goods out of
her income of $10,000. Fortunately, Pulmonian Airways
provides air service and has a frequent flyer program. A
round trip between the two cities normally costs $500,
but any customer who makes more than 10 trips a year
gets to make additional trips during the year for only
$200 per round trip.
a) On a graph with round trips on the horizontal axis and
“other goods” on the vertical axis, draw Toni’s budget
line. (Hint: This problem demonstrates that a budget line
need not always be a straight line.)
b) On the graph you drew in part (a), draw a set of indifference curves that illustrates why Toni may be better off
with the frequent flyer program.
c) On a new graph draw the same budget line you found
in part (a). Now draw a set of indifference curves that
illustrates why Toni might not be better off with the frequent flyer program.
4.14. A consumer has preferences between two
goods, hamburgers (measured by H ) and milkshakes
(measured by M ). His preferences over the two goods
are represented by the utility function U ⫽ 1H ⫹ 1M.
For this utility function MUH ⫽ 1/(21H) and
MUM ⫽ 1/(2/ 1M) .
a) Determine if there is a diminishing MRSH,M for this
utility function.
b) Draw a graph to illustrate the shape of a typical indifference curve. Label the curve U1. Does the indifference
curve intersect either axis? On the same graph, draw a
second indifference curve U2, with U2 ⬎ U1.
c) The consumer has an income of $24 per week. The
price of a hamburger is $2 and the price of a milkshake is
$1. How many milkshakes and hamburgers will he buy
each week if he maximizes utility? Illustrate your answer
on a graph.
4.15. Justin has the utility function U ⫽ xy, with the
marginal utilities MUx ⫽ y and MUy ⫽ x. The price of x
is 2, the price of y is py, and his income is 40. When he
maximizes utility subject to his budget constraint, he
purchases 5 units of y. What must be the price of y and
the amount of x consumed?
4.16. A student consumes root beer and a composite
good whose price is $1. Currently, the government imposes an excise tax of $0.50 per six-pack of root beer. The
student now purchases 20 six-packs of root beer per
month. (Think of the excise tax as increasing the price of
root beer by $0.50 per six-pack over what the price would
141
PROBLEMS
be without the tax.) The government is considering eliminating the excise tax on root beer and, instead, requiring
consumers to pay $10.00 per month as a lump-sum tax
(i.e., the student pays a tax of $10.00 per month, regardless
of how much root beer is consumed). If the new proposal
is adopted, how will the student’s consumption pattern (in
particular, the amount of root beer consumed) and welfare
be affected? (Assume that the student’s marginal rate of
substitution of root beer for other goods is diminishing.)
4.17. When the price of gasoline is $2.00 per gallon,
Joe consumes 1,000 gallons per year. The price increases
to $2.50, and to offset the harm to Joe, the government
gives him a cash transfer of $500 per year. Will Joe be better off or worse off after the price increase and cash transfer than he was before? What will happen to his gasoline
consumption? (Assume that Joe’s marginal rate of substitution of gasoline for other goods is diminishing.)
4.18. Paul consumes only two goods, pizza (P) and hamburgers (H), and considers them to be perfect substitutes,
as shown by his utility function: U(P, H) ⫽ P ⫹ 4H. The
price of pizza is $3 and the price of hamburgers is $6, and
Paul’s monthly income is $300. Knowing that he likes
pizza, Paul’s grandmother gives him a birthday gift certificate of $60 redeemable only at Pizza Hut. Though
Paul is happy to get this gift, his grandmother did not
realize that she could have made him exactly as happy by
spending far less than she did. How much would she have
needed to give him in cash to make him just as well off as
with the gift certificate?
4.19. Jack makes his consumption and saving decisions
two months at a time. His income this month is $1,000,
and he knows that he will get a raise next month, making
his income $1,050. The current interest rate (at which he
is free to borrow or lend) is 5 percent. Denoting this
month’s consumption by x and next month’s by y, for each
of the following utility functions state whether Jack would
choose to borrow, lend, or do neither in the first month.
(Hint: In each case, start by assuming that Jack would simply spend his income in each month without borrowing
or lending money. Would doing so be optimal?)
a) U(x, y) ⫽ xy2, MUx ⫽ y2, MUy ⫽ 2xy
b) U(x, y) ⫽ x2y, MUx ⫽ 2xy, MUy ⫽ x 2
c) U(x, y) ⫽ xy, MUx ⫽ y, MUy ⫽ x
4.20. The figure in this problem shows a budget set for
a consumer over two time periods, with a borrowing rate
rB and a lending rate rL, with rL ⬍ rB. The consumer purchases C1 units of a composite good in period 1 and C2
units in period 2. The following is a general fact about
consumers making consumption decisions over two time
periods: Let A denote the basket at which a consumer
spends exactly his income each period (the point at the kink
of the budget line). Then a consumer with a diminishing
MRSC1, C2 will choose to borrow in the first period if at
basket A MRSC1, C2 7 1 ⫹ rB and will choose to lend if at
basket A MRSC1, C2 6 1 ⫹ rL. If the MRS lies between
these two values, then he will neither borrow nor lend.
(You can try to prove this if you like. Keep in mind that
diminishing MRS plays an important role in the proof.)
Using this rule, consider the decision of Meg, who
earns $2,000 this month and $2,200 the next with a utility
function given by U(C1, C2) ⫽ C1C2, where the C’s denote
the value of consumption in each month. For this utility
function MUC1 ⫽ C2 and MUC2 ⫽ C1. Suppose rL ⫽ 0.05
(the lending rate is 5 percent) and rB ⫽ 0.12 (the borrowing
rate is 12 percent). Would Meg borrow, lend, or do neither
this month? What if the borrowing rate fell to 8 percent?
C2
2,200 +
2,000(1.05)
2,200
Slope = –1.05
A
Slope = –1.12
2,000 2,000 + 2,200/1.12 C1
4.21. Sally consumes housing (denote the number of
units of housing by h) and other goods (a composite good
whose units are measured by y), both of which she likes.
Initially she has an income of $100, and the price of a unit
of housing (Ph) is $10. At her initial basket she consumes
2 units of housing. A few months later her income rises
to $120; unfortunately, the price of housing in her city
also rises, to $15. The price of the composite good does
not change. At her later basket she consumes 1 unit of
housing. Using revealed preference analysis (without
drawing indifference curves), what can you say about
how she ranks her initial and later baskets?
4.22. Samantha purchases food (F ) and other goods
(Y ) with the utility function U ⫽ FY, with MUF ⫽ Y and
MUy ⫽ F. Her income is 12. The price of a food is 2 and
the price of other goods 1.
a) How many units of food does she consume when she
maximizes utility?
b) The government has recently completed a study suggesting that, for a healthy diet, every consumer should
consume at least F ⫽ 8 units of food. The government is
considering giving a consumer like Samantha a cash subsidy that would induce her to buy F ⫽ 8. How large would
the cash subsidy need to be? Show her optimal basket
with the cash subsidy on an optimal choice diagram with
F on the horizontal axis and Y on the vertical axis.
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CHAPTER 4
CONSUMER CHOICE
c) As an alternative to the cash subsidy in part (b), the
government is also considering giving consumers like
Samantha food stamps, that is, vouchers with a cash value
that can only be redeemed to purchase food. Verify that
if the government gives her vouchers worth $16, she will
choose F ⫽ 8. Illustrate her optimal choice on an optimal
choice diagram. (You may use the same graph you drew
in part (b).)
4.23. As shown in the following figure, a consumer
buys two goods, food and housing, and likes both goods.
When she has budget line BL1, her optimal choice is
basket A. Given budget line BL2, she chooses basket B,
and with BL3, she chooses basket C.
Housing
B
C
BL1
BL2
BL3
Food
a) What can you infer about how the consumer ranks
baskets A, B, and C? If you can infer a ranking, explain
how. If you cannot infer a ranking, explain why not.
b) On the graph, shade in (and clearly label) the areas
that are revealed to be less preferred to basket B, and explain why you indicated these areas.
c) On the graph, shade in (and clearly label) the areas
that are revealed to be (more) preferred to basket B, and
explain why you indicated these areas.
4.24. The following graph shows the consumption
decisions of a consumer over bundles of x and y, both of
which he likes. When faced with budget line BL1, he
chose basket A, and when faced with budget line BL2, he
chose basket B. If he were to face budget line BL3, what
possible set of baskets could he choose in order for his
behavior to be consistent with utility maximization?
18
16
12
A
BL3
8
6
4
B
2
BL2
BL1
0
2
Graph Darrell’s budget constraint under each of the two
plans. If Plan A is better for him, what is the set of baskets he may purchase if his behavior is consistent with
utility maximization? What baskets might he purchase if
Plan B is better for him?
4
6
8
4.27. Angela has a monthly income of $120, which she
spends on MP3s and a composite good (whose price you
may assume is $1 throughout this problem). Currently,
she does not belong to an MP3 club, so she pays the
retail price of an MP3 of $2; her optimal basket includes
20 MP3s monthly.
For the past several months Asteroid, a media company, has offered her the chance to join their “Premium
Club”; to join the club she would need to pay a membership fee of $60 per month, but then she could buy all the
MP3s she wants at a price of $0.50. She has decided not
to join the club.
Asteroid has now introduced an “Economy Club”;
to join, Angela would need to pay a membership fee of
$30 per month, but then she could buy all the MP3s she
wants at a price of $1. Draw a graph illustrating
(1) Angela’s budget line and optimal basket when she joins
no club, (2) the budget line she would have faced had
she joined the Premium Club, and (3) her budget line if
she joins the Economy Club. Will Angela surely want to
join the Economy Club? If she were to join the club,
how many MP3s per month might she buy? Show how
you arrive at your answers using a revealed preference
argument.
4.28. Alex buys two goods, food (F ) and clothing (C ).
He likes both goods. His preferences for the goods do
not change from month to month. The following table
shows his income, the baskets he selected, and the prices
of the goods over a two-month period.
14
10
• Plan A: Pay no monthly fee and make calls for $0.50
per minute.
• Plan B: Pay a $20 monthly fee and make calls for
$0.20 per minute.
4.26. Figure 4.17 illustrates the case in which a consumer is better off with a quantity discount. Can you
draw an indifference map for a consumer who would not
be better off with the quantity discount?
A
y
4.25. Darrell has a monthly income of $60. He spends
this money making telephone calls home (measured in
minutes of calls) and on other goods. His mobile phone
company offers him two plans:
10
x
12
14
16
18
20
Month
PF
PC
Income
Basket Chosen
1
2
3
2
2
4
48
48
F ⫽ 16, C ⫽ 0
F ⫽ 14, C ⫽ 5
143
A P P E N D I X 1 : T H E M AT H E M AT I C S O F C O N S U M E R C H O I C E
a) On the graph with F on the horizontal axis and C on
the vertical axis, plot and clearly label the budget lines
and consumption baskets during these two weeks. Label
the consumption bundle in week 1 by point A on the
graph and the consumption basket in week 2 by point B.
Using revealed preference analysis, what can you say
about Alex’s preferences for baskets A and B (i.e., how
does he rank them)?
b) In month 3 Alex’s income rises to 57. The prices of
food and clothing are both 3. Assuming his preferences
do not change, describe the set of baskets he might consume in month 3 if he continues to maximize utility.
Show this set of baskets in the graph.
4.29. Brian consumes units of electricity (E ) and a
composite good (Y ), whose price is always 1. He likes
both goods.
In period 1 the power company sets the price of
electricity at $7 per unit, for all units of electricity consumed. Brian consumes his optimal basket, 20 units of
electricity and 70 units of the composite good.
In period 2 the power company then revises its
pricing plan, charging $10 per unit for the first 5 units
and $4 per unit for each additional unit. Brian’s income
is unchanged. Brian’s optimal basket with this plan includes 30 units of electricity and 60 units of the composite good.
APPENDIX 1:
In period 3 the power company allows the consumer
to choose either the pricing plan in period 1 or the plan
in period 2. Brian’s income is unchanged. Which pricing
plan will he choose? Illustrate your answer with a clearly
labeled graph.
4.30. Carina consumes two goods, X and Y, both of
which she likes. In month 1 she chooses basket A given
budget line BL1. In month 2 she chooses B given budget
line BL2, and in month 3 she chooses C given budget line
BL3. Assume her indifference map is unchanged over
the three months. Use the theory of revealed preference
to show whether her choices are consistent with utilitymaximizing behavior. If so, show how she ranks the three
baskets. If it is not possible to infer how she ranks the
baskets, explain why not.
Y
B
A
The Mathematics of Consumer Choice
In this section we solve the consumer choice problem using the calculus technique of
Lagrange multipliers. Suppose the consumer buys two goods, where x measures the
amount of the first good and y the amount of the second good. The price of the first
good is Px and the price of the second is Py. The consumer has an income I.
Let’s assume that the marginal utilities of both goods are positive, so we know that he
will expend all of his income at his optimal basket. The consumer choice problem is then:
max U(x, y)
(x, y)
BL1
C
BL2
(A4.1)
subject to: Px x ⫹ Py y ⫽ I
We define the Lagrangian (¶) as ¶(x, y, l) ⫽ U(x, y) ⫹ l(I ⫺ Px x ⫺ Py y),where is
a Lagrange multiplier. The first-order necessary conditions for an interior optimum
(with x ⬎ 0 and y ⬎ 0) are
0U (x, y)
0¶
⫽01
⫽ lPx
0x
0x
(A4.2)
0U (x, y)
0¶
⫽01
⫽ lPy
0y
0y
(A4.3)
0¶
⫽ 0 1 I ⫺ Px x ⫺ Py y ⫽ 0
0l
(A4.4)
BL3
X
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CHAPTER 4
CONSUMER CHOICE
The partial derivative ⭸U(x, y)/⭸x is the mathematical expression for the marginal utility
of x (MUx ). It measures how much utility increases as x increases, holding y constant.
Similarly, the partial derivative ⭸U(x, y) Ⲑ⭸y is the mathematical expression for the marginal
utility of y (MUy). It measures how much utility increases as y increases, holding x constant.
We can combine equations (A4.2) and (A4.3) to eliminate the Lagrange multiplier, so our first-order conditions reduce to:
MUx
Px
⫽
MUy
Py
(A4.5)
Px x ⫹ Py y ⫽ I
(A4.6)
Equation (A4.5) is just the condition requiring that the marginal utility per dollar
spent be equal at an optimum (MUx ⲐPx ⫽ MUy ⲐPy), or equivalently, that the indifference curve and the budget line be tangent to one another (MUx ⲐMUy ⫽ Px ⲐPy).
Equation (A4.6) is the equation for the budget line. So the mathematical solution to
the consumer choice problem tells us that an optimal interior basket will satisfy the
tangency condition and be on the budget line. This verifies the conditions for an optimum we developed in the text, using a graphical approach.
For a further discussion of the use of Lagrange multipliers, see the Mathematical
Appendix in this text.
A P P E N D I X 2 : The Time Value of Money
Suppose you have won a raffle, and you are given a choice between two prizes: $100
in cash today or a $100 in cash a year from now. If you are like most people, you would
prefer the first prize. This illustrates an important property in economics: Money has
time value. Given a choice between a given amount of money received immediately
and the same amount of money received at some point in the future, individuals prefer the immediate sum to the same sum in the future.
The fact that money has time value is reflected by an important feature of realworld markets: the presence of interest rates. As a condition for loaning money, a
lender typically requires that the borrower not only repay the amount of money that
was lent, but also pay an interest rate on the borrowed money. The interest rate compensates the lender for sacrificing something (the use of a given amount of money
today) in return for something else that is worth less (a promise to return the same
amount of money at the date of repayment).
The fact that money has a time value complicates the comparison of different
amounts of money received at different points in time. For example, if the prizes in
the raffle had been $100 in cash immediately or $120 in cash one year from now, your
preference between the two prizes would not be as obvious. In this Appendix, we introduce you to techniques that can be used to compare amounts of money received at
different points in time.
F U T U R E VA L U E A N D P R E S E N T VA L U E
To illustrate how we might compare a prize of $100 received today with a prize of
$120 received a year from now, suppose that you could invest the $100 prize in an account that yielded an annual interest rate of 5 percent (r ⫽ 0.05) and there are no
A P P E N D I X 2 : T H E T I M E VA L U E O F M O N E Y
other investment options offering a better return. After one year, your account would
have grown in value to $100(1.05) ⫽ $105. This amount—$105—is the future value
of $100 one year from now at an interest rate of 5 percent. In general, the future
value of an amount C received t periods from now when the interest rate per period
is r is the amount of money that you would have t periods from now if you put $C into
an account that earned an interest rate of r each period. The formula for the future
value of an amount is
C (1 ⫹ r)t
This formula holds because your interest is compounded as you keep the money in
the account:
• During the first period, you earn interest equal to r on the $C in your account,
so by the end of the first period your account will have grown to C(1 ⫹ r).
• During the second period, you earn interest equal to r on the $(1 ⫹ r)C in your
account, so by the end of the second period your account will have grown to
C(1 ⫹ r) ⫹ rC(1 ⫹ r), which equals C(1 ⫹ r)2.
• During the third period, you earn interest equal to r on the $(1 ⫹ r)2C in your
account, so by the end of the second period your account will have grown to
C(1 ⫹ r)2 ⫹ rC(1 ⫹ r)2, which equals C(1 ⫹ r)2(1 ⫹ r) or C(1 ⫹ r)3.
Repeating this logic for t periods gives us the formula for future value.
Note that, in our example, the future value of the $100 prize in one year is less
than the $120 prize received in a year. Thus, we conclude that $120 received a year
from now is more valuable than $100 received immediately.
This approach is based on a comparison of future values. We can also compare
their values in the present. Let’s ask: How much would you need to invest in your account today at an interest rate of 5 percent in order to have exactly $120 one year from
now? The answer would be to solve the following equation for C:
C (1.05) ⫽ $120
or
C⫽
$120
(1.05)
⫽ $114.28
This amount—$114.28—is the present value of $120 received one year from now
at an interest rate of 5 percent. In general, the present value of an amount C received t periods from now when the interest rate per period is r is the amount of
money that you would need to invest today in an account that earns an interest rate
of r each period so that t periods from now you would have $C. The formula is
C
(1 ⫹ r)t
To compute a present value of an amount, one needs to know the number of
periods from now, t, at which the amount is received and the interest rate r, or what
145
146
CHAPTER 4
CONSUMER CHOICE
is called the discount rate. The discount rate is the interest rate used in a present
value calculation. Because, in our example, the present value of $120 a year from now
exceeds $100, we would conclude that a $120 prize received in a year is more valuable
than a $100 price received immediately, the same conclusion we reached by comparing future values.
Present value is an extremely useful concept because it enables an “apples to apples”
comparison in today’s dollars between amounts of money received at different points
in time. Because it is so useful, this concept is widely used in a variety of applications
including capital budgeting in firms, actuarial analysis in insurance, and cost-benefit
analysis in the public sector.
The concept of present value of an amount can be extended to the present value
of a stream of payments. The present value of a stream of amounts C1, C2, . . . , CT,
where the first payment is received one period from now, the second payment is received two periods from now, and so forth, is the sum of the present values of the
amounts in the stream, that is,
C1
CT
C2
⫹p⫹
⫹
2
(1 ⫹ r)
(1 ⫹ r)
(1 ⫹ r)T
For example, suppose a consulting firm expects to receive payments of $1 million one
year from now, $1.2 million two years from now, and $1.5 million three years from
now, from a three-year contract with a client. With a discount rate of 10 percent, the
present value of the revenue stream from this contract would thus be
$1,000,000
$1,500,000
$1,200,000
⫹
⫽ $3,027,799
⫹
2
1.10
1.10
1.103
Notice that this present value is less than the simple sum of the payments ($3.7
million). This is because the dollars received in one year, two years, and three years
from now are worth less than a dollar received immediately.
A special case of a stream of payments is an annuity. An annuity is a stream of
constant, equally spaced, payments over a certain period of time. The formula for the
present value of an annuity of C over T periods with a discount rate r is
C
C
C
⫹
⫹p⫹
(1 ⫹ r)
(1 ⫹ r)2
(1 ⫹ r)T
After several steps of algebra, this formula can be rewritten as follows:
C
1
c1 ⫺
d
r
(1 ⫹ r)T
A particular type of an annuity is a perpetuity. This is an annuity that lasts forever.
Examples of a perpetuity are the Consol Bonds issued by the British government in
1752, which promised to pay a fixed amount of money to the holder of the bonds forever. (Some of these bonds still exist today.) We can derive the formula for the present
value of a perpetuity from the formula for the present value of an annuity by noting that
A P P E N D I X 2 : T H E T I M E VA L U E O F M O N E Y
1
goes to zero. Thus, the formula for
(1 ⫹ r)T
the present value of a perpetuity is given by
as T becomes infinitely large, the term
C
r
For example, if you owned a bond from the British government that paid £1,000 a
year forever starting next year, and if your discount rate were 0.20, the present value
of this perpetuity would be
£1,000
⫽ £5,000
0.20
Thus even though the sum of an infinite stream of £1,000 payment is infinitely large,
the present value of a perpetuity of £1,000 is finite. This is because money has time
value. Thus, amounts of money to be received very far into the future, say, 1,000 years
from now, have a present value that is virtually zero.
N E T P R E S E N T VA L U E
An important use of present value is to compare benefits to costs. Suppose that a firm
is considering building a new plant, and suppose that the goods produced in the new
plant will increase the firm’s cash flows by $1.5 million per year over the 20-year lifetime of the plant. Suppose, further, that the plant costs $20 million to build. Finally,
suppose that the firm’s discount rate for new investments is 15 percent. Is the stream
of benefits from the new plant greater than the upfront cost of the plant? To answer
this question, we compute the net present value of the plant. The net present value
(NPV) is the difference between the present value of the stream of benefits and the
upfront cost that must be incurred to receive those benefits. The formula for NPV is
NPV ⫽ ⫺C0 ⫹
C1
C2
CT
⫹
⫹p⫹
2
(1 ⫹ r)
(1 ⫹ r)
(1 ⫹ r)T
where C0 is the initial upfront payment that must be made to receive the stream of
cash benefits, C1, . . . , CT. Applying this formula to our example, we see that the NPV
of the new plant (whose stream of benefits is an annuity) is
NPV ⫽ ⫺$20,000,000 ⫹
⫽ ⫺$20,000,000 ⫹
$1,500,000
$1,500,000
$1,500,000
⫹
⫹p⫹
2
1.15
1.15
1.1520
$1,500,000
1
c1 ⫺
d
0.15
1.1520
⫽ ⫺10,611,003
Since NPV ⬍ 0, we can see that the present value of the benefits from the new plant
is less than the upfront cost of the new plant. The new plant’s benefits are thus not
worth the cost.
147
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CHAPTER 4
CONSUMER CHOICE
P R E S E N T VA L U E , F U T U R E VA L U E , A N D T H E
OPTIMAL CONSUMPTION CHOICE PROBLEM
The concepts of present value and future value play a role in the analysis of optimal
consumption choice over time discussed in Section 4.3. First, let’s consider the consumer’s budget line. As Figure 4.15 shows, the horizontal intercept of the consumer’s
budget line is equal to
I1 ⫹
I2
1⫹r
This tells us that given the consumer’s anticipated flow of income, this year and next
year, the most that the consumer could spend this year is equal to the present value of
this year’s income and next year’s income. The consumer could achieve this level of
current consumption by borrowing an amount equal to his entire future income.
The vertical intercept of the consumer’s budget line is
I2 ⫹ I1(1 ⫹ r)
This tells us that the most the consumer could spend next year is the future value of
this year’s income and next year’s income. The consumer could achieve this level of
future consumption by saving all of his income this year and consuming an amount
next year equal to his next year’s income, plus his savings, plus his accumulated interest on that savings.
Note that the slope of the budget line is ⫺(1 ⫹ r). This tells us that the consumer
must give up 1 ⫹ r dollars of future consumption in order to achieve one additional
dollar of current consumption. In other words, one additional dollar of current consumption requires that the consumer sacrifice the future value of one dollar of future
consumption.
Now, let’s think about the consumer’s optimal level of current and future consumption and explore under what circumstances a consumer is likely to be a borrower
or a saver. The consumer would find it optimal to borrow money if the point of tangency defining its optimal basket was to the southeast of point A on the budget line,
as shown in Figure 4.15. To explore the circumstances under which this is likely to be
the case, we will make a simplifying assumption, namely, that the consumer’s utility
function is given by the formula
U(C1) ⫹
U(C2)
1⫹r
where U(C ) is a utility function that indicates the utility the consumer receives from
consuming C dollars worth of a composite good within a given year. In other words,
we assume that the consumer’s utility is the present value of the utility from consumption this year and next year using a discount rate of . This discount rate is
referred to as the consumer’s rate of time preference and is a measure of the consumer’s impatience. The higher the value of the consumer’s . the more impatient the
consumer is, that is, the smaller is the utility the consumer derives from consumption
in the future.
A P P E N D I X 2 : T H E T I M E VA L U E O F M O N E Y
The marginal rate of substitution of consumption this year for consumption next
year equals the ratio of the marginal utility of consumption this year to the marginal
utility of consumption next year. With the utility function above this equals:
MRSC1, C2 ⫽ (1 ⫹ r)
U¿(C1)
U¿(C2)
where U¿(C1) and U¿(C2) denote the marginal utility of consumption this year and
next year, respectively. The consumer’s optimal basket will occur to the right of point
A in Figure 4.15—that is, the consumer will borrow—if, at point A as shown in
Figure 4.15, MRSC1, C2 exceeds the absolute value of the slope of the budget line, that is,
U¿(I1)
1⫹r
7
U¿(I2)
1⫹r
This condition is more likely to hold if:
• the consumer is sufficiently impatient, that is, the consumer’s rate of time preference is greater than the market interest rate r.
and/or
• the consumer’s marginal utility of consumption given current-year income exceeds
his marginal utility of consumption given next year’s income. With diminishing
marginal utility of consumption, this would occur if the consumer expects a
growth in income from this year to next year, that is, I2 ⬎ I1.
This theory of optimal choice suggests, then, that for a given expectation of income growth, a more impatient individual will have a greater propensity to borrow
than a more patient individual. And for a given rate of time preference, an individual
with a higher expectation of income growth will have a greater propensity to borrow
than an individual with a lower expectation of income growth.
149
5
THE THEORY OF DEMAND
5.1
OPTIMAL CHOICE AND DEMAND
APPLICATION 5.1
What Would People Pay for
Cable?
APPLICATION 5.2
The Irish Potato Famine
APPLICATION 5.3
Rats Respond When Prices
5.2
CHANGE IN THE PRICE OF A GOOD:
SUBSTITUTION EFFECT AND INCOME
EFFECT
Change!
Have Economists Finally Found
a Giffen Good? Rice and Noodles in China
APPLICATION 5.4
5.3
CHANGE IN THE PRICE OF A GOOD:
THE CONCEPT OF CONSUMER
SURPLUS
How Much Would You Be Willing
to Pay to Have a Wal-Mart in Your Neighborhood?
APPLICATION 5.5
5.4
MARKET DEMAND
Externalities in Social
Networking Websites
APPLICATION 5.6
5.5
THE CHOICE OF LABOR AND LEISURE
The Backward-Bending Supply
of Nursing Services
APPLICATION 5.7
5.6
CONSUMER PRICE INDICES
150
The Substitution Bias in the
Consumer Price Index
APPLICATION 5.8
Why Understanding the Demand for Cigarettes Is Important for Public Policy
In 2009, the United States imposed the largest increase in the federal excise tax on cigarettes in history.
The federal tax rose to about $1.00 on each pack of 20 cigarettes. Together with excise taxes imposed
in varying amounts by the states, the national average of excise taxes increased from $0.57 per pack in
1995 to about $2.21 per pack in 2009.1
To understand the potential virtues and limitations of higher cigarette taxes, or indeed the wisdom of
taxing cigarettes at all, it is important to understand the nature of cigarette demand and how, in particular,
it responds to the price of cigarettes and consumer income.
For example, antismoking advocates sometimes propose higher cigarette excise tax rates as a way to discourage smoking. The higher cigarette prices induced by higher excise tax rates do discourage smoking, but
only to a limited extent because the demand for cigarettes is known to be rather price inelastic. Still, higher
cigarette excise tax rates may be helpful as way to discourage young people from smoking. In a summary of
the evidence about the price elasticity of demand for cigarettes, the U.S. Center for Disease Control suggests
that while a 10 percent increase in the price of cigarettes would result in only a 4 percent decline in cigarette
smoking among adults, it would be expected to lead to a 7 percent drop in smoking among young consumers.
Increases in cigarette excise taxes (as well as other excise taxes such as those for gasoline and alcohol)
have also been considered by states seeking to balance their budgets in the midst of an economic recession.
For example, Kentucky and Arkansas, states with historically low cigarette taxes, each recently increased its
cigarette excise tax rate by nearly 100 percent.2 The fact that the demand for cigarettes is relatively price
inelastic is good news for this strategy: For products with price inelastic demands, a higher excise tax rate
typically leads to higher tax receipts for the government imposing those taxes.3
Cigarette taxes might be fiscally beneficial to states during recessions for another reason. Evidence suggests that the demand for cigarettes is not only relatively insensitive to changes in the price of cigarettes, it is
also insensitive to changes in consumer income.4 Thus, reductions in aggregate income levels during a recession would not be expected to have much of an impact on a state’s receipts from a cigarette excise tax (holding the tax rate constant). In other words, a tax on cigarettes may be a relatively stable source of tax revenue
for states because it is less likely to be affected by an economic downturn than sales taxes on goods whose
demand is more cyclical, such as hotel rooms or new cars.
As you learned in Chapter 4, price and income play
a potentially important role in shaping the decisions of
consumers who are choosing among various goods and
1
The material in this discussion is drawn from Morbidity and Mortality
Weekly, U.S. Center for Disease Control, 58(19) (May 22, 2009),
pp. 524–527.
2
“States Look at Tobacco to Balance the Budget,” New York Times
(March 20, 2009).
3
In Chapter 10, you will learn more about how an excise tax affects the
price of a good and the amount of tax revenue the government receives.
4
See Joni Hersch, “Gender, Income Levels, and the Demand for
Smoking,” Journal of Risk and Uncertainty 21, no. 2/3 (2000), pp.
263–282.
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CHAPTER 5
T H E T H E O RY O F D E M A N D
services subject to a budget constraint. By studying the impact of changes in prices
and income levels on an individual’s consumption decisions, as we do in this chapter, we can gain insight into why some goods, such as cigarettes, have demands
that are relatively insensitive to changes in prices and income, while other goods,
such as automobiles, might have demands that are relatively more sensitive to
changes in prices, or income, or both.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Explain how a consumer’s demand for a good depends on the prices of all
goods and on income.
• Examine how a change in the price of a good affects a consumer through a
substitution effect and an income effect.
• Explain how a change in the price of a good affects three measures of consumers’ well-being: consumer surplus, compensating variation, and equivalent
variation.
• Derive market demand curves from individual demand curves.
• Discuss the effects of network externalities on demand curves.
• Explain how consumers choose to allocate their time between labor and
leisure and how this relates to the supply of labor in the market.
• Explain the biases in the Consumer Price Index.
5.1
OPTIMAL
CHOICE AND
DEMAND
Where do demand curves come from? In Chapter 4, we showed how to determine
a consumer’s optimal basket. Given the consumer’s preferences and income and the
prices of all goods, we could ask how much ice cream a consumer will buy each month
if the price of a gallon of ice cream is $5. This will be a point on the consumer’s demand
curve for ice cream. We can find more points on her demand curve by repeating the
exercise for different prices of ice cream, asking what her monthly consumption of ice
cream will be if the price is $4, $3, or $2 per gallon. Let’s see how to do this, using a simplified setting in which our consumer buys only two goods, food and clothing.
THE EFFECTS OF A CHANGE IN PRICE
What happens to the consumer’s choice of food when the price of food changes while
the price of clothing and the amount of income remain constant? We have two ways
to answer this question, one using the optimal choice diagram in Figure 5.1(a) and the
second using the demand curve in Figure 5.1(b).
Looking at an Optimal Choice Diagram
The graph in Figure 5.1(a) shows the quantity of food consumed (x) on the horizontal axis and the quantity of clothing ( y) on the vertical axis. It also shows three of the
consumer’s indifference curves (U1, U2, and U3). Suppose the consumer’s weekly income is $40 and the price of clothing is Py ⫽ $4 per unit.
5.1 OPTIMAL CHOICE AND DEMAND
10
A
y, units of clothing
8
C Price consumption curve
6
B
5
U3
U2
U1
BL1 (Px = $4)
0
2
10
20
BL3 (Px = $1)
40
x, units of food
(a)
A'
$4
Px, price of food
16
BL2 (Px = $2)
B'
2
C'
1
D Food
0
2
(b)
10
16
x, units of food
FIGURE 5.1
The Effects of Changes in the Price of a Good on Consumption
The consumer has a weekly income of $40. The price of clothing Py is $4 per unit.
(a) Optimal choice diagram. When the price of food is $4, the budget line is BL1. When
the price of food is $2 and $1, respectively, the budget lines are BL2 and BL3. The optimal
baskets are A, B, and C. The curve connecting the optimal baskets is called the price
consumption curve.
(b) Demand curve for food (based on optimal choice diagram above). The consumer buys
more food as its price falls, so the demand curve is downward sloping.
Consider the consumer’s choices of food and clothing for three different prices of
food. First, suppose the price of food is Px ⫽ $4. The budget line that the consumer
faces when Px ⫽ $4, Py ⫽ $4, and I ⫽ $40 is labeled BL1 in the figure. The slope of
BL1 is ⫺ Px ⲐPy ⫽ ⫺4 Ⲑ4 ⫽ ⫺1. The consumer’s optimal basket is A, indicating that her
optimal weekly consumption is 2 units of food and 8 units of clothing.
153
154
price consumption
curve The set of utilitymaximizing baskets as the
price of one good varies
(holding constant income
and the prices of other
goods).
CHAPTER 5
T H E T H E O RY O F D E M A N D
What happens when the price of food falls to Px ⫽ $2? The vertical intercept of
the budget line is the same because income and the price of clothing are unchanged.
However, as we saw in Chapter 4, the horizontal intercept moves to the right (to BL2).
The slope of BL2 is ⫺Px ⲐPy ⫽ ⫺2 Ⲑ4 ⫽ ⫺1Ⲑ2. Her optimal basket is B, with a weekly
consumption of 10 units of food and 5 units of clothing.
Finally, suppose the price of food falls to Px ⫽ $1. The budget line rotates out to
BL3, which has a slope of ⫺Px ⲐPy ⫽ ⫺1Ⲑ4. The consumer’s optimal basket is C, with a
weekly consumption of 16 units of food and 6 units of clothing.
One way to describe how changes in the price of food affect the consumer’s purchases of both goods is to draw a curve connecting all of the baskets that are optimal
as the price of food changes (holding the price of clothing and income constant). This
curve is called the price consumption curve.5 In Figure 5.1(a), the optimal baskets
A, B, and C lie on the price consumption curve.
Observe that the consumer is better off as the price of food falls. When the price
of food is $4 (and she chooses basket A ), she reaches the indifference curve U1. When
the price of food is $2 (and she chooses basket B), her utility rises to U2. If the price
of food falls to $1, her utility rises even farther, to U3.
Changing Price: Moving along a Demand Curve
We can use the optimal choice diagram of Figure 5.1(a) to trace out the demand curve
for food shown in Figure 5.1(b), where the price of food appears on the vertical axis
and the quantity of food on the horizontal axis.
Let’s see how the two graphs are related to each other. When the price of food is
$4, the consumer chooses basket A in Figure 5.1(a), containing 2 units of food. This corresponds to point A on her demand curve for food in Figure 5.1(b). Similarly, at basket
B in Figure 5.1(a), the consumer purchases 10 units of food when the price of food is
$2, matching point B on her demand curve in Figure 5.1(b). Finally, as basket C in
Figure 5.1(a) indicates, if the price of food falls to $1, the consumer buys 16 units of
food, corresponding to point C in Figure 5.1(b). In sum, a decrease in the price of food
leads the consumer to move down and to the right along her demand curve for food.
The Demand Curve Is Also a “Willingness to Pay” Curve
As you study economics, you will sometimes find it useful to think of a demand curve
as a curve that represents a consumer’s “willingness to pay” for a good. To see why this
is true, let’s ask how much the consumer would be willing to pay for another unit of
food when she is currently at the optimal basket A (purchasing 2 units of food) in
Figure 5.1(a). Her answer is that she would be willing to pay $4 for another unit of
food. Why? At basket A her marginal rate of substitution of food for clothing is
MRSx, y 1.6 Thus, at basket A one more unit of food is worth the same amount to
her as one more unit of clothing. Since the price of clothing is $4, the value of an
additional unit of food will also be $4. This reasoning helps us to understand why
point A on the demand curve in Figure 5.1(b) is located at a price of $4. When the
consumer is purchasing 2 units of food, the value of another unit of food to her (i.e.,
her “willingness to pay” for another unit of food) is $4.
Note that her MRSx, y falls to 1Ⲑ2 at basket B and to 1Ⲑ4 at basket C. The value of
an additional unit of food is therefore $2 at B (when she consumes 10 units of food)
5
In some textbooks the price consumption curve is called the “price expansion path.”
At A the indifference curve U 1, and the budget line BL1 are tangent to one another, so their slopes are
equal. The slope of the budget line is Px ⲐPy 1. Recall that the MRSx, y at A is the negative of the
slope of the indifference curve (and the budget line) at that basket. Therefore, MRSx, y 1.
6
5.1 OPTIMAL CHOICE AND DEMAND
155
and only $1 at basket C (when she consumes 16 units of food). In other words, her
willingness to pay for an additional unit of food falls as she buys more and more food.
THE EFFECTS OF A CHANGE IN INCOME
y, clothing
What happens to the consumer’s choices of food and clothing as income changes? Let’s
look at the optimal choice diagram in Figure 5.2(a), which measures the quantity of
food consumed (x) on the horizontal axis and the quantity of clothing ( y) on the vertical axis. Suppose the price of food is Px $2 and the price of clothing is Py $4 per
unit, with both prices held constant. The slope of her budget lines is Px ⲐPy 1 Ⲑ2.
In Chapter 4 we saw that an increase in income results in an outward, parallel shift
of the budget line. Figure 5.2(a) illustrates the consumer’s budget lines and optimal
choices of food and clothing for three different levels of income, as well as three of
Income consumption curve
C
11
U3
B
8
U2
A
5
U1
BL1 (I = $40)
10
18
BL2
(I = $68)
BL3 (I = $92)
24
x, units of food
(a)
Px, price of food
FIGURE 5.2
$2
A'
B'
C'
D3 (I = $92)
D2 (I = $68)
D1 (I = $40)
10
18
24
x, units of food
(b)
The Effects of
Changes in Income on Consumption
The consumer buys food at Px $2
per unit and clothing at Py $4 per
unit. Both prices are held constant
as income varies.
(a) Optimal choice diagram. The
budget lines reflect three different
levels of income. The slope of all
budget lines is Px ⲐPy 1Ⲑ2. BL1
is the budget line when the weekly
income is $40. BL2 and BL3 are the
budget lines when income is $68
and $92, respectively. We can draw
a curve connecting the baskets that
are optimal (A, B, and C ) as income
changes. This curve is called the
income consumption curve.
(b) Demand curves for food. The
consumer’s demand curve for food
shifts out as income rises.
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A P P L I C A T I O N
T H E T H E O RY O F D E M A N D
5.1
What Would People Pay for Cable?
The cable television industry is one of the most
important sources of programming for households
in the United States. The major competitor is direct
broadcast satellites (DBS). Recently, consumers have
had increasing access to a third choice: viewing many
television programs using Internet connections.
Public policy toward the cable television industry has
changed repeatedly during the last two decades. In
1984 the industry was deregulated, and cable systems
rapidly expanded the services they offered. However,
by the early 1990s, Congress had become concerned
that local cable operators were charging unacceptably high prices and that many home owners lacked
adequate access to alternative programming. In
1992, Congress passed a sweeping set of regulations
for the industry, but in 1996 Congress removed
regulation from much of the industry, recognizing
that competition to provide programming had
increased.
income consumption
curve The set of utilitymaximizing baskets as income varies (and prices are
held constant).
Public policy debates on this subject often focus on
the nature of the demand for cable television. How
much will consumers pay for basic cable television services? How sensitive are consumers to changes in
prices or to the availability of competing products?
One study recently estimated the elasticity of demand
for basic cable to be 1.5, while premium cable and
DBS were found to have more elastic demands.7
Thus, a 10 percent increase in the price of a basic subscription would lead to a loss of 15 percent of subscribers. This estimate is larger than estimates from
the 1990s, probably because consumers have more
substitutes available now than they did then.
Another study found that when the price of cable
television increases substantially, many subscribers
switch to DBS.8 However, that study also found that
the cross-price elasticity of demand for DBS with
respect to cable prices was smaller in markets where
cable television offered regional sports channels.
Presumably, those channels were often not available
on DBS as well.
her indifference curves (U1, U2, and U3). Initially, when the consumer’s weekly income
is I1 $40, her budget line is BL1. She chooses basket A, consuming 10 units of food
and 5 units of clothing per week. As her income rises to I2 $68, the budget line shifts
out to BL2. She then chooses basket B, with a weekly consumption of 18 units of food
and 8 units of clothing. If her income increases to I3 $92, she faces budget line BL3.
Her optimal basket is C, with 24 units of food and 11 units of clothing.
One way we can describe how changes in income affect the consumer’s purchases is by drawing a curve that connects all the baskets that are optimal as income
changes (keeping prices constant). This curve is called the income consumption
curve.9 In Figure 5.2(a), the optimal baskets A, B, and C lie on the income consumption curve.
Changing Income: Shifting a Demand Curve
In Figure 5.2(a) the consumer purchases more of both goods as her income rises. In
other words, an increase in income results in a rightward shift in her demand curve
for each good. In Figure 5.2(b) we illustrate this by seeing how a change in income
affects her demand curve for food. The price of food (held constant at $2) appears on
the vertical axis, and the quantity of food on the horizontal axis. When the consumer’s
7
Austan Goolsbee and Amil Petrin, “The Consumer Gains from Direct Broadcast Satellites and the
Competition with Cable TV.” Econometrica (2004), vol. 72, no. 2, pp. 359–381.
8
Andrew Wise and Kiran Duwadi, “Competition between Cable Television and Direct Broadcast Satellite:
The Importance of Switching Costs and Regional Sports Networks,” Journal of Competition Law &
Economics (2005), vol. 1, no. 4, pp. 679–705.
9
Some textbooks call the income consumption curve the “income expansion path.”
5.1 OPTIMAL CHOICE AND DEMAND
157
weekly income is $40, she buys 10 units of food each week, corresponding to point A
on demand curve D1 in Figure 5.2(b). If her income rises to $68, she buys 18 units of
food, corresponding to point B on demand curve D2. Finally, if her income rises to
$92, she buys 24 units of food, corresponding to point C on demand curve D3.
Using a similar approach, you can also show how the demand curves for clothing
shift as income changes (see Problem 5.1 at the end of this chapter).
Engel Curves
Another way of showing how a consumer’s choice of a particular good varies with income is to draw an Engel curve, a graph relating the amount of the good consumed
to the level of income. Figure 5.3 shows an Engel curve relating the amount of food
consumed to the consumer’s income. Here the amount of food (x) is on the horizontal axis and the level of income (I ) is on the vertical axis. Point A on the Engel curve
shows that the consumer buys 10 units of food when her weekly income is $40. Point
B indicates that she buys 18 units of food when her income is $68. When her weekly
income rises to $92, she buys 24 units of food (point C ). Note that we draw the
Engel curve holding constant the prices of all goods (the price of food is $2 and the
price of clothing is $4). For a different set of prices we would draw a different Engel
curve.
The income consumption curve in Figure 5.2(a) shows that the consumer purchases more food when her income rises. When this happens, the good (food) is said
to be a normal good. For a normal good the Engel curve will have a positive slope,
as in Figure 5.3.
From Figure 5.2(a) you can also see that clothing is a normal good. Therefore, if
you were to draw an Engel curve for clothing, with income on the vertical axis and
the amount of clothing on the horizontal axis, the slope of the Engel curve would be
positive. Learning-By-Doing Exercise 5.1 shows that a good with a positive income
elasticity of demand will have a positively sloped Engel curve.
As you might suspect, consumers don’t always purchase more of every good as income rises. If a consumer wants to buy less of a good when income rises, that good is
Engel curve A curve
that relates the amount of
a commodity purchased to
the level of income, holding
constant the prices of all
goods.
normal good A good
that a consumer purchases
more of as income rises.
I, weekly income
Engel curve
C''
$92
B''
$68
$40
A''
FIGURE 5.3
10
18
24
x, units of food
Engel Curve
The Engel curve relates the amount of a
good purchased (in this example, food) to
the level of income, holding constant the
prices of all goods. The price of a unit of
food is $2, and the price of a unit of
clothing is $4.
158
CHAPTER 5
inferior good A good
that a consumer purchases
less of as income rises.
termed an inferior good. Consider a consumer with the preferences for hot dogs and a
composite good (“other goods”) depicted in Figure 5.4(a). For low levels of income, this
consumer views hot dogs as a normal good. For example, as monthly income rises from
$200 to $300, the consumer would change his optimal basket from A to B, buying more
hot dogs. However, as income continues to rise, the consumer prefers to buy fewer hot
dogs and more of the other goods (such as steak or seafood). The income consumption
curve in Figure 5.4(a) illustrates this possibility between baskets B and C. Over this range
of the income consumption curve, hot dogs are an inferior good.
The Engel curve for hot dogs is shown in Figure 5.4( b). Note that the Engel
curve has a positive slope over the range of incomes for which hot dogs are a normal
good and a negative slope over the range of incomes for which hot dogs are an inferior good.
y, monthly consumption of other goods
T H E T H E O RY O F D E M A N D
C
U3
B
U2
BL3 (I = $400)
BL2 (I = $300)
U1
BL1 (I = $200)
A
13 16
18
x, hot dogs per month
FIGURE 5.4
Inferior Good
(a) As income rises from $200 to $300, the
consumer’s weekly consumption of hot dogs
increases from 13 (basket A) to 18 (basket B).
However, as income rises from $300 to $400,
the consumer’s weekly consumption of hot
dogs decreases from 18 to 16 (basket C ).
(b) Hot dogs are a normal good between
points A and B (i.e., over the income range
$200 to $300), where the Engel curve has a
positive slope. But between points B and C
(i.e., over the income range $300 to $400),
hot dogs are an inferior good, and the Engel
curve has a negative slope.
I, monthly income
(a)
C'
$400
B'
$300
A'
$200
Engel
curve
13 16 18
x, hot dogs per month
(b)
5.1 OPTIMAL CHOICE AND DEMAND
S
E
159
L E A R N I N G - B Y- D O I N G E X E R C I S E 5 . 1
D
A Normal Good Has a Positive Income Elasticity of Demand
Problem
A consumer likes to attend
rock concerts and consume other goods. Suppose x
measures the number of rock concerts he attends each
year, and I denotes his annual income. Show that the following statement is true: If he views rock concerts as a
normal good, then his income elasticity of demand for
rock concerts must be positive.
where all prices are held constant. If rock concerts are
a normal good, then x increases as income I rises, so
(x ⲐI ) 0. Since income I and the number of rock
concerts attended x are positive, it must also be true that
(IⲐx) 0. Therefore, ⑀x,I 0.
Similar Problems: 5.3, 5.5
Solution In Chapter 2 we learned that the income
elasticity of demand is defined as ⑀x,I (x ⲐI )( IⲐx),
This exercise demonstrates a general proposition: If a good is normal, its income
elasticity of demand is positive. The converse is also true: If a good’s income elasticity of demand is positive, the good is a normal good.
Using similar reasoning, you can demonstrate that the following statements are
also true: (1) An inferior good has a negative income elasticity of demand. (2) A good
with a negative income elasticity of demand is an inferior good.
A P P L I C A T I O N
5.2
The Irish Potato Famine
During the early nineteenth century, Ireland’s population grew rapidly. Nearly half of the Irish people lived
on small farms that produced little income. Many others
who were unable to afford their own farms leased
land from owners of big estates. But these landlords
charged such high rents that leased farms also were
not profitable.
Because they were poor, many Irish people depended on potatoes as an inexpensive source of nourishment. In Why Ireland Starved, noted economic historian
Joel Mokyr described the increasing importance of the
potato in the Irish diet by the 1840s:
It is quite unmistakable that the Irish diet was undergoing changes in the first half of the nineteenth
century. Eighteenth-century diets, the evergrowing
importance of potatoes notwithstanding, seem to
have been supplemented by a variety of vegetables,
dairy products, and even pork and fish. . . . Although
glowing reports of the Irish cuisine in the eighteenth
century must be deemed unrepresentative since they
pertain to the shrinking class of well-to-do farmers,
things were clearly worsening in the nineteenth.
There was some across-the-board deterioration of
diets, due to the reduction of certain supplies, such
as dairy products, fish, and vegetables, but the
main reason was the relative decline of the number
of people who could afford to purchase decent
food. The dependency on the potato, while it cut
across all classes, was most absolute among the
lower two-thirds of the income distribution.10
Mokyr’s account suggests that the income consumption curve for a typical Irish consumer might
have looked like the one in Figure 5.4 (with potatoes
on the horizontal axis instead of hot dogs). For people
with a low income, potatoes might well have been a
normal good. But consumers with higher incomes
could afford other types of food, and therefore consumed fewer potatoes.
Given the heavy reliance on potatoes as food and
as a source of income, it is not surprising that a crisis
occurred between 1845 and 1847, when a plant
disease caused the potato crop to fail. During the
Irish potato famine, about 750,000 people died of
starvation or disease, and hundreds of thousands of
others emigrated from Ireland to escape poverty and
famine.
10
Joel Mokyr, Why Ireland Starved: A Quantitative and Analytical History of the Irish Economy, 1800–1850
(London: George Allen and Unwin, 1983), pp. 11 and 12.
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CHAPTER 5
T H E T H E O RY O F D E M A N D
THE EFFECTS OF A CHANGE IN PRICE OR INCOME:
A N A L G E B R A I C A P P R OAC H
So far in this chapter, we have used a graphical approach to show how the amount of
a good consumed depends on the levels of prices and income. We have shown how to
find the shape of the demand curve when the consumer has a given level of income (as
in Figure 5.1), and how the demand curve shifts as the level of income changes (as in
Figure 5.2).
We can also describe the demand curve algebraically. In other words, given a utility
function and a budget constraint, we can find the equation of the consumer’s demand
curve. The next two exercises illustrate this algebraic approach.
The solution to part (a) of this exercise starts out looking very much like the
solution to Learning-By-Doing Exercise 4.2, where we were interested in finding the
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 5 . 2
D
Finding a Demand Curve (No Corner Points)
A consumer purchases two goods, food and
clothing. The utility function is U(x, y) xy, where x denotes the amount of food consumed and y the amount of
clothing. The marginal utilities are MUx y and MUy
x. The price of food is Px, the price of clothing is Py, and
income is I.
Problem
• Since the optimum is interior, the tangency condition,
equation (4.3), must also hold: MUx ⲐMUy Px ⲐPy,
or, with the marginal utilities given, y Ⲑx Px ⲐPy, or
y (Px ⲐPy)x.
We can now solve for x by substituting y (Px ⲐPy)x
into the equation for the budget line Px x Py y I.
This gives us:
(a) Show that the equation for the demand curve for
food is x I Ⲑ(2Px).
(b) Is food a normal good? Draw D1, the consumer’s demand curve for food when the level of income is I
$120. Draw D2, the demand curve when I $200.
Solution
(a) In Learning-By-Doing Exercise 3.3, we learned that
the indifference curves for the utility function U(x, y)
xy are bowed in toward the origin and do not intersect
the axes. So any optimal basket must be interior; that is,
the consumer buys positive amounts of both food and
clothing.
How do we determine the optimal choice of food?
We know that an interior optimum must satisfy two conditions:
• An optimal basket will be on the budget line. This
means that equation (4.1) must hold: Px x Py y I.
Px x Py a
Px
xb I
Py
or x I Ⲑ(2Px ).
This is the equation of the demand curve for food.
Given the consumer’s income and the price of food, we
can easily find the quantity of food the consumer will
purchase.
(b) If income is $120, the equation of the demand curve
for food D1 will be x 120 Ⲑ(2Px) 60 ⲐPx . We can
plot points on the demand curve, as we have done in
Figure 5.5.
An increase in income to $200 shifts the demand
curve rightward to D2, with the equation x 200Ⲑ(2Px)
100 ⲐPx . Thus, food is a normal good.
Similar Problems: 5.6, 5.8
161
5.1 OPTIMAL CHOICE AND DEMAND
$20
Px, price of food
15
A
B
10
C
D2 (I = $200)
5
D1 (I = $120)
0
5
10
15
20
x, units of food
FIGURE 5.5
Demand Curves for Food at Different Income Levels
The quantity of food demanded, x, depends on the price of food, Px , and on the level of
income, I. The equation representing the demand for food is x I Ⲑ(2Px ). When income is
$120, the demand curve is D1 in the graph. Thus, if the price of food is $15, the consumer
buys 4 units of food (point A). If the price of food drops to $10, she buys 6 units of food
(point B ). If income rises to $200, the demand curve shifts to the right, to D2. In this case,
if the price of food is $10, the consumer buys 10 units of food (point C ).
optimal consumption of food and clothing given a specific set of prices and level of
income. Learning-By-Doing Exercise 5.2, however, goes further. By using the exogenous variables (Px, Py , and I ) instead of actual numbers, we find the equation of the
demand curve, which lets us determine the quantity of food demanded for any price
and income.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 5 . 3
D
Finding a Demand Curve (with a Corner Point Solution)
A consumer purchases two goods, food
and clothing. He has the utility function U(x, y) xy
10x, where x denotes the amount of food consumed
and y the amount of clothing. The marginal utilities are
MUx y 10 and MUy x. The consumer’s income
is $100, and the price of food is $1. The price of clothing is Py.
Problem
Show that the equation for the consumer’s
demand curve for clothing is
y
100 10Py
2Py
y 0,
when Py 6 10
,
when Py
10
Use this equation to fill in the following table to show
how much clothing he will purchase at each price of
clothing (these are points on his demand curve):
Py
y
2
4
5
10
12
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CHAPTER 5
Solution
T H E T H E O RY O F D E M A N D
In Learning-By-Doing Exercise 4.3, we
learned that the indifference curves for the utility function U(x, y) xy 10x are bowed in toward the origin.
They also intersect the x axis, since the consumer could
have a positive level of utility with purchases of food
(x 0) but no purchases of clothing ( y 0). So he
might not buy any clothing (i.e., choose a corner point)
if the price of clothing is too high.
How do we determine the consumer’s optimal
choice of clothing? If he is at an interior optimum, we
know that his optimal basket will be on the budget line.
This means that equation (4.1) must hold with the price
of x and income given: x Py y 100. At an interior optimum, the tangency condition as expressed in equation
(4.4) must also hold: MUx ⲐMUy Px ⲐPy, or with the
marginal utilities given, ( y 10)Ⲑx 1ⲐPy, or more simply,
x Py y 10Py.
We can now solve for y by substituting x Py y
10Py into the equation for the budget line x Py y 100.
5.2
CHANGE IN
THE PRICE OF
A GOOD:
SUBSTITUTION
EFFECT AND
INCOME
EFFECT
substitution effect
The change in the amount
of a good that would be
consumed as the price of
that good changes, holding
constant all other prices
and the level of utility.
income effect The
change in the amount of a
good that a consumer
would buy as purchasing
power changes, holding all
prices constant.
This gives us 2Py y 10Py 100, or y (100
10Py)Ⲑ(2Py). Note that the value of this equation for the
consumer’s demand curve for clothing is positive when
Py 10. But if Py 10, then 100 10Py is zero or negative, and the consumer will demand no clothing (in
effect, y 0 when Py
10, since the consumer can’t
demand negative amounts of clothing). In other words,
when Py 10 the consumer will be at a corner point at
which he buys only food.
Using the equation for the demand curve, we can
complete the table as follows:
Py
2
4
5
10
12
y
20
7.5
5
0
0
Similar Problems: 5.12, 5.16
I
n the previous section, we analyzed the overall effect of a change in the price of a
good. Here, we refine our analysis by breaking this effect down into two components—
a substitution effect and an income effect:
• When the price of a good falls, the good becomes cheaper relative to other goods.
Conversely, a rise in price makes the good more expensive relative to other goods.
In either case, the consumer experiences the substitution effect—the change in
the quantity of the good the consumer would purchase after the price change to
achieve the same level of utility. For example, if the price of food falls, the consumer can achieve the same level of utility by substituting food for other goods
(i.e., by buying more food and less of other goods); similarly, if the price of food
rises, the consumer may substitute other goods for food to achieve the same level
of utility.
• When the price of a good falls, the consumer’s purchasing power increases,
since the consumer can now buy the same basket of goods as before the price
decrease and still have money left over to buy more goods. Conversely, a rise
in price decreases the consumer’s purchasing power (i.e., the consumer can no
longer afford to buy the same basket of goods). This change in purchasing
power is termed the income effect because it affects the consumer in much
the same way as a change in income would; that is, the consumer realizes a
higher or lower level of utility because of the increase or decrease in purchasing power and therefore purchases a higher or lower amount of the good
whose price has changed. The income effect accounts for the part of the total
difference in the quantity of the good purchased that isn’t accounted for by the
substitution effect.
5.2 CHANGE IN THE PRICE OF A GOOD: SUBSTITUTION EFFECT AND INCOME
The substitution effect and the income effect occur at the same time when the
price of a good changes, resulting in an overall movement of the consumer from an
initial basket (before the price change) to a final basket (after the price change). To
better understand this overall effect of a price change, we will show how to break it
down (decompose it) into its two components—the substitution effect and the income effect.
In the following sections, we perform this analysis in relation to price decreases.
(Learning-By-Doing Exercise 5.5, on page 170, shows a corresponding analysis in
relation to a price increase.)
THE SUBSTITUTION EFFECT
Suppose that a consumer buys two goods, food and clothing, that both goods have a
positive marginal utility, and that the price of food decreases. The substitution effect
is the amount of additional food the consumer would buy to achieve the same level of
utility. Figure 5.6 shows three optimal choice diagrams that illustrate the steps involved in finding the substitution effect associated with this price change.
Step 1. Find the initial basket (the basket the consumer chooses at the initial price
Px1). As shown in Figure 5.6(a), when the price of food is Px1, the consumer
faces budget line BL1 and maximizes utility by choosing basket A on indifference curve U1. The quantity of food she purchases is xA.
Step 2. Find the final basket (the basket the consumer chooses after the price falls
to Px2). As shown in Figure 5.6(b), when the price of food falls to Px2, the
budget line rotates outward to BL2, and the consumer maximizes utility by
choosing basket C on indifference curve U2. The quantity of food she purchases is xC. Thus, the overall effect of the price change on the quantity of
food purchased is xC xA. Predictably, the consumer realizes a higher level
of utility as a result of the price decrease, as shown by the fact that the initial basket A lies inside the new budget line BL2.
Step 3. Find an intermediate decomposition basket that will enable us to identify the
portion of the change in quantity due to the substitution effect. We can
find this basket by keeping two things in mind. First, the decomposition
basket reflects the price decrease, so it must lie on a budget line that is
parallel to BL2. Second, the decomposition basket reflects the assumption
that the consumer achieves the initial level of utility after the price decrease, so the basket must be at the point where the budget line is tangent to
indifference curve U1. As shown in Figure 5.6(c), these two conditions are
fulfilled by basket B (the decomposition basket) on budget line BLd (the decomposition budget line). At basket B, the consumer purchases the quantity of
food xB. Thus, the substitution effect accounts for the consumer’s movement
from basket A to basket B—that is, the portion of the overall effect on the
quantity of food purchased that can be attributed to the substitution effect
is xB xA.
THE INCOME EFFECT
Still looking at Figure 5.6, suppose the consumer has income I. When the price of
food is Px1, she can buy any basket on BL1, and when the price of food is Px2, she can
buy any basket on BL2. Note that the decomposition budget line BLd lies inside BL2,
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CHAPTER 5
T H E T H E O RY O F D E M A N D
y, clothing
164
Step 1: Find the initial
basket A.
Px
Slope of BL1 = – 1
Py
A
U1
BL1
xA
(a)
x, food
y, clothing
Step 2: Find the final basket C.
Px
Slope of BL1 = – 1
Py
A
C
Slope of BL2 = –
BL2
Px
2
Py
U2
U1
BL1
xA
xC
(b)
x, food
Step 3: Find the decomposition
basket B.
Px
Slope of BL1 = – 1
Py
Px
Slope of BL2 = – 2
Py
Income and
Substitution Effects: Case 1 (x Is a
Normal Good)
As the price of food drops from Px1 to
Px2, the substitution effect leads to
an increase in the amount of food
consumed from xA to xB (so the substitution effect is xB xA). The income
effect also leads to an increase in food
consumption, from xB to xC (so the
income effect is xC xB). The overall
increase in food consumption is xC xA.
When a good is normal, the income
and substitution effects reinforce each
other.
y, clothing
FIGURE 5.6
A
BLd
C
Slope of BLd = –
B
U2
BL2
U1
BL1
xA
(c)
xB
xC
x, food
Px
2
Py
5.2 CHANGE IN THE PRICE OF A GOOD: SUBSTITUTION EFFECT AND INCOME
which means that the income Id that would be needed to buy a basket on BLd is less
than the income I needed to buy a basket on BL2. Also note that basket A (on BL1) and
basket B (on BLd) are on the same indifference curve U1 (i.e., the consumer would be
equally satisfied by baskets A and B), which means that the consumer would be indifferent between the following two situations: (1) having a higher income I when the
price of food is higher at Px1 (i.e., buying basket A) and (2) having a lower income Id
when the price of food is lower at Px2 (i.e., buying basket B). Another way of saying
this is that the consumer would be willing to have her income reduced to Id if she can
buy food at the lower price Px2.
With this in mind, let’s find the income effect, the change in the amount of a
good consumed as the consumer’s utility changes. In the example illustrated by
Figure 5.6, the movement from basket A to basket B (i.e., the movement due to the
substitution effect) doesn’t involve any change in utility, and as we have just seen,
we can view this movement as the result of a reduction in income from I to Id as the
price falls from Px1 to Px2. In reality, however, the consumer’s income doesn’t fall
when the price of food decreases, so her level of utility increases, and we account for
this by “restoring” the “lost” income. When we do this, the budget line shifts from
BLd to BL2, and the consumer’s optimal basket shifts from basket B (on BLd) to basket
C (on BL2). Thus, the income effect accounts for the consumer’s movement from
the decomposition basket B to the final basket C—that is, the portion of the overall
effect on the quantity of food purchased that can be attributed to the income effect
is xC xB.
In sum, when the price of food falls from Px1 to Px2, the total change on food consumption is (xC xA). This can be decomposed into the substitution effect (xB xA)
and the income effect (xC xB). When we add the substitution effect and the income
effect, we get the total change in consumption.
INCOME AND SUBSTITUTION EFFECTS
W H E N G O O D S A R E N OT N O R M A L
As we noted earlier, the graphs in Figure 5.6 are drawn for the case (we call it Case 1)
in which food is a normal good. As the price of food falls, the income effect leads to
an increase in food consumption. Also, because the marginal rate of substitution is
diminishing, the substitution effect leads to increased food consumption as well.
Thus, the income and substitution effects work in the same direction. The demand
curve for food will be downward sloping because the quantity of food purchased will
increase when the price of food falls. (Similarly, if the price of food were to rise, both
effects would be negative. At a higher price of food, the consumer would buy less food.)
However, the income and substitution effects do not always work in the same
direction. Consider Case 2, in Figure 5.7 (instead of drawing three graphs like those
in Figure 5.6, we have only drawn the final graph [like Figure 5.6(c)] with the initial,
final, and decomposition baskets). Note that basket C, the final basket, lies directly
above basket B, the decomposition basket. As the budget line shifts out from BLd to
BL2, the quantity of food consumed does not change. The income effect is therefore
zero (xC xB 0). Here a decrease in the price of food leads to a positive substitution effect on food consumption (xB xA 0) and a zero income effect. The demand
curve for food will still be downward sloping because more food is purchased at the
lower price (xC xA 0).
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T H E T H E O RY O F D E M A N D
Slope of BL1 = –
FIGURE 5.7
Income and
Substitution Effects: Case 2 (x Is Neither
a Normal Good nor an Inferior Good)
As the price of food drops from Px1 to Px2,
the substitution effect leads to an
increase in the amount of food consumed
from xA to xB (so the substitution effect is
xB xA). The income effect on food consumption is zero because xB is the same
as xC (so the income effect is xC xB 0).
The overall effect on food consumption is
xC xA.
y, clothing
Slope of BL2 = –
Slope of BLd = –
C
A
Px
1
Py
Px
2
Py
Px
2
Py
U2
B
BL2
U1
BL1
xA
xB = xC
BLd
x, food
The income and substitution effects might even work in opposite directions, as in
Case 3, in Figure 5.8, where food is an inferior good. When a good is inferior, the indifference curves will show that the income effect is negative (i.e., the final basket C
will be to the left of the decomposition basket B); as the budget line shifts out from
BLd to BL2, the quantity of food consumed decreases (xC xB 0). In contrast, the
substitution effect is still positive (xB xA 0). In this case, because the substitution
effect is larger than the income effect, the total change in the quantity of food consumed is also still positive (xC xA 0), and, therefore, the demand curve for food
will still be downward sloping.
Slope of BL1 = –
Slope of BL2 = –
Income and
Substitution Effects: Case 3 (x Is an
Inferior Good) with a Downward-Sloping
Demand Curve
As the price of food drops from Px1 to
Px2, the substitution effect leads to an
increase in the amount of food consumed
from xA to xB (so the substitution effect is
xB xA). The income effect on food consumption is negative (xC xB 0). The
overall effect on food consumption is
xC xA 0. When a good is inferior, the
income and substitution effects work in
opposite directions.
C
y, clothing
FIGURE 5.8
Slope of BLd = –
A
Px
Px
Px
U1
xA xC
xB
x, food
2
Py
B
BLd
2
Py
U2
BL1
1
Py
BL2
5.2 CHANGE IN THE PRICE OF A GOOD: SUBSTITUTION EFFECT AND INCOME
Slope of BL1 = –
Slope of BL2 = –
C
y, clothing
U2
Slope of BLd = –
A
Px
167
1
Py
Px
2
Py
Px
2
Py
FIGURE 5.9
B
U1
BL1
xC xA
xB
x, food
BLd
BL2
Income and
Substitution Effects: Case 4 (x Is a
Giffen Good)
As the price of food drops from Px1
to Px2, the substitution effect leads to
an increase in the amount of food
consumed from xA to xB (so the substitution effect is xB xA). The income
effect on food consumption is negative
(xC xB 0). The overall effect on
food consumption is xC xA 0.
Case 4, in Figure 5.9, illustrates the case of a so-called Giffen good. In this case,
the indifference curves indicate that food is a strongly inferior good, with the final
basket C lying not only to the left of the decomposition basket B, but also to the left
of the initial basket A. The income effect is so strongly negative that it more than cancels
out the positive substitution effect.
What about the demand curve for food in the case illustrated by Figure 5.9?
When the price of food drops from Px1 to Px2, the quantity of food actually decreases
from xA to xC, so the demand curve for food will be upward sloping over that range of
prices. A Giffen good has a demand curve with a positive slope over part of the curve.
As we have already noted, some goods are inferior over some price ranges for
some consumers. For instance, your consumption of hot dogs may fall if your income rises, because you decide to eat more steaks and fewer hot dogs. But expenditures on inferior goods typically represent only a small part of a consumer’s income.
Income effects for individual goods are usually not large, and the largest income
effects are usually associated with goods that are normal rather than inferior, such
as food and housing. For an inferior good to have an income effect large enough to
offset the substitution effect, the income elasticity of demand would have to be negative and the expenditures on the good would need to represent a large part of the
consumer’s budget. Thus, while the Giffen good is intriguing, it is not of much
practical concern.
Researchers have long searched to confirm the existence of a Giffen good for
human beings. Some economists have suggested that the Irish potato famine (see
Application 5.2) came close to creating the right environment. However, as Joel
Mokyr observed, “For people with a very low income, potatoes might have well been
a normal good. But consumers with higher levels of income could afford other types
of food, and therefore consumed fewer potatoes.” Thus, while expenditures on potatoes did constitute a large part of consumer expenditures, potatoes may not have been
inferior at low incomes. This may explain why researchers have not shown the potato
to have been a Giffen good at that time.
Giffen good A good so
strongly inferior that the income effect outweighs the
substitution effect, resulting
in an upward-sloping
demand curve over some
region of prices.
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A P P L I C A T I O N
T H E T H E O RY O F D E M A N D
5.3
Rats Respond When Prices Change!
In Chapter 2 we cited studies showing that people
have negatively sloped demand curves for goods and
services and that many goods are adequate substitutes for one another. In the early 1980s several economists conducted experiments designed to show how
rats would respond to changes in relative prices. In
one famous experiment, white rats were offered root
beer and collins mix in different containers. To extract
a unit of the beverage, a rat had to “pay a price” by
pushing a lever a certain number of times. The researchers allowed the rat a specified number of
pushes per day. This was the rat’s income.
Each rat was then able to choose its initial basket
of the beverages. Then the experimenters altered the
relative prices of the beverages by changing the number
of times the rat needed to push the lever to extract a
unit of each beverage. The rat’s income was adjusted
so that it would allow a rat to consume its initial basket.
The researchers found that the rats altered their consumption patterns to choose more of the beverage
with the lower relative price. The choices the rats
made indicated that they were willing to substitute
one beverage for the other when the relative prices of
the beverages changed.
S
E
In another experiment, rats were offered a similar set of choices between food and water. When
relative prices were changed, the rats were willing to
engage in some limited substitution toward the good
with the lower relative price. But the cross-price elasticities of demand were much lower in this experiment
because food and water are not good substitutes for
one another.
In a third study, researchers designed an experiment to see if they could confirm the existence of a
Giffen good for rats. When the rats were offered a
choice between quinine water and root beer, researchers discovered that quinine water was an inferior
good. They reduced the rats’ incomes to low levels
and set prices so that the rats spent most of their
budget on quinine water. This was the right environment for the potential discovery of a Giffen good.
Theory predicts that we are most likely to observe a
Giffen good when an inferior good (quinine water)
also comprises a large part of a consumer’s expenditures. When researchers lowered the price of quinine
water, they found that the rats did in fact extract less
quinine water, using their increased wealth to choose
more root beer. The researchers concluded that for
rats, quinine water was a Giffen good.11
L E A R N I N G - B Y- D O I N G E X E R C I S E 5 . 4
D
Finding Income and Substitution Effects Algebraically
In Learning-By-Doing Exercises 4.2 and
5.2, we met a consumer who purchases two goods, food
and clothing. He has the utility function U(x, y) xy,
where x denotes the amount of food consumed and y the
amount of clothing. His marginal utilities are MUx y
and MUy x. Now suppose that he has an income of
$72 per week and that the price of clothing is Py $1
per unit. Suppose that the price of food is initially
Px1 $9 per unit and that the price subsequently falls to
Px2 $4 per unit.
11
Problem Find the numerical values of the income
and substitution effects on food consumption, and graph
the results.
Solution To find the income and substitution
effects, we follow the procedure explained earlier in this
section on pages 163–165.
Step 1. Find the initial consumption basket A when the
price of food is $9. We know that two conditions must be
satisfied at an optimum. First, an optimal basket will be
See J. Kagel, R. Battalio, H. Rachlin, L. Green, R. Basmann, and W. Klemm, “Experimental Studies
of Consumer Demand Behavior,” Economic Inquiry (March 1975): 22–38; and J. Kagel, R. Battalio, H.
Rachlin, and L. Green, “Demand Curves for Animal Consumers,” Quarterly Journal of Economics
(February 1981): 1–16; and R. Battalio, J. Kagel, and C. Kogut, “Experimental Confirmation of the
Existence of a Giffen Good,” American Economic Review (September 1991): 961–970.
5.2 CHANGE IN THE PRICE OF A GOOD: SUBSTITUTION EFFECT AND INCOME
on the budget line. This means that Px x Py y I, or
with the given information, 9x y 72.
Second, since the optimum is interior, the tangency
condition must hold. From equation (4.3), we know that
at a tangency, MUx ⲐMUy Px ⲐPy, which, with the given
information, simplifies to y 9x.
When we solve these two equations with two unknowns, we find that x 4 and y 36. So at basket A
the consumer purchases 4 units of food and 36 units of
clothing each week.
Step 2. Find the final consumption basket C when the
price of food is $4. We repeat step 1, but now with the
price of a unit of food of $4, which again yields two
equations with two unknowns:
4x y 72 (coming from the budget line)
y 4x (coming from the tangency condition)
When we solve these two equations, we find that x 9
and y 36. So at basket C, the consumer purchases
9 units of food and 36 units of clothing each week.
Step 3. Find the decomposition basket B. The decomposition basket must satisfy two conditions. First, it must lie
on the original indifference curve U1 along with basket A.
Recall that this consumer’s utility function is U(x, y) xy,
so at basket A, utility U1 4(36) 144. At basket B the
amounts of food and clothing must also satisfy xy 144.
Second, the decomposition basket must be at the point
169
where the decomposition budget line is tangent to the
indifference curve. Remember that the price of food Px
on the decomposition budget line is the final price of $4.
The tangency occurs when MUx ⲐMUy Px ⲐPy, that is,
when yⲐx 4 Ⲑ1, or y 4x. When we solve the two equations xy 144 and y 4x, we find that, at the decomposition basket, x 6 units of food and y 24 units of
clothing.
Now we can find the income and substitution effects.
The substitution effect is the increase in food purchased
as the consumer moves along initial indifference curve
U1 from basket A (at which he purchases 4 units of food)
to basket B (at which he purchases 6 units of food). The
substitution effect is therefore 6 4 2 units of food.
The income effect is the increase in food purchased as
he moves from basket B (at which he purchases 6 units of
food) to basket C (at which he purchases 9 units of food).
The income effect is therefore 9 6 3 units of food.
Figure 5.10 graphs the income and substitution
effects. In this exercise food is a normal good. As expected,
the income and substitution effects have the same sign.
The consumer’s demand curve for food is downward
sloping because the quantity of food he purchases increases when the price of food falls.
Similar Problem:
5.20
Slope of BL1 = –9
Slope of BL2 = –4
y, clothing
Slope of BLd = –4
A
36
C
B
24
U2 = 324
U1 = 144
BL1
4
6
9
BLd
BL2
x, food
FIGURE 5.10 Income and
Substitution Effects
As the price of food drops from $9 to
$4, the substitution effect leads to an
increase in food consumption from 4
(at the initial basket A) to 6 (at the
decomposition basket B). The substitution effect is therefore 6 4 2.
The income effect is the change in
food consumption as the consumer
moves from the decomposition basket B (where 6 units of food are purchased) to the final basket C (where 9
units of food are bought). The income effect is therefore 9 6 3.
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A P P L I C A T I O N
T H E T H E O RY O F D E M A N D
5.4
Have Economists Finally Found
a Giffen Good? Rice and Noodles
in China
For over 100 years, economists have sought convincing evidence of the existence of a Giffen good. The
search should focus on inferior goods on which consumers spend a large portion of their income. As
already noted, perhaps the reason researchers have
not been able to conclude that potatoes in Ireland in
the late 1840s were a Giffen good is that potatoes
may not have been inferior for the low-income consumers who spent the largest portion of their income
on potatoes.
Economist David McKenzie analyzed the demand
for tortillas in Mexico from 1994 to 1996, a period
when tortilla prices increased dramatically while
average income fell.12 McKenzie noted that poor consumers often spent much of their income on tortillas.
But he found tortillas to be a normal good for consumers with very low levels of income; thus, tortillas
were not a Giffen good for these consumers. He did
find that tortillas were an inferior good for those
with higher incomes; the Engel curve for tortillas
thus resembled the one shown in Figure 5.4(b). But
he was still unable to conclude that tortillas were
a Giffen good, even for consumers with higher
incomes.
A recent study by two economists claims to have
found the first evidence for Giffen goods. Robert
Jensen and Nolan Miller conducted a field study in
the Chinese provinces of Hunan and Gansu in 2006.13
In Hunan, rice is the staple food in people’s diets,
while in Gansu wheat (eaten as bread or noodles)
is the staple. Jensen and Miller randomly selected
households, which were given vouchers to subsidize
the price of rice or wheat flour for five months. Data
were collected from households that received vouchers as well as those that did not. The researchers suggested that rice appears to be a Giffen good for some
consumers in Hunan, with weaker evidence that
wheat flour is a Giffen good in Gansu. Like tortillas,
rice and wheat flour may be normal goods at some
levels of income and inferior at others. Jensen and
Miller point out that many attempts to find Giffen
goods use aggregate consumption and price data
that may not separate subsets of consumers with inferior demands. The use of less aggregated data may
help in finding Giffen goods.
To this point, all our discussions and examples of the substitution and income effects have been in relation to price decreases. Learning-By-Doing Exercise 5.5 shows
how these effects work with a price increase.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 5 . 5
D
Income and Substitution Effects with a Price Increase
The indifference curves in Figure 5.11
depict a consumer’s preferences for housing x and a
composite good y. The consumer’s marginal utilities for
both goods are positive.
Problem On the graph, show what the income and
substitution effects on housing would be if the current
12
price of housing were to increase so that the consumer’s
budget line shifted from BL1 to BL2.
Solution
At the initial price of housing, the consumer’s budget line is BL1 and the consumer’s optimal
basket is A. This enables the consumer to reach indifference curve U1. When the price of housing increases, the
This example draws from David McKenzie, “Are Tortillas a Giffen Good in Mexico?” Economics Bulletin
15, no. 1, (2002): 1–7.
13
Robert Jensen and Nolan Miller, “Giffen Behavior: Theory and Evidence,” National Bureau of Economic
Research, Working Paper, July 2007.
5.2 CHANGE IN THE PRICE OF A GOOD: SUBSTITUTION EFFECT AND INCOME
171
to the decomposition basket B, housing consumption
decreases from xA to xB. The substitution effect is therefore xB xA. The income effect is measured by the
change in housing consumption as the consumer moves
from the decomposition basket B to the final basket C.
The income effect is therefore xC xB.
consumer’s budget line is BL2. The consumer purchases
basket C and reaches the indifference curve U2.
To draw the decomposition budget line BLd, remember that BLd is parallel to the final budget line BL2
and that the decomposition basket B is located where
BLd is tangent to the initial indifference curve U1.
(Students often err by placing the decomposition basket
on the final indifference curve instead of on the initial
indifference curve.) As we move from the initial basket A
Similar Problems:
5.9, 5.21, 5.33
B
y, other goods
C
A
U1
FIGURE 5.11
U2
BL2
xC xB
S
E
xA
BLd
BL1
x, housing
Income and
Substitution Effects with a Price
Increase
At the initial basket A on budget line
BL1, the consumer purchases xA units of
food. At the final basket C on budget
line BL2, the consumer purchases xC
units of food. At the decomposition
basket B on budget line BLd, the consumer purchases xB units of food. The
substitution effect is xB xA. The income
effect is xC xB.
L E A R N I N G - B Y- D O I N G E X E R C I S E 5 . 6
D
Income and Substitution Effects with a Quasilinear Utility Function
A college student who loves chocolate has
a budget of $10 per day, and out of that income she purchases chocolate x and a composite good y. The price of
the composite good is $1.
The quasilinear utility function U(x, y) 21x y
represents the student’s preferences. (See Chapter 3 for
discussion of this kind of utility function.) For this utility function, MUx 1 1x and MUy 1.
Problem
(a) Suppose the price of chocolate is initially $0.50 per
ounce. How many ounces of chocolate and how many
units of the composite good are in the student’s optimal
consumption basket?
(b) Suppose the price of chocolate drops to $0.20 per ounce.
How many ounces of chocolate and how many units of the
composite good are in the optimal consumption basket?
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T H E T H E O RY O F D E M A N D
16
14
Utility function U(x, y) = 2 x + y
y, other goods
12
FIGURE 5.12 Income and Substitution
Effects with a Quasilinear Utility Function
At the initial basket A on budget line BL1, the
consumer purchases 4 ounces of chocolate at
a price of $0.50 per ounce. At the final basket
C on budget line BL2, the consumer purchases
25 ounces of chocolate at a price of $0.20 per
ounce. At the decomposition basket B on
budget line BLd, the consumer also purchases
25 ounces of chocolate at a price of $0.20 per
ounce. The substitution effect is 25 4
21 ounces. The income effect is 25 25
0 ounces.
10
8
A
U2 = 15
U1 = 12
6
C
4
BL2
2
B
BL1
0
(c) What are the substitution and income effects that
result from the decline in the price of chocolate? Illustrate
these effects on a graph.
Solution
(a) At an interior optimum, MUx ⲐMUy Px ⲐPy, or
1 Ⲑ 1x ⫽ Px. The student’s demand curve for chocolate is
therefore x 1Ⲑ(Px)2. When the price of chocolate is $0.50
per ounce, she buys 1Ⲑ(0.5)2 4 ounces of chocolate per day.
We can find the number of units of the composite
good from the equation for the budget line, Px x Py y
I. With the information given, the budget line equation
is (0.5)(4) (1)y 10, so the student buys y 8 units of
the composite good.
(b) We use the consumer’s demand curve for chocolate
from part (a) to find her demand for chocolate when the
price falls to $0.20 per ounce. She buys x 1Ⲑ(0.2)2
25 ounces of chocolate at the lower price. Her budget
line equation now becomes (0.2)(25) (1)y 10, so she
buys y 5 units of the composite good.
(c) In the first two parts of this problem we found all we
need to know about the initial basket A and the final basket C. Figure 5.12 shows these baskets.
5
10
15
20
BLd
25
30
35
x, ounces of chocolate
To find the income and substitution effects, we need to
find the decomposition basket B. We know two things
about basket B. First, the consumer’s utility at basket B
must be the same as at the initial basket A, where x ⫽ 4,
y ⫽ 8, and, therefore, utility is U1 214 8 12.
Thus, at basket B, 21x y 12. Second, the slope of
the decomposition budget line at basket B must be the
same as the slope of the final budget line at basket C—
that is, MUx ⲐMUy ⫽ Px ⲐPy. Given that MUx 1/ 1x,
that MUy ⫽ 1, and that, at basket C, Px ⫽ 0.20 and Py ⫽ 1,
this equation simplifies to 1/ 1x 0.20. When we solve
these two equation with two unknowns, we find that at
basket B, x ⫽ 25 and y ⫽ 2. Basket B is also shown on
Figure 5.12.
The substitution effect is the change in the quantity
of chocolate purchased as the consumer moves from the
initial basket A (where she consumes 4 ounces of chocolate) to the decomposition basket B (where she consumes
25 ounces of chocolate). The substitution effect on
chocolate is therefore 25 4 21 ounces. The income
effect is the change in the quantity of chocolate purchased
as the consumer moves from the decomposition basket B
to the final basket C. Because she consumes the same
amount of chocolate at B and C, the income effect is zero.
173
5.3 CHANGE IN THE PRICE OF A GOOD: THE CONCEPT OF CONSUMER SURPLUS
Learning-By-Doing Exercise 5.6 illustrates one of the properties of a quasilinear
utility function with a constant marginal utility of y and indifference curves that are
bowed in toward the origin. When prices are constant, at an interior optimum the consumer will purchase the same amount of x as income varies. In other words, the income
consumption curve will be a vertical line in the graph, and the income effect associated
with a price change on x will be zero, as in Figure 5.7.
C
onsumer surplus is the difference between the maximum amount a consumer is
willing to pay for a good and the amount he must actually pay to purchase the good in
the marketplace. Thus, it measures how much better off the consumer will be when he
purchases the good and can, therefore, be a useful tool for representing the impact of
a price change on consumer well-being. In this section, we will view this impact from
two different perspectives: first, by looking at the demand curve, and second, by looking at the optimal choice diagram.
U N D E R S TA N D I N G C O N S U M E R S U R P L U S
F R O M T H E D E M A N D C U RV E
In the previous section, we saw how changes in price affect consumer decision making
and utility in cases where we know the utility function. If we do not know the utility
function, but do know the equation for the demand curve, we can use the concept of
consumer surplus to measure the impact of a price change on the consumer.
Let’s begin with an example. Suppose you are considering buying a particular automobile and that you are willing to pay up to $15,000 for it. But you can buy that automobile for $12,000 in the marketplace. Because the amount you are willing to pay
exceeds the amount you actually have to pay, you will buy it. When you do, you will
have a consumer surplus of $3,000 from that purchase. Your consumer surplus is your
net economic benefit from making the purchase, that is, the maximum amount you
would be willing to pay ($15,000) less the amount you actually pay ($12,000).
Of course, for many types of commodities you might want to consume more than
one unit. You will have a demand curve for such a commodity, which, as we have already
pointed out, represents your willingness to pay for the good. For example, suppose
that you like to play tennis and that you must rent the tennis court for an hour each
time you play. Your demand curve for court time appears in Figure 5.13. It shows that
you would be willing to pay up to $25 for the first hour of court time each month, $23
for the second hour, $21 for the third hour, and so on. Your demand curve is downward sloping because you have a diminishing marginal utility for playing tennis.
Suppose you must pay $10 per hour to rent the court. At that price your demand
curve indicates that you will play tennis for 8 hours during the month because you are
willing to pay $11 for the eighth hour, but only $9 for the ninth hour, and even less
for additional hours.
How much consumer surplus do you get from playing tennis 8 hours each
month? To find out, you add the surpluses from each of the units you consume. Your
consumer surplus from the first hour is $15 (the $25 you are willing to pay minus the
$10 you actually must pay). The consumer surplus from the second hour is $13. The
consumer surplus from using the court for the 8 hours during the month is then $64
(the sum of the consumer surpluses for each of the 8 hours, or $15 $13 $11 $9
$7 $5 $3 $1).
5.3
CHANGE IN
THE PRICE OF
A GOOD: THE
CONCEPT OF
CONSUMER
SURPLUS
consumer surplus
The
difference between the
maximum amount a consumer is willing to pay for
a good and the amount he
or she must actually pay
when purchasing it.
FIGURE 5.13
Consumer Surplus and
the Demand Curve
The dark-shaded area under the demand
curve, but above the $10 per hour price
the consumer must pay, indicates the consumer surplus for each additional hour of
court time. The consumer will receive a
consumer surplus of $64 from purchasing
8 hours of court time.
T H E T H E O RY O F D E M A N D
$25
time (dollars per hour)
CHAPTER 5
P, price consumer is willing to pay for court
174
Consumer surplus =
23
21
19
Actual price of court time = $10/hour
17
15
13
10
11
9
7
Demand curve
1
2
3 4
5 6 7 8 9 10 11 12 13
Q, hours of court time
As the example illustrates, the consumer surplus is the area below the demand
curve and above the price that the consumer must pay for the good. We represented
the demand curve here as a series of “steps” to help us illustrate the consumer surplus
from each unit purchased. In reality, however, a demand curve will usually be smooth
and can be represented as an algebraic equation. The concept of consumer surplus is
the same for a smooth demand curve.
As we shall show, the area under a demand curve exactly measures net benefits for
a consumer only if the consumer experiences no income effect over the range of price
change. This may often be a reasonable assumption, but if it is not satisfied, then the
area under the demand curve will not measure the consumer’s net benefits exactly. For
the moment, let’s assume that there is no income effect, so we need not worry about
this complication.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 5 . 7
D
Consumer Surplus: Looking at the Demand Curve
Suppose the equation Q 40 4P represents a consumer’s monthly demand curve
for milk, where Q is the number of gallons of milk purchased when the price is P dollars per gallon.
Problem
(a) What is the consumer surplus per month if the price
of milk is $3 per gallon?
(b) What is the increase in consumer surplus if the price
falls to $2 per gallon?
Solution
(a) Figure 5.14 shows the demand curve for milk. When
the price is $3, the consumer will buy 28 gallons of milk.
The consumer surplus is the area under the demand
curve and above the price of $3—that is, the area of triangle G, or (1Ⲑ2)(10 3)(28) $98.
(b) If the price drops from $3 to $2, the consumer will
buy 32 gallons of milk. Consumer surplus will increase
by the areas H ($28) and I ($2), or by $30. The total consumer surplus will now be $128 (G H I ).
Similar Problems: 5.18, 5.19
5.3 CHANGE IN THE PRICE OF A GOOD: THE CONCEPT OF CONSUMER SURPLUS
175
$10
P, price of milk
DMilk: Q = 40 – 4P
G
3
2
H
I
DMilk
28
Q, gallons of milk
32
40
FIGURE 5.14 Consumer
Surplus and the Demand Curve
When the price of milk is $3 per
gallon, consumer surplus area
of triangle G $98. If the price
drops to $2 per gallon, the increase
in consumer surplus sum of
areas H ($28) and I ($2) $30.
Total consumer surplus when
the price is $2 per gallon $98
$30 $128.
U N D E R S TA N D I N G C O N S U M E R S U R P L U S F R O M T H E
O P T I M A L C H O I C E D I AG R A M : C O M P E N S AT I N G
VA R I AT I O N A N D E Q U I VA L E N T VA R I AT I O N
We have shown how a price change affects the level of utility for a consumer.
However, there is no natural measure for the units of utility. Economists therefore
often measure the impact of a price change on a consumer’s well-being in monetary
terms. How can we estimate the monetary value that a consumer would assign to a
change in the price of a good? In this section, we use optimal choice diagrams to study
two equally valid ways of answering this question:
• First, we see how much income the consumer would be willing to give up after a
price reduction, or how much additional income the consumer would need after
a price increase, to maintain the level of utility she had before the price change.
We call this change in income the compensating variation (because it is the
change in income that would exactly compensate the consumer for the impact
of the price change). The compensating variation for a price reduction is
positive because the price reduction makes the consumer better off. For a price
increase, the compensating variation is negative because the price increase makes
the consumer worse off.
• Second, we see how much additional income the consumer would need before a
price reduction, or how much less income the consumer would need before a
price increase, to give the consumer the level of utility she would have after the
price change. We call this change in income the equivalent variation (because
it is the change in income that would be equivalent to the price change in its
impact on the consumer). The equivalent variation for a price reduction is
positive because the price reduction makes the consumer better off. For a price
increase, the equivalent variation is negative because the price increase makes
the consumer worse off.
compensating variation
A measure of how much
money a consumer would
be willing to give up after
a reduction in the price of
a good to be just as well
off as before the price
decrease.
equivalent variation
A measure of how much
additional money a consumer would need before
a price reduction to be as
well off as after the price
decrease.
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T H E T H E O RY O F D E M A N D
Slope of BL1 = slope of J E = – Px
1
Slope of BL2 = slope of LB = – Px
FIGURE 5.15
y, clothing
J
Compensating and
Equivalent Variations with a Positive
Income Effect
The price change from Px1 to Px2 has a
positive income effect, so the compensating variation (the length of the
segment KL) and the equivalent
variation (the length of the segment JK )
are not equal. In this case, JK KL.
K
L
E
A
C
U2
B
U1
BL2
BL1
O
2
x, food
The optimal choice diagram shown in Figure 5.15 illustrates a case where the
consumer buys two goods, food x and clothing y. The price of clothing is $1. The
price of food is initially Px1 and then decreases to Px2. With the consumer’s income remaining fixed, the budget line moves from BL1 to BL2 and the consumer’s optimal
basket moves from A to C.
The compensating variation is the difference between the income necessary to
buy basket A at the initial price Px1 and the income necessary to buy the decomposition basket B at the new price Px2. Basket B lies at the point where a line parallel to the
final budget line BL2 is tangent to the initial indifference curve U1.
The equivalent variation is the difference between the income necessary to buy
basket A at the initial price Px1 and the income necessary to buy basket E at the initial
price Px1. Basket E lies at the point where a line parallel to the initial budget line BL1
is tangent to the final indifference curve U2.
In graphical terms, the compensating and equivalent variations are simply two
different ways of measuring the distance between the initial and final indifference
curves. Since the price of clothing y is $1, the segment OK measures the consumer’s
income. The segment OL measures the income needed to buy basket B at the new
price of food Px2. The difference (the segment KL) is the compensating variation.
Baskets B and A are on the same indifference curve U1, so the consumer would accept
a reduction in income of KL if she could buy food at the lower price.
To find the equivalent variation, note that, as before, the segment OK measures
the consumer’s income because Py $1. The segment OJ measures the income
needed to buy basket E at the old price of food Px1. The difference (the segment KJ )
is the equivalent variation. Baskets E and C are on the same indifference curve, so the
consumer would require an increase in income of KJ to be equally well off buying
food at the initial higher price as at the lower final price.
In general, the sizes of the compensating variation (the segment KL) and the
equivalent variation (the segment KJ ) will not be the same because the price change
5.3 CHANGE IN THE PRICE OF A GOOD: THE CONCEPT OF CONSUMER SURPLUS
177
Slope of BL1 = slope of J E = – Px
1
Slope of BL2 = slope of LB = – Px
y, clothing
J
2
E
K
L
A
C
B
BL1
O
U2
U1
x, food
BL2
FIGURE 5.16 Compensating and
Equivalent Variations with No Income
Effect (Utility Function Is Quasilinear)
The utility function is quasilinear, so
indifference curves U1 and U2 are parallel,
and there is no income effect (C lies
directly above B, and E lies directly
above A). The compensating variation
(KL) and equivalent variation (JK) are
equal.
would have a nonzero income effect (in Figure 5.15, C lies to the right of B, so the income effect is positive). That is why one must be careful when trying to measure the
monetary value that a consumer associates with a price change.
As illustrated in Figure 5.16, however, if the utility function is quasilinear, the
compensating and equivalent variations will be the same because the price change
would have a zero income effect (as we saw in Learning-By-Doing Exercise 5.6).
Graphically, this is represented by the fact that the indifference curves associated with
a quasilinear utility function are parallel, which means that the vertical distance between
any two curves is the same at all values of x.14 Thus, in Figure 5.16, where basket C
lies directly above basket B, and basket E lies directly above basket A, the vertical distance CB is equal to the vertical distance EA. Now note that the compensating variation in this figure is represented by the length of the line segment JK (which is equal
to EA), and the equivalent variation is represented by the length of the line segment
KL (which is equal to CB). If JK EA and KL CB and EA CB, then JK KL—
that is, the compensating variation and the equivalent variation must be equal.
Furthermore, if there is no income effect, not only are the compensating variation and
the equivalent variation equal to each other, they are also equal to the change in the consumer
surplus (the change in the area under the demand curve as a result of the price change). This
important point is illustrated by Learning-By-Doing Exercise 5.8 and the discussion
following that exercise.
14
Suppose the utility function U(x, y) is quasilinear, so that U(x, y) f(x) ky, where k is some positive
constant. Since U always increases by k units whenever y increases by 1 unit, we know that MUy k.
Therefore, the marginal utility of y is constant. For any given level of x, U ky. So the vertical distance between indifference curves will be y2 y1 (U2 U1) Ⲑk. Note that this vertical distance between
indifference curves is the same for all values of x. That is why the indifference curves are parallel.
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D
Compensating and Equivalent Variations with No Income Effect
As in Learning-By-Doing Exercise 5.6, a
student consumes chocolate and “other goods” with the
quasilinear utility function U(x, y) 21x ⫹ y. She has
an income of $10 per day, and the price of the composite good y is $1 per unit. For this utility function,
MUx 1/ 1x and MUy 1. Suppose the price of
chocolate is $0.50 per ounce and that it then falls to
$0.20 per ounce.
Problem
(a) What is the compensating variation of the reduction
in the price of chocolate?
(b) What is the equivalent variation of the reduction in
the price of chocolate?
Solution
(a) Consider the optimal choice diagram in Figure 5.17.
The compensating variation is the difference between
her income ($10) and the income she would need to purchase the decomposition basket B at the new price of
chocolate of $0.20. At basket B she buys 25 units of
chocolate and 2 units of the composite good, so she would
need Px x ⫹ Py y ⫽ ($0.20)(25) ⫹ ($1)(2) ⫽ $7. She would
be willing to have her income reduced from $10 to $7
16
Utility function U(x, y) = 2 x + y
Slope of line through E = slope of
BL1 = – 0.50
Slope of line through B = slope of
BL2 = – 0.20
14
12
E
y, other goods
10
8
A
7
6
C
U2 = 15
4
BL2
2
B
U1 = 12
BL1
0
5
10
15
20
25
30
35
x, ounces of chocolate
FIGURE 5.17 Compensating and Equivalent Variations with No Income Effect
The consumer’s income is $10, and the price of the composite good y is $1 per unit. When
the price of chocolate is $0.50 per ounce, the consumer’s budget line is BL1 and she buys
basket A, with utility U1. After the price of chocolate falls to $0.20 per ounce, her budget
line is BL2 and she buys basket C, with utility U2. To reach utility U1 after the price decrease, she could buy basket B for $7, so her compensating variation is $10 ⫺ $7 ⫽ $3. To
reach utility U2 before the price decrease, she could buy basket E for $13, so her equivalent
variation is $13 ⫺ $10 ⫽ $3. When there is no income effect (as here, because the utility
function is quasilinear), the compensating variation and the equivalent variation are equal.
5.3 CHANGE IN THE PRICE OF A GOOD: THE CONCEPT OF CONSUMER SURPLUS
(a change of $3) if the price of chocolate falls from $0.50
to $0.20 per ounce. Thus, the compensating variation
equals $3.
(b) In Figure 5.17, the equivalent variation is the difference
between the income she would need to buy basket E at the
initial price of $0.50 per ounce of chocolate and her actual
income ($10). To find the equivalent variation, we need to
determine the location of basket E. We know that basket E
lies on the final indifference curve U2, which has a value of
15. Therefore, at basket E, 21x y 15. We also know
that at basket E the slope of the final indifference curve
U2 (MUx /MUy ) must equal the slope of the initial
179
budget line BL1 (Px ⲐPy), or (1/ 2x)/1 0.5/1, which
reduces to x 4. When we substitute this value of x into
the equation 21x y 15, we find that y 11. Thus, at
basket E the consumer purchases 4 units of chocolate and
11 units of the composite good. To purchase basket E at the
initial price of $0.50 per ounce of chocolate, she would
need an income of Px x Py y $0.50(4) $1(11) $13.
The equivalent variation is the difference between this
amount ($13) and her income ($10), or $3. Thus, the
equivalent variation and the compensating variation are
equal.
Similar Problem: 5.27
Still considering the consumer in Learning-By-Doing Exercise 5.8, let’s see what
happens if we try to measure the change in the consumer surplus by looking at the
change in the area under her demand curve for chocolate. In Learning-By-Doing
Exercise 5.6, we showed that her demand function for chocolate is x 1Ⲑ(Px)2. Figure 5.18
shows the demand curve for chocolate. As the price of chocolate falls from $0.50 per
ounce to $0.20 per ounce, her daily consumption of chocolate rises from 4 ounces to
25 ounces. The shaded area in the figure illustrates the increase in consumer surplus
as the price of chocolate falls. The size of that shaded area is $3, exactly the same as
both the compensating and equivalent variations. Thus, the change in the area under
the demand curve exactly measures the monetary value of a price change when the
utility function is quasilinear (i.e., when there is no income effect).
As we have already noted, if there is an income effect, the compensating variation and
equivalent variation will give us different measures of the monetary value that a consumer
would assign to the reduction in price of the good. Moreover, each of these measures will
generally be different from the change in the area under the demand curve. However, if
the income effect is small, the equivalent and compensating variations may be close to
one another, and then the area under the demand curve will be a good approximation
(though not an exact measure) of the compensating and equivalent variations.
Px, price of chocolate (dollars per ounce)
1.0
Demand, x =
0.8
1
(Px ) 2
0.6
0.5
0.4
0.2
0
4 5
10
15
20
x, ounces of chocolate per day
25
30
FIGURE 5.18 Consumer Surplus
with No Income Effect
When the price of chocolate falls
from $0.50 per ounce to $0.20 per
ounce, the consumer increases
consumption from 4 ounces to
25 ounces per day. Her consumer
surplus increases by the shaded area,
or $3 per day.
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D
Compensating and Equivalent Variations with an Income Effect
As in Learning-By-Doing Exercise 5.4, a
consumer purchases two goods, food x and clothing y.
He has the utility function U(x, y) xy. He has an income of $72 per week, and the price of clothing is $1 per
unit. His marginal utilities are MUx y and MUy x.
Suppose the price of food falls from $9 to $4 per unit.
Problem
(a) What is the compensating variation of the reduction
in the price of food?
(b) What is the equivalent variation of the reduction in
the price of food?
Solution
(a) Consider the optimal choice diagram in Figure 5.19.
The compensating variation is the difference between
his income ($72) and the income he would need to purchase the decomposition basket B at the new price of
food of $4. At basket B he buys 6 units of food and
24 units of clothing, so he would need Px x Py y
$4(6) $1(24) $48. The consumer would be willing
to have his income reduced from $72 to $48 (a change of
$24) if the price of food falls from $9 to $4. Therefore,
the compensating variation associated with the price
reduction is $24.
(b) In Figure 5.19, the equivalent variation is the difference between the income he would need to buy basket E
at the initial price of $9 per unit of food and his actual
income ($72). To find the equivalent variation, we need
to determine the location of basket E. We know that
basket E lies on the final indifference curve U2, which
has a value of 324. Therefore, at basket E, xy 324. We
also know that at basket E the slope of the final indifference curve U2 (MUx ⲐMUy) must equal the slope of the
initial budget line BL1 (Px ⲐPy), or y/x 9Ⲑ1, which
reduces to y 9x. When we solve these two equations
with two unknowns, we find that x 6 and y 54.
Thus, at basket E the consumer purchases 6 units of
food and 54 units of clothing. To purchase basket E at
the initial price of $9 per unit of food, he would need
income equal to Px x Py y $9(6) $1(54) $108.
The equivalent variation is the difference between this
amount ($108) and his income ($72), or $36. Thus, the
equivalent variation ($36) and the compensating variation ($24) are not equal.
Similar Problems: 5.20, 5.21, 5.32, 5.33
72
BL2
E
54
y, clothing
FIGURE 5.19 Compensating and
Equivalent Variation with an Income Effect
The consumer’s income is $72, and the
price of the clothing y is $1 per unit. When
the price of food is $9 per unit, the consumer’s budget line is BL1, and he buys
basket A, with utility U1. After the price of
food falls to $4 per unit, his budget line is
BL2, and he buys basket C, with utility U2.
To reach utility U1 after the price decrease,
he would need an income of $48 to buy
basket B, so his compensating variation is
$72 $48 $24. To reach utility U2 before
the price decrease, he would need an
income of $108 to buy basket E, so his
equivalent variation is $108 $72 $36.
When there is an income effect (basket E is
not directly above basket A, and basket C
is not directly above basket B), the compensating variation and the equivalent
variation are generally not equal.
BL1
C
A
36
Utility function U(x, y) = x y
Slope of BL1 = slope of line through
E = –9
Slope of BL2 = slope of line through
B = –4
B
24
U2 = 324
U1 = 144
4
6
9
x, Food
5.3 CHANGE IN THE PRICE OF A GOOD: THE CONCEPT OF CONSUMER SURPLUS
P, price of food (dollars per unit)
$10
Demand for food, x =
8
181
36
Px
6
4
2
0
2
4
6
8
x, units of food
10
FIGURE 5.20 Consumer Surplus with an
Income Effect
When the price of food falls from $9 per unit
to $4 per unit, the consumer increases his food
consumption from 4 units to 9 units. His consumer
surplus increases by the shaded area, or $29.20.
Still considering the consumer in Learning-By-Doing Exercise 5.9, let’s see what
happens if we measure consumer surplus using the area under the demand curve for
food. In Learning-By-Doing Exercise 5.4, we showed that his demand function for
food is x IⲐ(2Px). Figure 5.20 shows his demand curve when his income is $72. As
the price of food falls from $9 to $4 per unit, his consumption rises from 4 units to
9 units. The shaded area in Figure 5.20, which measures the increase in consumer surplus, equals $29.20. Note that this increase in consumer surplus ($29.20) is different
from both the compensating variation ($24) and the equivalent variation ($36). Thus,
the change in the area under the demand curve will not exactly measure either the compensating variation or the equivalent variation when the income effect is not zero.
A P P L I C A T I O N
5.5
How Much Would You Be Willing
to Pay to Have a Wal-Mart in Your
Neighborhood?
In the last 20 years, “big-box” mass-merchandise stores
such as Wal-Mart, Costco, and Target have proliferated
throughout the United States. In contrast to traditional
retailers such as grocery stores, which concentrate on
one line of merchandise, big-box mass merchandisers
sell a wide variety of consumer goods, including food,
clothing, CDs, books, housewares, toys, sporting goods,
and much more. In addition to wide variety, the bigbox mass merchandisers usually sell at discount prices.
These stores often create controversy when they open.
Competing stores often resist them aggressively, fearing
15
their effects on price levels and profits. Labor unions
sometimes also resist them over concerns about the impact of these stores on the wages in local labor markets.
Still, the wide variety and low prices offered by these
stores are presumably good for consumers. For example,
U.S. consumers make at least 25 percent of their food
expenditures at such stores. Thus, these stores could
potentially have a large impact on consumer welfare.
An important question is how big this impact is likely
to be.
In a recent study, economists Jerry Hausman and
Ephraim Leibtag shed light on this question by estimating the benefits to consumers from the opening of
new Wal-Mart supercenters in local retail markets.15
Using data on food expenditures of approximately
61,500 households from 1998 to 2001 in a variety of
Jerry Hausman and Ephraim Leibtag, “Consumer Benefits from Increased Competition in Shopping
Outlets: Measuring the Effect of Wal-Mart,” Journal of Applied Econometrics 22 (2007): 1157–1187.
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food expenditures. To put these amounts in perspective,
in Hausman and Leibtag’s sample of households, the
average food expenditure was about $150 per
month, or $1,800 a year. The combined compensating
variations from low prices and enhanced product
variety thus amounted to 25 percent of this amount,
or $450 per year. This represents the maximum
amount of income a typical U.S. household would
have been willing to forgo in the late 1990s in exchange for the lower prices and greater product variety
engendered by the entry of a Wal-Mart supercenter.
U.S. cities, Hausman and Leibtag estimate the compensating variation due to low food prices induced by the
entry of a new Wal-Mart supercenter in a local retail
market. They also estimate the compensating variation due to the increased product variety provided by
the presence of the new Wal-Mart.
Hausman and Leibtag estimate that the compensating variation due to low food prices is equal to
an amount that is approximately 5 percent of household food expenditures. The compensating variation
resulting from increased variety is even larger,
amounting to about 20 percent of total household
5.4
MARKET
DEMAND
I
n the previous sections of this chapter, we showed how to use consumer theory to
derive the demand curve of an individual consumer. But business firms and policy
makers are often more concerned with the demand curve for an entire market of consumers. Since markets might consist of thousands, or even millions, of individual consumers, where do market demand curves come from?
In this section, we illustrate an important principle: The market demand curve is the
horizontal sum of the demands of the individual consumers. This principle holds whether
two consumers, three consumers, or a million consumers are in the market.
Let’s work through an example of how to derive a market demand from individual consumer demands. To keep it simple, suppose only two consumers are in the market for orange juice. The first is “health conscious” and likes orange juice because of
its nutritional value and its taste. In Table 5.1, the second column tells us how many
liters of orange juice he would buy each month at the prices listed in the first column.
The second user (a “casual consumer” of orange juice) also likes its taste, but is less
concerned about its nutritional value. The third column of Table 5.1 tell us how many
liters of orange juice she would buy each month at the prices listed in the first column.
To find the total amount consumed in the market at any price, we simply add the
quantities that each consumer would purchase at that price. For example, if the market price is $5 per liter, neither consumer will buy orange juice. If the price is $3 or
$4, only the health-conscious consumer will buy it. Thus, if the price is $4 per liter,
he will buy 3 liters, and the market demand will also be 3 liters; if the price is $3 per
liter, the market demand will be 6 liters. Finally, if the market price is below $3, both
consumers will purchase orange juice. Thus, if the price is $2 per liter, the market demand will be 11 liters; if the price is $1 the market demand will be 16 liters.
TABLE 5.1
Market Demand for Orange Juice
Price
($/Liter)
Health Conscious
(Liters/Month)
Casual
(Liters/Month)
Market Demand
(Liters/Month)
5
4
3
2
1
0
3
6
9
12
0
0
0
2
4
0
3
6
11
16
5.4 MARKET DEMAND
Dc = Demand of casual consumer
Dh = Demand of health-conscious consumer
Dm = Market demand
$5
P, (dollars per liter)
183
4
A
3
2
1
Dc
0
4
6
Dh
12
1516
Dm
21
Q, (liters of orange juice per month)
FIGURE 5.21 Market and
Segment Demand Curves
The market demand curve Dm (the dark
curve) is found by adding the demand
curves Dh and Dc for the individual consumers horizontally.
In Figure 5.21 we show both the demand curve for each consumer (Dh and Dc)
and the market demand (the thick line, Dm).
Finally, we can describe the three demand curves algebraically. Let Qh be the quantity demanded by the health-conscious consumer, Qc the quantity demanded by the
casual consumer, and Qm the quantity demanded in the whole market (which contains
only the two consumers). What are the three demand functions Qh(P ), Qc(P), and Qm(P )?
As you can see in Figure 5.21, the demand curve Dh for the health-conscious consumer is a straight line; he buys orange juice only when the price is below $5 per liter.
You can verify that the equation of his demand curve is
Qb (P) ⫽ e
15 ⫺ 3P, when P 6 5
0, when P ⱖ 5
The demand curve for the casual consumer is also a straight line; she buys orange
juice only when the price is below $3 per liter. The equation of her demand curve Dc is
Qc(P) ⫽ e
6 ⫺ 2P, when P 6 3
0, when P ⱖ 3
As shown in Figure 5.21, when the price is higher than $5, neither consumer buys
orange juice; when the price is between $3 and $5, only the health-conscious consumer buys it. Therefore, over this range of prices, the market demand curve is the
same as the demand curve for the health-conscious consumer. Finally, when the price
is below $3, both consumers buy orange juice. (This explains why the market demand
curve Dm is kinked at point A, which is where the casual consumer’s demand kicks in.)
So the market demand Qm(P ) is just the sum of the segment demands Qh(P ) ⫹ Qc (P )
(15 ⫺ 3P) ⫹ (6 ⫺ 2P ) 21 ⫺ 5P. Therefore, the market demand Qm(P ) is
21 ⫺ 5P, when P 6 3
Qm (P) e15 ⫺ 3P, when 3 ⱕ P 6 5
0, when P ⱖ 5
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The discussion demonstrates that you must be careful when you add segment
demands to get a market demand curve. First, since the construction of a market demand
curve involves adding quantities, you must write the demand curves in the normal form
(with Q expressed as a function of P ) before adding them, rather than using the inverse form of the demand (with P written as a function of Q).
Second, you must pay attention to how the underlying individual demands vary
across the range of prices. In the example above, if you simply add the equations for
the individual demands to get the market demand Qm Qh(P ) Qc(P ) 21 5P, this
expression is not valid for a price above $3. For example, if the price is $4, the expression Qm 21 5P would tell you that the quantity demanded in the market would be
1 liter. Yet, as we can see by Table 5.1, the actual quantity demanded in the market at
that price is 3 liters. See if you can figure out why this approach leads to an error. (If
you give up, look at the footnote.)16
MARKET DEMAND WITH NETWORK EXTERNALITIES
network externalities
A demand characteristic
present when the amount
of a good demanded by one
consumer depends on the
number of other consumers
who purchase the good.
Thus far we have been assuming that each person’s demand for a good is independent
of everyone else’s demand. For example, the amount of chocolate a consumer wants
to purchase depends on that consumer’s income, the price of chocolate, and possibly
other prices, but not on anyone else’s demand for chocolate. This assumption enables
us to find the market demand curve for a good by adding up the demand curves of all
of the consumers in the market.
For some goods, however, a consumer’s demand does depend on how many other
people purchase the good. In that case, we say there are network externalities. If one
consumer’s demand for a good increases with the number of other consumers who buy
the good, the externality is positive. If the amount a consumer demands increases when
fewer other consumers have the good, the externality is negative. Many goods and
services have network externalities.
Although we can often find network externalities related to physical networks (like
telephone networks), we may also see them in other settings (sometimes called virtual
networks because there is no physical connection among consumers). For example, the
computer software Microsoft Word would have some value in preparing written documents even if that software had only one user. However, the product becomes more
valuable to each user when it has many users. The virtual network of users makes it
possible for each user to exchange and process documents with many other users.
A virtual network may also be present if a good or service requires two complementary components to have value. For example, a computer operating system, such
as Microsoft Windows, has value only if software applications exist that can run on
the operating system. The operating system becomes more valuable as the number of
applications that can run on it increases. A software application also has a higher value
if it runs on a widely accepted operating system. Thus, more people using an operating
system leads to more software applications, raising the demand for the operating system,
and so on.
16
The error arises because we derived the market demand equation Qm 21 5P by adding Qh(P )
15 3P and Qc (P ) 6 2P. According to these individual demand equations, when P 4, Qh(P ) 3
and Qc (P ) 2. Sure enough, the sum is 1. But you are assuming that the casual consumer demands a
negative quantity of orange juice (2 liters) when the price is $4, and this is economic nonsense! The
expression for the demand of the casual consumer Qc (P ) 6 2P is not valid at a price of $4. At this
price, Qc (P ) 0, not 2.
5.4 MARKET DEMAND
A P P L I C A T I O N
185
5.6
Externalities in Social Networking
Websites
Many products exhibit positive network externalities.
An obvious example is telephones. A consumer would
find little value in having a telephone unless there were
other people with telephones. For most people, a telephone becomes more useful as the number of other
people with telephones increases. To some extent, a
software application like Microsoft Word provides
another example. Consumers value using the most popular document formats, since doing so makes it easier to
share created documents with others. Instant messaging services offer a further example. As a specific messaging service becomes more popular, it also creates
more value to a given consumer because the service can
be used to communicate with more people.
In recent years we have witnessed a dramatic
increase in social networking sites such as Facebook,
LinkedIn, and Twitter. Consider the experience of
LinkedIn, a site that allows businesspeople to post
information about their credentials and career experience. Many professionals use LinkedIn to search for
jobs, develop contacts within their industry, or find
new customers for their services. LinkedIn was
founded in 2002. By the end of 2003 it had 83,000
users. Two years later it had 4 million users, and by
mid-2009 it had 43 million users.
Facebook has seen an even more dramatic rise.
Founded in 2004, the site had over 300 million users
worldwide by late 2009, and it was available in over 50
languages. Facebook is popular with a wider population than LinkedIn, as its design is more flexible and
encourages different types of users to use the site in
different ways. For example, alumni from a specific
high school and year can locate each other, become
Facebook “Friends,” and set up a group to post information related to their school. A member can set up or
join many groups simultaneously, with different purposes. Many Facebook users treat the site as a blog,
posting information about their current activities, interests, or links to articles on the Internet related to a
particular theme. This flexibility has enabled Facebook
to grow extremely rapidly in popularity.
Such explosive growth is quite common in goods
with positive network externalities because bandwagon effects often get stronger as a particular
product becomes more popular. A positive network
externality can make it very difficult for a new entrant in the market, even when a new rival offers advantages in quality, availability, or price.
Finally, positive network externalities can occur if a good or service is a fad. We
often see fads for goods and services that affect lifestyles, such as fashions of clothing,
children’s toys, or beer. Advertisers and marketers often try to highlight the popularity
of a product as part of its image.
Figure 5.22 illustrates the effects of a positive network externality. The graph
shows a set of market demand curves for connections to the Internet. For this example,
let’s assume that a connection to the Internet refers to a subscription to a provider of
access to the Internet, such as America Online or Microsoft Network. The curve D30
represents the demand if consumers believe that 30 million subscribers have access to
the Internet. The curve D60 represents the demand if consumers believe that 60 million
subscribers have access. Suppose that access initially costs $20 per month and that
there are 30 million subscribers (point A in the graph).
What happens if the monthly price of access drops to $10? If there were no positive network externality, the quantity demanded would simply change to some other
point on D30. In this case, the quantity of subscriptions would grow to 38 million (point
B in the graph). However, there is a positive network externality; as more people use
e-mail, instant messaging, and other Internet features, even more people want to sign up.
Therefore, at the lower price, the number of consumers wanting access will be even
greater than a movement along D30 to point B would indicate. The total number of
subscriptions actually demanded at a price of $10 per month will grow to 60 million
186
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T H E T H E O RY O F D E M A N D
D60
bandwagon effect A
positive network externality
that refers to the increase
in each consumer’s demand
for a good as more consumers buy the good.
snob effect A negative
network externality that
refers to the decrease in
each consumer’s demand
as more consumers buy
the good.
(dollars per month)
FIGURE 5.22 Positive Network
Externality: Bandwagon Effect
What happens to the demand
for access to the Internet if the
monthly charge for access falls from
$20 to $10? Without network externalities, the quantity demanded
would increase from 30 to 38 million
subscribers because of the pure
price effect. But this increase in
subscribers leads even more people
to want access. This positive network
externality (a bandwagon effect)
adds another 22 million subscribers
to the Internet.
P, price for Internet access
D30
$25
$20
A
$15
$05
0
C
B
$10
Pure
price
effect
30
38
Bandwagon
effect
Demand
60
Q (millions of subscribers)
(point C in the graph). The total effect of the price decrease is an increase of 30 million subscribers. The total effect is the pure price effect of 8 million new subscribers
(moving from point A to point B) plus a bandwagon effect of 22 million new subscribers
(moving from point B to point C). This bandwagon effect refers to the increased
quantity demanded as more consumers are connected to the Internet. Thus, a demand
curve with positive network externalities (such as the heavy demand curve in Figure 5.22)
is more elastic than a demand curve with no network externalities (such as D30).
For some goods, there is a negative network externality—the quantity demanded
decreases when more people have the good. Rare items, such as Stradivarius violins,
Babe Ruth baseball cards, and expensive automobiles are examples of such goods.
These goods enjoy a snob effect, a negative network externality that refers to the decrease in the quantity of a good that is demanded as more consumers buy it. A snob
effect may arise because consumers value being one of the few to own a particular type
of good. We might also see the snob effect if the value of a good or service diminishes
because congestion increases when more people purchase that good or service.
Figure 5.23 shows the effects of a snob effect. The graph illustrates a set of market
demand curves for membership in a health and fitness club. The curve D1000 represents
the demand if consumers believe the club has 1,000 members. The curve D1300 shows
the demand if consumers believe it has 1,300 members. Suppose a membership initially
costs $1,200 per year and that the club has 1,000 members (point A in the graph).
What happens if the membership price decreases to $900? If consumers believed
that the number of members would stay at 1,000, 1,800 would actually want to join
the club (point B in the graph). However, consumers know that the fitness club will
become more congested as more members join, and this will shift the demand curve
inward. The total number of memberships actually demanded at a price of $900 per
month will grow only to 1,300 (point C in the graph). The total effect of the price decrease is the pure price effect of 800 new members (moving from point A to point B)
plus a snob effect of 500 members (moving from point B to point C ), or an increase
of only 300 members. A demand curve with negative network externalities (such as the
demand curve connecting points A and C in Figure 5.23) is less elastic than a demand
curve without network externalities (such as D1000).
187
5.5 THE CHOICE OF LABOR AND LEISURE
Annual membership price
Demand
$1,500
A
$1,200
B
$1,900
C
$1,600
$1,300
0
D1000
D1300
Snob
effect
1000
1300
1800
Pure price effect
Q, number of memberships
FIGURE 5.23 Negative Network
Externality: Snob Effect
What happens to the demand for
membership in a fitness club if the
annual membership charge falls from
$1,200 to $900? Without network
externalities, the pure price effect
would increase the membership by
800 (from 1,000 to 1,800). But this
increase in membership would discourage some people from joining. This
negative externality (a snob effect)
leads to a reduction of 500 members
(from 1,800 to 1,300). The net effect
of the price reduction is therefore an
increase of 300 members.
A
s we have already seen, the model of optimal consumer choice has many everyday
applications. In this section, we use that model to examine a consumer’s choice of how
much to work.
A S WAG E S R I S E , L E I S U R E F I R S T D E C R E A S E S,
THEN INCREASES
Let’s divide the day into two parts: the hours when an individual works and the hours
when he pursues leisure. Why does the consumer work at all? Because he works, he
earns an income, and he uses the income to pay for the activities he enjoys in his
leisure time. The term leisure includes all nonwork activities, such as eating, sleeping,
recreation, and entertainment. We assume that the consumer likes leisure activities.
Suppose the consumer chooses to enjoy leisure for L hours per day. Since a day
has 24 hours, the time available for work will be the time that remains after leisure,
that is, 24 L hours.
The consumer is paid an hourly wage rate w. Thus, his total daily income will be
w(24 L). He uses the income to purchase units of a composite good at a price of
$1 per unit.
The consumer’s utility U depends on the amount of leisure time and the number
of units of the composite good he can buy. We can represent the consumer’s decision
on the optimal choice diagram in Figure 5.24. The horizontal axis represents the
number of hours of leisure each day, which can be no greater than 24 hours. The vertical axis represents the number of units of the composite good that he purchases from
his income. Since the price of the composite good is $1, the vertical axis also measures
the consumer’s income.
To find an optimal choice of leisure and other goods, we need a set of indifference
curves and a budget constraint. Figure 5.24 shows a set of indifference curves for
which the marginal utility of leisure and the composite good are both positive. Thus
U5 U4 U3 U2 U1.
5.5
THE CHOICE
OF LABOR
AND LEISURE
188
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T H E T H E O RY O F D E M A N D
$600
FIGURE 5.24
Optimal Choice
of Labor and Leisure
As the wage rate w rises from
$5 to $10 to $15, the consumer
chooses progressively less leisure
and more work: He moves from
basket E (16 hours of leisure, 8 of
work) to basket F (14 hours of
leisure, 10 of work) to basket G
(13 hours of leisure, 11 of work).
But as the wage rate rises from
$15 to $20 to $25, he chooses
progressively more leisure and less
work, moving from basket G to
basket H to basket I (at basket I,
he is working only 9 hours, with
15 hours of leisure).
Daily income and units of the composite good
w = 25
$480
w = 20
$360
w = 15
U4
$240
U3
w = 10
U2
w=5
$120
B
U5
H
I
G
U1
F
E
0
13 14 15 16
A
24
Hours of leisure
The consumer’s budget line for this problem will tell us all the combinations
of the composite good and hours of leisure (L) that the consumer can choose. If the
consumer does no work, he will have 24 hours of leisure but no income to spend
on the composite good. This corresponds to point A on the budget line in the
graph.
The location of the rest of the budget line depends on the wage rate w. Suppose
the wage rate is $5 per hour. This means that for every hour of leisure the consumer
gives up to work, he can buy 5 units of the composite good. The budget line thus has
a slope of 5. If the consumer were to work 24 hours per day, his income would be
$120 and he would be able to buy 120 units of the composite good, corresponding to
basket B on the budget line. The consumer’s optimal choice will then be basket E;
thus, when the wage rate is $5, the consumer will work 8 hours.
For any wage rate, the slope of the budget line is w. The figure shows budget
lines for five different wage rates ($5, $10, $15, $20, and $25), along with the optimal
choice for each wage rate. As the wage rate rises from $5 to $15, the number of hours
of leisure falls. However, as the wage rate continues to rise, the consumer begins to
increase his amount of leisure time.
The next section discusses a phenomenon that is directly related to this change in
the consumer’s choice of labor versus leisure as wage rates rise.
5.5 THE CHOICE OF LABOR AND LEISURE
T H E BAC K WA R D - B E N D I N G S U P P LY O F L A B O R
Wage rate (dollars per hour)
Since a day has only 24 hours, the consumer’s choice about the amount of leisure time
is also a choice about the amount of labor he will supply. The optimal choice diagram
in Figure 5.24 contains enough information to enable us to construct a curve showing
how much labor the consumer will supply at any wage rate. In other words, we can
draw the consumer’s supply of labor curve, as shown in Figure 5.25.
The points E, F, G, H, and I in Figure 5.25 correspond, respectively, to points
E, F, G, H, and I in Figure 5.24. When the wage rate is $5, the consumer supplies
8 hours of labor (points E and E ). As the wage rate goes up from $5 to $15, the labor
supply rises too—at a wage rate of $15, the labor supply is 11 hours (points G and G ).
But when the wage rate continues to rise past $15, the labor supply begins to fall,
until, finally, at a wage rate of $25, the consumer works only 9 hours (points I and I ).
For most goods and services, a higher price stimulates supply; in this case, however, a
higher wage rate decreases the labor supply. (Remember, the wage rate is the price of
labor.) To understand this phenomenon, which is reflected in the backward-bending
shape of the supply of labor curve in Figure 5.25, let’s examine the income and substitution effects associated with a change in the wage rate.
Look again at the optimal choice diagram in Figure 5.24. Instead of having a fixed income, our consumer has a fixed amount of time in the day, 24 hours. That is why the horizontal intercept of the budget line stays at 24 hours, regardless of the wage rate. An hour
of work always “costs” the consumer an hour of leisure, no matter what the wage rate is.
However, an increase in the wage rate makes a unit of the composite good look less
expensive to the consumer. If the wage rate doubles, the consumer needs to work only
half as long to buy as much of the composite good as before. That is why the vertical
intercept of the budget line moves up as the wage rate rises. The increase in the wage
rate therefore leads to an upward rotation of the budget line, as Figure 5.24 shows.
An increase in the wage rate reduces the amount of work required to buy a unit
of the composite good, and this leads to both a substitution effect and an income effect. The substitution effect on the labor supply is positive—it induces the consumer
to substitute more of the composite good for leisure, leading to less leisure and more
labor. In contrast, the income effect on labor supply is negative—it leads to more
leisure and less labor because leisure is a normal good for most people (i.e., the
consumer wants more leisure as his income rises).
I′
$25
H′
$20
FIGURE 5.25
G′
$15
$10
F′
$05
0
E′
1
2
3
Supply of labor
4 5 6 7 8 9 10 11
Hours worked per day
Backward-Bending Supply
of Labor
The points E, F, G, H,
and I correspond, respectively, to points E, F, G, H,
and I in Figure 5.24. The
supply of labor curve is
backward bending for
wage rates above $15.
189
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T H E T H E O RY O F D E M A N D
$600
Daily income and units of the composite good
190
BL2, w = 25
$480
$360
BLd
$240
BL1, w = 15
I
J
G
U5
U3
$120
A
0
12 13
15
24
Hours of leisure
FIGURE 5.26 Optimal Choice of Labor and Leisure
At the initial basket G on budget line BL1, the consumer has 13 hours of leisure (and works for
11 hours). At the final basket I on budget line BL2, the consumer has 15 hours of leisure (and
works for 9 hours). At the decomposition basket J on budget line BLd, the consumer has 12 hours
of leisure (and works for 12 hours). The substitution effect on leisure is 1 (the change in leisure
between G and J ). The income effect on leisure is 3 (the change in leisure between J and I).
Thus, the total effect on leisure is 2, and the corresponding total effect on labor is 2.
Now let’s examine the income and substitution effects of a wage increase from
$15 to $25. Figure 5.26 shows the initial budget line BL1 (with the wage rate of $15)
and the optimal initial basket G, with 13 hours of leisure and, therefore, 11 hours of
work. The figure also shows the final budget line BL2 (with the wage rate of $25) and
the optimal final basket I, with 15 hours of leisure and 9 hours of work. Finally, the
figure shows the decomposition budget line BLd (which is tangent to the initial indifference curve U3 and parallel to the final budget line BL2) and the decomposition
basket J, with 12 hours of leisure and 12 hours of work.
The substitution effect on leisure is thus 1 hour (the change in leisure as we
move from G to J ). The income effect on leisure is 3 hours (the change in leisure as
we move from J to I ). Since the income effect outweighs the substitution effect, the
net effect of the change in the wage rate on the amount of leisure is 2 hours. Thus,
the net effect of the increase in the wage rate on the amount of labor is 2 hours. This
accounts for the backward-bending shape of the labor supply curve in Figure 5.25 as
the wage rate rises above $15.
In sum, the labor supply curve slopes upward over the region where the substitution effect associated with a wage increase outweighs the income effect, but bends
backward over the region where the income effect outweighs the substitution effect.
5.5 THE CHOICE OF LABOR AND LEISURE
S
E
191
L E A R N I N G - B Y- D O I N G E X E R C I S E 5 . 1 0
D
The Demand for Leisure and the Supply of Labor
Problem
Jan’s utility for leisure (L)
and a composite good ( Y ) is U LY. The marginal utility of leisure is MUL Y, and the marginal utility of the
composite good is MUY L. The price of the composite good is $1. When she enjoys L hours of leisure per
day, Jan works (24 L) hours per day. Her wage rate is
w, so her daily income is w(24 L). Show that, for any
positive wage rate, the optimal number of hours of
leisure that Jan enjoys is always the same. What is the
number of hours of leisure she would demand, and how
many hours of labor will she supply each day?
Solution With the Cobb–Douglas utility function,
there will be an interior optimum, with positive values of
Y and L. Once we find Jan’s optimal choice for leisure
each day (L), we know she will work (24 L) hours.
At her optimal choices of Y and L, Jan will need to
satisfy two conditions. First, the tangency condition requires that the ratio of the marginal utility of leisure to the
price of leisure must equal the ratio of the marginal utility of the composite good to the price of that good. The
price of leisure is the wage rate; that represents how much
A P P L I C A T I O N
MUY
MUL
w
1
The tangency condition tells us that Y/w L, or that
Y wL.
Jan must also satisfy her budget constraint. She receives an income equal to the wage rate times the numbers of hours she works; she therefore earns an income
equal to w(24 L). She buys Y units of the composite
good at a price of $1; she therefore spends $Y. So her
budget constraint is just w(24 L) Y.
Together the tangency condition and the budget
line require that w(24 L) wL. In this example, Jan’s
optimal demand for leisure is L 12 hours per day, and
she will supply 12 hours of labor per day, independent of
the wage rate. Of course, for many other utility functions, her demand for leisure (and thus, her supply of
labor), will depend on the wage rate.
Similar Problems: 5.29, 5.30, 5.31
5.7
The Backward-Bending Supply
of Nursing Services
Medical groups and hospitals have long had difficulty
attracting enough workers. In response, they have
often increased the pay of medical workers, but this
may not always increase the amount of labor supplied.
In 1991 the Wall Street Journal described some of
these difficulties in an article titled “Medical Groups
Use Pay Boosts, Other Means to Find More Workers.”
According to the article, the American Hospital
Association concluded that “Pay rises may have worsened the nursing shortage in Massachusetts by enabling nurses to work fewer hours.”17
Why might this have happened? As we saw in our
discussion related to Figure 5.26, a higher wage may
induce a consumer to pursue more leisure and thus
supply less labor. In other words, many nurses may be
on the backward-bending region of their supply curve
17
income she loses when she enjoys an extra hour of leisure
instead of working for that hour. Thus, at an optimum
for labor. Using data from 2000, an academic study
estimated the labor supply of nurses in the United
States.18 The study concluded that the short-run labor
supply curve was backward bending. It appears that
the labor market for nurses may continue to experience the short-run problems it suffered from in 1991.
Since wage increases alone do not always attract
more workers, employers have resorted to other
strategies. For example, the article in the Wall Street
Journal states that the M.D. Anderson Cancer Center
at the University of Texas gave employees a $500
bonus if they referred new applicants who took
“hard-to-fill” jobs. The Texas Heart Institute in
Houston recruited nurses partly by showcasing
prospects for promotion. The University of Pittsburgh
Medical Center started an “adopt-a-high-school” program to encourage students to enter the health care
sector, and reimbursed employees’ tuition fees when
they enrolled in programs to increase their skills.
Albert R., Karr, “Medical Groups Use Pay Boosts, Other Means to Find More Workers,” The Wall Street
Journal, August 27, 1991, p. A1.
18
Lynn Unruh and Joanne Spetz, “Can Wage Increases End Nursing Shortages? A Reexamination of the Supply
Curve of Registered Nurses.” Academy of Health Meetings Abstracts, 2005: vol. 22, abstract no. 4480.
5.6
CONSUMER
PRICE
INDICES
CHAPTER 5
T H E T H E O RY O F D E M A N D
T
he Consumer Price Index (CPI) is one of the most important sources of information
about trends in consumer prices and inflation in the United States. It is often viewed
as a measure of the change in the cost of living and is used extensively for economic
analysis in both the private and public sectors. For example, in contracts among individuals and firms, the prices at which goods are exchanged are often adjusted over
time to reflect changes in the CPI. In negotiations between labor unions and employers,
adjustments in wage rates often reflect past or expected future changes in the CPI.
The CPI also has an important impact on the budget of the federal government. On
the expenditure side, the government uses the CPI to adjust payments to Social Security
recipients, to retired government workers, and for many entitlement programs such as
food stamps and school lunches. As the CPI rises, the government’s payments increase.
And changes in the CPI also affect how much money the government collects through
taxes. For example, individual income tax brackets are adjusted for inflation using the
CPI. As the CPI increases, tax revenues decrease.
Measuring the CPI is not easy. Let’s construct a simple example to see what factors
might be desirable in designing a CPI. Suppose we consider a representative consumer,
who buys only two goods, food and clothing, as illustrated in Figure 5.27. In year 1, the
price of food was PF1 $3 and the price of clothing was PC1 $8. The consumer had
BL1 has slope – (3/8)
BL2 has slope – (2/3)
BL3 has slope – (2/3)
C, clothing
192
40
E
B
A
30
U1
60
80
F, food
FIGURE 5.27 Substitution Bias in the Consumer Price Index
In year 1 the consumer has an income of $480, the price of food is $3, and the price of
clothing is $8. The consumer chooses basket A. In year 2 the price of food rises to $6, and
the price of clothing rises to $9. The consumer could maintain his initial level of utility U1
at the new prices by purchasing basket B, costing $720. An ideal cost of living index would
be 1.5 (ⴝ$720/$480), telling us that the cost of living has increased by 50 percent. However,
the actual CPI assumes the consumer does not substitute clothing for food as relative prices
change, but continues to buy basket A at the new prices, for which he would need an income
of $750. The CPI ($750/$480 ⴝ 1.56) suggests that the consumer’s cost of living has increased
by about 56 percent, which overstates the actual increase in the cost of living. In fact, if
the consumer’s income in year 2 were $750, he could choose a basket such as E on BL3 and
achieve a higher level of utility than U1.
5.6 CONSUMER PRICE INDICES
an income of $480 and faced the budget line BL1 with a slope of PF1/PC1 3/8. He
purchased the optimal basket A, located on indifference curve U1 and containing
80 units of food and 30 units of clothing.
In year 2 the prices of food and clothing increase to PF2 $6 and PC2 $9. How
much income will the consumer need in year 2 to be as well off as in year 1, that is, to
reach the indifference curve U1? The new budget line BL2 must be tangent to U1 and
have a slope reflecting the new prices, PF2/PC2 2/3. At the new prices, the least
costly combination of food and clothing on the indifference curve is at basket B, with
60 units of food and 40 units of clothing. The total expenditure necessary to buy basket
B at the new prices is PF2F PC2C ($6)(60) ($9)(40) $720.
In principle, the CPI should measure the percentage increase in expenditures that
would be necessary for the consumer to remain as well off in year 2 as he was in year
1. In the example, the necessary expenditures increased from $480 in year 1 to $720
in year 2. The “ideal” CPI would be the ratio of the new expenses to the old expenses––that is $720Ⲑ$480 1.5. In other words, at the higher prices, it would take
50 percent more income in year 2 to make the consumer as well off as he was in year 1.
In this sense the “cost of living” in year 2 is 50 percent greater than it was in year 1.
In calculating this ideal CPI, we would need to recognize that the consumer would
substitute more clothing for food when the price of food rises relative to the price of
clothing, moving from the initial basket A to basket B.
Note that to determine the ideal CPI, the government would need to collect data on
the old prices and the new prices and on changes in the composition of the basket (how
much food and clothing are consumed). But considering the huge number of goods and
services in the economy, this is an enormous amount of data to collect! It is hard enough
to collect data on the way so many prices change over time, and even more difficult to
collect information on the changes in the baskets that consumers actually purchase.
In practice, therefore, to simplify the measurement of the CPI, the government has
historically calculated the change in expenditures necessary to buy a fixed basket as prices
change, where the fixed basket is the amount of food and clothing purchased in year 1.
In our example, the fixed basket is A. The income necessary to buy basket A at the new
prices is PF2F PC2C ($6)(80) ($9)(30) $750. If he were given $750 with the
new prices, he would face the budget line BL3. If we were to calculate a CPI using the
fixed basket A, the ratio of the new expenses to the old expenses is $750Ⲑ$480 1.5625.
This index tells us that the consumer’s expenditures would need to increase by 56.25
percent to buy the fixed basket (i.e., the basket purchased in year 1) at the new prices.19
As the example shows, the index based on the fixed basket overcompensates the
consumer for the higher prices. Economists refer to the overstatement of the increase
in the cost of living as the “substitution bias.” By assuming that the consumer’s basket
is fixed at the initial levels of consumption, the index ignores the possible substitution
that consumers will make toward goods that are relatively less expensive in a later year.
In fact, if the consumer were given an income of $750 instead of $720 in year 2, he
could choose a basket such as E on BL3 and make himself better off than he was at A.
19
An index that measures the expenditure necessary to buy the fixed basket at the prices in year 2 divided
by the expenditure necessary to purchase the same basket at the prices in year 1 is called a Laspeyres index.
Let’s see how to calculate this index with the example in the text. Denote the prices of food in years 1 and 2
as PF1 and PF2, and the prices of clothing in years 1 and 2 as PC1 and PC2. The fixed basket is the quantity of
food F and clothing C consumed in year 1. Then the Laspeyres index L is
L
PF2 F PC2C
PF1 F PC1C
193
194
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A P P L I C A T I O N
T H E T H E O RY O F D E M A N D
5.8
The Substitution Bias in the
Consumer Price Index
While economists have long argued that the Consumer
Price Index overstates changes in the cost of living, the
bias in the CPI took center stage in the 1990s when
Congress tried to balance the budget. In 1995 Alan
Greenspan, the chairman of the Federal Reserve,
brought this controversy to the fore when he told
Congress that the official CPI might be overstating the
true increase in the cost of living by perhaps 0.5 to 1.5
percent. The Senate Finance Committee appointed a
panel chaired by economist Michael Boskin to study the
magnitude of the bias. The panel concluded that the
CPI overstates the cost of living by about 1.1 percent.
While estimates of the impact of the substitution
bias are necessarily imprecise, they are potentially very
important. Greenspan estimated that if the annual
level of inflation adjustments to indexed programs and
taxes were reduced by 1 percentage point, the annual
level of the deficit would be lowered by as much as
$55 billion after five years. The Office of Management
and Budget estimated that in fiscal year 1996, a 1 percent increase in the index led to an increase in government expenditures of about $5.7 billion, as well as a
decrease in tax revenues of about $2.5 billion.
The government has long been aware of the need
to periodically update the “fixed basket” used in the
CPI calculation. In fact, the basket has been revised
approximately every 10 years, with the most recent
revision taking place in 2002.20 In light of the potential
biases of the CPI, the government continues to investigate ways to improve how it is calculated. For example,
in January 1999 the government began to use a new
formula to calculate many of the component indices
that form the CPI. The use of this new formula is
intended to counteract the substitution bias and was
expected to reduce the annual rate of increase in the
CPI by about 0.2 percentage points a year.
20
See, for example, John S. Greenless and Charles C. Mason, “Overview of the 1998 Revision of the
Consumer Price Index,” Monthly Labor Review (December 1996): 3–9, and Brent R. Moulton, “Bias in the
Consumer Price Index: What Is the Evidence?’’ Journal of Economic Perspectives (Fall 1996): 159–177.
CHAPTER SUMMARY
• We can derive an individual’s demand curve for a
good from her preferences and the budget constraint. A
consumer’s demand curve shows how the optimal choice
of a commodity changes as the price of the good varies.
We can also think of a demand curve as a schedule of
the consumer’s “willingness to pay” for a good. (LBD
Exercises 5.2, 5.3)
• A good is normal if the consumer purchases more of
that good as income rises. A good is inferior if he purchases less of that good as income increases. (LBD
Exercise 5.1)
• We can separate the effect of a price change on the
quantity of a good demanded into two parts: a substitution effect and an income effect. The substitution effect
is the change in the amount of a good that would be consumed as the price of that good changes, holding constant the level of utility. When the indifference curves
are bowed in toward the origin (because of diminishing
marginal rate of substitution), the substitution effect will
move in the opposite direction from the price change. If the
price of the good decreases, its substitution effect will be
positive. If the price of the good increases, its substitution
effect will be negative. (LBD Exercises 5.4, 5.5, 5.6)
• The income effect for a good is the change in the
amount of that good that a consumer would buy as her
purchasing power changes, holding prices constant. If
the good is normal, the income effect will reinforce the
substitution effect. If the good is inferior, the income
effect will oppose the substitution effect.
• If the good is so strongly inferior that the income
effect outweighs the substitution effect, the demand curve
will have an upward slope over some range of prices.
Such a good is called a Giffen good.
• Consumer surplus is the difference between what a
consumer is willing to pay for a good and what he must
pay for it. Without income effects, consumer surplus
provides a monetary measure of how much better off the
consumer will be when he purchases a good. On a graph
REVIEW QUESTIONS
the consumer surplus will be the area under an ordinary
demand curve and above the price of the good. Changes
in consumer surplus can measure how much better off or
worse off a consumer is if the price changes. (LBD
Exercise 5.7)
• Using optimal choice diagrams, we can look at the
monetary impact of a price change from two perspectives: compensating variation and equivalent variation.
The compensating variation measures how much money
the consumer would be willing to give up after a reduction in the price of a good to make her just as well off as
she was before the price change.
• The equivalent variation measures how much money
we would have to give the consumer before a price reduction to keep her as well off as she would be after the price
change.
• If there is an income effect, the compensating variation and equivalent variation will differ, and these measures will also be different from the change in the area
under the ordinary demand curve. (LBD Exercise 5.9)
• If the income effect is small, the equivalent and
compensating variations may be close to one another,
and the change in the area under an ordinary demand
curve will be a good approximation (although not an
exact measure) of the monetary impact of the price
change.
195
• Without an income effect, the compensating variation and equivalent variation will give us the same measure of the monetary value that a consumer would assign
to a change in the price of the good. The change in the
area under an ordinary demand curve will be equal to
the compensating variation and equivalent variation.
(LBD Exercise 5.8)
• The market demand curve for a good is the horizontal sum of the demands of all of the individual consumers in the market (assuming there are no network
externalities).
• The bandwagon effect is a positive network externality. With a bandwagon effect, each consumer’s demand
for a good increases as more consumers buy it. The snob
effect is a negative network externality. With a snob effect each consumer’s demand for a good decreases as
more consumers buy it.
• The consumer choice model also helps us to understand how much an individual chooses to work. A consumer’s happiness depends on the amount of time she
spends in leisurely activities, as well as on the amounts of
goods and services she can purchase. She must work
(forego leisure) to earn income to buy the goods and
services she desires. Thus, when she determines her demand for leisure, she is also determining her supply of
labor. (LBD Exercise 5.10)
REVIEW QUESTIONS
1.
What is a price consumption curve for a good?
2. How does a price consumption curve differ from an
income consumption curve?
3. What can you say about the income elasticity of demand of a normal good? of an inferior good?
4. If indifference curves are bowed in toward the origin
and the price of a good drops, can the substitution effect
ever lead to less consumption of the good?
5. Suppose a consumer purchases only three goods,
food, clothing, and shelter. Could all three goods be
normal? Could all three goods be inferior? Explain.
6. Does economic theory require that a demand curve
always be downward sloping? If not, under what circumstances might the demand curve have an upward slope
over some region of prices?
7.
What is consumer surplus?
8. Two different ways of measuring the monetary value
that a consumer would assign to the change in price of
the good are (1) the compensating variation and (2) the
equivalent variation. What is the difference between
the two measures, and when would these measures be
equal?
9. Consider the following four statements. Which
might be an example of a positive network externality?
Which might be an example of a negative network
externality?
(i) People eat hot dogs because they like the taste, and hot
dogs are filling.
(ii) As soon as Zack discovered that everybody else was
eating hot dogs, he stopped buying them.
(iii) Sally wouldn’t think of buying hot dogs until
she realized that all her friends were eating them.
(iv) When personal income grew by 10 percent, hot dog
sales fell.
10. Why might an individual supply less labor (demand
more leisure) as the wage rate rises?
196
CHAPTER 5
T H E T H E O RY O F D E M A N D
PROBLEMS
5.1. Figure 5.2(a) shows a consumer’s optimal choices
of food and clothing for three values of weekly income:
I1 $40, I2 $68, and I3 $92. Figure 5.2(b) illustrates
how the consumer’s demand curve for food shifts as income changes. Draw three demand curves for clothing
(one for each level of income) to illustrate how changes
in income affect the consumer’s purchases of clothing.
5.2. Use the income consumption curve in Figure 5.2(a)
to draw the Engel curve for clothing, assuming the price
of food is $2 and the price of clothing is $4.
5.3. Show that the following statements are true:
a) An inferior good has a negative income elasticity of
demand.
b) A good whose income elasticity of demand is negative
will be an inferior good.
5.4. If the demand for a product is perfectly price inelastic, what does the corresponding price consumption
curve look like? Draw a graph to show the price consumption curve.
5.5. Ann consumes five goods. The prices of all goods
are fixed. The price of good x is px. She spends 25 percent of her income on good x, regardless of the size of
her income.
a) Show that her income elasticity of demand of good x
is the same for any level of income, and determine its
value.
b) Would the value of the income elasticity of demand
for x be different if Ann always spends 60 percent of her
income on good x?
a) Derive Karl’s demand curve for beer as a function of
the exogenous variables.
b) Which affects Karl’s consumption of beer more: a one
dollar increase in PH or a one dollar increase in PB?
5.8. David has a quasilinear utility function of the form
U(x, y ) 1x y, with associated marginal utility functions MUx 1/(21x ) and MUy 1.
a) Derive David’s demand curve for x as a function of the
prices, Px and Py. Verify that the demand for x is independent of the level of income at an interior optimum.
b) Derive David’s demand curve for y. Is y a normal
good? What happens to the demand for y as Px increases?
5.9. Rick purchases two goods, food and clothing. He
has a diminishing marginal rate of substitution of food
for clothing. Let x denote the amount of food consumed
and y the amount of clothing. Suppose the price of food
increases from Px1 to Px2. On a clearly labeled graph, illustrate the income and substitution effects of the price
change on the consumption of food. Do so for each of
the following cases:
a) Case 1: Food is a normal good.
b) Case 2: The income elasticity of demand for food is zero.
c) Case 3: Food is an inferior good, but not a Giffen good.
d) Case 4: Food is a Giffen good.
5.10. Reggie consumes only two goods: food and
shelter. On a graph with shelter on the horizontal axis
and food on the vertical axis, his price consumption curve
for shelter is a vertical line. Draw a pair of budget lines
and indifference curves that are consistent with this description of his preferences. What must always be true
about Reggie’s income and substitution effects as the result
of a change in the price of shelter?
5.6. Suzie purchases two goods, food and clothing. She
has the utility function U(x, y) xy, where x denotes the
amount of food consumed and y the amount of clothing. The marginal utilities for this utility function are
MUx y and MUy x.
a) Show that the equation for her demand curve for
clothing is y I Ⲑ(2Py).
b) Is clothing a normal good? Draw her demand curve
for clothing when the level of income is I 200. Label
this demand curve D1. Draw the demand curve when I
300 and label this demand curve D2.
c) What can be said about the cross-price elasticity of
demand of food with respect to the price of clothing?
5.11. Ginger’s utility function is U(x, y) x 2y, with associated marginal utility functions MUx 2xy and
MUy x 2. She has income I 240 and faces prices Px
$8 and Py $2.
a) Determine Ginger’s optimal basket given these prices
and her income.
b) If the price of y increases to $8 and Ginger’s income
is unchanged, what must the price of x fall to in order
for her to be exactly as well off as before the change
in Py?
5.7. Karl’s preferences over hamburgers (H ) and beer
(B) are described by the utility function: U(H, B)
min(2H, 3B). His monthly income is I dollars, and he
only buys these two goods out of his income. Denote the
price of hamburgers by PH and of beer by PB.
5.12. Ann’s utility function is U(x, y) x y, with associated marginal utility functions MUx 1 and MUy 1.
Ann has income I 4.
a) Determine all optimal baskets given that she faces
prices Px 1 and Py 1.
197
PROBLEMS
b) Determine all optimal baskets given that she faces
prices Px 1 and Py 2.
c) What is demand for y when Px 1 and Py 1? What
is demand for y when Px 1 and Py 1? What is demand for y when Px 1 and Py 1? Plot Ann’s demand
for y as a function of Py.
d) Repeat the exercises in (a), (b) and (c) for U(x, y)
2x y, with associated marginal utility functions MUx
2 and MUy 1, and with the same level of income.
5.17. The accompanying figure illustrates the change
in consumer surplus, given by Area ABEC, when the
price decreases from P1 to P2. This area can be divided
into the rectangle ABDC and the triangle BDE. Briefly
describe what each area represents, separately, keeping in
mind the fact that consumer surplus is a measure of how
well off consumers are (therefore the change in consumer
surplus represents how much better off consumers are).
(Hint: Note that a price decrease also induces an increase
in the quantity consumed.)
5.13. Some texts define a “luxury good” as a good for
which the income elasticity of demand is greater than 1.
Suppose that a consumer purchases only two goods. Can
both goods be luxury goods? Explain.
5.14. Scott consumes only two goods, steak and ale.
When the price of steak falls, he buys more steak and more
ale. On an optimal choice diagram (with budget lines and
indifference curves), illustrate this pattern of consumption.
5.15. Dave consumes only two goods, coffee and
doughnuts. When the price of coffee falls, he buys the
same amount of coffee and more doughnuts.
a) On an optimal choice diagram (with budget lines and
indifference curves), illustrate this pattern of consumption.
b) Is this purchasing behavior consistent with a quasilinear utility function? Explain.
5.16. (This problem shows that an optimal consumption choice need not be interior and may be at a corner
point.) Suppose that a consumer’s utility function is
U(x, y) xy 10y. The marginal utilities for this utility
function are MUx y and MUy x 10. The price of
x is Px and the price of y is Py, with both prices positive.
The consumer has income I.
a) Assume first that we are at an interior optimum. Show
that the demand schedule for x can be written as x
I Ⲑ(2Px) 5.
b) Suppose now that I 100. Since x must never be negative, what is the maximum value of Px for which this
consumer would ever purchase any x?
c) Suppose Py 20 and Px 20. On a graph illustrating
the optimal consumption bundle of x and y, show that
since Px exceeds the value you calculated in part (b), this
corresponds to a corner point at which the consumer
purchases only y. (In fact, the consumer would purchase
y IⲐPy 5 units of y and no units of x.)
d) Compare the marginal rate of substitution of x for y
with the ratio (Px ⲐPy) at the optimum in part (c). Does
this verify that the consumer would reduce utility if she
purchased a positive amount of x?
e) Assuming income remains at 100, draw the demand
schedule for x for all values of Px. Does its location depend on the value of Py?
Demand
P1
P2
A
B
C
D
E
q1
q2
5.18. The demand function for widgets is given by
D(P) 16 2P. Compute the change in consumer
surplus when the price of a widget increases from $1 to $3.
Illustrate your result graphically.
5.19. Jim’s preferences over cookies (x) and other
goods ( y) are given by U(x, y) xy with associated marginal utility functions MUx y and MUy x. His income is $20.
a) Find Jim’s demand schedule for x when the price of y
is Py $1.
b) Illustrate graphically the change in consumer surplus
when the price of x increases from $1 to $2.
5.20. Lou’s preferences over pizza (x) and other goods
( y) are given by U(x, y) xy, with associated marginal
utilities MUx y and MUy x. His income is $120.
a) Calculate his optimal basket when Px 4 and Py 1.
b) Calculate his income and substitution effects of a decrease in the price of food to $3.
c) Calculate the compensating variation of the price
change.
d) Calculate the equivalent variation of the price
change.
5.21. Carina buys two goods, food F and clothing C,
with the utility function U FC F. Her marginal utility of food is MUF C 1 and her marginal utility of
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CHAPTER 5
T H E T H E O RY O F D E M A N D
clothing is MUC F. She has an income of 20. The price
of clothing is 4.
a) Derive the equation representing Carina’s demand for
food, and draw this demand curve for prices of food
ranging between 1 and 6.
b) Calculate the income and substitution effects on
Carina’s consumption of food when the price of food
rises from 1 to 4, and draw a graph illustrating these
effects. Your graph need not be exactly to scale, but it
should be consistent with the data.
c) Determine the numerical size of the compensating
variation (in monetary terms) associated with the increase in the price of food from 1 to 4.
5.22. Suppose the market for rental cars has two segments, business travelers and vacation travelers. The demand curve for rental cars by business travelers is Qb
35 0.25P, where Qb is the quantity demanded by business travelers (in thousands of cars) when the rental price
is P dollars per day. No business customers will rent cars
if the price exceeds $140 per day.
The demand curve for rental cars by vacation travelers is Qv 120 1.5P, where Qv is the quantity demanded by vacation travelers (in thousands of cars) when
the rental price is P dollars per day. No vacation customers will rent cars if the price exceeds $80 per day.
a) Fill in the table to find the quantities demanded in the
market at each price.
Price
($/day)
Business
(thousands of
cars/day)
Vacation
(thousands of
cars/day)
Market
Demand
(thousands of
cars/day)
100
90
80
70
60
50
b) Graph the demand curves for each segment, and draw
the market demand curve for rental cars.
c) Describe the market demand curve algebraically. In
other words, show how the quantity demanded in the
market Qm depends on P. Make sure that your algebraic
equation for the market demand is consistent with your
answers to parts (a) and (b).
d) If the price of a rental car is $60, what is the consumer
surplus in each market segment?
5.23. There are two types of consumers in a market for
sheet metal. Let P represent the market price.
The total quantity demanded by Type I consumers is
Q1 100 2P, for 0 P 50.
The total quantity demanded by Type II consumers is
Q2 40 P, for 0 P 40.
Draw the total market demand on a clearly labeled
graph.
5.24. There are two consumers on the market: Jim and
Donna. Jim’s utility function is U(x, y) xy, with associated
marginal utility functions MUx y and MUy x. Donna’s
utility function is U(x, y) x2y, with associated marginal
utility functions MUx 2xy and MUy x2. Income of Jim
is IJ 100 and income of Donna is ID 150.
a) Find optimal baskets of Jim and Donna when price of
y is Py 1 and price of x is P.
b) On separate graphs plot Jim’s and Donna’s demand
schedule for x for all values of P.
c) Compute and plot aggregate demand when Jim and
Donna are the only consumers.
d) Plot aggregate demand when there is one more consumer that has identical utility function and income as
Donna.
5.25. One million consumers like to rent movie videos
in Pulmonia. Each has an identical demand curve for
movies. The price of a rental is $P. At a given price, will
the market demand be more elastic or less elastic than the
demand curve for any individual? (Assume there are no
network externalities.)
5.26. Suppose that Bart and Homer are the only people
in Springfield who drink 7-UP. Moreover their inverse
demand curves for 7-UP are, respectively, P 10 4QB
and P 25 2QH, and, of course, neither one can consume a negative amount. Write down the market demand
curve for 7-UP in Springfield, as a function of all possible prices.
5.27. Joe’s income consumption curve for tea is a vertical line on an optimal choice diagram, with tea on the
horizontal axis and other goods on the vertical axis.
a) Show that Joe’s demand curve for tea must be downward sloping.
b) When the price of tea drops from $9 to $8 per pound,
the change in Joe’s consumer surplus (i.e., the change in
the area under the demand curve) is $30 per month.
Would you expect the compensating variation and the
equivalent variation resulting from the price decrease to
be near $30? Explain.
5.28. Consider the optimal choice of labor and leisure
discussed in the text. Suppose a consumer works the first
199
PROBLEMS
8 hours of the day at a wage rate of $10 per hour, but
receives an overtime wage rate of $20 for additional time
worked.
a) On an optimal choice diagram, draw the budget constraint. (Hint: It is not a straight line.)
b) Draw a set of indifference curves that would make it
optimal for him to work 4 hours of overtime each day.
5.29. Terry’s utility function over leisure (L) and other
goods (Y ) is U(L, Y ) Y LY. The associated marginal
utilities are MUY 1 L and MUL Y. He purchases
other goods at a price of $1, out of the income he earns
from working. Show that, no matter what Terry’s wage
rate, the optimal number of hours of leisure that he consumes is always the same. What is the number of hours
he would like to have for leisure?
5.30. Consider Noah’s preferences for leisure (L) and
other goods (Y ), U(L, Y ) 1L 1Y . The associated
marginal utilities are MUL 1/(21L) and MUY
1/(21Y ). Suppose that PY $1. Is Noah’s supply of
labor backward bending?
5.31. Raymond consumes leisure (L hours per day) and
other goods (Y units per day), with preferences described
by U(L, Y ) L 21Y. The associated marginal utilities are MUY 1 and MUL 1 / 1L. The price of other
goods is 1 euro per unit. The wage rate is w euros per
hour.
a) Show how the number of units of leisure Raymond
chooses depends on the wage rate.
b) How does Raymond’s daily income depend on the
wage rate?
c) Does Raymond work more when the wage rate rises?
5.32. Julie buys food and other goods. She has an income of $400 per month. The price of food is initially
$1.00 per unit. It then rises to $1.20 per unit. The prices
of other goods do not change. To help Julie out, her
mother offers to send her a check each month to supplement her income. Julie tells her mother, “Thanks, Mom.
If you would send me a check for $50 per month, I would
be exactly as happy paying $1.20 per unit as I would have
been paying $1.00 per unit and not receiving the $50
from you.” Which of the following statements is true?
Explain.
The increased price of food has:
a) an income effect of $50 per month
b) an income effect of $50 per month
c)
d)
e)
f)
a compensating variation of $50 per month
a compensating variation of $50 per month
an equivalent variation of $50 per month
an equivalent variation of $50 per month
5.33. Gina lives in Chicago and very much enjoys traveling by air to see her mother in Italy. On the accompanying graph, x denotes her number of round trips to Italy
each year. The composite good y measures her annual
consumption of other goods; the price of the composite
good is py, which is constant in this problem. Several indifference curves from her preference map are drawn,
U2
U3
U4
U5. If she
with levels of utility U1
spends all her income on the composite good, she can
purchase y* units, as shown in the graph. When the initial price of air travel is $1,000, she can purchase as many
as 18 round trips if she spends all her income on air travel
to Italy.
a) Make a copy of the graph, and use it to determine the
income and substitution effects on the number of round
trips Gina makes as the price of a round trip increases
from $1,000 to $3,000. Clearly label these effects on the
graph.
b) Using the graph, estimate the numerical size of the
compensating variation associated with the price increase.
You may refer to the graph to explain your answer.
c) Will the consumer surplus measured using Gina’s demand for air travel to Italy provide an exact measure of
the monetary value she associates with the price increase?
In a sentence, explain why or why not.
y
y*
U5
U4
U3
U2
U1
0
2
4
6
8
10
12
14
16
18
x
6
INPUTS AND
PRODUCTION
FUNCTIONS
6.1
I N T R O D U C T I O N TO I N P U T S
AND PRODUCTION FUNCTIONS
APPLICATION 6.1
Competition Breeds Efficiency
6.2
PRODUCTION FUNCTIONS
WITH A SINGLE INPUT
The Resurgence of Labor
Productivity in the United States
APPLICATION 6.2
6.3
PRODUCTION FUNCTIONS
WITH MORE THAN ONE INPUT
High-Tech Workers versus
Low-Tech Workers
APPLICATION 6.3
6.4
S U B S T I T U TA B I L I T Y A M O N G
INPUTS
Elasticities of Substitution
in German Industries
APPLICATION 6.5 Measuring Productivity
APPLICATION 6.6 Estimating a CES Production
Function for U.S. Industries
APPLICATION 6.4
6.5
R E T U R N S TO S C A L E
APPLICATION 6.6
Returns to Scale in Electric
Power Generation
6.6
TECHNOLOGICAL PROGRESS
APPENDIX
THE ELASTICITY OF SUBSTITUTION
FOR A COBB–DOUGLAS PRODUCTION
FUNCTION
200
Technological Progress . . .
and Educational Progress
APPLICATION 6.7
Can They Do It Better and Cheaper?
In his classic collection of stories, I, Robot, Isaac Asimov explores a world in which humans and robots
coexist. Asimov, the grand master of science fiction, published I, Robot in 1950 and depicted a world
inhabited by intelligent robots who could lead, laugh, and scheme and who occasionally even needed robot
psychologists. At a time that was, by today’s standards, distinctively low tech (e.g., the first commercially
available computer, UNIVAC I, was still a year away), Asimov’s stories were indeed science fiction. But little
more than 60 years later, the notion of a world in which robots play a central role is no longer so far fetched.
Significant industrial applications of robots go back at least 25 years, when automobile manufacturers
such as General Motors began installing robots along their assembly lines in order to save labor costs. In
the 1990s, producers of semiconductor chips began adding robots to their “fabs,” expensive factories that
can cost more than $3 billion to construct. For chip manufacturers, robots were an attractive alternative to
human workers because in order to avoid contaminating chips, fabs must be 1,000 times cleaner than a
hospital operating room, a standard that was easier to attain with robots than with humans. Today, with
worldwide sales of robots booming, robots are taking on ever more imaginative roles: Robots can perform
prostate surgery, drive a car, assume the role of lifeguard at a swimming pool; robots can even milk a cow!
Robots that perform sophisticated tasks are not cheap. This means that a business, such as a semiconductor maker, that contemplates employing robots faces an important trade-off: Are the production cost
savings that result from using robots worth the investment needed to acquire the robots in the first place?
With the sophisticated and self-sufficient robots that are available today, many businesses have concluded
that the answer to this question is “yes.”
This chapter lays the foundation for studying this type of economic trade-off.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Explain how a production function represents the various technological recipes the firm
can choose.
• Illustrate the difference between technologically efficient combinations of inputs and outputs
and technologically inefficient combinations of inputs and outputs.
• Distinguish between the concepts of total product,
marginal product, and average product for a production function with a single input.
• Describe the concept of diminishing marginal returns.
• Illustrate graphically how the graphs of the marginal product and average product functions relate
to the graph of the total product function.
• Demonstrate how a production function with two
variable inputs can be represented by isoquants.
• Derive the equation of an isoquant from the
equation of the production function.
201
202
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
• Explain how the concept of marginal rate of the technical substitution is related to the concept of
marginal product.
• Show graphically how a firm’s input substitution opportunities determine the shape of the firm’s isoquants.
• Describe how the concept of elasticity of substitution measures the firm’s input substitution opportunities.
• Compare and contrast a number of special production functions that are frequently used in microeconomic analysis: the linear production function, the Leontief production function, the Cobb–Douglas
production function, and the CES production function.
• Determine whether a production function exhibits increasing, constant, or decreasing returns to scale.
• Verify whether a change in a production function represents technological progress, and if it does,
determine whether the technological progress is labor-saving, neutral, or capital-saving.
6.1
INTRODUCTION
TO I N P U T S
AND
PRODUCTION
FUNCTIONS
inputs Resources, such
as labor, capital equipment,
and raw materials, that are
combined to produce
finished goods.
factors of production
Resources that are used to
produce a good.
output The amount of a
good or service produced
by a firm.
production function
A mathematical representation that shows the
maximum quantity of output
a firm can produce given
the quantities of inputs that
it might employ.
P
roduction of goods and services involves transforming resources—such as labor
power, raw materials, and the services provided by facilities and machines—into finished products. Semiconductor producers, for example, combine the labor services
provided by their employees and the capital services provided by fabs, robots, and
processing equipment with raw materials, such as silicon, to produce finished chips.
The productive resources, such as labor and capital equipment, that a firm uses to
manufacture goods and services are called inputs or factors of production, and the
amount of goods and services produced is the firm’s output.
As our semiconductor example suggests, real firms can often choose one of several
combinations of inputs to produce a given volume of output. A semiconductor firm can
produce a given number of chips using workers and no robots or using fewer workers
and many robots. The production function is a mathematical representation of the
various technological recipes from which a firm can choose to configure its production
process. In particular, the production function tells us the maximum quantity of output
the firm can produce given the quantities of the inputs that it might employ. We will
write the production function this way:
Q f (L, K )
(6.1)
where Q is the quantity of output, L is the quantity of labor used, and K is the quantity of capital employed. This expression tells us that the maximum quantity of output the firm can get depends on the quantities of labor and capital it employs. We
could have listed more categories of inputs, but many of the important trade-offs that
real firms face involve choices between labor and capital (e.g., robots and workers for
semiconductor firms). Moreover, we can develop the main ideas of production theory
using just these two categories of inputs.
The production function in equation (6.1) is analogous to the utility function
in consumer theory. Just as the utility function depends on exogenous consumer
tastes, the production function depends on exogenous technological conditions.
Over time, these technological conditions may change, an occurrence known as
6 . 1 I N T R O D U C T I O N TO I N P U T S A N D P R O D U C T I O N F U N C T I O N S
203
Technically efficient
Q = f (L)
Q, units of output per year
D
C
B
A
FIGURE 6.1
Technically inefficient
L, units of labor per year
Technical Efficiency and
Inefficiency
At points C and D the firm is technically
efficient. It is producing as much output as it
can with the production function Q f (L)
given the quantity of labor it employs. At
points A and B the firm is technically
inefficient. It is not getting as much output
as it could with its labor.
technological progress, and the production function may then shift. We discuss
technological progress in Section 6.6. Until then, we will view the firm’s production function as fixed and unchangeable.
The production function in equation (6.1) tells us the maximum output a firm
could get from a given combination of labor and capital. Of course, inefficient management could reduce output from what is technologically possible. Figure 6.1 depicts
this possibility by showing the production function for a single input, labor: Q f (L).
Points on or below the production function make up the firm’s production set, the
set of technically feasible combinations of inputs and outputs. Points such as A and B
in the production set are technically inefficient (i.e., at these points the firm gets less
output from its labor than it could). Points such as C and D, on the boundary of the
production set, are technically efficient. At these points, the firm produces as much
output as it possibly can given the amount of labor it employs.
If we invert the production function, we get a function L g (Q), which tells us
the minimum amount of labor L required to produce a given amount of output Q. This
function is the labor requirements function. If, for example, Q 1L is the production function, then L Q2 is the labor requirements function; thus, to produce an
output of 7 units, a firm will need at least 72 49 units of labor.
Because the production function tells us the maximum attainable output from a
given combination of inputs, we will sometimes write Q ⱕ f (L, K ) to emphasize that
the firm could, in theory, produce a quantity of output that is less than the maximum
level attainable given the quantities of inputs it employs.
production set The
set of technically feasible
combinations of inputs
and outputs.
technically inefficient
The set of points in the
production set at which the
firm is getting less output
from its labor than it could.
technically efficient
The set of points in the production set at which the firm
is producing as much output
as it possibly can given the
amount of labor it employs.
labor requirements
function A function that
indicates the minimum
amount of labor required to
produce a given amount of
output.
204
CHAPTER 6
A P P L I C A T I O N
INPUTS AND PRODUCTION FUNCTIONS
6.1
Competition Breeds Efficiency
Does more competition make firms more efficient?
Economists have long attempted to answer this question. A classic study by Richard Caves and David Barton
examined the extent of technical inefficiency among
U.S. manufacturers.1 For the typical manufacturer, they
estimated that the ratio of actual output to the maximum output that would be attainable given the firm’s
labor and capital was 63 percent. (In the notation used
in the text, we would say that Q/f(L,K) 0.63 for the
typical firm.) This finding implies that the typical U.S.
manufacturer was technically inefficient.
According to Caves and Barton, an important
determinant of technical efficiency is the extent to
which a firm faces competition from other firms. They
found that firms in industries facing less competition
from foreign firms tended to be less technically efficient. In addition, technical efficiency was lower in industries where sales were concentrated in relatively
few firms. These findings suggest that the pressure of
competition—whether from imports or other firms in
the industry—tends to motivate firms to search for
ways to get as much output as they can from their
existing combinations of inputs, thus moving them
closer to the boundaries of their production sets.
A recent study by David Brown and John Earle
examined the effects of an abrupt transition to
greater competition on firm efficiency.2 Prior to 1992,
the Russian economy was centrally planned, with
most firms managed by government agencies. Prices,
labor markets, and most other aspects of the economy
6.2
PRODUCTION
FUNCTIONS
WITH A
SINGLE INPUT
were strictly regulated. Many companies had statesponsored monopolies. On January 1, 1992, the
Russian government implemented economic “Shock
Therapy” by simultaneously deregulating prices,
labor markets, foreign trade, and entry into industries. For the first time, Russian firms were forced to
compete with each other and with foreign firms.
Brown and Earle studied the impact of this dramatic transition on efficiency in nearly 15,000 Russian
firms during the 1990s. Their findings were similar to
those of Caves and Barton: Domestic product market
competition and foreign competition had strong positive effects on firm efficiency. Brown and Earle also
found that better transportation infrastructure increased efficiency because it facilitated competition
among firms across Russia’s large territory. They estimated that the positive impact of domestic product
market competition on technical efficiency was 45 to
60 percent greater in regions where transportation
infrastructure was good than in regions where transportation infrastructure was poor.
Brown and Earle’s study found that private Russian
firms that were part of joint ventures with foreign companies performed better than state-run Russian companies at the beginning of the transition. This could have
been caused by the transfer of management techniques
from more efficient foreign firms, as well as better incentives and greater flexibility in private firms. However,
these relative advantages declined over time. One interpretation for this decline is that the competition from
such firms motivated greater efficiency from firms that
were state-run or did not have foreign partners.
T
he business press is full of discussions of productivity, which broadly refers to the
amount of output a firm can get from the resources it employs. We can use the production function to illustrate a number of important ways in which the productivity
of inputs can be characterized. To illustrate these concepts most clearly, we will start
our study of production functions with the simple case in which the quantity of output depends on a single input, labor.
1
Richard Caves and David Barton, Efficiency in U.S. Manufacturing Industries (Cambridge, MA: MIT
Press, 1990).
2
David Brown and John Earle, “Market Competition and Firm Performance in Russia,” Russian Economic
Trends 9, no. 1 (March 2000): 13–18.
205
6.2 PRODUCTION FUNCTIONS WITH A SINGLE INPUT
TABLE 6.1 Total
Product Function
Q, thousands of chips per day
200
Total
product
function
150
100
Q
0
6
12
18
24
30
0
30
96
162
192
150
*L is expressed in thousands of man-hours per
day, and Q is expressed
in thousands of semiconductor chips per day.
50
Increasing
marginal
returns
0
L*
6
Diminishing
marginal
returns
12
18
Diminishing
total
returns
24
30
36
L, thousands of man-hours per day
FIGURE 6.2
Total Product Function
The total product function shows the relationship between the quantity of labor (L) and
the quantity of output (Q). Here the function has three regions: a region of increasing
marginal returns (L 12); a region of diminishing marginal returns (12 L 24); and a
region of diminishing total returns (L 24).
TOTA L P R O D U C T F U N C T I O N S
Single-input production functions are sometimes called total product functions.
Table 6.1 shows a total product function for a semiconductor producer. It shows the
quantity of semiconductors Q the firm can produce in a year when it employs various
quantities L of labor within a fab of a given size with a given set of machines.
Figure 6.2 shows a graph of the total product function in Table 6.1. This graph
has four noteworthy properties. First, when L 0, Q 0. That is, no semiconductors can be produced without using some labor. Second, between L 0 and L 12,
output rises with additional labor at an increasing rate (i.e., the total product function
is convex). Over this range, we have increasing marginal returns to labor. When
there are increasing marginal returns to labor, an increase in the quantity of labor
increases total output at an increasing rate. Increasing marginal returns are usually
thought to occur because of the gains from specialization of labor. In a plant with a
small work force, workers may have to perform multiple tasks. For example, a worker
might be responsible for moving raw materials within the plant, operating the
machines, and inspecting the finished goods once they are produced. As more workers are added, workers can specialize—some will be responsible only for moving raw
materials in the plant; others will be responsible only for operating the machines; still
others will specialize in inspection and quality control. Specialization enhances the
marginal productivity of workers because it allows them to concentrate on the tasks at
which they are most productive.
Third, between L 12 and L 24, output rises with additional labor but at a
decreasing rate (i.e., the total product function is concave). Over this range we have
total product function
A production function. A
total product function with
a single input shows how
total output depends on
the level of the input.
increasing marginal
returns to labor The
region along the total product function where output
rises with additional labor
at an increasing rate.
206
CHAPTER 6
diminishing marginal
returns to labor The
diminishing marginal returns to labor. When there are diminishing marginal returns to labor, an increase in the quantity of labor still increases total output but at a
decreasing rate. Diminishing marginal returns set in when the firm exhausts its ability to increase labor productivity through the specialization of workers.
Finally, when the quantity of labor exceeds L 24, an increase in the quantity of
labor results in a decrease in total output. In this region, we have diminishing total
returns to labor. When there are diminishing total returns to labor, an increase in the
quantity of labor decreases total output. Diminishing total returns occur because of the
fixed size of the fabricating plant: if the quantity of labor used becomes too large, workers
don’t have enough space to work effectively. Also, as the number of workers employed
in the plant grows, their efforts become increasingly difficult to coordinate.3
region along the total product function in which output
rises with additional labor
but at a decreasing rate.
diminishing total
returns to labor The
region along the total
product function where
output decreases with
additional labor.
INPUTS AND PRODUCTION FUNCTIONS
M A R G I N A L A N D AV E R AG E P R O D U C T
average product of
labor The average
amount of output per unit
of labor.
We are now ready to characterize the productivity of the firm’s labor input. There are two
related, but distinct, notions of productivity that we can derive from the production function. The first is the average product of labor, which we write as APL. The average product of labor is the average amount of output per unit of labor.4 This is usually what commentators mean when they write about, say, the productivity of U.S. workers as compared
to their foreign counterparts. Mathematically, the average product of labor is equal to:
APL
total product
quantity of labor
Q
L
Table 6.2 and Figure 6.3 show the average product of labor for the total product function in Table 6.1. They show that the average product varies with the amount of labor
the firm uses. In our example, APL increases for quantities of labor less than L 18
and falls thereafter.
Figure 6.4 shows the graphs of the total product and average product curves
simultaneously. The average product of labor at any arbitrary quantity L0 corresponds
to the slope of a ray drawn from the origin to the point along the total product function corresponding to L0. For example, the height of the total product function at
point A is Q0, and the amount of labor is L0. The slope of the line segment connecting the origin to point A is Q0 ⲐL0, which is the average product APL0 per the equation
displayed above. At L 18, the slope of a ray from the origin attains its maximal value,
indicating that APL reaches its peak at this quantity of labor.
TABLE 6.2
3
Average Product of Labor
L
Q
6
12
18
24
30
30
96
162
192
150
Q
APL
L
5
8
9
8
5
We could also have diminishing total returns to other inputs, such as materials. For example, adding
fertilizer to an unfertilized field will increase crop yields. But too much fertilizer will burn out the crop,
and output will be zero.
4
The average product of labor is also sometimes called the average physical product of labor and is then
written as APPL.
207
6.2 PRODUCTION FUNCTIONS WITH A SINGLE INPUT
APL, MPL, chips per man-hour
10
A
5
APL is increasing
so MPL > APL
0
6
12
APL is decreasing
so MPL < APL
18
24
30
APL
36
L, thousands of man-hours per day
−5
−10
MPL
FIGURE 6.3
Average and Marginal Product Functions
APL is the average product function. MPL is the marginal product function. The marginal product
function rises in the region of increasing marginal returns (L 12) and falls in the region of
diminishing marginal returns (12 L 24). It becomes negative in the region of diminishing
total returns (L 24). At point A, where APL is at a maximum, APL MPL.
The other notion of productivity is the marginal product of labor, which we
write as MPL. The marginal product of labor is the rate at which total output changes
as the firm changes its quantity of labor:
MPL
change in total product
change in quantity of labor
¢Q
¢L
The marginal product of labor is analogous to the concept of marginal utility from consumer theory, and just as we could represent that curve graphically, we can also represent the marginal product curve graphically, as shown in Figure 6.3. Marginal product,
like average product, is not a single number but varies with the quantity of labor. In the
region of increasing marginal returns, where 0 ⱕ L 12, the marginal product function is increasing. When diminishing marginal returns set in, at L 12, the marginal
product function starts decreasing. When diminishing total returns set in, at L 24, the
marginal product function cuts through the horizontal axis and becomes negative. As
shown in the upper panel in Figure 6.4, the marginal product corresponding to any particular amount of labor L1 is the slope of the line that is tangent to the total product
function at L1 (line BC in the figure). Since the slopes of these tangent lines vary as we
move along the production function, the marginal product of labor must also vary.
In most production processes, as the quantity of one input (e.g., labor) increases,
with the quantities of other inputs (e.g., capital and land) held constant, a point will
be reached beyond which the marginal product of that input decreases. This phenomenon, which reflects the experience of real-world firms, seems so pervasive that economists call it the law of diminishing marginal returns.
marginal product of
labor The rate at which
total output changes as the
quantity of labor the firm
uses is changed.
law of diminishing
marginal returns
Principle that as the usage
of one input increases, the
quantities of other inputs
being held fixed, a point
will be reached beyond
which the marginal product
of the variable input will
decrease.
208
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
Q, thousands of chips per day
C
Marginal product at
L1 equals slope of
line BC
Average product at
L0 equals slope of
ray 0A
Q0
FIGURE 6.4
Relationship
among Total, Average, and
Marginal Product Functions
The marginal product of labor
at any point equals the slope of
the total product curve at that
point. The average product at
any point is equal to the slope
of the ray from the origin to
the total product curve at that
point.
A P P L I C A T I O N
APL, MPL , chips per man-hour
0
B
Total
product
function
A
L0
18 L1 24
L, thousands of man-hours per day
APL
0
L0
18 L1
24
L, thousands of man-hours per day
MPL
6.2
The Resurgence of Labor
Productivity in the United States
When the average product of labor is computed for
an entire economy—say, that of the United States—
what we get is a measure of overall labor productivity
in the economy. Labor productivity is an important
indicator of the overall well-being of an economy.
Rising labor productivity implies that more output
can be produced from a given amount of labor, and
when that is the case, the standard of living in the
economy rises over time. By contrast, when the
growth of labor productivity stalls, improvements in
the standard of living will slow down as well.
6.2 PRODUCTION FUNCTIONS WITH A SINGLE INPUT
The accompanying table shows the average annual
growth in labor productivity in the United States between 1947 and 2009.5 The table reveals a striking pattern: from 1947 though the mid-1970s, labor productivity
grew at a rate of about 2.5 to 3 percent per year.
However, from the mid-1970s through the mid-1990s,
the growth of labor productivity slowed significantly,
falling to a rate of about 1.4 to 1.5 percent annually.
Beginning in the late 1990s, there was a resurgence of
labor productivity, with annual growth rates between
1995 and 2005 averaging over 2.9 percent. Keeping in
mind that the 1995–2005 period encompassed 9/11, the
“Dot Bomb” technology crash, the recession of 2001, and
numerous corporate governance scandals, the growth of
labor productivity over this period is impressive indeed.
Growth in Labor Productivity in the
United States, 1947–2009
Years
Annual Growth Rate in Labor
Productivity
1947–1955
1955–1965
1965–1975
1975–1985
1985–1995
1995–2005
2005–2009
3.21%
2.61%
2.18%
1.38%
1.51%
2.94%
1.90%
What explains the slowdown in labor productivity
beginning in the mid-1970s? Based on the study of
detailed industry-level data on labor productivity,
William Nordhaus finds that the largest slowdowns in
productivity growth were in energy-reliant industries
such as pipelines, oil and gas extraction, and automobile repair services.6 This suggests, then, that the primary culprits in the slowdown of productivity growth
in the United States were the oil shocks of 1973 and
1979. As Nordhaus puts it, “In a sense, the energy
5
209
shocks were the earthquake, and the industries with
the largest slowdown were nearest the epicenter of
the tectonic shifts in the economy.”
To explain the resurgence of labor productivity
since 1995, it is useful to identify factors that
would tend to make workers more productive. One
important factor that can affect labor productivity is
the amount of sophistication of the capital equipment available to workers. The period between 1995
and 2005 was one of rapid growth in the sophistication and ubiquity of information and communications technologies. Thus the hypothesis that the
post–1995 resurgence of labor productivity is attributable to increases in the quantity and quality of
capital (what economists call “capital deepening”) is
quite plausible.
A second factor affecting the productivity of
labor is the increase in the quality of labor itself.
Improvements in aggregate labor quality occur primarily when the ratio of high-skill to lower-skill
workers increases, which in turn occurs as firms demand higher levels of experience and education from
their workers (which, of course, is related to the increased sophistication of the capital that workers use
in their jobs).
So what does explain the resurgence of U.S. productivity growth since 1995? According to an analysis
by Dale Jorgenson, Mun Ho, and Kevin Stiroh (JHS),
the most important factor was capital deepening.7
Indeed, JHS find that capital deepening explains
more than half of the jump in the labor productivity
growth rate in the period after 1975. As one might
expect, much of the capital deepening was due to
improvements in information and communications
technology. On the other hand, JHS find that changes
in labor quality played a relatively small role in driving
productivity growth upward, suggesting that changes
in the mix between high- and low-skill workers have
not been responsible for the increases in the growth
of labor productivity since 1995.
The growth rates were calculated from changes in output per hour in all nonfarm business in the United
States, using data from the Bureau of Labor Statistics website www.bls.gov/data/. Data for 2005–2009 are
calculated through the second quarter of 2009 only.
6
William Nordhaus, “Retrospective on the 1970s Productivity Slowdown,” NBER Working Paper
No. W10950 (December 2004), available at SSRN, http://ssrn.com/abstract=629592.
7
Dale Jorgenson, Mun Ho, and Kevin Stiroh, “Will the U.S. Productivity Resurgence Continue?” Current
Issues in Economics & Finance 10, no. 13, Federal Reserve Bank of New York (December 2004): 1–7.
210
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
R E L AT I O N S H I P B E T W E E N M A R G I N A L
A N D AV E R AG E P R O D U C T
As with other average and marginal concepts you will study in this book (e.g., average
cost versus marginal cost), there is a systematic relationship between average product
and marginal product. Figure 6.3 illustrates this relationship:
• When average product is increasing in labor, marginal product is greater than
average product. That is, if APL increases in L, then MPL APL.
• When average product is decreasing in labor, marginal product is less than average
product. That is, if APL decreases in L, then MPL APL.
• When average product neither increases nor decreases in labor because we are at a
point at which APL is at a maximum (point A in Figure 6.3), then marginal
product is equal to average product.
The relationship between marginal product and average product is the same as
the relationship between the marginal of anything and the average of anything. To
illustrate this point, suppose that the average height of students in your class is 160 cm.
Now Mike Margin joins the class, and the average height rises to 161 cm. What do
we know about Mike’s height? Since the average height is increasing, the “marginal
height” (Mike Margin’s height) must be above the average. If the average height had
fallen to 159 cm, it would have been because his height was below the average. Finally,
if the average height had remained the same when Mike joined the class, his height
would have had to exactly equal the average height in the class.
The relationship between average and marginal height in your class is the same as
the relationship between average and marginal product shown in Figure 6.3. It is also the
relationship between average and marginal cost that we will study in Chapter 8 and the
relationship between average and marginal revenue that we will see in Chapter 11.
6.3
PRODUCTION
FUNCTIONS
WITH MORE
THAN ONE
INPUT
The single-input production function is useful for developing key concepts, such as
marginal and average product, and building intuition about the relationships between these concepts. However, to study the trade-offs facing real firms, such as
semiconductor companies thinking about substituting robots for humans, we need to
study multiple-input production functions. In this section, we will see how to describe a multiple-input production function graphically, and we will study a way to
characterize how easily a firm can substitute among the inputs within its production
function.
TOTA L P R O D U C T A N D M A R G I N A L P R O D U C T
WITH TWO INPUTS
To illustrate a production function with more than one input, let’s consider a situation
in which the production of output requires two inputs: labor and capital. This might
broadly illustrate the technological possibilities facing a semiconductor manufacturer
contemplating the use of robots (capital) or humans (labor).
211
6.3 PRODUCTION FUNCTIONS WITH MORE THAN ONE INPUT
TABLE 6.3
Production Function for Semiconductors*
K **
L**
0
6
12
18
24
30
0
6
12
18
24
30
0
0
0
0
0
0
0
5
15
25
30
23
0
15
48
81
96
75
0
25
81
137
162
127
0
30
96
162
192
150
0
23
75
127
150
117
*Numbers in table equal the output that can be produced with various combinations of labor
and capital.
**L is expressed in thousands of man-hours per day; K is expressed in thousands of machinehours per day; and Q is expressed in thousands of semiconductor chips per day.
Table 6.3 shows a production function (or, equivalently, the total product function) for semiconductors, where the quantity of output Q depends on the quantity of
labor L and the quantity of capital K employed by the semiconductor firm. Figure 6.5
shows this production function as a three-dimensional graph. The graph in Figure 6.5
is called a total product hill––a three-dimensional graph that shows the relationship
between the quantity of output and the quantity of the two inputs employed by the
firm.8
total product hill A
three-dimensional graph of
a production function.
C
Q (thousands of
B
semiconductor chips
per day) = height of
hill at any point
thousands
K
of machine30
hours
per day
North
A
24
18
12
6
0
6
12
18
L, thousands
30 of man-hours
per day
24
East
FIGURE 6.5
Total Product Hill
The height of the hill at any point is equal to the quantity of output Q attainable from the
quantities of labor L and capital K corresponding to that point.
8
In Figure 6.5, we show the “skeleton,” or frame, of the total product hill, so that we can draw various
lines underneath it. Figure 6.6 shows the same total product hill as a solid surface.
212
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
The height of the hill at any point is equal to the quantity of output Q the firm
produces from the quantities of inputs it employs. We could move along the hill in any
direction, but it is easiest to imagine moving in either of two directions. Starting from
any combination of labor and capital, we could move eastward by increasing the quantity of labor, or we could move northward by increasing the quantity of capital. As we
move either eastward or northward, we move to different elevations along the total
product hill, where each elevation corresponds to the particular quantity of output.
Let’s now see what happens when we fix the quantity of capital at a particular
level, say K 24, and increase the quantity of labor. The outlined column in Table 6.3
shows that when we do this, the quantity of output initially increases but then
begins to decrease (when L 24). In fact, notice that the values of Q in Table 6.3 are
identical to the values of Q for the total product function in Table 6.1. This shows that
the total product function for labor can be derived from a two-input production function by holding the quantity of capital fixed at a particular level (in this case, at K 24)
and varying the quantity of labor.
We can make the same point with Figure 6.5. Let’s fix the quantity of capital at
K 24 and move eastward up the total product hill by changing the quantity of labor.
As we do so, we trace out the path ABC, with point C being at the peak of the hill.
This path has the same shape as the total product function in Figure 6.2, just as the
K 24 column in Table 6.3 corresponds exactly to Table 6.1.
Just as the concept of total product extends directly to the multiple input case, so
too does the concept of marginal product. The marginal product of an input is the rate
at which output changes as the firm changes the quantity of one of its inputs, holding
the quantities of all other inputs constant. The marginal product of labor is given by:
change in quantity of output Q
change in quantity of labor L
Q
L K is held constant
MPL
K is held constant
(6.2)
Similarly, the marginal product of capital is given by:
change in quantity of output Q
change in quantity of capital K
Q
K L is held constant
MPK
L is held constant
(6.3)
The marginal product tells us how the steepness of the total product hill varies as
we change the quantity of an input, holding the quantities of all other inputs fixed.
The marginal product at any particular point on the total product hill is the steepness
of the hill at that point in the direction of the changing input. For example, in Figure 6.5,
the marginal product of labor at point B—that is, when the quantity of labor is 18 and
the quantity of capital is 24—describes the steepness of the total product hill at point
B in an eastward direction.
I S O Q UA N T S
To illustrate economic trade-offs, it helps to reduce the three-dimensional graph of
the production function (the total product hill) to two dimensions. Just as we used a
contour plot of indifference curves to represent utility functions in consumer theory,
213
6.3 PRODUCTION FUNCTIONS WITH MORE THAN ONE INPUT
TABLE 6.4
Production Function for Semiconductors*
K**
0
6
12
18
24
30
L**
0
6
12
18
24
30
0
0
0
0
0
0
0
5
15
25
30
23
0
15
48
81
96
75
0
25
81
137
162
127
0
30
96
162
192
150
0
23
75
127
150
117
*Numbers in table equal the output that can be produced with various combinations of labor
and capital.
**L is expressed in thousands of man-hours per day; K is expressed in thousands of machine-hours
per day; and Q is expressed in thousands of semiconductor chips per day.
we can also use a contour plot to represent the production function. However, instead
of calling the contour lines indifference curves, we call them isoquants. Isoquant
means “same quantity”: any combination of labor and capital along a given isoquant
allows the firm to produce the same quantity of output.
To illustrate, let’s consider the production function described in Table 6.4 (the
same function as in Table 6.3). From this table we see that two different combinations
of labor and capital—(L 6, K 18) and (L 18, K 6)—result in an output of
Q 25 units (where each “unit” of output represents a thousand semiconductors).
Thus, each of these input combinations is on the Q 25 isoquant.
The same isoquant is shown in Figure 6.6 (equivalent to Figure 6.5), illustrating the
total product hill for the production function in Table 6.4. Suppose that you started
isoquant A curve that
shows all of the combinations of labor and capital
that can produce a given
level of output.
All combinations of L and K
along path ABCDE produce
25 units of output, where
each "unit" represents a
thousand semiconductor
chips.
thousands
K
of machine30
hours
per day North
A
B
E
C
24
D
18
12
6
0
FIGURE 6.6
6
12
18
24
Isoquants and the Total Product Hill
If we start at point A and walk along the hill so that our elevation remains unchanged at
25 units of output, then we will trace out the path ABCDE. This is the 25-unit isoquant for
this production function.
L, thousands
30 of man-hours
per day
East
214
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
walking along the total product hill from point A with the goal of maintaining a constant
elevation (i.e., a constant quantity of output). Line segment ABCDE is the path you
should follow. At each input combination along this path, the height of the total product hill is Q 25 (i.e., each of these input combinations is on the Q 25 isoquant).
From this example, we can see that an isoquant is like a line on a topographical
map, such as the one of Mount Hood, in Oregon, in Figure 6.7. A line on this
8250
7250
7000
7750
7500
7500
00
65
62
50
60
00
(a)
Glisan
Glacier
7750
000
82
50
8
Pulpit
Rock Coe
Glacier
8500
7750
50
50
1050
1075 0
0
00
The
Chimney Newton Clark
00
Glacier
0
Coalman
Glacier
11
Newton Clark
Glacier
8000
Zigzag
Glacier
(b)
FIGURE 6.7
Mississippi
82
5
0
87
50
8000
7500
7250
Steel
Cliff
9250
775
0
82
95
Hot
Rocks
Eliot
Glacier
7500
50
67
00
65
Illumination
Rock
Cooper
Spur
97
0
00
10 50
2
10
Reid
Glacier
Eliot
Glacier
9000
8750
8500
9
0
0
0
9
95 250
00
Mt. Hood Wilderness Area
Sandy
Glacier
White River 8500
Glacier Newton Clark
Glacier
7000
Three-Dimensional and Topographic Map for Mount Hood
Panel (a) is a three-dimensional map of Mount Hood. The product hill in Figure 6.6 is analogous to
this kind of map. Panel (b) shows a topographic map of Mount Hood. A graph of isoquants (as in
Figure 6.8) is analogous to this topographic map.
Source: www.delorme.com.
72
50
K, thousands of machine-hours per day
6.3 PRODUCTION FUNCTIONS WITH MORE THAN ONE INPUT
B
18
Q3 > Q2
Q2 > 25
D
6
0
215
Q1 = 25
6
18
L, thousands of man-hours per day
FIGURE 6.8
Isoquants
for the Production Function
in Table 6.4 and Figure 6.6
Every input combination of
labor and capital along the
Q1 25 isoquant (in particular, combinations B and D)
produces the same output,
25,000 semiconductor chips
per day. As we move to the
northeast, the isoquants
correspond to progressively
higher outputs.
topographical map shows points in geographic space at which the elevation of the land
is constant. The total product hill in Figure 6.6 is analogous to the three-dimensional
map of Mount Hood in panel (a) of Figure 6.7, and the isoquants of the total product
hill (see Figure 6.8) are analogous to the lines on the topographical map of Mount
Hood in panel (b) of Figure 6.7.
Figure 6.8 shows isoquants for the production function in Table 6.4 and
Figure 6.6. The fact that the isoquants are downward sloping in Figure 6.8 illustrates
an important economic trade-off: A firm can substitute capital for labor and keep its
output unchanged. If we apply this idea to a semiconductor firm, it tells us that the
firm could produce a given quantity of semiconductors using lots of workers and a
small number of robots or using fewer workers and more robots. Such substitution is
always possible whenever both labor and capital (e.g., robots) have positive marginal
products.
Any production function has an infinite number of isoquants, each one corresponding to a particular level of output. In Figure 6.8, isoquant Q1 corresponds to
25 units of output. Notice that points B and D along this isoquant correspond to the
highlighted input combinations in Table 6.4. When both inputs have positive marginal products, using more of each input increases the amount of output attainable.
Hence, isoquants Q2 and Q3, to the northeast of Q1 in Figure 6.8, correspond to larger
and larger quantities of output.
An isoquant can also be represented algebraically, in the form of an equation, as
well as graphically (like the isoquants in Figure 6.8). For a production function like the
ones we have been considering, where quantity of output Q depends on two inputs
(quantity of labor L and quantity of capital K ), the equation of an isoquant would express K in terms of L. Learning-By-Doing Exercise 6.1 shows how to derive such an
equation.
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INPUTS AND PRODUCTION FUNCTIONS
L E A R N I N G - B Y- D O I N G E X E R C I S E 6 . 1
D
Deriving the Equation of an Isoquant
Problem
(a) Consider the production function whose equation is
given by the formula Q 1K L. What is the equation of
the isoquant corresponding to Q 20?
(b) For the same production function, what is the general equation of an isoquant, corresponding to any level
of output Q?
Solution
(a) The Q 20 isoquant represents all of the combinations of labor and capital that allow the firm to produce
20 units of output. For this isoquant, the production
function satisfies the following equation:
20 1KL
To find the equation of the 20-unit isoquant, we solve
this equation for K in terms of L. The easiest way to do
this is to square each side of equation (6.4) and then
solve for K in terms of L. Doing this yields K 400/L.
This is the equation of the 20-unit isoquant.
(b) In the general case, we begin with the production
function itself: Q 1K L. To find the general equation
of an isoquant, we again square each side and solve for
K in terms of L. Doing this yields K Q2/L. (If you substitute Q 20 into this equation, you get the equation
of the 20-unit isoquant that we solved for above.)
Similar Problems: 6.9, 6.10, 6.11
(6.4)
ECONOMIC AND UNECONOMIC REGIONS
OF PRODUCTION
uneconomic region of
production The region
of upward-sloping or
backward-bending isoquants.
In the uneconomic region,
at least one input has a
negative marginal product.
economic region of
production The region
where the isoquants are
downward sloping.
The isoquants in Figure 6.8 are downward sloping: In the range of values of labor and
capital shown in the graph, as we increase the amount of labor we use, we can hold
output constant by reducing the amount of capital. But now look at Figure 6.9, which
shows the same isoquants when we expand the scale of Figure 6.8 to include quantities
of labor and capital greater than 24,000 man-hours and machine-hours per day. The isoquants now have upward-sloping and backward-bending regions. What does this mean?
The upward-sloping and backward-bending regions correspond to a situation in
which one input has a negative marginal product, or what we earlier called diminishing total returns. For example, the upward-sloping region in Figure 6.9 occurs
because there are diminishing total returns to labor (MPL 0), while the backwardbending region arises because of diminishing total returns to capital (MPK 0). If we
have diminishing total returns to labor, then as we increase the quantity of labor, holding the quantity of capital fixed, total output goes down. Thus, to keep output constant (remember, this is what we do when we move along an isoquant), we must also
increase the amount of capital to compensate for the diminished total returns to labor.
A firm that wants to minimize its production costs should never operate in a region
of upward-sloping or backward-bending isoquants. For example, a semiconductor producer should not operate at a point such as A in Figure 6.9 where there are diminishing total returns to labor. The reason is that it could produce the same output but at a
lower cost by producing at a point such as E. By producing in the range where the marginal product of labor is negative, the firm would be wasting money by spending it on
unproductive labor. For this reason, we refer to the range in which isoquants slope upward or bend backward as the uneconomic region of production. By contrast, the
economic region of production is the region of downward-sloping isoquants. From
now on, we will show only the economic region of production in our graphs.
217
6.3 PRODUCTION FUNCTIONS WITH MORE THAN ONE INPUT
K, thousands of machine-hours per day
Uneconomic region
Isoquants are
backward bending
(MPK < 0)
24
B
18
Economic
region
Isoquants are
upward sloping
(MPL < 0)
E
A
D
6
0
Q = 25
6
18
24
L, thousands of man-hours per day
FIGURE 6.9
Economic and Uneconomic Regions of Production
The backward-bending and upward-sloping regions of the isoquants make up the uneconomic
region of production. In this region, the marginal product of one of the inputs is negative.
A cost-minimizing firm would never produce in the uneconomic region.
M A R G I N A L R AT E O F T E C H N I C A L S U B S T I T U T I O N
A semiconductor firm that is contemplating investments in sophisticated robotics
would naturally be interested in the extent to which it can replace humans with robots.
That is, the firm will need to consider the question: How many robots will it need to
invest in to replace the labor power of one worker? Answering this question will be
crucial in determining whether an investment in robotics would be worthwhile.
The “steepness” of an isoquant determines the rate at which the firm can substitute between labor and capital in its production process. The marginal rate of technical substitution of labor for capital, denoted by MRTSL,K, measures how steep an
isoquant is. The MRTSL,K tells us the following:
• The rate at which the quantity of capital can be decreased for every one-unit
increase in the quantity of labor, holding the quantity of output constant, or
• The rate at which the quantity of capital must be increased for every one-unit
decrease in the quantity of labor, holding the quantity of output constant.
The marginal rate of technical substitution is analogous to the marginal rate of
substitution from consumer theory. Just as the marginal rate of substitution of good X
for good Y is the negative of the slope of an indifference curve drawn with X on the
marginal rate of
technical substitution
of labor for capital
The rate at which the
quantity of capital can be
reduced for every one-unit
increase in the quantity of
labor, holding the quantity
of output constant.
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CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
Slope of tangent line A = –2.5
Slope of tangent line B = –0.4
Marginal Rate of
Technical Substitution
of Labor for Capital
(MRTSL,K) along an
Isoquant
At point A, the MRTSL,K
is 2.5. Thus, the firm can
hold output constant by
replacing 2.5 machinehours of capital services
with an additional manhour of labor. At point
B, the MRTSL,K is 0.4.
Here, the firm can hold
output constant by replacing 0.4 machine-hours
of capital with an additional man-hour of labor.
diminishing marginal
rate of technical
substitution A feature
of a production function in
which the marginal rate of
technical substitution of
labor for capital diminishes
as the quantity of labor
increases along an isoquant.
K, machine-hours per day
FIGURE 6.10
MRTSL,K at A = 2.5
MRTSL,K at B = 0.4
50
A
B
20
Q = 1000
0
20
50
L, man-hours per day
horizontal axis and Y on the vertical axis, the marginal rate of technical substitution of
labor for capital is the negative of the slope of an isoquant drawn with L on the horizontal axis and K on the vertical axis. The slope of an isoquant at a particular point is
the slope of the line that is tangent to the isoquant at that point, as Figure 6.10 shows.
The negative of the slope of the tangent line is the MRTSL,K at that point.
Figure 6.10 illustrates the MRTSL,K along the Q 1000 unit isoquant for a particular production function. At point A, the slope of the line tangent to the isoquant is
⫺2.5. Thus, MRTSL,K 2.5 at point A, which means that, starting from this point, we
can substitute 1.0 man-hour of labor for 2.5 machine-hours of capital, and output will
remain unchanged at 1,000 units. At point B, the slope of the isoquant is ⫺0.4. Thus,
MRTSL,K 0.4 at point B, which means that, starting from this point, we can substitute 1.0 man-hour of labor for 0.4 machine-hour of capital without changing output.
As we move down along the isoquant in Figure 6.10, the slope of the isoquant increases (i.e., becomes less negative), which means that the MRTSL,K gets smaller and
smaller. This property is known as diminishing marginal rate of technical substitution. When a production function exhibits diminishing marginal rate of technical substitution (i.e., when the MRTSL,K along an isoquant decreases as the quantity of labor
L increases), the isoquants are convex to the origin (i.e., bowed in toward the origin).
We can show that there is a precise connection between MRTSL,K and the marginal products of labor (MPL) and capital (MPK). Note that when we change the quantity of labor by ⌬L units and the quantity of capital by ⌬K units of capital, the change
in output that results from this substitution would be as follows:
¢Q change in output from change in quantity of capital
change in output from change in quantity of labor
From equations (6.2) and (6.3), we know that
change in output from change in quantity of capital (¢K )(MPK )
change in output from change in quantity of labor (¢L)(MPL )
6 . 4 S U B S T I T U TA B I L I T Y A M O N G I N P U T S
219
Thus, ⌬Q (⌬K )(MPK) ⫹ (⌬L)(MPL). Along a given isoquant, output is unchanged
(i.e., ⌬Q 0). So, 0 (⌬K )(MPK ) ⫹ (⌬L)(MPL), or ⫺ (⌬K )(MPK) (⌬L)(MPL),
which can be rearranged to
⫺
MPL
¢K
¢L
MPK
But ⫺⌬K/⌬L is the negative of the slope of the isoquant, which is equal to the
MRTSL,K. Thus,
MPL
MRTSL, K
MPK
(6.5)
This shows that the marginal rate of technical substitution of labor for capital is equal
to the ratio of the marginal product of labor (MPL) to the marginal product of capital
(MPK). (This is analogous to the relationship between marginal rate of substitution
and marginal utility that we saw in consumer theory.)
To illustrate why this relationship is significant, consider semiconductor production. Suppose that, at the existing input combination, an additional unit of labor would
increase output by 10 units, while an additional unit of capital (robots) would increase
output by just 2 units (i.e., MPL 10, while MPK 2). Thus, at our current input combination, labor has a much higher marginal productivity than capital. Equation (6.5)
tells us that the MRTSL,K 10/2 5, which means that the firm can substitute 1 unit
of labor for 5 units of capital without affecting output. Clearly, a semiconductor firm
would want to know the marginal productivity of both inputs before making an investment decision involving the mix between robots and human workers.
S
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L E A R N I N G - B Y- D O I N G E X E R C I S E 6 . 2
D
Relating the Marginal Rate of Technical Substitution to Marginal Products
Problem At first glance, you might
think that when a production function has a diminishing
marginal rate of technical substitution of labor for capital, it must also have diminishing marginal products of
capital and labor. Show that this is not true, using the
production function Q K L, with the corresponding
marginal products MPK L and MPL K.
Solution
First, note that MRTSL,K MPL ⲐMPK
K ⲐL, which diminishes as L increases and K falls as we move
along an isoquant. So the marginal rate of technical substitution of labor for capital is diminishing. However, the
6.4
S U B S T I T U TABILITY
AMONG
INPUTS
marginal product of capital MPK is constant (not diminishing)
as K increases (remember, the amount of labor is held fixed
when we measure MPK ). Similarly, the marginal product of
labor is constant (again, because the amount of capital is
held fixed when we measure MPL ). This exercise demonstrates that it is possible to have a diminishing marginal rate
of technical substitution even though both of the marginal
products are constant. The distinction is that in analyzing
MRTSL,K , we move along an isoquant, while in analyzing
MPL and MPK, total output can change.
Similar Problems:
6.13, 6.14
A semiconductor manufacturer considering the choice between robots and workers
would want to know how easily it can substitute between these inputs. The answer to
this question will determine, in part, a firm’s ability to shift from one mode of production (e.g., a high ratio of labor to capital) to another (e.g., a low ratio of labor to capital)
as the relative prices of labor and capital change. In this section, we explore how to
describe the ease or difficulty with which a firm can substitute between different inputs.
220
CHAPTER 6
A P P L I C A T I O N
INPUTS AND PRODUCTION FUNCTIONS
6.3
High-Tech Workers versus
Low-Tech Workers
Over the last 20 years computers have become a
ubiquitous part of the business landscape. As this has
happened, firms have changed the composition of
their work force, replacing “low-tech” workers with
“high-tech” workers with greater knowledge about
and experience in using computers.
Using data on employment and computer usage
over the period 1988–1991, Frank Lichtenberg has
estimated the extent to which computer equipment
and computer-oriented personnel have contributed
to output in U.S. businesses.9 As part of this study,
Lichtenberg estimated the marginal rate of technical
substitution of high-tech labor—computer and information systems personnel—for low-tech labor—workers
employed in activities other than information systems
and technology. If we hold a typical U.S. firm’s output
fixed, and also assume that its stock of computer
equipment remains fixed, then the MRTS of high-tech
labor for low-tech labor is about 6. That is, once the
firm has determined its stock of computers, 1 hightech worker can be substituted for 6 low-tech workers
and output will remain unchanged. The reason that
this MRTS is so large is that once the firm has invested
in the acquisition of computer equipment, the marginal product of high-tech, computer-literate workers
is much higher than the marginal product of low-tech
workers with fewer computer skills.
Lichtenberg notes that his estimate of the MRTS
of low-tech and high-tech workers is consistent with
the experience of real firms. He notes, for example,
that when a large U.S. telecommunications company
decided to automate and computerize its responses
to customer service inquiries, it hired 9 new computer
programmers and information systems workers.
These new workers displaced 75 low-tech service
workers who had handled customer inquiries under
the old system. For every additional high-tech worker
the firm hired, it was able to replace more than 8
low-tech workers (75/9 ⬇ 8.3).
When the information technology revolution
first began, many feared that computers would lead
to mass unemployment as workers were replaced
by machines. However, that never happened. To see
why, note that computers sometimes are substitutes
for employees in production, but sometimes they are
complements.10 In the example of customer service
at a telecommunications company, computers were
used to substitute for employees. However, in many
jobs computers make employees more productive,
leading to greater demand for such high-skill workers. Computers are very good at tasks that are repetitive, use rules-based logic, are predictable, and can
be standardized. By contrast, employees are better at
tasks that require creativity, are unpredictable, and
require abstraction. Computers can improve the productivity of workers who perform such tasks in many
ways. For example, software such as spreadsheets
or relational databases, which can organize, process,
and analyze large quantities of data extremely
quickly, can greatly expand the scope and complexity
of analyses that workers in a business can do, potentially making them more productive in dealing with
difficult analytical issues.
DESCRIBING A FIRM’S INPUT SUBSTITUTION
O P P O R T U N I T I E S G R A P H I C A L LY
Let’s consider two possible production functions for the manufacture of semiconductors. Figure 6.11(a) shows the 1-million-chip-per-month isoquant for the first production function, while Figure 6.11(b) shows the 1-million-chip-per-month isoquant for
the second production function.
9
F. Lichtenberg, “The Output Contributions of Computer Equipment and Personnel: A Firm-Level
Analysis,” Economics of Innovation and New Technology 3, no. 3–4 (1995): 201–217.
10
Frank Levy and Richard Murnane, The New Division of Labor: How Computers Are Creating the Next Job
Market (Princeton, NJ: Princeton University Press, 2004).
221
50
45
0
A
B
Q=1
million
100
L, man-hours per month
(a) Production Function
with Limited Input
Substitution Opportunities
400
K, machine-hours per month
K, machine-hours per month
6 . 4 S U B S T I T U TA B I L I T Y A M O N G I N P U T S
50
A
B
20
0
Q=1
million
100
L, man-hours per month
(b) Production Function
with Abundant Input
Substitution Opportunities
FIGURE 6.11 Input Substitution Opportunities and the Shape of Isoquants
In panel (a), start from point A and move along the isoquant Q 1 million (i.e., holding output
constant). If the firm increases one input significantly (either L or K), it will only be able to
reduce the other input by a small amount. The firm is in a position where there is virtually no
substitutability between labor and capital. By contrast, in panel (b) the firm has abundant
substitution opportunities—that is, a significant increase in one input would allow the firm to
reduce the other input by a significant amount, holding output constant.
These two production functions differ in terms of how easy it is for the firm to
substitute between labor and capital. In Figure 6.11(a), suppose the firm operates at
point A, with 100 man-hours of labor and 50 machine-hours of capital. At this point,
it is hard for the firm to substitute labor for capital. Even if the firm quadruples its use
of labor, from 100 to 400 man-hours per month, it can reduce its quantity of capital
by only a small amount—from 50 to 45 machine-hours—to keep monthly output at
1 million chips. Figure 6.11(a) also indicates that the firm would face a similar difficulty in substituting capital for labor. A large increase in the number of machine-hours
(i.e., moving up the isoquant from point A) would yield only a small decrease in the
number of man-hours.
By contrast, with the production function illustrated in Figure 6.11(b), the
firm’s substitution opportunities are more abundant. Starting from the input combination at point A, the firm can reduce its employment of capital significantly—
from 50 to 20 machine-hours—if it increases the quantity of labor from 100 to 400
man-hours per month. Similarly, it could achieve significant reductions in manhours by increasing machine-hours. Of course, whether it would want to do either
would depend on the relative cost of labor versus capital (an issue we will study in
the next chapter), but the point is that the firm can potentially make substantial
labor-for-capital (or capital-for-labor) substitutions. In contrast to Figure 6.11(a),
the production function in Figure 6.11(b) gives the firm more opportunities to substitute between labor and capital.
400
222
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
A semiconductor firm would probably want to know whether its opportunities to substitute labor for capital are limited or abundant. But what distinguishes
one situation from the other? Note that in Figure 6.11(a), the MRTSL,K changes
dramatically as we move through point A on the 1-million-unit isoquant. Just
above point A on the isoquant, MRTSL,K is quite large, almost infinite, but just
beyond point A, the MRTSL,K abruptly shifts and becomes practically equal to 0.
By contrast, as we move along the isoquants in Figure 6.11(b), the MRTSL,K
changes gradually.
This suggests that the ease or difficulty with which a firm can substitute among
inputs depends on the curvature of its isoquants. Specifically,
• When the production function offers limited input substitution opportunities,
the MRTSL,K changes substantially as we move along an isoquant. In this case,
the isoquants are nearly L-shaped, as in Figure 6.11(a).
• When the production function offers abundant input substitution opportunities,
the MRTSL,K changes gradually as we move along an isoquant. In this case, the
isoquants are nearly straight lines, as in Figure 6.11(b).
elasticity of substitution
ELASTICITY OF SUBSTITUTION
The concept of elasticity of substitution is a numerical measure that can help us
describe the firm’s input substitution opportunities based on the relationships we just
derived in the previous section. Specifically, the elasticity of substitution measures
how quickly the marginal rate of technical substitution of labor for capital changes as
we move along an isoquant. Figure 6.12 illustrates elasticity of substitution. As labor
is substituted for capital, the ratio of the quantity of capital to the quantity of labor,
FIGURE 6.12 Elasticity of
Substitution of Labor for Capital
As the firm moves from point A to
point B, the capital–labor ratio K/L
changes from 4 to 1 (⫺75%), as does
the MRTSL,K. Thus, the elasticity of
substitution of labor for capital over
the interval A to B equals 1.
K, machine-hours per month
A measure of how easy it
is for a firm to substitute
labor for capital. It is equal
to the percentage change
in the capital–labor ratio
for every 1 percent change
in the marginal rate of
technical substitution of
labor for capital as we
move along an isoquant.
K/L at A = slope of ray 0A = 4
MRTSL,K at A = 4
K/L at B = slope of ray 0B = 1
MRTSL,K at B = 1
A
20
B
10
Q = 1 million
0
5
10
L, man-hours per month
223
6 . 4 S U B S T I T U TA B I L I T Y A M O N G I N P U T S
known as the capital–labor ratio, K ⲐL, must fall. The marginal rate of substitution of
capital for labor, MRTSL,K, also falls, as we saw in the previous section. The elasticity of
substitution, often denoted by , measures the percentage change in the capital–labor
ratio for each 1 percent change in MRTSL,K as we move along an isoquant:
s⫽
⫽
capital–labor ratio
The ratio of the quantity
of capital to the quantity
of labor.
percentage change in capital–labor ratio
percentage change in MRTSL, K
%¢ (KL )
(6.6)
% ¢MRTSL,K
Figure 6.12 illustrates the elasticity of substitution. Suppose a firm moves from
the input combination at point A (L 5 man-hours per month, K 20 machinehours per month) to the combination at point B (L 10, K 10). The capital–labor
ratio K ⲐL at A is equal to the slope of a ray from the origin to A (slope of ray 0A 4);
the MRTSL,K at A is equal to the negative of the slope of the isoquant at A (slope of
isoquant ⫽ ⫺4; thus, MRTSL,K ⫽ 4). At B, the capital–labor ratio equals the slope of
ray 0B, or 1; the MRTSL,K equals the negative of the slope of the isoquant at B, also 1.
The percent change in the capital–labor ratio from A to B is ⫺75 percent (from 4
down to 1), as is the percent change in the MRTSL,K between those points. Thus, the
elasticity of substitution over this interval is 1 (⫺75%Ⲑ⫺75% ⫽ 1).
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 6 . 3
D
Calculating the Elasticity of Substitution from a Production Function
Consider a production function whose equation is given by the formula Q ⫽ 1K L,
which has corresponding marginal products, MPL ⫽
Problem
1 K
1 L
ⴢ Show that the elasticity of suband MPK ⫽
2 BL
2B K
stitution for this production function is exactly equal to 1,
no matter what the values of K and L are.
Solution
First note that MRTSL, K ⫽
MPL
. In this
MPK
case that implies,
MRTSL, K ⫽
1 K
2 BL
1 L
2 BK
s⫽
K
L
Now recall that the definition of the elasticity of substitution is
%¢MRTSL, K
K
, it follows that %⌬MRTSL,K will
L
K
be exactly equal to %¢ a b. In other words, since the
L
marginal rate of substitution of labor for capital equals
the capital–labor ratio, the percentage change in the
marginal rate of substitution of labor for capital must
equal the percentage change in the capital–labor ratio.
K
Since %¢MRTSL,K ⫽ %¢a b, then using the definiL
tion of the elasticity of substitution, it follows that
Since MRTSL, K ⫽
which simplifies to
MRTSL, K ⫽
K
%¢a b
L
s⫽
K
%¢a b
L
K
%¢a b
L
⫽1
Similar Problems: 6.22, 6.23
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CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
In general, the elasticity of substitution can be any number greater than or equal to 0.
What is the significance of the elasticity of substitution?
• If the elasticity of substitution is close to 0, there is little opportunity to substitute between inputs. We can see this from equation (6.6), where will be close
to 0 when the percentage change in MRTSL,K is large, as in Figure 6.11(a).
• If the elasticity of substitution is large, there is substantial opportunity to substitute
between inputs. In equation (6.6), this corresponds to the fact that will be large if
the percentage change in MRTSL,K is small, as illustrated in Figure 6.11(b).
A P P L I C A T I O N
6.4
Elasticities of Substitution
in German Industries11
Industry
Chemicals
Stone and earth
Iron
Motor vehicles
Paper
Food
0.37
0.21
0.50
0.10
0.35
0.66
K, units of
capital per year
greater extent than they can in the production of
motor vehicles (elasticity of substitution 0.10). Figure
6.13 shows this graphically. Isoquants in iron production would have the shape of Figure 6.13(a), while
the isoquants in vehicle production would have the
shape of Figure 6.13(b).
L, units of labor per year
(a) Isoquants for German Iron Production
L, units of labor per year
(b) Isoquants for German Motor Vehicle Production
FIGURE 6.13 Isoquants for Iron and Motor Vehicle Production in Germany
The higher elasticity of substitution of labor for capital in the iron industry [panel (a)] implies
that labor and capital inputs are more easily substitutable in this industry than they are in the
production of motor vehicles [panel (b)].
11
Elasticity of
Substitution
K, units of
capital per year
Using data on output and input quantities over the
period 1970–1988, Claudia Kemfert has estimated the
elasticity of substitution between capital and labor in
a number of manufacturing industries in Germany.
Table 6.5 shows the estimated elasticities.
The results in Table 6.5 show two things. First,
the fact that the estimated elasticity of substitution is
less than 1 in all industries tells us that, generally
speaking, labor and capital inputs are not especially
substitutable in these industries. Second, the ease of
substitutability of capital for labor is higher in some
industries than in others. For example, in the production of iron (elasticity of substitution equal to 0.50),
labor and capital can be substituted to a much
TABLE 6.5 Elasticities of Substitution in
German Manufacturing Industries, 1970–1988
This example is based on “Estimated Substitution Elasticities of a Nested CES Production Function
Approach for Germany,” Energy Economics 20 (1998): 249–264.
6 . 4 S U B S T I T U TA B I L I T Y A M O N G I N P U T S
225
SPECIAL PRODUCTION FUNCTIONS
The relationship between the curvature of isoquants, input substitutability, and the
elasticity of substitution is most apparent when we compare and contrast a number of
special production functions that are frequently used in microeconomic analysis. In
this section, we will consider four special production functions: the linear production
function, the fixed-proportions production function, the Cobb–Douglas production
function, and the constant elasticity of substitution production function.
Linear Production Function (Perfect Substitutes)
H, quantity of high-capacity computers
In some production processes, the marginal rate of technical substitution of one input
for another may be constant. For example, a manufacturing process may require energy
in the form of natural gas or fuel oil, and a given amount of natural gas can always be
substituted for each liter of fuel oil. In this case, the marginal rate of technical substitution of natural gas for fuel oil is constant. Sometimes a firm may find that one type of
equipment may be perfectly substituted for another type. For example, suppose that a
firm needs to store 200 gigabytes of company data and is choosing between two types
of computers for that purpose. One has a high-capacity hard drive that can store 20 gigabytes of data, while the other has a low-capacity hard drive that can store 10 gigabytes
of data. At one extreme, the firm could purchase 10 high-capacity computers and no
low-capacity computers (point A in Figure 6.14). At the other extreme, it could purchase
no high-capacity computers and 20 low-capacity computers (point B in Figure 6.14). Or,
in the middle, it could purchase 5 high-capacity computers and 10 low-capacity computers (point C in Figure 6.14) because (5 ⫻ 20) ⫹ (10 ⫻ 10) 200.
In this example, the firm has a linear production function whose equation
would be Q 20H 10L, where H is the number of high-capacity computers the
firm employs, L is the number of low-capacity computers the firm employs, and Q is
the total gigabytes of data the firm can store. A linear production function is a production function whose isoquants are straight lines. Thus, the slope of any isoquant is
constant, and the marginal rate of technical substitution does not change as we move
along the isoquant.
linear production
function A production
function of the form Q
aL bK, where a and b are
positive constants.
Slope of isoquants = – 1/2, a constant
10
5
A
C
Q = 200 gigabytes
B
0
10
L, quantity of low-capacity computers
20
FIGURE 6.14 Isoquants for a Linear
Production Function
The isoquants for a linear production
function are straight lines. The MRTSL,H at
any point on an isoquant is thus a constant.
226
CHAPTER 6
perfect substitutes
Because MRTSL,H does not change as we move along an isoquant, ⌬MRTSL,H 0.
Using equation (6.6), this means that the elasticity of substitution for a linear production
function must be infinite ( q). In other words, the inputs in a linear production function are infinitely (perfectly) substitutable for each other. When we have a linear production function, we say that the inputs are perfect substitutes. In our computer example,
the fact that low-capacity and high-capacity computers are perfect substitutes means that
in terms of data storage capabilities, two low-capacity computers are just as good as one
high-capacity computer. Or, put another way, the firm can perfectly replicate the productivity of one high-capacity computer by employing two low-capacity computers.
(in production) Inputs in a
production function with a
constant marginal rate of
technical substitution.
INPUTS AND PRODUCTION FUNCTIONS
Fixed-Proportions Production Function (Perfect Complements)
fixed-proportions
production function
A production function
where the inputs must be
combined in a constant
ratio to one another.
Figure 6.15 illustrates a dramatically different case: isoquants for the production of
water, where the inputs are atoms of hydrogen (H ) and atoms of oxygen (O). Since
each molecule of water consists of two hydrogen atoms and one oxygen atom, the inputs must be combined in that fixed proportion. A production function where the inputs must be combined in fixed proportions is called a fixed-proportions production
function, and the inputs in a fixed-proportions production function are called perfect
complements.12 Adding more hydrogen to a fixed number of oxygen atoms gives us
no additional water molecules; neither does adding more oxygen to a fixed number of
hydrogen atoms. Thus, the quantity Q of water molecules that we get is given by:
H
Q min a , Ob
2
perfect complements
(in production) Inputs in a
fixed-proportions production
function.
where the notation min means “take the minimum value of the two numbers in the
parentheses.”
Isoquant for 1 molecule of water
Isoquant for 2 molecules of water
FIGURE 6.15
Isoquants for a
Fixed-Proportions Production Function
Two atoms of hydrogen (H) and one
atom of oxygen (O) are needed to
make one molecule of water. The
isoquants for this production function are L-shaped, which indicates
that each additional atom of oxygen
produces no additional water unless
two additional atoms of hydrogen
are also added.
12
O, quantity of oxygen atoms
Isoquant for 3 molecules of water
C
3
B
2
1
0
A
2
4
6
H, quantity of hydrogen atoms
The fixed-proportions production function is also called the Leontief production function, after the economist Wassily Leontief, who used it to model relationships between sectors in a national economy.
6 . 4 S U B S T I T U TA B I L I T Y A M O N G I N P U T S
227
When inputs are combined in fixed proportions, the elasticity of substitution
is zero (i.e., 0), because the marginal rate of technical substitution along the
isoquant of a fixed-proportions production function changes from q to 0 when we
pass through the corner of an isoquant (e.g., point A, B, or C ). A firm facing a fixedproportions production function has no flexibility in its ability to substitute among
inputs. We can see this in Figure 6.15: to produce a single molecule of water, there
is only one sensible input combination—two atoms of hydrogen and one atom of
oxygen.
We often observe production processes with fixed proportions. The production
of certain chemicals requires the combination of other chemicals, and sometimes heat,
in fixed proportions. Every bicycle must always have two tires and one frame. An automobile requires one engine, one chassis, and four tires, and these inputs cannot be
substituted for one another.
Cobb–Douglas Production Function
Figure 6.16 illustrates isoquants for the Cobb–Douglas production function, which
is intermediate between a linear production function and a fixed-proportions production function. The Cobb–Douglas production function is given by the formula Q
AL␣K , where A, ␣, and  are positive constants (in Figure 6.16, their values are 100,
0.4, and 0.6, respectively). With the Cobb–Douglas production function, capital and
labor can be substituted for each other. Unlike a fixed-proportions production function, capital and labor can be used in variable proportions. Unlike a linear production
function, though, the rate at which labor can be substituted for capital is not constant
as you move along an isoquant. This suggests that the elasticity of substitution for a
Cobb–Douglas production function falls somewhere between 0 and q. In fact, it turns
out that the elasticity of substitution along a Cobb–Douglas production function is
always equal to 1. (This result is derived in the Appendix to this chapter.)
Cobb–Douglas production function A production function of the
form Q AL␣K , where Q
is the quantity of output
from L units of labor and K
units of capital and where
A, ␣, and  are positive
constants.
50
K, units of capital per year
40
30
20
10
0
10
20
30
L, units of labor per year
40
50
FIGURE 6.16 Isoquants for
a Cobb–Douglas Production
Function
The isoquants for a Cobb–Douglas
production function are nonlinear
downward-sloping curves.
228
CHAPTER 6
A P P L I C A T I O N
INPUTS AND PRODUCTION FUNCTIONS
6.5
Measuring Productivity
Because the Cobb–Douglas production function is
thought to be a plausible way of characterizing many
real-world production processes, economists often
use it to study issues related to input productivity. For
example, Nicholas Bloom, Raffaella Sadun, and John
Van Reenen estimated Cobb–Douglas production
functions to study the ability of U.S. and European
companies to exploit information technology (IT) to
raise productivity.13 Specifically, they estimated production functions of the general form
Q ALaKbIT g
constant elasticity of
substitution (CES) production function A
type of production function
that includes linear production functions, fixedproportions production
functions, and Cobb–Douglas
production functions as
special cases.
where IT denotes a firm’s spending on computers and
other types of information technology. They explored
whether the production function coefficients (especially ␥) differed between different types of firms.
The United States experienced productivity
growth in the late 1990s, especially in industries that
use IT intensively, but the same did not occur in Europe.
The researchers compared U.S.-owned firms operating
in the United Kingdom to domestic U.K. firms and
non-U.S.-based multinationals. U.S.-owned firms had
higher productivity than those that were not, and this
difference was primarily due to their more effective
use of IT. They also found that non-U.S. firms that were
taken over by U.S. multinationals increased productivity
from IT, relative to firms that were not taken over.
Constant Elasticity of Substitution Production Function
Each of the three production functions we have discussed is a special case of a production function called the constant elasticity of substitution (CES) production
function, which is given by the equation:
Q [aL
s⫺1
s
⫹ bK
]
s⫺1 s
s s⫺1
where a, b, and are positive constants ( is the elasticity of substitution). Figure 6.17
shows that as varies between 0 and q, the shape of the isoquants of the CES
FIGURE 6.17 Isoquants for the
CES Production Function
This figure depicts the Q 1 isoquant
for five different CES production
functions, each corresponding to a
different value of the elasticity of
substitution . At 0, the isoquant
is that of a fixed-proportions
production function. At 1, the
isoquant is that of a Cobb–Douglas
production function. At q, the
isoquant is that of a linear production
function.
13
K, units of capital per year
2
σ=0
σ = 0.1
1
σ=1
σ=5
σ=∞
0
1
2
L, units of labor services per year
Nicholas Bloom, Raffaella Sadun, and John Van Reenen, “Americans Do I.T. Better: U.S.
Multinationals and the Productivity Miracle,” NBER Working Paper W13085 (May 2007), available at
SSRN, http://ssrn.com/abstract986935.
229
6 . 4 S U B S T I T U TA B I L I T Y A M O N G I N P U T S
TABLE 6.6
Characteristics of Production Functions
Elasticity of
Substitution ( )
Production Function
Linear production function
Fixed-proportions production function
Cobb–Douglas production function
CES production function
0
1
0
Other Characteristics
Inputs are perfect substitutes
Isoquants are straight lines
Inputs are perfect complements
Isoquants are L-shaped
Isoquants are curves
Includes other three production
functions as special cases
Shape of isoquants varies
production function changes from the L-shape of the fixed-proportions production
function to the curve of the Cobb–Douglas production function to the straight line of
the linear production function.
Table 6.6 summarizes the characteristics of these four specific production
functions.
A P P L I C A T I O N
6.6
Estimating a CES Production
Function for U.S. Industries
Using data from the Bureau of Economic Analysis for
1947–1998, economists Edward Balistreri, Christine
McDaniel, and Eina Vivian Wong (BMW) estimated
the constant in a CES production function relating
the quantity of output to the quantities of labor and
capital in each of 28 U.S. industries.14 Because, as
discussed in the text, represents the elasticity of substitution, BMW’s estimates provide insight into the
opportunities for substituting between labor and capital in
these industries.
Table 6.7 shows the estimates of s for a subset
of the 28 industries BMW studied. The table shows
two types of estimates for each industry: a long-run
elasticity of substitution and a short-run elasticity of
14
TABLE 6.7 Estimates of s for Selected
U.S. Industries
Estimated Value of
Industry
Agricultural services,
forestry, and fishing
Coal mining
Furniture and fixtures
Fabricated metal products
Industrial machinery and
equipment
Motor vehicles and
equipment
Textile mill products
Apparel and other textile
products
Edward Balistreri, Christine McDaniel, and Eina Vivian Wong, “An Estimation of US Industry-Level
Capital–Labor Substitution Elasticities: Support for Cobb-Douglas,” North American Journal of Economics
and Finance 14 (2003): 343–356.
Short Run
Long Run
0.23
0.10
0.10
0.11
0.36
1.27
1.01
1.39
0.23
0.82
0.05
0.05
0.40
1.14
0.13
2.05
230
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
substitution. The long-run elasticity of substitution is
the elasticity of substitution when a firm has enough
time to fully adjust its mix of labor and capital to its
desired level (which, as we will see in Chapter 7, depends on the relative prices of these inputs as well as
the quantity of output a firm wants to produce). The
short-run elasticity of substitution, by contrast, reflects the firm’s substitution opportunities within a
given period of time (in this case, a year).
BMW’s estimates have three noteworthy aspects.
First, the opportunities for substituting capital for
labor are higher in some industries than in others. For
example, the textile mill products industry has lower
short-run and long-run elasticities of substitution
than the apparel and other textile products industry.
The textile mill products industry makes raw textiles,
largely through mass production factories. These employ technology that is difficult to adjust in the short
run and, to some extent, in the long run as well. The
apparel and other textile products industry uses those
raw textiles to produce clothing and other goods.
6.5
RETURNS
TO S C A L E
Production methods are more likely to involve individual employees working with sewing machines and
tend to be less capital intensive. Apparently, firms in
that industry have more flexibility in substituting capital for labor.
Second, it is clear from Table 6.7 that the shortrun elasticities of substitution are much smaller than
long-run elasticities of substitution. This makes
sense: firms have a greater ability to change their
methods of production the more time that they have
to adjust.
Third, although some of the estimates of the
long-run elasticities of substitution are below 1 and
others are above 1, in 20 of the 28 industries studied,
BMW were unable to statistically reject the hypothesis that 1. Since, as noted in Table 6.6, the case of
1 corresponds to a Cobb–Douglas production
function, BMW’s analysis suggests that the Cobb–
Douglas production function may be a plausible representation of production technology in many sectors
of the U.S. economy.
I
n the previous section, we explored the extent to which inputs could be substituted
for each other to produce a given level of output. In this section, we study how increases in all input quantities affect the quantity of output the firm can produce.
DEFINITIONS
returns to scale The
concept that tells us the
percentage by which output will increase when all
inputs are increased by a
given percentage.
When inputs have positive marginal products, a firm’s total output must increase
when the quantities of all inputs are increased simultaneously—that is, when a firm’s
scale of operations increases. Often, though, we might want to know by how much output will increase when all inputs are increased by a given percentage amount. For
example, by how much would a semiconductor firm be able to increase its output if it
doubled its man-hours of labor and its machine-hours of robots? The concept of
returns to scale tells us the percentage increase in output when a firm increases all
of its input quantities by a given percentage amount:
Returns to scale
%¢ (quantity of output)
% ¢ (quantity of all inputs)
Suppose that a firm uses two inputs, labor L and capital K, to produce output Q. Now
suppose that all inputs are “scaled up” by the same proportionate amount , where 1
(i.e., the quantity of labor increases from L to L, and the quantity of capital increases
231
6 . 5 R E T U R N S TO S C A L E
from K to K).15 Let represent the resulting proportionate increase in the quantity
of output Q (i.e., the quantity of output increases from Q to Q). Then:
increasing returns to
scale A proportionate
• If , we have increasing returns to scale. In this case, a proportionate increase in all input quantities results in a greater than proportionate increase in
output.
• If , we have constant returns to scale. In this case, a proportionate
increase in all input quantities results in the same proportionate increase in
output.
• If , we have decreasing returns to scale. In this case, a proportionate
increase in all input quantities results in a less than proportionate increase in
output.
increase in all input quantities resulting in a greater
than proportionate increase
in output.
constant returns to
scale A proportionate
increase in all input quantities simultaneously that results in the same percentage increase in output.
2
2
Q=3
Q=2
Q=1
1
0
1
2
Q=3
Q=2
Q=1
1
0
1
2
L
(a) Increasing Returns to Scale
decreasing returns to
scale A proportionate
increase in all input quantities resulting in a less than
proportionate increase in
output.
K
K
K
Figure 6.18 illustrates these three cases.
Why are returns to scale important? When a production process exhibits increasing returns to scale, there are cost advantages from large-scale operation. In particular, a single large firm will be able to produce a given amount of output at a lower cost
per unit than could two equal-size smaller firms, each producing exactly half as much
output. For example, if two semiconductor firms can each produce 1 million chips at
$0.10 per chip, one large semiconductor firm could produce 2 million chips for less
than $0.10 per chip. This is because, with increasing returns to scale, the large firm
needs to employ less than twice as many units of labor and capital as the smaller firms
to produce twice as much output. When a large firm has such a cost advantage over
smaller firms, a market is most efficiently served by one large firm rather than several
smaller firms. This cost advantage of large-scale operation has been the traditional
justification for allowing firms to operate as regulated monopolists in markets such as
electric power and oil pipeline transportation.
Q=3
2
1
0
Q=2
1
L
L
(b) Constant Returns to Scale
(c) Decreasing Returns to Scale
FIGURE 6.18 Increasing, Constant, and Decreasing Returns to Scale
In panel (a), doubling the quantities of capital and labor more than doubles output. In panel (b),
doubling the quantities of capital and labor exactly doubles output. In panel (c), doubling the
quantities of capital and labor less than doubles output.
15
Therefore, the percentage change in all input quantities is ( 1)
Q=1
2
100 percent.
232
CHAPTER 6
S
E
INPUTS AND PRODUCTION FUNCTIONS
L E A R N I N G - B Y- D O I N G E X E R C I S E 6 . 4
D
Returns to Scale for a Cobb–Douglas Production Function
Problem
Does a Cobb–Douglas
production function, Q AL␣K , exhibit increasing,
decreasing, or constant returns to scale?
Solution Let L1 and K1 denote the initial quantities
of labor and capital, and let Q1 denote the initial output,
so Q1 ALa1 K 1b. Now let’s increase all input quantities
by the same proportional amount l, where l 7 1, and
let Q 2 denote the resulting volume of output:
Q2 A(lL1 ) a (lK1 ) b labALa1 K b1 labQ1. From
this, we can see that if:
•
a b 7 1, then lab 7 l, and so Q2 7
(increasing returns to scale).
A P P L I C A T I O N
Q1
a b 1, then lab l, and so Q2 lQ1
(constant returns to scale).
•
␣  6 1, then l␣ 6 l, and so Q2 6 lQ1 (decreasing returns to scale).
This shows that the sum of the exponents ␣  in the
Cobb–Douglas production function determines whether
returns to scale are increasing, constant, or decreasing.
For this reason, economists have paid considerable attention to estimating this sum when studying production functions in specific industries.
Similar Problems:
6.19, 6.20, 6.21, 6.23
6.7
Returns to Scale in Electric Power
Generation
Returns to scale have been thoroughly studied in electric power generation, where the pioneering work
was done by economist Marc Nerlove.16 Using data
from 145 electric utilities in the United States during
the year 1955, Nerlove estimated the exponents of a
Cobb–Douglas production function and found that
their sum was greater than 1. As illustrated in
Learning-By-Doing Exercise 6.4, this implies that electricity generation is subject to increasing returns to
scale. Other studies in this same industry using data
from the 1950s and 1960s also found evidence of
increasing returns to scale. However, studies using
more recent data (and functional forms for the production function other than Cobb–Douglas) have
16
•
found that electricity generation in large plants is
probably now characterized by constant returns to
scale.17
It is possible that both conclusions are correct. If
generation was characterized by increasing returns
to scale in the 1950s and 1960s but constant returns
to scale thereafter, we should expect to see a growth
in the scale of generating units throughout the 1950s
and 1960s followed by smaller growth in later years.
This is exactly what we observe. The average capacity
of all units installed between 1960 and 1964 was
151.7 megawatts. By the period 1970–1974, the average capacity of new units had grown to 400.3
megawatts. Over the next 10 years, the average
capacity of new units continued to grow, but more
slowly: Of all units installed between 1980 and 1982,
the average capacity was 490.3 megawatts.18
Marc Nerlove, “Returns to Scale in Electricity Supply,” Chapter 7 in Carl F. Christ, ed., Measurement in
Economics: Studies in Honor of Yehuda Grunfeld (Stanford, CA: Stanford University Press, 1963): 167–198.
17
See T. G. Cowing and V. K. Smith, “The Estimation of a Production Technology: A Survey of
Econometric Analyses of Steam Electric Generation,” Land Economics (May 1978): 157–170, and L. R.
Christensen and W. Greene, “Economies of Scale in U.S. Electric Power Generation,” Journal of Political
Economy (August 1976): 655–676.
18
These data come from Table 5.3 (p. 50) in P. L. Joskow and R. Schmalensee, Markets for Power:
An Analysis of Electric Utility Deregulation (Cambridge, MA: MIT Press, 1983).
233
K, units of capital per year
6.6 TECHNOLOGICAL PROGRESS
E
30
D
20
Q = 300
10
0
A
B
C
Q = 200
Q = 170
Q = 140
Q = 100
10
20
30
L, units of labor per year
FIGURE 6.19 Diminishing Marginal
Returns versus Returns to Scale
This production function exhibits constant
returns to scale but diminishing marginal returns
to labor.
R E T U R N S TO S C A L E V E R S U S D I M I N I S H I N G
MARGINAL RETURNS
It is important to understand the distinction between the concepts of returns to scale
and marginal returns (see Section 6.2). Returns to scale pertains to the impact of an
increase in all input quantities simultaneously, while marginal returns (i.e., marginal
product) pertains to the impact of an increase in the quantity of a single input, such as
labor, holding the quantities of all other inputs fixed.
Figure 6.19 illustrates this distinction. If we double the quantity of labor, from 10
to 20 units per year, holding the quantity of capital fixed at 10 units per year, we move
from point A to point B, and output goes up from 100 to 140 units per year. If we then
increase the quantity of labor from 20 to 30, we move from B to C, and output goes
up to 170. In this case, we have diminishing marginal returns to labor: The increase
in output brought about by a 10-unit increase in the quantity of labor goes down as
we employ more and more labor.
By contrast, if we double the quantity of both labor and capital from 10 to 20 units
per year, we move from A to D, and output doubles from 100 to 200. If we triple the
quantity of labor and capital from 10 to 30, we move from A to E, and output triples
from 100 to 300. For the production function in Figure 6.19 we have constant returns
to scale but diminishing marginal returns to labor.
S
o far, we have treated the firm’s production function as fixed over time. But as
knowledge in the economy evolves and as firms acquire know-how through experience and investment in research and development, a firm’s production function will
6.6
TECHNOLOGICAL PROGRESS
234
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
K, units of capital per year
A
Q = 100 isoquant before
technological progress
Q = 100 isoquant after
technological progress
0
L, units of labor per year
FIGURE 6.20
Neutral Technological Progress (MRTSL,K Remains the Same)
Under neutral technological progress, an isoquant corresponding to any particular level of output shifts inward, but the MRTSL,K (the negative of the slope of a line tangent to the isoquant)
along any ray from the origin, such as 0A, remains the same.
technological progress
A change in a production
process that enables a firm
to achieve more output
from a given combination
of inputs or, equivalently,
the same amount of output
from less inputs.
neutral technological
progress Technological
progress that decreases the
amounts of labor and capital needed to produce a
given output, without affecting the marginal rate of
technical substitution of
labor for capital.
labor-saving technological progress
Technological progress that
causes the marginal product
of capital to increase relative
to the marginal product
of labor.
change. The notion of technological progress captures the idea that production
functions can shift over time. In particular, technological progress refers to a situation
in which a firm can achieve more output from a given combination of inputs, or equivalently, the same amount of output from lesser quantities of inputs.
We can classify technological progress into three categories: neutral technological
progress, labor-saving technological progress, and capital-saving technological
progress.19 Figure 6.20 illustrates neutral technological progress. In this case, an
isoquant corresponding to a given level of output (100 units in the figure) shifts inward (indicating that lesser amounts of labor and capital are needed to produce a given
output), but the shift leaves MRTSL,K, the marginal rate of technical substitution of
labor for capital, unchanged along any ray (e.g., 0A) from the origin. Under neutral
technological progress, each isoquant corresponds to a higher level of output than before,
but the isoquants themselves retain the same shape.
Figure 6.21 illustrates labor-saving technological progress. In this case, too,
the isoquant corresponding to a given level of output shifts inward, but now along any
ray from the origin, the isoquant becomes flatter, indicating that the MRTSL,K is less
than it was before. You should recall from Section 6.3 that MRTSL,K MPL/MPK, so
the fact that the MRTSL,K decreases implies that under this form of technological
progress the marginal product of capital increases more rapidly than the marginal
product of labor. This form of technological progress arises when technical advances
19
J. R. Hicks, The Theory of Wages (London: Macmillan, 1932).
235
6.6 TECHNOLOGICAL PROGRESS
K, units of capital per year
A
Q = 100 isoquant before
technological progress
Q = 100 isoquant after
technological progress
0
L, units of labor per year
FIGURE 6.21 LaborSaving Technological
Progress (MRTSL,K
Decreases)
Under labor-saving technological progress, an isoquant
corresponding to any particular level of output shifts
inward, but the MRTSL,K
(the negative of the slope
of a line tangent to the
isoquant) along any ray
from the origin, such as
0A, goes down.
in capital equipment, robotics, or computers increase the marginal productivity of
capital relative to the marginal productivity of labor.
Figure 6.22 depicts capital-saving technological progress. Here, as an isoquant
shifts inward, MRTSL,K increases, indicating that the marginal product of labor increases more rapidly than the marginal product of capital. This form of technological
progress arises if, for example, the educational or skill level of the firm’s actual (and
potential) work force rises, increasing the marginal productivity of labor relative to
the marginal product of capital.
S
E
capital-saving technological progress
Technological progress that
causes the marginal product
of labor to increase relative
to the marginal product of
capital.
L E A R N I N G - B Y- D O I N G E X E R C I S E 6 . 5
D
Technological Progress
A firm’s production function requires that
it use at least one unit of labor and one unit of capital, i.e.,
L 1 and K 1. Initially the production function is
Q 1KL, with MPK 0.5( 1L/ 1K ) and MPL
0.5( 1K / 1L). Over time, the production function
changes to Q L 1K, with MPK 0.5(L/ 1K ) and
MPL 1K.
Problem
a) Verify that this change represents technological
progress.
b) Show whether this change is labor-saving, capitalsaving, or neutral.
Solution
a) With any quantities of K and L greater than or equal
to 1, more Q can be produced with the final production
function. So there is technological progress.
b) With the initial production function, MRTSL,K
MPL ⲐMPK K ⲐL. With the final production function,
MRTSL,K MPL ⲐMPK (2K) ⲐL. For any ratio of capital to labor (i.e., along any ray from the origin),
MRTSL,K is higher with the second production function.
Thus, the technological progress is capital saving.
Similar Problems: 6.26, 6.27, 6.28, 6.29
236
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
Q = 100
isoquant after
technological
progress
Q = 100 isoquant before
FIGURE 6.22 CapitalSaving Technological Progress
(MRTSL,K Increases)
Under capital-saving technological progress, an isoquant corresponding to any particular level
of output shifts inward, but the
MRTSL,K (the negative of the
slope of a line tangent to the
isoquant) along any ray from
the origin, such as 0A, goes up.
A P P L I C A T I O N
A
K, units of capital per year
technological progress
0
L, units of labor per year
6.8
Technological Progress . . .
and Educational Progress
One of the striking developments of the last 30 years
in the United States has been the growing inequality
in the wages earned by individuals with different
educational attainments. Figure 6.23 shows the trend
in real (i.e., inflation-adjusted) hourly wages of U.S.
workers, according to the worker’s level of educational attainment. (The wages are normalized so that
1973 100.) Between 1973 and 2005, the hourly
wage of individuals with a bachelor’s degree increased
nearly 20 percent. However, for those with only a
high school education, real wages in 2005 were
slightly lower than they were in 1973: This group
experienced no wage growth over this roughly 30-year
period. The result: The “salary premium” for receiving
an undergraduate or graduate degree has increased
dramatically over the last 30 years.
Economists Claudia Goldin and Lawrence Katz, in
a comprehensive historical study of income inequality
and education in the United States titled The Race
between Education and Technology, present compelling
evidence that wage and income inequality in the
United States during the 20th and early 21st century is
the result of two powerful forces: (1) the nature of
technological progress, and in particular, whether it
favors workers with advanced skill sets; and (2) the supply of skills provided by workers in the marketplace,
which reflects the level of educational attainment in
the work force.20 They argue that technological
progress in the United States throughout the 20th century tended to favor highly skilled workers rather than
unskilled workers, what economists call skill-biased
technological change. In other words, technological
20
Claudia Goldin and Lawrence F. Katz, The Race Between Education and Technology,
(Cambridge, MA: Belknap Press 2008).
6.6 TECHNOLOGICAL PROGRESS
progress tended to increase the marginal product of
skilled workers more than it did the marginal product
of unskilled workers. Expressed in the terminology of
this chapter, skill-biased technological change is
unskilled labor-saving technological progress.
Technological progress of this form would be
expected to increase the demand for skilled workers
relative to unskilled workers. Absent any changes in
the relative supply of workers of each type, unskilled
labor-saving technological progress would tend to
drive up the wages of skilled workers relative to
unskilled workers.
But the relative supply of skilled and unskilled
workers in the United States did not remain the same
throughout the 20th century. From roughly 1915
though 1980, the supply of skilled workers entering
the work force grew much faster than the supply of
unskilled workers, a phenomenon due primarily to
the rise of mass high school education in the United
States in the later 19th and early 20th centuries. Further
more, this rate of increase in the relative supply of
skilled workers was greater than the increase in the
relative demand for skilled workers due to skillbiased technological change. For example, Goldin
and Katz estimate that between 1915 and 1940, the
relative supply of college-educated workers increased
at a rate of 3.19 percent per year, compared to an
annual increase of 2.27 percent in the relative demand
for college-educated workers. Similarly, between 1940
and 1960, the relative supply of college-educated
workers increased by 2.63 percent, while the relative
demand for college-educated workers increased by
only 1.79 percent annually. Given these changes in
supply and demand, the wages of college-educated
workers relative to non–college-educated workers
actually fell between 1915 and the 1970s, a pattern
very different from the one depicted in Figure 6.23.
Similar trends occurred with respect to the wages of
high school-educated workers relative to those with
less than a high school education. As a result, from
1915 through the late 1970s, wage inequality and
income inequality in the United States declined.
All workers: 1973 = 100
130
Normalized real hourly wage of income group
237
Advanced degree
120
College
110
100
Some college
High school
90
Some high school
80
70
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05
60
FIGURE 6.23 Real Wages by Educational Attainment in the United States, 1973–2005
The figure shows the trend between 1973 and 2005 in real (i.e., inflation-adjusted) hourly
wages of U.S. workers, according to the worker’s level of educational attainment. The wages
are normalized so that 1973 100.
Source: Economic Policy Institute, http://www.epi.org/content.cfm/datazone_dznational
(accessed November 10, 2008).
238
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
This picture changed in the 1980s, 1990s, and the
2000s. During this period, skill-biased technological
change continued, and perhaps even accelerated with
rapid advances in information technology and computing power. However, the relative supply of skilled
workers began to shrink. This was partly due to immigration, which increased the relative supply of unskilled workers in the United States. But as Goldin and
Katz demonstrate, changes in the educational landscape in the United States were far more important
than immigration in explaining the reduction in the
relative supply of skilled workers (by a factor of about
9 to 1). The high school graduation rate in the United
States peaked just short of 80 percent around 1970
and actually declined until the mid-1990s. Goldin and
Katz point out that a child born in 1945 would
achieve two more years of education than his or her
parents, but a child born in 1975, by contrast, would
achieve only 0.50 more years of education than his
or her parents. Because of the dramatic slowdown in
educational progress in the United States and the
probable acceleration in the rate of unskilled laborsaving technological progress, the relative supply of
skilled workers has grown more slowly than the relative demand for skilled workers. The pattern of real
wage growth that you see in Figure 6.23 is a consequence of this change in the race between education
and technology.
CHAPTER SUMMARY
• The production function tells us the maximum quantity of output a firm can get as a function of the quantities of various inputs that it might employ.
• Single-input production functions are total product
functions. A total product function typically has three
regions: a region of increasing marginal returns, a region
of diminishing marginal returns, and a region of diminishing total returns.
• The average product of labor is the average amount
of output per unit of labor. The marginal product of
labor is the rate at which total output changes as the
quantity of labor a firm uses changes.
• The law of diminishing marginal returns says that as
the usage of one input (e.g., labor) increases—the quantities of other inputs, such as capital or land, being held
fixed—then at some point the marginal product of that
input will decrease.
• Isoquants depict multiple-input production functions in a two-dimensional graph. An isoquant shows all
combinations of labor and capital that produce the same
quantity of output. (LBD Exercise 6.1)
• For some production functions, the isoquants have
an upward-sloping and backward-bending region. This
region is called the uneconomic region of production.
Here, one of the inputs has a negative marginal product.
The economic region of production is the region of
downward-sloping isoquants.
• The marginal rate of technical substitution of labor
for capital tells us the rate at which the quantity of capital can be reduced for every one-unit increase in the
quantity of labor, holding the quantity of output constant. Mathematically, the marginal rate of technical
substitution of labor for capital is equal to the ratio of
the marginal product of labor to the marginal product of
capital. (LBD Exercise 6.2)
• Isoquants that are bowed in toward the origin exhibit
a diminishing marginal rate of technical substitution.
When the marginal rate of technical substitution of
labor for capital diminishes, fewer and fewer units of
capital can be sacrificed as each additional unit of labor
is added along an isoquant.
• The elasticity of substitution measures the percentage rate of change of K/L for each 1 percent change in
MRTSL,K. (LBD Exercise 6.3)
• Three important special production functions are
the linear production function (perfect substitutes), the
fixed-proportions production function (perfect complements), and the Cobb–Douglas production function.
Each of these is a member of a class of production functions known as constant elasticity of substitution production functions.
• Returns to scale tell us the percentage by which
output will increase when all inputs are increased by a
given percentage. If a given percentage increase in the
PROBLEMS
quantities of all inputs increases output by more than
that percentage, we have increasing returns to scale. If a
given percentage increase in the quantities of all inputs
increases output by less than that percentage, we have
decreasing returns to scale. If a given percentage increase in the quantities of all inputs increases output by
the same percentage, we have constant returns to scale.
(LBD Exercise 6.4)
239
• Technological progress refers to a situation in which
a firm can achieve more output from a given combination of inputs, or equivalently, the same amount of output from smaller quantities of inputs. Technological
progress can be neutral, labor saving, or capital saving,
depending on whether the marginal rate of technical
substitution remains the same, decreases, or increases
for a given capital-to-labor ratio. (LBD Exercise 6.5)
REVIEW QUESTIONS
1. We said that the production function tells us the
maximum output that a firm can produce with its quantities
of inputs. Why do we include the word maximum in this
definition?
2. Suppose a total product function has the “traditional
shape” shown in Figure 6.2. Sketch the shape of the corresponding labor requirements function (with quantity of
output on the horizontal axis and quantity of labor on the
vertical axis).
3. What is the difference between average product and
marginal product? Can you sketch a total product function such that the average and marginal product functions coincide with each other?
4. What is the difference between diminishing total
returns to an input and diminishing marginal returns to an
input? Can a total product function exhibit diminishing
marginal returns but not diminishing total returns?
6. Could the isoquants corresponding to two different
levels of output ever cross?
7. Why would a firm that seeks to minimize its expenditures on inputs not want to operate on the uneconomic
portion of an isoquant?
8. What is the elasticity of substitution? What does it
tell us?
9. Suppose the production of electricity requires just
two inputs, capital and labor, and that the production
function is Cobb–Douglas. Now consider the isoquants
corresponding to three different levels of output: Q
100,000 kilowatt-hours, Q 200,000 kilowatt-hours,
and Q 400,000 kilowatt-hours. Sketch these isoquants
under three different assumptions about returns to scale:
constant returns to scale, increasing returns to scale, and
decreasing returns to scale.
5. Why must an isoquant be downward sloping when
both labor and capital have positive marginal products?
PROBLEMS
6.1. A firm uses the inputs of fertilizer, labor, and hothouses to produce roses. Suppose that when the quantity
of labor and hothouses is fixed, the relationship between
the quantity of fertilizer and the number of roses produced is given by the following table:
Tons of
Fertilizer
per Month
Number of
Roses
per Month
Tons of
Fertilizer
per Month
Number of
Roses
per Month
0
1
2
3
4
0
500
1000
1700
2200
5
6
7
8
2500
2600
2500
2000
a) What is the average product of fertilizer when 4 tons
are used?
b) What is the marginal product of the sixth ton of
fertilizer?
c) Does this total product function exhibit diminishing
marginal returns? If so, over what quantities of fertilizer
do they occur?
d) Does this total product function exhibit diminishing
total returns? If so, over what quantities of fertilizer do
they occur?
6.2. A firm is required to produce 100 units of output
using quantities of labor and capital (L, K ) (7, 6). For
each of the following production functions, state whether
it is possible to produce the required output with the given
input combination. If it is possible, state whether the input
combination is technically efficient or inefficient.
a) Q 7L ⫹ 8K
b) Q ⫽ 20 1K L
240
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
c) Q min(16L, 20K)
d) Q 2(KL L 1)
6.3. For the production function Q 6L2 ⫺ L3, fill in
the following table and state how much the firm should
produce so that:
a) average product is maximized
b) marginal product is maximized
c) total product is maximized
d) average product is zero
L
if there were increasing marginal returns to labor, you or
I could produce all the steel in the world in a backyard
blast furnace. Using numerical arguments based on the
production function shown in the following table, show
that this (logically absurd) conclusion is correct. The fact
that it is correct shows that marginal returns to labor cannot be everywhere increasing when the production function exhibits constant returns to scale.
Q
0
1
2
L
K
Q
0
1
2
4
8
16
32
100
100
100
100
100
100
100
0
1
4
16
64
256
1024
3
4
5
6.7. The following table shows selected input quantities, total products, average products, and marginal products. Fill in as much of the table as you can:
6
6.4. Suppose that the production function for DVDs
is given by Q KL2 ⫺ L3, where Q is the number of
disks produced per year, K is machine-hours of capital,
and L is man-hours of labor.
a) Suppose K 600. Find the total product function and
graph it over the range L 0 to L 500. Then sketch the
graphs of the average and marginal product functions. At
what level of labor L does the average product curve appear
to reach its maximum? At what level does the marginal
product curve appear to reach its maximum?
b) Replicate the analysis in (a) for the case in which
K 1200.
c) When either K 600 or K 1200, does the total product function have a region of increasing marginal returns?
6.5. Are the following statements correct or incorrect?
a) If average product is increasing, marginal product
must be less than average product.
b) If marginal product is negative, average product must
be negative.
c) If average product is positive, total product must be rising.
d) If total product is increasing, marginal product must
also be increasing.
6.6. Economists sometimes “prove” the law of diminishing marginal returns with the following exercise:
Suppose that production of steel requires two inputs,
labor and capital, and suppose that the production function is characterized by constant returns to scale. Then,
Labor, L
Total Product, Q
0
1
2
3
4
5
6
7
APL
MPL
0
—
19
0
19
36
8
9
10
11
12
13
14
15
256
375
64
103
637
91
129
133
96
891
100
1089
89
96
⫺7
⫺15
75
6.8. Widgets are produced using two inputs, labor, L,
and capital, K. The following table provides information
on how many widgets can be produced from those inputs:
L
K
0
0
0
1
2
2
4
3
6
4
8
1
2
4
6
8
10
2
4
6
8
10
12
3
6
8
10
12
14
4
8
10
12
14
16
241
PROBLEMS
a) Use data from the table to plot sets of input pairs that
produce the same number of widgets. Then, carefully,
sketch several of the isoquants associated with this production function.
b) Find marginal products of K and L for each pair of inputs in the table.
c) Does the production function in the table exhibit decreasing, constant, or increasing returns to scale?
6.9. Suppose the production function for automobiles
is Q LK where Q is the quantity of automobiles produced per year, L is the quantity of labor (man-hours),
and K is the quantity of capital (machine-hours).
a) Sketch the isoquant corresponding to a quantity of
Q 100.
b) What is the general equation for the isoquant corresponding to any level of output Q?
c) Does the isoquant exhibit diminishing marginal rate
of technical substitution?
6.10. Suppose the production function is given by the
equation Q L2K. Graph the isoquants corresponding
to Q 10, Q 20, and Q 50. Do these isoquants exhibit
diminishing marginal rate of technical substitution?
a) Sketch a graph of the isoquants for this production
function.
b) Does this production function have an uneconomic
region? Why or why not?
6.12. Suppose the production function is given by the
following equation (where a and b are positive constants):
Q aL bK. What is the marginal rate of technical substitution of labor for capital (MRTSL,K) at any point along
an isoquant?
6.13. You might think that when a production function
has a diminishing marginal rate of technical substitution
of labor for capital, it cannot have increasing marginal
products of capital and labor. Show that this is not true,
using the production function Q K2L2, with the corresponding marginal products MPK 2KL2 and MPL
2K2L.
6.14. Consider the following production functions and
their associated marginal products. For each production
function, determine the marginal rate of technical substitution of labor for capital, and indicate whether the isoquants for this production function exhibit diminishing
marginal rate of technical substitution.
6.11. Consider again the production function for
DVDs: Q KL2 ⫺ L3.
Production
Function
MPL
MPK
QLK
MPL 1
MPK 1
Q 2LK
MPL
Q 2L 2K
MPL
1 2K
2 2L
1 1
2 2L
MPK
MPK
MRTSL,K
Diminishing
Marginal
Product of
Labor?
Diminishing
Marginal
Product of
Capital?
Diminishing
Marginal
Rate of
Technical
Substitution?
1 2L
2 2K
1 1
2 2K
Q L3K3
MPL 3L2K3
MPK 3L3K2
Q L2 K2
MPL 2L
MPK 2K
6.15. Suppose that a firm’s production function is given
by Q KL K, with MPK L 1 and MPL K. At
point A, the firm uses K 3 units of capital and L 5
units of labor. At point B, along the same isoquant, the
firm would only use 1 unit of capital.
a) Calculate how much labor is required at point B.
b) Calculate the elasticity of substitution between A
and B. Does this production function exhibit a higher or
lower elasticity of substitution than a Cobb–Douglas
function over this range of inputs?
6.16. Two points, A and B, are on an isoquant drawn
with labor on the horizontal axis and capital on the vertical axis. The capital–labor ratio at B is twice that at A,
and the elasticity of substitution as we move from A to B
is 2. What is the ratio of the MRTSL,K at A versus that
at B?
242
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
b) Is the marginal product of labor ever diminishing for
this production function? If so, when? Is it ever negative,
and if so, when?
6.17. Let B be the number of bicycles produced from F
bicycle frames and T tires. Every bicycle needs exactly
two tires and one frame.
a) Draw the isoquants for bicycle production.
b) Write a mathematical expression for the production
function for bicycles.
6.22. Consider a production function whose equation is
given by the formula Q ⫽ LK 2, which has corresponding marginal products, MPL ⫽ K 2 and MPK ⫽ 2LK.
Show that the elasticity of substitution for this production
function is exactly equal to 1, no matter what the values of
K and L are.
6.18. To produce cake, you need eggs E and premixed ingredients I. Every cake needs exactly one egg and one package of ingredients. When you add two eggs to one package
of ingredients, you produce only one cake. Similarly, when
you have only one egg, you can’t produce two cakes even
though you have two packages of ingredients.
a) Draw several isoquants of the cake production function.
b) Write a mathematical expression for this production
function. What can you say about returns to scale for this
function?
6.23. A firm’s production function is Q ⫽ 5L2/3 K 1/3
with MPK ⫽ (5/3)L2/3K⫺2/3 and MPL ⫽ (10/3)L⫺1/3K 1/3
a) Does this production function exhibit constant, increasing, or decreasing returns to scale?
b) What is the marginal rate of technical substitution
of L for K for this production function?
c) What is the elasticity of substitution for this production function?
6.19. What can you say about the returns to scale of the
linear production function Q aK ⫹ bL, where a and b
are positive constants?
6.24. Consider a CES production function given by Q ⫽
(K 0.5 ⫹ L0.5)2.
a) What is the elasticity of substitution for this production function?
b) Does this production function exhibit increasing, decreasing, or constant returns to scale?
c) Suppose that the production function took the form
Q ⫽ (100 ⫹ K 0.5 ⫹ L0.5)2. Does this production function
exhibit increasing, decreasing, or constant returns to scale?
6.20. What can you say about the returns to scale of the
Leontief production function Q ⫽ min(aK, bL), where a
and b are positive constants?
6.21. A firm produces a quantity Q of breakfast cereal
using labor L and material M with the production function Q ⫽ 50 1ML ⫹ M ⫹ L. The marginal product
functions for this production function are
MPL ⫽ 25
M
⫹1
BL
MPM ⫽ 25
6.25. Consider the following production functions and
their associated marginal products. For each production function, indicate whether (a) the marginal product of each input is diminishing, constant, or increasing
in the quantity of that input; (b) the production function exhibits decreasing, constant, or increasing returns
to scale.
L
⫹1
BM
a) Are the returns to scale increasing, constant, or decreasing for this production function?
Production
Function
MPL
MPK
Q⫽L⫹K
MPL ⫽ 1
MPK ⫽ 1
Q ⫽ 2LK
MPL ⫽
Q ⫽ 2L ⫹ 2K
MPL ⫽
1 2K
2 2L
1 1
2 2L
MPK ⫽
MPK ⫽
1 2L
2 2K
1 1
2 2K
Q ⫽ L3K 3
MPL ⫽ 3L2K 3
MPK ⫽ 3L3K 2
Q ⫽ LK
MPL ⫽ K
MPK ⫽ L
Marginal
Product of
Labor?
Marginal
Product of
Capital?
Returns to
Scale?
A P P E N D I X : THE ELASTICITY OF SUBSTITUTION FOR A COBB–DOUGLAS PRODUCTION FUNCTION
6.26. The following table presents information on how
many cookies can be produced from eggs and a mixture
of other ingredients (measured in ounces):
Mix
Eggs
0
1
2
3
4
0
4
8
12
16
0
0
0
0
0
0
8
8
8
8
0
8
16
16
16
0
8
16
24
24
0
8
16
24
32
Recently, you found a new way to mix ingredients with eggs.
The same amount of ingredients and eggs produces different numbers of cookies, as shown in the following table:
Mix
Eggs
0
1
2
3
4
0
4
8
12
16
0
0
0
0
0
0
9
10
11
12
0
10
19
22
23
0
11
20
25
26
0
12
21
26
33
a) Verify that the change to the new production function
represents technological progress.
b) For each production function find the marginal products
of eggs when mixed ingredients is held fixed at 8. Verify
that when mixed ingredients is held fixed at 8, the technological progress increases the marginal product of eggs.
6.27. Suppose a firm’s production function initially took
the form Q ⫽ 500(L 3K ). However, as a result of a
manufacturing innovation, its production function is now
Q 1,000(0.5L 10K ).
APPENDIX:
243
a) Show that the innovation has resulted in technological
progress in the sense defined in the text.
b) Is the technological progress neutral, labor saving, or
capital saving?
6.28. A firm’s production function is initially Q ⫽ 1KL,
with MPK ⫽ 0.5(1LⲐ 1K ) and MPL ⫽ 0.5(1KⲐ 1L ).
Over time, the production function changes to Q ⫽ KL,
with MPK ⫽ L and MPL ⫽ K . (Assume, as in LearningBy-Doing Exercise 6.5, that for this production process,
L and K must each be greater than or equal to 1.)
a) Verify that this change represents technological progress.
b) Is this change labor saving, capital saving, or neutral?
6.29. A firm’s production function is initially Q ⫽ 2KL,
with MPK ⫽ 0.5(1LⲐ 1K ) and MPL ⫽ 0.5(1KⲐ 1L ).
Over time, the production function changes to
Q ⫽ K1L, with MPK ⫽ 1L and MPL 0.5(K Ⲑ 1L).
(Assume, as in Learning-By-Doing Exercise 6.5, that for
this production process, L and K must each be greater
than or equal to 1.)
a) Verify that this change represents technological progress.
b) Is this change labor saving, capital saving, or neutral?
6.30. Suppose that in the 21st century the production of
semiconductors requires two inputs: capital (denoted by
K ) and labor (denoted by L). The production function
takes the form Q 2KL. However, in the 23rd century,
suppose the production function for semiconductors will
take the form Q K. In other words, in the 23rd century
it will be possible to produce semiconductors entirely
with capital (perhaps because of robots).
a) Does this change in the production function change
the returns to scale?
b) Is this change in the production function an illustration of technological progress?
The Elasticity of Substitution for a Cobb–Douglas Production Function
In this appendix we derive the elasticity of substitution for a Cobb–Douglas production function, f (L, K ) ⫽ ALaK b. The marginal product of labor and capital are found by taking the
partial derivatives of the production function with respect to labor and capital, respectively
(for a discussion of partial derivatives, see the Mathematical Appendix in this book):
0f
MPL ⫽
0L
0f
MPK
0K
⫽ aALa⫺1K b
bALaK b⫺1
Now, recall that, in general,
MRTSL,K
MPL
MPK
244
CHAPTER 6
INPUTS AND PRODUCTION FUNCTIONS
Thus, for this Cobb–Douglas production function,
MRTSL,K
aALa⫺1K b
bALaK b⫺1
aK
bL
Rearranging terms yields
b
K
MRTSL,K
L
a
(A6.1)
Therefore, ¢(K L) ( ba)¢MRTSL,K or:
K
¢a b
L
¢MRTSL,K
b
a
(A6.2)
a
b
(A6.3)
Also, from (A6.1),
MRTSL,K
K
a b
L
Now, using the definition of the elasticity of substitution in equation (6.6):
s
K
%¢a b
L
%¢MRTSL,K
°
K
¢a b
L
¢MRTSL,K
K K
¢a b^
L
L
¢MRTSL,K
a
b
MRTSL,K
¢°
MRTSL,K
K
L
¢
(A6.4)
Substituting (A6.2) and (A6.3) into (A6.4) yields
s
b
a
⫻ 1
a
b
That shows that the elasticity of substitution along a Cobb–Douglas production function is
equal to 1 for all values of K and L.
7
COSTS AND COST
MINIMIZATION
7.1
COST CONCEPTS FOR
DECISION MAKING
APPLICATION 7.1
APPLICATION 7.2
To Smelt or Not to Smelt?
The Mark-to-Market
Controversy
Who Is More Likely to Avoid
the Sunk Cost Fallacy?
APPLICATION 7.3
7.2
T H E C O S T- M I N I M I Z AT I O N P R O B L E M
APPLICATION 7.4
Self-Checkout or Cashier?
APPLICATION 7.5
The End of Meter Maids?
Reducing Costs by Offshoring
Input Demand in Alabama
7.3
C O M PA R AT I V E S TAT I C S A N A LYS I S O F
T H E C O S T- M I N I M I Z AT I O N P R O B L E M
APPLICATION 7.6
APPLICATION 7.7
7.4
S H O RT- RU N C O S T M I N I M I Z AT I O N
What Fraction of a Capital
Investment Is Sunk Cost?
APPLICATION 7.8
APPENDIX
A DVA N C E D TO P I C S I N C O S T
M I N I M I Z AT I O N
What’s Behind the Self-Service Revolution?
In the last quarter of the 20th century, self-service became a pervasive feature of certain parts of the
American retail landscape.1 Customers have grown so used to pumping their own gasoline or withdrawing
money from an ATM (automated teller machine) that it is hard to remember a time when those services
were provided by human beings. But in recent years the pace of automation in retail and service
1
This introduction draws from “More Consumers Reach Out to Touch the Screen,” New York Times
(November 17, 2003): A1 and A12; “Self-Checkout Transactions to Approach $450 billion Annually
by 2008,” Progressive Grocer (August 8, 2005); “Self-Checkout Drops Sales of Impulse Items by More
Than 45 Percent, Says New Study from IHL Consulting Group,” Business Wire ( July 25, 2006).
245
businesses seems to have increased. In most large airports these days, you can obtain your boarding
pass at an automated check-in machine. Large retail chains such as Kroger and Home Depot have
deployed machines that allow customers to scan, bag, and pay for their merchandise themselves. And
fast-food restaurants, such as McDonald’s and Jack-in-the-Box, have begun to install automatic ordering
machines that eliminate the need to order from a human being. In 2005, consumers spent nearly
$111 billion on retail transactions that took place through self-checkout systems, an amount expected
to exceed $1 trillion by 2010.
What has driven the growth of self-service machines in recent years? Experts believe that one factor is
that as consumers have grown more comfortable with personal technologies such as laptop computers, cell
phones, and PDAs, they have become increasingly willing to place their faith in machines when they travel,
shop, or purchase fast food. But another key reason is that improvements in technology have made it possible for firms to install self-service machines that allow consumers to perform functions such as scanning
groceries or transmitting a food order just as fast and accurately as cashiers can, but at a fraction of the
cost to the firm. For example, one estimate in The Economist suggests that a transaction carried out
through a kiosk may cost only a tenth as much as a transaction handled by an employee. “The savings
come chiefly from replacing employees with machines, which do not require health benefits or a salary.”2
In effect, retailers and other service firms are finding that they can lower their costs by substituting capital
(e.g., self-checkout systems) for labor (e.g., cashiers).
This chapter studies costs and cost minimization. In this chapter, we will introduce concepts that
will help you think more clearly and systematically about what costs are and how they factor into the
analysis of decisions, such as the one to adopt self-checkout systems. With the tools that we present in
this chapter, we can better understand the nature of the trade-offs that retailers such as Kroger or fastfood restaurants such as McDonald’s face as they contemplate the appropriate degree to which they
should automate their service operations.
CHAPTER PREVIEW
After
reading and studying this chapter,
you will be able to:
• Identify and apply different
concepts of costs that figure
in a firm’s decision making,
including explicit versus implicit
costs, opportunity cost, economic
versus accounting costs, and
sunk versus nonsunk costs.
• Describe a firm’s cost-minimization
problem in the long run, using
2
“Help Yourself: The Recession Spurs Self
Service,” The Economist, 392, no. 8638
( July 4 –10, 2009): 63.
246
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7.1 COST CONCEPTS FOR DECISION MAKING
the concept of isocost lines (the combinations of inputs such as labor and capital that have the same
total cost).
• Employ comparative statics analysis to explain how changes in the prices of inputs and the level of
output affect a firm’s choices of inputs and its costs of production.
• Describe the firm’s cost-minimization problem in the short run and analyze the firm’s choice of inputs
when the firm has at least one fixed factor of production and one or more variable factors.
Managers are most experienced with cost presented as monetary expenses in an 7.1
income statement. Politicians and policy analysts are more familiar with costs as an
expense item in a budget statement. Consumers think of costs as their monthly bills
and other expenses.
But economists use a broader concept of cost. To an economist, cost is the value
of sacrificed opportunities. What is the cost to you of devoting 20 hours every week
to studying microeconomics? It is the value of whatever you would have done instead
with that 20 hours (leisure activities, perhaps). What is the cost to an airline of using
one of its planes in scheduled passenger service? In addition to the money the airline
spends on items such as fuel, flight-crew salaries, maintenance, airport fees, and food
and drinks for passengers, the cost of flying the plane also includes the income the
airline sacrifices by not renting out its jet to other parties (e.g., another airline) that
would be willing to lease it. What is the cost to repair an expressway in Chicago?
Besides the money paid to hire construction workers, purchase materials, and rent
equipment, it would also include the value of the time that drivers sacrifice as they sit
immobilized in traffic jams.
Viewed this way, costs are not necessarily synonymous with monetary outlays.
When the airline flies the planes that it owns, it does pay for the fuel, flight-crew
salaries, maintenance, and so forth. However, it does not spend money for the use of
the airplane itself (i.e., it does not need to lease it from someone else). Still, in most
cases, the airline incurs a cost when it uses the plane because it sacrifices the opportunity to lease that airplane to others who could use it.
Because not all costs involve direct monetary outlays, economists distinguish
between explicit costs and implicit costs. Explicit costs involve a direct monetary
outlay, whereas implicit costs do not. For example, an airline’s expenditures on fuel
and salaries are explicit costs, whereas the income it forgoes by not leasing its jets is
an implicit cost. The sum total of the explicit costs and the implicit costs represents
what the airline sacrifices when it makes the decision to fly one of its planes on a
particular route.
COST
CONCEPTS
FOR DECISION
MAKING
explicit costs Costs
that involve a direct monetary outlay.
implicit costs Costs
that do not involve outlays
of cash.
OPPORTUNITY COST
The economist’s notion that cost is the value of sacrificed opportunities is based on the
concept of opportunity cost. To understand opportunity cost, consider a decision maker,
such as a business firm, that must choose among a set of mutually exclusive alternatives,
each of which entails a particular monetary payoff. The opportunity cost of a particular
alternative is the payoff associated with the best of the alternatives that are not chosen.
opportunity cost The
value of the next best alternative that is forgone when
another alternative is chosen.
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The opportunity cost of an alternative includes all of the explicit and implicit
costs associated with that alternative. To illustrate, suppose that you own and manage
your own business and that you are contemplating whether you should continue to
operate over the next year or go out of business. If you remain in business, you will
need to spend $100,000 to hire the services of workers and $80,000 to purchase supplies; if you go out of business, you will not need to incur these expenses. In addition,
the business will require 80 hours of your time every week. Your best alternative to
managing your own business is to work the same number of hours in a corporation
for an income of $75,000 per year. In this example, the opportunity cost of continuing in business over the next year is $255,000. This amount includes an explicit cost
of $180,000—the required cash outlays for labor and materials; it also includes an implicit cost of $75,000—the income that you forgo by continuing to manage your own
firm as opposed to choosing your best available alternative.
The concept of opportunity cost is forward looking in that it measures the value
that the decision maker sacrifices at the time the decision is made and beyond. To
illustrate this point, consider an automobile firm that has an inventory of sheet steel
that it purchased for $1 million. It is planning to use the sheet steel to manufacture
automobiles. As an alternative, it can resell the steel to other firms. Suppose that the
price of sheet steel has gone up since the firm made its purchase, so if it resells its steel
the firm would get $1.2 million. The opportunity cost of using the steel to produce
automobiles is thus $1.2 million. In this illustration, opportunity cost differs from the
original expense incurred by the firm.
After reading this last example, students sometimes ask, “Why isn’t the opportunity
cost of the steel $200,000: the difference between the market value of the steel
($1.2 million) and its original cost ($1 million)?” After all, the firm has already spent
$1 million to buy the steel. Why isn’t the opportunity cost the amount above and
beyond that original cost ($200,000 in this example)? The way to answer this question
is to remember that the notion of opportunity cost is forward looking, not backward
looking. To assess opportunity cost we ask: “What does the decision maker give up at
the time the decision is being made?” In this case, when the automobile company uses
the steel to produce cars, it gives up more than just $200,000. It forecloses the opportunity to receive a payment of $1.2 million from reselling the steel. The opportunity
cost of $1.2 million measures the full amount the firm sacrifices at the moment it
makes the decision to use the steel to produce cars rather than to resell it in the open
market.
Opportunity Costs Depend on the Decision Being Made
The forward-looking nature of opportunity costs implies that opportunity costs can
change as time passes and circumstances change. To illustrate this point, let’s return
to our example of the automobile firm that purchased $1 million worth of sheet steel.
When the firm first confronted the decision to “buy the steel” or “don’t buy the steel,”
the relevant opportunity cost was the purchase price of $1 million. This is because the
firm would save $1 million if it did not buy the steel.
But—moving ahead in time—once the firm purchases the steel and the market
price of steel changes, the firm faces a different decision: “use the steel to produce
cars” or “resell it in the open market.” The opportunity cost of using the steel is the
$1.2 million payment that the firm sacrifices by not selling the steel in the open
market. Same steel, same firm, but different opportunity cost! The opportunity
costs differ because there are different opportunity costs for different decisions under different
circumstances.
7.1 COST CONCEPTS FOR DECISION MAKING
249
Opportunity Costs and Market Prices
Note that the unifying feature of this example is that the relevant opportunity cost was,
in both cases, the current market price of the sheet steel. This is no coincidence. From
the firm’s perspective, the opportunity cost of using the productive services of an input is
the current market price of the input. The opportunity cost of using the services of an
input is what the firm’s owners would save or gain by not using those services. A firm
can “not use” the services of an input in two ways. It can refrain from buying those
services in the first place, in which case the firm saves an amount equal to the market
price of the input. Or it can resell unused services of the input in the open market, in
which case it gains an amount equal to the market price of the input. In both cases, the
opportunity cost of the input services is the current market price of those services.
A P P L I C A T I O N
7.1
To Smelt or Not to Smelt?3
We have said that the opportunity cost of an alternative is the payoff associated with the best of the alternatives that are not chosen. Sometimes that payoff
becomes so large that the optimal course of action is
to choose the best alternative instead. Such was the
case with Kaiser Aluminum in 2000.
For many years, Kaiser operated two aluminum
smelters (giant plants used to manufacture raw aluminum ingots) near the cities of Spokane and Tacoma,
Washington. The production of aluminum requires a
substantial amount of electric power, so one of the
most important determinants of the cost of producing aluminum is the price of electricity.
In 2000, Kaiser was purchasing electricity at about
$23 per megawatt hour under a long-term contract
with the Bonneville Power Administration (BPA), the
federal agency that produces electricity from dams
along the Columbia River. Kaiser signed the contract
with BPA in 1996 when the spot market price (the
current price on the open market) was low. However,
in late 2000 and early 2001 the spot market price of
electricity skyrocketed, on some days averaging over
$1,000 per megawatt hour.4
Kaiser had a great deal because its contract enabled it to buy electricity at far below the market
3
price. But the sharply rising electricity prices also
created a sharply rising opportunity cost for Kaiser as
long as it used that electricity to operate its aluminum plants, because its contract with BPA gave it
the right to resell the electricity if market prices escalated. (The BPA had offered this option to induce
Kaiser to sign a long-term contract in the first place.)
If Kaiser used the electricity purchased from the BPA
to smelt aluminum, it sacrificed the opportunity to
resell that electricity in the open market. The profit
that Kaiser would forgo by not reselling electricity
was huge. In December 2000, Kaiser decided to shut
down both smelters. Kaiser then resold the electricity
to BPA at $550 per megawatt hour, which at the time
was somewhat below the prevailing spot price of
electricity, but far above Kaiser’s cost of $23.
Kaiser did not reopen the smelters, even when
the market price of electricity declined in the spring
and summer of 2001. The market price of aluminum
fell to a 2-year low in 2001. As a result, Kaiser decided
that it was uneconomical to reopen its two plants. In
2003 the Tacoma plant was sold to the Port of
Tacoma, which razed it in 2006 to create room for
more capacity at the port. In 2004 the Spokane plant
was sold for only $4 million. The price was low because the company that purchased it also assumed
responsibility for cleaning up pollution at the site.
This example draws from “Plants Shut Down and Sell the Energy” Washington Post (December 21, 2000),
and “Kaiser Will Mothball Mead Smelter,” Associated Press ( January 14, 2003).
4
The reason that the price of electricity in the Pacific Northwest rose so sharply in the fall of 2000 and
winter of 2001 is bound up in events that were taking place in California’s electric power markets. The markets
for electricity in the Pacific Northwest and California were interrelated, since California relied on imports
of electricity generated by hydroelectric dams in the Pacific Northwest to satisfy part of its demand for
electricity. Application 2.8 discusses the factors responsible for the California power crisis of 2000 and 2001.
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7.2
The Mark-to-Market Controversy
During the financial crisis of 2008, the accounting
practice of “mark-to-market” (MTM) became controversial. Some banks argued that this rule caused the
financial crisis to become far worse than it needed
to be. MTM derives from rules established by the
Financial Analysts Standards Board about how public
companies value capital assets in their accounting
statements. The rule requires that certain assets be
valued at their current “fair market value.”
Consider a bank that lends money to home owners by issuing mortgages. Each mortgage is an asset
to the bank. The bank can expect monthly payments
from the home owners, unless the mortgage goes
into default. Even if the mortgage defaults, the bank
can foreclose on the home and sell it, recouping some
value. Therefore, the bank needs to account for the
value of these assets on its accounting statements.
The value of a specific mortgage falls if the
home’s value falls below what it was when the mortgage was issued. The probability of a default on the
mortgage rises dramatically, so the bank’s expected
receipts fall. Even if the bank forecloses and sells the
home, it is likely to receive less than the amount it
loaned in the first place. This is exactly what happened in 2008, but in very large numbers. Housing
prices fell across the entire United States, and default
rates on mortgages skyrocketed. Banks foreclosed on
many homes but found it difficult to sell those homes.
When houses were sold, prices were often far below
their previous values.
During this crisis banks had to revalue their
mortgage-based assets—mark them to market value.
The market values of these assets plummeted, so the
value of bank accounting statements also dropped
sharply. This had an important consequence. The
amount of money that a bank is allowed to lend to
customers depends on the value of the bank’s capital.
If the capital falls in value, it must reduce lending. In
2008, lending by banks plummeted to levels that were
nearly unprecedented. An important reason for this is
that MTM greatly reduced their capital values. Banks
play a critical role in the economy, loaning money to
businesses to help them maintain operations. As lending dried up, many businesses were forced to close.
Thus, the housing crisis became a banking crisis, leading to a deep recession for the whole economy.
A more complete examination of the controversy
surrounding the merits of MTM in banking would
go well beyond the scope of our discussion here.
However, the purpose of MTM is relevant to our discussion of opportunity costs. Suppose that banks were
not required to revalue assets if their market value fell.
If so, bank accounting statements would overstate the
economic value of their assets in a situation such as the
housing crisis of 2008. The accounting value of mortgage-based assets would be above their current opportunity cost, which is the market value of those assets
if the bank attempted to sell them. In other words, the
point of the MTM rule is to try to have accounting
statements reflect economic costs as well as possible.
This example also illustrates another point.
Sometimes accounting costs can be greater than
economic costs. In other words, just because accounting costs exclude implicit costs, while economic costs
include implicit costs, it does not follow that accounting costs are always less han economic costs.
E C O N O M I C V E R S U S AC C O U N T I N G C O S T S
economic costs The
sum of the firm’s explicit
costs and implicit costs.
accounting costs The
total of explicit costs that
have been incurred in the
past.
Closely related to the distinction between explicit and implicit costs is the distinction
between economic costs and accounting costs. Economic costs are synonymous with
opportunity costs and, as such, are the sum of all decision-relevant explicit and implicit costs. Accounting costs—the costs that would appear on accounting statements—
are explicit costs that have been incurred in the past. Accounting statements are designed
to serve an audience outside the firm, such as lenders and equity investors, so accounting costs must be objectively verifiable. That’s why accounting statements typically
include historical expenses only—that is, explicit cash outlays already made (e.g., the
amounts the firm actually spent on labor and materials in the past year). An accounting statement would not include implicit costs such as the opportunity costs associated
7.1 COST CONCEPTS FOR DECISION MAKING
251
with the use of the firm’s factories because such costs are often hard to measure in an
objectively verifiable way. For that reason, an accounting statement for an owneroperated small business would not include the opportunity cost of the owner’s time.
And because accounting statements use historical costs, not current market prices, to
compute costs, the costs on the profit-and-loss statement of the automobile firm that
purchased that sheet steel would reflect the $1 million purchase price of that steel, but
it would not reflect the $1.2 million opportunity cost that it incurs when the firm actually uses that steel to manufacture automobiles.
In contrast, economic costs include all these decision-relevant costs. To an economist, all decision-relevant costs (whether explicit or implicit) are opportunity costs
and are therefore included as economic costs.
S U N K ( U N AVO I DA B L E ) V E R S U S N O N S U N K
( AVO I DA B L E ) C O S T S
To analyze costs, we also need to distinguish between sunk and nonsunk costs. When assessing the costs of a decision, the decision maker should consider only those costs that
the decision actually affects. Some costs have already been incurred and therefore cannot
be avoided, no matter what decision is made. These are called sunk costs. By contrast,
nonsunk costs are costs that will be incurred only if a particular decision is made and
are thus avoided if the decision is not made (for this reason, nonsunk costs are also called
avoidable costs). When evaluating alternative decisions, the decision maker should ignore
sunk costs and consider only nonsunk costs. Why? Consider the following example.
You pay $7.50 to go see a movie. Ten minutes into the movie, it is clear that the movie
is awful. You face a choice: Should you leave or stay? The relevant cost of staying is that
you could more valuably spend your time doing just about anything else. The relevant
cost of leaving is the enjoyment that you might forgo if the movie proves to be better than
the first 10 minutes suggest. The relevant cost of leaving does not include the $7.50 price
of admission. That cost is sunk. No matter what you decide to do, you’ve already paid
the admission fee, and its amount should be irrelevant to your decision to leave.
The next example further illustrates the distinction between sunk costs and nonsunk costs. Consider a sporting goods firm that manufactures bowling balls. Let’s assume that a bowling ball factory costs $5 million to build and that, once it is built, the
factory is so highly specialized that it has no alternative uses. Thus, if the sporting
goods firm shuts the factory down and produces nothing, it will not “recover” any of
the $5 million it spent to build the factory.
• In deciding whether to build the factory, the $5 million is a nonsunk cost. It is a cost
the sporting goods firm incurs only if it builds the factory. At the time the decision
is being considered, the decision maker can avoid spending the $5 million.
• After the factory is built, the $5 million is a sunk cost. It is a cost the sporting goods
firm incurs no matter what it later chooses to do with the factory, so this cost is
unavoidable. When deciding whether to operate the factory or shut it down, the sporting
goods firm therefore should ignore this cost.
This example illustrates an important point: Whether a cost is sunk or nonsunk depends on the decision that is being contemplated. To identify what costs are sunk and what
costs are nonsunk in a particular decision, you should always ask which costs would
change as a result of making one choice as opposed to another. These are the nonsunk
costs. The costs that do not change no matter what choice we make are the sunk costs.
sunk costs Costs that
have already been incurred
and cannot be recovered.
nonsunk costs Costs
that are incurred only if a
particular decision is made.
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7.3
Who Is More Likely to Avoid
the Sunk Cost Fallacy?
Suppose that several months ago you purchased a
ticket to an outdoor concert or sporting event being
held today. However, it turns out that the event is not
likely to be very enjoyable because the weather is
cold and rainy. Should you go to the event? If you
decide to go because you don’t want to “waste the
cost of the ticket,” you are not properly ignoring a
sunk cost. Psychologists and economists refer to such
behavior as the “sunk cost fallacy.”
A recent study by several psychologists analyzed
whether older or younger people are more likely to
commit a sunk cost fallacy.5 They presented college
students and senior citizens with two stories to test
the likelihood that both groups would decide to
watch a movie. The first vignette read, “You paid
$10.95 to see a movie on pay TV. After 5 minutes, you
S
E
are bored and the movie seems pretty bad.” The
other vignette did not include a cost. Participants
selected their time commitment from these options:
stop watching, watch 10 minutes more, 20 minutes
more, 30 more minutes, or watch until the end.
They found that senior citizens expected to
spend the same amount of time watching the movie,
regardless of whether they had paid for the movie
or incurred no cost. This is the rational behavior
suggested by economic theory. By contrast, college
students chose to watch the movie longer if they had
paid for it than if it was free.
The psychologists interpreted this as meaning
that young adults have a “negativity bias,” weighing
negative information more heavily than positive
information—in this case, trying to “recover” their
cost by watching the movie longer. Regardless of interpretation, the results suggest that college students are
more likely to engage in the sunk cost fallacy. Do you?
L E A R N I N G - B Y- D O I N G E X E R C I S E 7 . 1
D
Using the Cost Concepts for a College Campus Business
Imagine that you have started a snack food
delivery business on your college campus. Students send
you orders for snacks, such as potato chips and candy bars,
via the Internet. You shop at local grocery stores to fill these
orders and then deliver the orders. To operate this business,
you pay $500 a month to lease computer time from a local
Web-hosting company to use its server to host and maintain your website. You also own a sports utility vehicle
(SUV) that you use to make deliveries. Your monthly car
payment is $300, and you pay $100 a month in insurance
costs. Each order that you fill takes, on average, a half hour
and consumes $0.50 worth of gasoline.6 When you fill an
order, you pay the grocer for the merchandise. You then
collect a payment, including a delivery fee, from the students to whom you sell. If you did not operate this business,
you could work at the campus dining hall, earning $6 an
hour. Right now, you operate your business five days a
week, Monday through Friday. On weekends, your business is idle, and you work in the campus dining hall.
5
Problem
(a) What are your explicit costs, and what are your implicit costs? What are your accounting costs and your
economic costs, and how would they differ?
(b) Last week you purchased five large cases of Fritos
for a customer who, as it turned out, did not accept delivery. You paid $100 for these cases. You have a deal
with your grocers that they will pay you $0.25 for each
dollar of returned merchandise. Just this week, you
found a fraternity on campus that will buy the five cartons for $55 (and will pick them up from your apartment, relieving you of the need to deliver them to the
frat house). What is the opportunity cost of filling this
order (i.e., selling these cartons to the fraternity)?
Should you sell the Fritos to the fraternity?
(c) Suppose you are thinking of cutting back your operation from five days to four days a week. (You will not
JoNell Strough, Clare Mehta, Joseph McFall, and Kelly Schuller, “Are Older Adults Less Subject to the
Sunk-Cost Fallacy Than Younger Adults?” Psychological Science (2008): 650–652.
6
For simplicity, let’s ignore other costs such as wear and tear on your vehicle.
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7 . 2 T H E C O S T- M I N I M I Z AT I O N P R O B L E M
operate on Monday and instead will work in the campus
dining hall.) What costs are nonsunk with respect to this
decision? What costs are sunk?
(d) Suppose you contemplate going out of business
altogether. What costs are nonsunk with respect to this
decision? What costs are sunk?
Solution
(a) Your explicit costs are those that involve direct monetary
outlays. These include your car payment, insurance, leasing
computer time, gasoline, and the money you pay grocers
for the merchandise you deliver. Your main implicit cost is
the opportunity cost of your time—$6 per hour.
Your economic costs are the sum of these explicit
and implicit costs. Your accounting costs would include
all of the explicit costs but not the implicit opportunity
cost of your time. Moreover, your accounting costs
would be historical (e.g., the actual costs you incurred
last year). Thus, if gasoline prices have gone down since
last year, your current gasoline costs would not equal
your historical accounting costs.
(b) The opportunity cost of filling the order is $25. This
is what you could have gotten for the Fritos if you had
resold them to your grocer and thus represents what you
sacrifice if you sell the Fritos to the fraternity instead.
Because you can sell the Fritos at a price that exceeds
this opportunity cost, you should fill the order.
What, then, does the $75 difference between your
$100 original cost and the $25 opportunity cost represent?
It is the cost you incurred in trying to satisfy a customer
who proved to be unreliable. It is a sunk cost of doing
business.
(c) Your nonsunk costs with respect to this decision are
those costs that you will avoid if you make this decision.
These include the cost of gasoline and the cost of purchased merchandise. (Of course, you also “avoid” receiving the revenue from delivering this merchandise.) In
addition, though, you avoid one day of the implicit opportunity cost of your time ( you no longer sacrifice the
opportunity to work in the dining hall on Mondays).
Your sunk costs are those that you cannot avoid by
making this decision. Because you still need your SUV
for deliveries, your car and insurance payments are sunk.
Your leasing of computer time is also sunk, since you still
need to maintain your website.
(d) You certainly will avoid your merchandising costs
and gasoline costs if you cease operations. These costs
are thus nonsunk with respect to the shutdown decision.
You also avoid the opportunity cost of your time, so this
too is a nonsunk cost. And you avoid the cost of leasing
computer time. Thus, while the computer leasing cost
was sunk with respect to the decision to scale back operations by one day, it is nonsunk with respect to the decision to cease operations altogether.
What about the costs of your SUV? Suppose you
plan to get rid of it, which means that you can avoid your
$100 a month insurance bill, so your insurance costs are
nonsunk. But suppose that you customized the SUV by
painting your logo on it. Because of this and because
people are wary about buying used vehicles, you can recover only 30 percent of the cost you paid for it. This
means that 70 percent of your car payment is sunk, while
30 percent is nonsunk.
Similar Problems:
7.1, 7.2, 7.3
Now that we have introduced a variety of different cost concepts, let’s apply them 7.2
to analyze an important decision problem for a firm: How to choose a combination
of inputs to minimize the cost of producing a given quantity of output. We saw in
Chapter 6 that firms can typically produce a given amount of output using many
different input combinations. Of all the input combinations that can be chosen, a
firm that wants to make its owners as wealthy as possible should choose the one
that minimizes its costs of production. The problem of finding this input combination is called the cost-minimization problem, and a firm that seeks to minimize
the cost of producing a given amount of output is called a cost-minimizing firm.
LONG RUN VERSUS SHORT RUN
We will study the firm’s cost-minimization problem in the long run and in the
short run. Although the terms long run and short run seem to connote a length
of time, it is more useful to think of them as pertaining to the degree to which
the firm faces constraints in its decision-making flexibility. A firm that makes a
T H E C O S TMINIMIZATION
PROBLEM
cost-minimization
problem The problem of
finding the input combination that minimizes a firm’s
total cost of producing a
particular level of output.
cost-minimizing firm
A firm that seeks to minimize the cost of producing
a given amount of output.
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CHAPTER 7
long run The period of
time that is long enough
for the firm to vary the
quantities of all of its inputs as much as it desires.
long-run decision faces a blank slate (i.e., no constraints): Over the long run it will
be able to vary the quantities of all its inputs as much as it desires. When our sporting goods firm in the previous section decides whether to build a new bowling ball
factory, it faces a long-run decision. It is free to choose whether to build the factory
and, if so, how large to make it. As it does this, it can simultaneously choose other
input quantities, such as the size of the work force and the amount of land for the
factory. Because the firm can, in principle, avoid the costs of all inputs by choosing
not to build, the costs associated with this long-run decision are necessarily nonsunk.
By contrast, a firm facing a short-run decision is subject to constraints: Over the
short run, it will not be able to adjust the quantities of some of its inputs and/or reverse the consequences of past decisions that it has made regarding those inputs. For
example, once our bowling ball firm builds a factory, it will, at least for a while, face
short-run decisions, such as how many workers it should employ given the physical
constraints of its capacity.
In microeconomics, the concept of short run and long run are convenient analytical
simplifications to help us focus our attention on the interesting features of the problem
at hand. In reality, firms face a continuum of “runs”; some decisions involve “blanker
slates” than others. In this section, we first focus on long-run cost minimization in order
to study carefully the trade-offs that firms can make in input choices when they start
with a blank slate. In the next section, we turn to short-run cost minimization to highlight how constraints on input usage can limit the firm’s ability to minimize costs.
short run The period of
time in which at least one
of the firm’s input quantities
cannot be changed.
C O S T S A N D C O S T M I N I M I Z AT I O N
T H E L O N G - R U N C O S T- M I N I M I Z AT I O N P R O B L E M
The cost-minimization problem is an example of constrained optimization, first
discussed in Chapter 1. We want to minimize the firm’s total costs, subject to the
requirement that the firm produce a given amount of output. In Chapter 4, we encountered two other examples of constrained optimization: the problem of maximizing utility subject to a budget constraint (utility maximization) and the problem of minimizing
consumption expenditures, subject to achieving a minimum level of utility (expenditure
minimization). You will see that the cost-minimization problem closely resembles the
expenditure-minimization problem from consumer choice theory.
Let’s study the long-run cost-minimization problem for a firm that uses two inputs:
labor and capital. Each input has a price. The price of a unit of labor services—also
called the wage rate—is w. This price per unit of capital services is r. The price of labor
could be either an explicit cost or an implicit cost. It would be an explicit cost if the firm
(as most firms do) hires workers in the open market. It would be an implicit cost if the
firm’s owner provides her own labor to run the firm and, in so doing, sacrifices outside
employment opportunities. Similarly, the price of capital could either be an explicit cost
or an implicit cost. It would be an explicit cost if the firm leased capital services from another firm (e.g., a firm that leases computer time on a server to host its website). It would
be an implicit cost if the firm owned the physical capital and, by using it in its own business, sacrificed the opportunity to sell capital services to other firms.7
The firm has decided to produce Q0 units of output during the next year. In later
chapters we will study how the firm makes such an output decision. For now, the quantity Q0 is exogenous (e.g., as if the manufacturing manager of the firm has been told how
much to produce). The long-run cost-minimization problem facing the manufacturing
manager is to figure out how to produce that amount in the cost-minimizing way. Thus,
7
In the Appendix, we discuss the factors that would determine the price of capital services.
7 . 2 T H E C O S T- M I N I M I Z AT I O N P R O B L E M
255
the manager must choose a quantity of capital K and a quantity of labor L that minimize
the total cost TC wL ⫹ rK of producing Q0 units of output. This total cost is the sum
of all the economic costs the firm incurs when it uses labor and capital services to produce output.
ISOCOST LINES
Let’s now try to solve the firm’s cost-minimization problem graphically. Our first step
is to draw isocost lines. An isocost line represents a set of combinations of labor and
capital that have the same total cost (TC ) for the firm. An isocost line is analogous to
a budget line from the theory of consumer choice.
Consider, for example, a case in which w ⫽ 10 per labor-hour, r ⫽ 20 per machinehour, and TC ⫽ $1 million per year. The $1 million isocost line is described by the equation 1,000,000 ⫽ 10L ⫹ 20K, which can be rewritten as K ⫽ 1,000,000/20 ⫺ (10/20)L.
The $2 million and $3 million isocost lines have similar equations: K ⫽ 2,000,000/20 ⫺
(10/20)L and K ⫽ 3,000,000/20 ⫺ (10/20)L.
More generally, for an arbitrary level of total cost TC, and input prices w and r,
the equation of the isocost line is K ⫽ TC/r ⫺ (w/r)L.
Figure 7.1 shows graphs of isocost lines for three different total cost levels, TC0,
TC1, and TC2, where TC2 TC1 TC0. In general, there are an infinite number of
isocost lines, one corresponding to every possible level of total cost. Figure 7.1 illustrates that the slope of every isocost line is the same: With K on the vertical axis and
L on the horizontal axis, that slope is ⫺w/r (the negative of the ratio of the price of
labor to the price of capital). The K-axis intercept of any particular isocost line is the
cost level for that isocost line divided by the price of capital (e.g., for the TC0 isocost
K, capital services per year
TC2
r
TC1
r
isocost line The set of
combinations of labor and
capital that yield the same
total cost for the firm.
Directions of
increasing total cost
Slope of each isocost line = – w
r
TC0
r
FIGURE 7.1
TC0
TC1
TC2
w
w
w
L, labor services per year
Isocost Lines
As we move to the northeast in the isocost
map, isocost lines correspond to higher
levels of total cost. All isocost lines have the
same slope.
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C O S T S A N D C O S T M I N I M I Z AT I O N
line, the K-axis intercept is TC0 /r). Similarly, the L-axis intercept of the TC0 isocost
line is TC0 /w. Notice that as we move to the northeast in the isocost map in Figure 7.1,
isocost lines correspond to higher levels of cost.
G R A P H I C A L C H A R AC T E R I Z AT I O N O F T H E S O L U T I O N
TO T H E L O N G - R U N C O S T- M I N I M I Z AT I O N P R O B L E M
Figure 7.2 shows two isocost lines and the isoquant corresponding to Q0 units of output.
The solution to the firm’s cost-minimization problem occurs at point A, where the
isoquant is just tangent to an isocost line. That is, of all the input combinations along
the isoquant, point A provides the firm with the lowest level of cost.
To verify this, consider other points in Figure 7.2, such as E, F, and G:
• Point G is off the Q0 isoquant altogether. Although this input combination could
produce Q0 units of output, in using it the firm would be wasting inputs (i.e.,
point G is technically inefficient). This point cannot be optimal because input
combination A also produces Q0 units of output but uses fewer units of labor
and capital.
• Points E and F are technically efficient, but they are not cost-minimizing because they are on an isocost line that corresponds to a higher level of cost than
the isocost line passing through the cost-minimizing point A. By moving from
point E to A or from F to A, the firm can produce the same amount of output,
but at a lower total cost.
Note that the slope of the isoquant at the cost-minimizing point A is equal
to the slope of the isocost line. In Chapter 6, we saw that the negative of the
slope of the isoquant is equal to the marginal rate of technical substitution of labor
for capital, MRTSL,K, and that MRTSL,K MPL ⲐMPK. As we just illustrated,
the slope of an isocost line is ⫺wⲐr. Thus, the cost-minimizing condition occurs
when:
FIGURE 7.2
Cost-Minimizing Input
Combination
The cost-minimizing input combination occurs at
point A. Point G is technically inefficient. Points
E and F are technically efficient, but they do not
minimize cost (the firm can lower cost from TC1
to TC0 by moving to input combination A).
K, capital services per year
Slope of each isocost line = – w
r
TC1
r
TC0
r
G
E
A
F
Q0 isoquant
L, labor services per year
7 . 2 T H E C O S T- M I N I M I Z AT I O N P R O B L E M
257
slope of isoquant ⫽ slope of isocost line
w
⫺MRTSL, K ⫽ ⫺
r
MPL
w
⫽
r
MPK
ratio of marginal products ⫽ ratio of input prices
(7.1)
In Figure 7.2, the optimal input combination A is an interior optimum. An interior
optimum involves positive amounts of both inputs (L 0 and K 0), and the optimum
occurs at a tangency between the isoquant and an isocost line. Equation (7.1) tells us
that at an interior optimum, the ratio of the marginal products of labor and capital
equals the ratio of the price of labor to the price of capital. We could also rewrite
equation (7.1) to state the optimality condition in this form:
MPL
MPK
⫽
w
r
(7.2)
Expressed this way, this condition tells us that at a cost-minimizing input combination, the additional output per dollar spent on labor services equals the additional
output per dollar spent on capital services. Thus, if we are minimizing costs, we get
equal “bang for the buck” from each input. (Recall that we obtained a similar condition at the solution to a consumer’s utility-maximization problem in Chapter 4.)
To see why equation (7.2) must hold, consider a non–cost-minimizing point in
Figure 7.2, such as E. At point E, the slope of the isoquant is more negative than the
slope of the isocost line. Therefore, ⫺(MPL /MPK) ⬍ ⫺(w/r), or MPL /MPK ⬎ w/r, or
MPL Ⲑw ⬎ MPK /r.
This condition implies that a firm operating at E could spend an additional dollar
on labor and save more than one dollar by reducing its employment of capital services
in a manner that keeps output constant. Since this would reduce total costs, it follows
that an interior input combination, such as E, at which equation (7.2) does not hold cannot
be cost-minimizing.
A P P L I C A T I O N
7.4
Self-Checkout or Cashier?
In the opening section of this chapter we described
how the self-service revolution has swept across the
American retail landscape as firms find that they can
lower their costs by substituting capital (like selfcheckout systems) for labor (like cashiers). In this
section we have examined how a business should
choose the mix of capital and labor if it wants to minimize its cost.
Let’s consider an example from the article in The
Economist cited in footnote 2 at the beginning of this
chapter. The article states, “According to Francie
Mendelsohn, the president of Summit Research
Associates, each self-service checkout at a grocery
store replaces around 2.5 employees.” As we learned
in Chapter 6, the marginal rate of technical substitution of capital for labor measures the number of workers the firm would be able to give up if it were able to
hire one more machine, holding output constant.
Thus, for the kind of enterprise described in the article,
the MRTSK,L would be 2.5 because the firm can give up
2.5 workers when it rents one more machine at the
checkout counter. In its reciprocal form, equation (7.1)
tells us that when a firm is minimizing the total cost of
production, the marginal rate of technical substitution
of machines for a labor should equal the ratio of the
rental price of a machine to the wage rate.
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C O S T S A N D C O S T M I N I M I Z AT I O N
When might such a firm reduce costs by renting
more machines and hiring less labor? If the rental price
of a machine is less than 2.5 times the wage rate, the
firm could handle the same number of transactions at
a lower cost if it rents another checkout machine and
hires 2.5 fewer cashiers. We can see this reasoning
using equations (7.1) and (7.2). If MPK / MPL ⫽ 2.5 and
r / w ⬍ 2.5, then MPK / MPL ⬎ r / w. This inequality can be
rewritten as MPK / r ⬎ MPL /w. Thus, the “bang for the
buck” with capital is higher than that for labor, so the
firm could reduce costs by increasing K and reducing L.
S
E
Of course, in most grocery stores and drug stores
you will find both automated checkout systems and
cashiers, and in most banks you can choose to make
your deposits or withdrawals with a teller or an
ATM. One of the reasons for this is that self-checkout
machines and employees are often not perfectly
interchangeable for one another. A self-checkout
machine may not be capable of carrying out every
kind of transaction that an employee can handle.
L E A R N I N G - B Y- D O I N G E X E R C I S E 7 . 2
D
Finding an Interior Cost-Minimization Optimum
Problem The optimal input combination satisfies equation (7.1) [or, equivalently, equation
(7.2)]. But how would you calculate it? To see how, let’s
consider a specific example. Suppose that the firm’s production function is of the form Q ⫽ 50 1LK. For this
production function, the equations of the marginal
products of labor and capital are MPL ⫽ 25 1K /L and
MPK ⫽ 25 1L/K. Suppose, too, that the price of labor
w is $5 per unit and the price of capital r is $20 per unit.
What is the cost-minimizing input combination if the
firm wants to produce 1,000 units per year?
Thus, our tangency condition [equation (7.1)] is K/L ⫽
5/20, which simplifies to L ⫽ 4K.
In addition, the input combination must lie on the
1,000-unit isoquant (i.e., the input combination must
allow the firm to produce exactly 1,000 units of output).
This means that 1,000 ⫽ 50 1K L, or, simplifying, L ⫽
400/K.
When we solve these two equations with two
unknowns, we find that K ⫽ 10 and L ⫽ 40. The costminimizing input combination is 10 units of capital and
40 units of labor.
Solution
Similar Problems: 7.8, 7.9
The ratio of the marginal products of labor
and capital is MPL /MPK ⫽ (251K /L)/(251L/K) ⫽ K/L.
CORNER POINT SOLUTIONS
In discussing the theory of consumer behavior in Chapter 4, we studied corner point
solutions: optimal solutions at which we did not have a tangency between a budget
line and an indifference curve. We can also have corner point solutions to the costminimization problem. Figure 7.3 illustrates this case. The cost-minimizing input
combination for producing Q0 units of output occurs at point A, where the firm uses
no capital.
At this corner point, the isocost line is flatter than the isoquant. Mathematically,
this says ⫺(MPL/MPK) ⬍ ⫺(w/r), or equivalently, MPL/MPK ⬎ w/r. Another way to
write this would be
MPL
MPK
7
w
r
(7.3)
Thus, at the corner solution at point A, the marginal product per dollar spent on
labor exceeds the marginal product per dollar spent on capital services. If you look
7 . 2 T H E C O S T- M I N I M I Z AT I O N P R O B L E M
MPL
MPK
w
Slope of isocost lines = – r
Q0 isoquant
K, capital services per year
259
Slope of isoquant = –
Isocost lines
E
F
A
L, labor services per year
FIGURE 7.3
Corner Point Solution to the Cost-Minimization Problem
The cost-minimizing input combination occurs at point A, where the firm uses no capital.
Points such as E and F cannot be cost minimizing, because the firm can lower costs and
keep output the same by substituting labor for capital.
closely at other points along the Q0 unit isoquant, you see that isocost lines are always
flatter than the isoquant. Hence, condition (7.3) holds for all input combinations
along the Q0 isoquant. A corner solution at which no capital is used can be thought of
as a response to a situation in which every additional dollar spent on labor yields more
output than every additional dollar spent on capital. In this situation, the firm should
substitute labor for capital until it uses no capital at all, as illustrated in Learning-ByDoing Exercise 7.3.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 7 . 3
D
Finding a Corner Point Solution with Perfect Substitutes
Problem In Chapter 6 we saw that a
linear production function implies that the inputs are
perfect substitutes. Suppose that we have the linear production function Q 10L ⫹ 2K. For this production
function MPL ⫽ 10 and MPK ⫽ 2. Suppose, too, that the
price of labor w is $5 per unit and that the price of capital services r is $2 per unit. Find the optimal input combination given that the firm wishes to produce 200 units
of output.
Solution
Figure 7.4 shows that the optimal input
combination is a corner point solution at which K ⫽ 0.
The following argument tells us that we must have a
corner point solution. We know that when inputs are
perfect substitutes, MRTSL,K ⫽ MPL ⲐMPK is constant
along an isoquant; in this particular example, it is equal
to 5. But wⲐr ⫽ 2.5, so there is no point that can satisfy
MPL /MPK ⫽ wⲐr. This tells us that we cannot have an
interior solution.
But what corner point will we end up at? In this
case, MPL / w ⫽ 10 Ⲑ5 ⫽ 2, and MPK Ⲑr ⫽ 2Ⲑ2 ⫽ 1, so the
marginal product per dollar of labor exceeds the marginal product per dollar of capital. This implies that the
firm will substitute labor for capital until it uses no capital. Hence the optimal input combination involves K ⫽ 0.
Since the firm is going to produce 200 units of output,
200 ⫽ 10L ⫹ 2(0), or L ⫽ 20.
Similar Problems:
7.10, 7.15, 7.16, 7.33
CHAPTER 7
C O S T S A N D C O S T M I N I M I Z AT I O N
K, capital services per year
260
200-unit isoquant
(slope = –5)
Isocost lines
(slope = –2.5)
FIGURE 7.4
Corner Point Solution to the
Cost-Minimization Problem
The solution to the cost-minimization problem when
capital and labor are perfect substitutes may be a
corner point. In this case, the solution occurs when
L 20 and K 0.
0
20
L, labor services per year
The cost-minimization problem we have been studying in this chapter should
strike you as familiar because it is analogous to the expenditure-minimization problem that we studied in Chapter 4. In the expenditure-minimization problem, a consumer seeks to minimize his or her total expenditures, subject to attaining a given level
of utility. In the cost-minimization problem, a firm seeks to minimize its expenditures
on goods and services, subject to producing a given level of output. Both the graphical analysis and the mathematics of the two problems are identical.
7.3
N
ow that we have characterized the solution to the firm’s cost-minimization problem, let’s explore how changes in input prices and output affect this solution.
COMPARATIVE
S TAT I C S
C O M PA R AT I V E S TAT I C S A N A LYS I S O F C H A N G E S
A N A LYS I S O F
IN INPUT PRICES
T H E C O S TFigure 7.5 shows a comparative statics analysis of the cost-minimization problem as the
MINIMIZATION price of labor w changes, with the price of capital r held constant at 1 and the quantity
of output held constant at Q0. As w increases from 1 to 2, the cost-minimizing quanPROBLEM
tity of labor goes down (from L1 to L2) while the cost-minimizing quantity of capital
goes up (from K1 to K2). Thus, the increase in the price of labor causes the firm to
substitute capital for labor.
In Figure 7.5, we see that the increase in w makes the isocost lines steeper, which
changes the position of the tangency point between the isocost line and the isoquant.
When w 1, the tangency is at point A, where the optimal input combination is
(L1, K1); when w 2, the tangency is at point B, where the optimal combination is
(L2, K2). Thus, with diminishing MRTSL,K, the tangency between the isocost line and
7 . 3 C O M PA R AT I V E S TAT I C S A N A LYS I S O F T H E C O S T- M I N I M I Z AT I O N P R O B L E M
261
Slope of isocost line C1 = –1
K, capital services per year
Slope of isocost line C2 = –2
B
K2
FIGURE 7.5
C1
A
K1
0
Q0 isoquant
C2
L2
L1
L, labor services per year
Comparative Statics Analysis
of Cost-Minimization Problem with Respect to the
Price of Labor
The price of capital r 1 and the quantity of
output Q0 are held constant. When the price of
labor is w 1, the isocost line is C1 and the ideal
input combination is at point A (L1, K1). When the
price of labor is w 2, the isocost line is C2 and
the ideal input combination is at point B (L2, K2).
Increasing the price of labor causes the firm to
substitute capital for labor.
the isoquant occurs farther up the isoquant (i.e., less labor, more capital). To produce
the required level of output, the firm uses more capital and less labor because labor
has become more expensive relative to capital (w/r has increased). By similar logic,
when w/r decreases, the firm uses more labor and less capital, so the tangency moves
farther down the isoquant.
This relationship relies on two important assumptions. First, at the initial input
prices, the firm must be using a positive quantity of both inputs. That is, we do not
start from a corner point solution. If this did not hold—if the firm were initially using
a zero quantity of an input—and the price of that input went up, the firm would continue to use a zero quantity of the input. Thus, the cost-minimizing input quantity
would not go down as in Figure 7.5, but instead would stay the same. Second, the isoquants must be “smooth” (i.e., without kinks). Figure 7.6 shows what happens when a
firm has a fixed-proportions production function and thus has isoquants with a kink
in them. As in the case where we start with a corner point, an increase in the price of
labor leaves the cost-minimizing quantity of labor unchanged.
Let’s summarize the results of our comparative statics analysis:
• When the firm has smooth isoquants with a diminishing marginal rate of
technical substitution, and is initially using positive quantities of an input, an
increase in the price of that input (holding output and other input prices fixed)
will cause the cost-minimizing quantity of that input to go down.
• When the firm is initially using a zero quantity of the input or the firm has a fixedproportions production function (as in Figure 7.6), an increase in the price of
the input will leave the cost-minimizing input quantity unchanged.
Note that these results imply that an increase in the input price can never cause the
cost-minimizing quantity of the input to go up.
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C O S T S A N D C O S T M I N I M I Z AT I O N
FIGURE 7.6
Comparative Statics Analysis
of the Cost-Minimization Problem with Respect
to the Price of Labor for a Fixed-Proportions
Production Function
The price of capital r 1, and the quantity of
output Q0 are held constant. When the price of
labor w 1, the isocost line is C1 and the ideal
input combination is at point A (L 1, K 1).
When the price of labor w 2, the isocost line
is C2 and the ideal input combination is still at
point A. Increasing the price of labor does not
cause the firm to substitute capital for labor.
A P P L I C A T I O N
K, capital services per year
Slope of isocost line C1 = –1
Slope of isocost line C2 = –2
C2
Q0 isoquant
C1
1
L, labor services per year
7.5
The End of Meter Maids? 8
Parking meters have been used to charge for parking
on city streets since the 1930s. From that time until the
present, the technology in parking meters has hardly
changed. Drivers put coins in the machines, which
then counted the time the car was allowed to remain
in the space. Typically, one meter was put at each
parking space, or in the last decade or so a double
meter was placed to cover two adjacent spaces.
Enforcement and collection of the coins was done by
“meter maids” (first hired by New York City in the
1960s, and initially all women). Digital meters were
introduced in the 1980s, replacing much of the mechanical works on the inside, but they otherwise operated exactly as those from the 1930s. Meters often
now have wireless communications, so that they can
report problems to the maintenance department.
A notable change implemented beginning in
2008–2009 was the introduction of a single meter for
an entire city block. Over time the rows of meters
dedicated to specific spaces are disappearing. Instead,
8
A
1
drivers park and walk to the single parking meter
on the block. The driver uses his or her credit card,
chooses the parking time desired, and the meter
prints a paper receipt that is placed on the car’s dashboard as proof of payment. While it is still possible to
use coins, most transactions at these machines are
done by credit card. The new meters dramatically reduce the need for meter maids or other parking
meter staff.
The move toward more sophisticated meters
makes sense. Their cost has fallen dramatically as information and telecommunications technologies have
fallen in cost, while even low-skilled wages have risen.
Wages have been particularly high in Europe (where
the meters originated). In the United States city government employees are usually unionized and have
relatively high wages as well. The high price of labor
compared to capital motivated the shift. As Figure 7.7
shows, a cost-minimizing firm (or city government)
faced with this situation has an incentive to operate
with a higher capital-labor ratio than does a firm facing a lower price of labor and a higher price of capital.
Based on Daniel Hamermesh, “Bad News for Meter-Maids,” Freakonomics blog, New York Times (September 11, 2009).
K, capital services per year
7 . 3 C O M PA R AT I V E S TAT I C S A N A LYS I S O F T H E C O S T- M I N I M I Z AT I O N P R O B L E M
CN = New isocost line
faced by parking agency
(w/r is larger)
KN
CO = Old isocost line
faced by parking agency
(w/r is smaller)
A
FIGURE 7.7
B
KO
CN
LN
CO
Q0 isoquant
LO
L, labor services per year
A P P L I C A T I O N
The Shift toward Modern
Parking Meters
The price of labor has risen over time, while
the cost of capital (advanced parking meters)
has fallen. The new isocost line has become
more steeply sloped than the original (slope of
CN CO). Therefore, the parking agency must
operate at a higher capital-labor ratio (KN / LN
KO / LO), so its cost-minimizing input combination is farther up the isoquant (point A)
than that of parking agencies in earlier eras
(point B).
7.6
Reducing Costs by Offshoring
In the last decade there has been an increase in “offshoring” of services by firms in the United States. For
example, a survey published by the Conference Board
in 2009 found that roughly half of the companies surveyed used offshoring, an increase of 22 percent compared to 2005.9 Offshoring refers to the outsourcing
of services such as software programming, accounting, or call center operations from firms overseas, instead of having the firm’s own employees provide
those services. The Conference Board’s survey found
that use of offshoring was accelerating. The industry
with the largest use of the practice was financial services. Survey firms reported that offshoring often resulted in cost savings, service or quality improvement,
improved relations with supplier firms, or was an effective way to overcome resistance to organizational
9
263
change by the firm’s own employees. The most common reason cited was cost savings.
To see how cost savings might arise for an individual firm, consider Figure 7.7. Suppose that a firm
uses only capital and skilled labor (for example, computer programmers) to produce Q0 units of output. In
the absence of offshoring, when the firm must pay a
relatively high wage rate for programmers, it would
choose input basket A, and its total cost would be CN.
Now suppose it becomes possible for the firm to
hire the same quality of skilled labor abroad, at a
lower wage rate. Assume that the firm does not care
whether the programming is done here or in a foreign
country because the programming services can be
“shipped” to the firm at essentially zero cost over the
Internet. If the firm still wishes to produce Q0 units of
output and the factor price of capital is unchanged,
the firm will now produce using basket B, at a total
Fifth Annual Conference Board / Duke Offshoring Research Network Survey (2009).
264
CHAPTER 7
C O S T S A N D C O S T M I N I M I Z AT I O N
cost of CO. We know that CO ⬍ CN because the vertical
intercept of the isocost line labeled CO lies below the
vertical intercept of the isocost line CN. Thus offshoring
results in a lower total cost.
Of course, this discussion provides only a very
narrow view of the effects of a reduction in a factor
price, such as a lower wage rate. We have assumed
that the firm produces a given amount of output and
that the prices of other factors of production remain
unchanged. Although it is well beyond the scope of
our discussion here, a change in the availability of a
resource (like the number of skilled laborers) in one
country can ultimately affect the prices of all goods
and factors of production in both domestic and
foreign economies. General equilibrium analysis and
international trade models might be used to understand more completely the effects of an activity such
as offshoring.10
C O M PA R AT I V E S TAT I C S A N A LYS I S
OF CHANGES IN OUTPUT
normal input An input
whose cost-minimizing
quantity increases as the
firm produces more output.
FIGURE 7.8
Now let’s do a comparative statics analysis of the cost-minimization problem for
changes in output quantity Q, with the prices of inputs (capital and labor) held constant. Figure 7.8 shows the isoquants for Q as output increases from 100 to 200 to
300. It also shows the tangent isocost lines for those three levels of output. As Q increases, the cost-minimizing combination of inputs moves to the northeast, from
point A to point B to point C, along the expansion path, the line connecting the
cost-minimizing combinations as quantity changes. Note that as quantity of output
increases, the quantity of each input also increases, indicating that, in this case, both
labor and capital are normal inputs. An input is normal if the firm uses more of it
Comparative Statics
Analysis of Cost-Minimization Problem with
Respect to Quantity: Normal Inputs
The price of capital and the price of labor
are held constant. When the quantity of
output increases from 100 to 200 to 300, the
cost-minimizing combination of inputs
moves along the expansion path, from point
A to point B to point C. When both inputs
are normal, the quantities of both increase as
the quantity of output increases (L1 ⬍ L2 ⬍
L3, and K1 ⬍ K2 ⬍ K3), and the expansion
path is upward sloping.
K, capital services per year
expansion path A line
that connects the costminimizing input combinations as the quantity of
output, Q, varies, holding
input prices constant.
Expansion path
C
K3
K2
B
A
K1
Q = 300
Q = 200
Q = 100
0
L1 L2 L3
L, labor services per year
10
Jagdish Bhagwhati, Arvind Panagariya, and T.N. Srinivasan, “The Muddles over Outsourcing,” Journal
of Economic Perspectives 18, no. 4 (Fall 2004): 96–114.
265
K, capital services per year
7 . 3 C O M PA R AT I V E S TAT I C S A N A LYS I S O F T H E C O S T- M I N I M I Z AT I O N P R O B L E M
Expansion path
K2
B
FIGURE 7.9
Q = 200
K1
A
Q = 100
L2 L1
L, labor services per year
Comparative Statics Analysis
of Cost-Minimization Problem with Respect to
Quantity: Labor Is an Inferior Input
The price of capital and the price of labor are
held constant. When the quantity of output
increases from 100 to 200, the cost-minimizing
combination of inputs moves along the expansion path, from point A to point B. If one input
(capital) is normal but the other (labor) is inferior,
then as the quantity of output increases, the
quantity of the normal input also increases
(K1 ⬍ K2). However, the quantity of the inferior
input decreases (L1 L2), and the expansion
path is downward sloping.
when producing more output. When both inputs are normal, the expansion path is
upward sloping.
What if one of the inputs is not normal, but is an inferior input—that is, the firm
uses less of it as output increases? This situation can arise if the firm drastically automates its production process to increase output, using more capital but less labor, as
shown in Figure 7.9 (in this case, labor is an inferior input). When one of the inputs
is inferior, the expansion path is downward sloping, as the figure shows.
When a firm uses just two inputs, can both inputs be inferior? Suppose they were;
then both inputs would decrease as output increases. But if the firm is minimizing
costs, it must be technically efficient, and if it is technically efficient, a decrease in
both inputs would decrease output (see Figure 6.1, on page 203). Thus, both inputs
cannot be inferior (one or both must be normal). This analysis demonstrates what we
can see intuitively: Inferiority of all inputs is inconsistent with the idea that the firm
is getting the most output from its inputs.
S U M M A R I Z I N G T H E C O M PA R AT I V E S TAT I C S
A N A LYS I S : T H E I N P U T D E M A N D C U RV E S
We’ve seen that the solution to the cost-minimization problem is an optimal input
combination: a quantity of capital and a quantity of labor. We’ve also seen that this
input combination depends on how much output the firm wants to produce and the
prices of labor and capital. Figure 7.10 shows one way to summarize how the costminimizing quantity of labor varies with the price of labor.
The top graph shows a comparative statics analysis for a firm that initially produces 100 units. The price of capital r is $1 and remains fixed in the analysis. The initial price of price of labor w is $1, and the cost-minimizing input combination is at
point A.
inferior input An
input whose costminimizing quantity
decreases as the firm
produces more output.
C O S T S A N D C O S T M I N I M I Z AT I O N
FIGURE 7.10
Comparative Statics Analysis
and the Labor Demand Curve
The labor demand curve
shows how the firm’s costminimizing amount of labor
varies as the price of labor
varies. For a fixed output of
100 units, an increase in the
price of labor from $1 to $2
per unit moves the firm
along its labor demand curve
from point A to point B.
Holding the price of labor
fixed at $1 per unit, an
increase in output from 100
to 200 units per year shifts
the labor demand curve
rightward and moves the firm
from point A to point C .
labor demand curve
A curve that shows how
the firm’s cost-minimizing
quantity of labor varies
with the price of labor.
K, capital services per year
CHAPTER 7
B
C
Q = 200
A
Q = 100
L, labor services per year
w, dollars per unit labor
266
$2
$1
B′
A′
C′
Labor demand, Q = 200
Labor demand, Q = 100
L, labor services per year
First let’s see what happens when the price of labor increases from $1 to $2, holding
output constant at 100 units. The cost-minimizing combination of inputs is at point B
in the top graph. The bottom graph shows the firm’s labor demand curve: how the
firm’s cost-minimizing quantity of labor varies with the price of labor. The movement
from point A to point B in the top graph corresponds to a movement from point A to
point B on the curve showing the demand for labor when output is 100. Thus, the
change in the price of labor induces the firm to move along the same labor demand
curve. As Figure 7.10 shows, the labor demand curve is generally downward sloping.11
Now let’s see why a change in the level of output (holding input prices constant)
leads to a shift in the labor demand curve. Once again, the firm initially chooses basket
A when the price of labor is $1 and the firm produces 100 units. If the firm needs to
increase production to 200 units, and the prices of capital and labor do not change, the
cost-minimizing combination of inputs is at point C in the top graph. The movement
from combination A to combination C in the top graph corresponds to a movement
from point A to point C in the bottom graph. Point C lies on the curve showing the
demand for labor when output is 200. Thus, the change in the level of output leads to
shift from the labor demand curve when output is 100 to the labor demand curve when
output is 200. If output increases and an input is normal, the demand for that input will
shift to the right, as shown in Figure 7.10. If output increases and an input is inferior,
the demand for that input will shift to the left.
11
As already noted, exceptions to this occur when the firm has a fixed-proportions production function
or when the cost-minimizing quantity of labor is zero. In these cases, as we saw, the quantity of labor
demanded does not change as the price of labor goes up.
267
7 . 3 C O M PA R AT I V E S TAT I C S A N A LYS I S O F T H E C O S T- M I N I M I Z AT I O N P R O B L E M
The firm’s capital demand curve (showing how the firm’s cost-minimizing quantity
of capital varies with the price of capital) could be illustrated in exactly the same way.
Learning-By-Doing Exercise 7.4 shows how to find input demand curves from a production function.
S
E
capital demand curve
A curve that shows how
the firm’s cost-minimizing
quantity of capital varies
with the price of capital.
L E A R N I N G - B Y- D O I N G E X E R C I S E 7 . 4
D
Deriving the Input Demand Curves from a Production Function
Problem Suppose that a firm faces the
production function Q 50 1LK. What are the demand
curves for labor and capital?
Solution We begin with the tangency condition
expressed by equation (7.1): MPL ⲐMPK wⲐr. As shown
in Learning-By-Doing Exercise 7.2, MPL ⲐMPK K ⲐL.
Thus, K ⲐL wⲐr, or L (rⲐw)K. This is the equation of
the expansion path (see Figure 7.8).
Let’s now substitute this for L in the production
function and solve for K in terms of Q, w, and r:
Q 50
or
K
r
a KbK
B w
Q
w
50 A r
This is the demand curve for capital. Since L ⫽ (r Ⲑw)K,
K ⫽ (w Ⲑr)L. Thus,
Q
w
r
L⫽
r
50 A w
or
L⫽
Q
r
50 A w
This is the demand curve for labor. Note that the demand
for labor is a decreasing function of w and an increasing
function of r. This is consistent with the graphical analysis
in Figures 7.5 and 7.10. Note also that both K and L
increase when Q increases. Therefore, both capital and
labor are normal inputs.
Similar Problems: 7.13, 7.23, 7.24
THE PRICE ELASTICITY OF DEMAND FOR INPUTS
We have just seen how we can summarize the solution to the cost-minimization problem
with input demand curves. In Chapter 2, we learned that we can describe the sensitivity
of the demand for any product to its price using the concept of price elasticity of demand.
Now let’s apply this concept to input demand curves. The price elasticity of demand
for labor ⑀L, w is the percentage change in the cost-minimizing quantity of labor with
respect to a 1 percent change in the price of labor:
⑀L, w
¢L
L
¢w
w
⫻ 100%
⫻ 100%
price elasticity of
demand for labor The
percentage change in the
cost-minimizing quantity
of labor with respect to a
1 percent change in the
price of labor.
or, rearranging terms and canceling the 100%s,
⑀L, w
¢L w
¢w L
Similarly, the price elasticity of demand for capital ⑀K,r is the percentage change in
the cost-minimizing quantity of capital with respect to a 1 percent change in the price
of capital:
¢K r
⑀K, r
¢r K
price elasticity of
demand for capital
The percentage change in
the cost-minimizing quantity
of capital with respect to a
1 percent change in the
price of capital.
268
Low elasticity of substitution implies . . .
High elasticity of substitution implies . . .
K, capital services per year
C O S T S A N D C O S T M I N I M I Z AT I O N
K, capital services per year
CHAPTER 7
15
10
5
0
B
Q = 100 isoquant
B
5
Q = 100 isoquant
0
10
A
(c)
2.2
5
L, labor services per year
inelastic demand for labor.
elastic demand for labor.
Wage rate, $ per unit labor
4.6 5
L, labor services per year
10
Wage rate, $ per unit labor
(a)
A
15
(b)
$2
$1
Labor
demand
0
4.6 5
L, labor services per year
10
(d)
10
$2
$1
0
Labor
demand
2.2
5
10
L, labor services per year
FIGURE 7.11 The Price Elasticity of Demand for Labor Depends on the Elasticity
of Substitution between Labor and Capital
The price of labor decreases from $2 to $1, with the price of capital and quantity of output
held constant. In panels (a) and (b), the elasticity of substitution is low (0.25), so the 50 percent
decrease in the price of labor results in only an 8 percent increase in the quantity of labor (i.e.,
demand for labor is relatively insensitive to price of labor; the cost-minimizing input combination moves only from point A to point B). In panels (c) and (d), the elasticity of substitution is
high (2), so the same 50 percent decrease in the price of labor results in a 127 percent increase
in the quantity of labor (i.e., demand for labor is much more sensitive to price of labor; the
movement of the cost-minimizing input combination from point A to point B is much greater).
An important determinant of the price elasticity of demand for inputs is the elasticity of substitution (see Chapter 6). In Figure 7.11, panels (a) and (b) show that when
the elasticity of substitution is small—that is, when the firm faces limited opportunities to substitute among inputs—large changes in the price of labor result in small
changes in the cost-minimizing quantity of labor. In panel (a), we see a comparative
statics analysis of a firm that faces a constant elasticity of substitution (CES) production function whose elasticity of substitution is 0.25. With this production function,
the firm’s opportunities to substitute between labor and capital are limited. As a result,
a 50 percent decrease in the price of labor, from w $2 to w $1 (holding the price
of capital fixed at r 1) results in an 8 percent increase in the cost-minimizing quantity of labor, from 4.6 to 5, shown both in panel (a), where the cost-minimizing input
combination moves from point A to point B, and by the labor demand curve in panel
(b). In this case, where the price elasticity of demand for labor is quite small, the demand for labor is relatively insensitive to the price of labor.
269
7 . 4 S H O RT- RU N C O S T M I N I M I Z AT I O N
A P P L I C A T I O N
7.7
Input Demand in Alabama
How elastic or inelastic are input demands in real industries? Research by A. H. Barnett, Keith Reutter, and
Henry Thompson suggests that input demands in
manufacturing industries might be relatively inelastic.12
Using data on input quantities, input prices, and outputs over the period 1971–1991, they estimated how
the cost-minimizing quantities of capital, labor, and
electricity varied with the prices of these inputs in
four industries in the state of Alabama: textiles,
paper, chemicals, and metals.
Table 7.1 shows their findings. To see how to interpret these numbers, consider the textile industry.
Table 7.1 tells us that the price elasticity of demand
for production labor in the textile industry is ⫺0.50.
This means that faced with a 1 percent increase in the
wage rate for production workers, a typical Alabama
textile firm will reduce the cost-minimizing quantity
of labor by 0.50 percent. This implies that the demand for production labor in Alabama’s textile industry is price inelastic, which means that the costminimizing quantity of labor is not that sensitive to
changes in the price of labor. All but one of the price
elasticities of input demand in Table 7.1 are between
0 and ⫺1, which suggests that in the four industries
studied, firms do not aggressively substitute among
inputs as input prices change. That is, firms in these
industries face situations more akin to panels (a) and
(b) in Figure 7.11 than to panels (c) and (d).
TABLE 7.1 Price Elasticities of Input Demand for Manufacturing
Industries in Alabama
Input
Industry
Textiles
Paper
Chemicals
Metals
Capital
0.41
0.29
0.12
0.91
Production
Labor
0.50
0.62
0.75
0.41
Nonproduction
Labor
1.04
0.97
0.69
0.44
Electricity
0.11
0.16
0.25
0.69
Source: Table 1 in A. H. Barnett, K. Reutter, and H. Thompson, “Electricity Substitution: Some
Local Industrial Evidence,” Energy Economics 20 (1998): 411–419.
By contrast, in panel (c) of Figure 7.11, we see a comparative statics analysis of a
firm that faces a CES production function whose elasticity of substitution is 2. With
this production function, the firm has relatively abundant opportunities to substitute
capital for labor. As a result, a 50 percent decrease in the price of labor, from w $2
to w $1, increases the firm’s cost-minimizing quantity of labor from 2.2 to 5, an increase of 127 percent, as shown both in panel (c), where the cost-minimizing input
combination moves from point A to point B, and in panel (d) by the labor demand
curve. With a greater flexibility to substitute between capital and labor, the firm’s demand for labor is more sensitive to the price of labor.
T
he cases we have studied so far in this chapter all involve long-run cost minimization, when the firm is free to vary the quantity of its inputs. In this section, we study
the firm’s cost-minimization problem in the short run, when the firm faces the constraint that one or more of the firm’s inputs cannot be changed (perhaps because past
12
A. H. Barnett, K. Reutter, and H. Thompson, “Electricity Substitution: Some Local Industrial
Evidence,” Energy Economics 20 (1998): 411–419.
7.4
S H O R T- R U N
COST
MINIMIZATION
270
CHAPTER 7
C O S T S A N D C O S T M I N I M I Z AT I O N
decisions make change impossible). For instance, consider a firm that, as in previous
examples, uses just two inputs, capital and labor. Suppose that the firm is unable
to alter its quantity of capital K, even if it produces zero output, but can alter its quantity of labor L (e.g., by hiring or firing workers). Thus, the firm’s total costs are
wL ⫹ rK.
C H A R AC T E R I Z I N G C O S T S I N T H E S H O R T R U N
Fixed versus Variable Costs; Sunk versus Nonsunk Costs
total variable cost
The sum of expenditures on
variable inputs, such as
labor and materials, at the
short-run cost-minimizing
input combination.
total fixed cost The
cost of fixed inputs; it does
not vary with output.
The two components of the firm’s total cost, wL and rK, differ from each other in two
important ways. First, they differ in the extent to which they are sensitive to output.
As we will see, the firm’s expenditures on labor wL go up or down as the firm produces
more or less output. The firm’s labor cost thus constitutes its total variable cost, the
output-sensitive component of its costs. By contrast, the firm’s capital cost, rK will
not go up or down as the firm produces more or less output. (The firm’s capital cost
might be the payment that it makes to lease factory space from another firm, or it
might be a mortgage payment if the firm borrowed money to build its own plant. In
either case, these costs would not change if the firm varies the amount of output it
produces within its plant.) The capital cost thus constitutes the firm’s total fixed
cost, the component of the firm’s cost that is output insensitive.
Second, the firm’s two categories of costs differ in the extent to which they are
sunk or nonsunk with respect to the decision to suspend operations by producing
zero output. This decision can be couched in terms of the question: Should the firm
produce no output, or should it produce some positive level of output? With respect
to this shutdown decision, the firm’s total expenditure on labor, wL, is a nonsunk
cost. If the firm produces no output, it can avoid its labor costs altogether. Since variable costs are completely avoidable, they are always nonsunk. By contrast, the firm’s
fixed capital cost rK may be sunk or nonsunk. The fixed cost will be sunk if there are
no alternative uses for the plant—that is, if the firm cannot find anyone else willing
to pay to use the plant. Because the firm cannot adjust the quantity of its capital in
the short run, the firm cannot avoid the cost associated with this capital, even if it
were to produce no output (e.g., if the firm has borrowed money to build its plant, it
must still make its mortgage payments, even if it does not operate the plant to produce output).
Are Fixed Costs and Sunk Costs the Same?
As we have just seen, variable costs are completely avoidable if the firm produces no
output. Therefore, variable costs are always nonsunk. However, fixed costs are not
necessarily sunk. For example, the firm’s capital may be fixed, and it may be obligated
to pay the bank a monthly fixed cost of rK (think of this as a mortgage payment). But
the firm may know that, instead of using the plant itself, it can rent the plant to someone else for a monthly rental payment of rK. Since the rental proceeds will cover the
mortgage payment, the firm can avoid all of the fixed cost by renting its plant. In that
case, the firm’s fixed cost is avoidable (nonsunk).
As another example, consider the cost of heating a factory. As long as the firm operates, the heating bill will be about the same, no matter how much output the firm
produces (thus, the heating cost is fixed). But if the firm temporarily shuts down its
7 . 4 S H O RT- RU N C O S T M I N I M I Z AT I O N
271
COST OF
INPUT
Cost is avoidable
if firm produces
zero output
Cost is not avoidable
if firm produces
zero output
FIGURE 7.12
Input usage goes
up or down as firm
produces more or
less output
Input usage doesn't
go up or down as
firm produces more
or less output
Input usage doesn't
go up or down as
firm produces more
or less output
Cost is
Variable (output
sensitive)
Nonsunk
Cost is
Fixed (output
insensitive)
Nonsunk
Cost is
Fixed (output
insensitive)
Sunk
Examples:
Examples:
Examples:
Some utilities (e.g.,
heating and lighting
for the plant)
Capital (plant and
equipment) under
some circumstances
Labor, materials
Classifying Costs in the
Short Run
A cost is variable (output
sensitive) and nonsunk if
the firm can avoid it by
producing zero output and
if it varies when output
varies. A cost is fixed (output
insensitive) and nonsunk if
the firm can avoid it by
producing zero output but
it does not vary when output varies. A cost is fixed
(output insensitive) and
sunk if the firm cannot
avoid it by producing zero
output (such costs do not
vary when output varies).
factory, producing no output, it can turn off the heat and the heating cost would go
away. The heating cost is then avoidable (nonsunk).13
Figure 7.12 summarizes these conclusions. Short-run costs can be
• Variable and nonsunk. (Such costs are, by definition, output sensitive.)
• Fixed and nonsunk. (Such costs are output insensitive, but avoidable if the firm
produces zero output. We will explore such costs in more detail in Chapter 9,
where we consider their impact on a firm’s decision to produce zero output in
the short run.)
• Fixed and sunk. (Such costs are output insensitive and unavoidable, even if the
firm produces zero output.)
13
Of course, this might not be the case if, by eliminating a shift from the plant, the firm could turn down
the heat during the period in which workers are not in the plant. But in many real-world factories, heating costs will not change much as the volume of output changes, either because of the need to keep the
plant at a constant temperature in order to maintain equipment in optimal operating condition or because
of the time it takes to adjust temperature up and down.
272
CHAPTER 7
A P P L I C A T I O N
C O S T S A N D C O S T M I N I M I Z AT I O N
7.8
What Fraction of a Capital
Investment Is Sunk Cost?
were to dispose of their machine tools. Some types
of machines were designed so that they could be used
by other firms; thus, a portion of the investment
costs could be recovered by selling the machines in a
secondhand market. But other machines could not
be sold in a secondhand market because they were
designed to perform tasks useful only to the firm that
originally purchased the capital. In those cases, the
nonsunk costs would be the scrap value from the sale
of the used machines.
Asplund found that the four manufacturing firms
could “only expect to get back 20–50 percent of the
initial price of a ‘new’ machine once it is installed.”
This means that 50 to 80 percent of the investment
costs were sunk, leading Asplund to conclude that
“capital investments in metalworking machinery (machine tools) appear to be largely sunk costs.”14
In the short run some portion of the costs associated
with an investment may be sunk. The fraction of the
investment cost that is sunk will depend on the possible alternative uses of the capital.
Marcus Asplund has analyzed capital investments
for four Swedish manufacturing firms that used machine tools (capital input) primarily to produce metal
products and nonelectrical machinery. Using data
from the decade prior to 1991, he examined the cost
structure of these firms to determine what portion of
the investments in machine tools is sunk. He indicated
that there were two main ways in which these firms
might recover some of the investment cost if they
C O S T M I N I M I Z AT I O N I N T H E S H O R T R U N
K, capital services per year
Let’s now consider the firm’s cost-minimization problem in the short run. Figure 7.13
shows the firm’s problem when it seeks to produce a quantity of output Q0 but is unable to change the quantity of capital from its fixed level K. The firm’s only technically
efficient combination of inputs occurs at point F, where the firm uses the minimum
quantity of labor that, in conjunction with the fixed quantity of K, allows the firm to
produce exactly the desired output Q0.
FIGURE 7.13 Short-Run Cost Minimization
with One Fixed Input
When the firm’s capital is fixed at K, the short-run
cost-minimizing input combination is at point F. If
the firm were free to adjust all of its inputs, the
cost-minimizing combination would be at point A.
A
K
F
Q0 isoquant
L, labor services per year
14
Marcus Asplund, “What Fraction of a Capital Investment Is Sunk Costs,” Journal of Industrial Economics
47, no. 3 (September 2000): 287–304.
K, capital services per year
7 . 4 S H O RT- RU N C O S T M I N I M I Z AT I O N
273
Long-run
expansion path
C
K
B
D
E
Short-run expansion path
A
Q2 isoquant
Q1 isoquant
Q0 isoquant
L, labor services per year
FIGURE 7.14 Short-Run Input Demand
versus Long-Run Input Demand
In the long run, as the firm’s output changes, its
cost-minimizing quantity of labor varies along
the long-run expansion path. In the short run,
as the firm’s output changes, its cost-minimizing
quantity of labor varies along the short-run expansion path. These expansion paths cross at
point B, where the input combination is
cost-minimizing in both the long run and the
short run.
This short-run cost-minimizing problem has only one variable factor (labor).
Because the firm cannot substitute between capital and labor, the determination of the
optimal amount of labor does not involve a tangency condition (i.e., no isocost line is
tangent to the Q0 isoquant at point F ). By contrast, in the long run, when the firm can
adjust the quantities of both inputs, it will operate at point A, where an isocost line is
tangent to the isoquant. Figure 7.13 thus illustrates that cost minimization in the short
run will not, in general, involve the same combination of inputs as cost minimization
in the long run; in the short run, the firm will typically operate with higher total costs
than it would if it could adjust all of its inputs freely.
There is, however, one exception, illustrated in Figure 7.14. Suppose the firm is
required to produce Q1. In the long run, it will operate at point B, freely choosing
K units of capital. However, if the firm is told that in the short run it must produce
with the amount of capital fixed at K, it will also operate at point B. In this case the
amount of capital the firm would choose in the long run just happens to be the same
as the amount of capital fixed in the short run. Therefore, the total cost the firm incurs in the short run is the same as the total cost in the long run.
C O M PA R AT I V E S TAT I C S : S H O R T- R U N I N P U T
DEMAND VERSUS LONG-RUN INPUT DEMAND
As we have seen, in the case of a firm that uses just two inputs, labor and capital, the
long-run cost-minimizing demand for labor will vary with the price of both inputs (as
discussed in Section 7.3). By contrast, in the short run, if the firm cannot vary its
quantity of capital, its demand for labor will be independent of input prices (as explained earlier and illustrated in Figure 7.13).
The firm’s demand for labor in the short run will, however, vary with the quantity of output. Figure 7.14 shows this relationship using the concept of an expansion
path (also discussed in Section 7.3). As the firm varies its output from Q0 to Q1 to Q2,
the long-run cost-minimizing input combination moves from point A to point B to
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point C, along the long-run expansion path. But in the short run, when the quantity
of capital is fixed at K, the cost-minimizing input combination moves from point D
to point B to point E, along the short-run expansion path. (As noted above, point B
illustrates a cost-minimizing input combination that is the same both in the long run
and in the short run, if the quantity of output is Q1.)
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 7 . 5
D
contains only one unknown, L: Q 50 2LK. Solving
this equation for L gives us L Q2/(2500 K). This is
the cost-minimizing quantity of labor in the short run.
Short-Run Cost Minimization with One Fixed Input
Problem Suppose that the firm’s production function is given by the production function in
Learning-By-Doing Exercises 7.2 and 7.4: Q 50 1LK.
The firm’s capital is fixed at K. What amount of labor
will the firm hire to minimize cost in the short run?
Solution
Similar Problems: 7.27, 7.28
Since output is given as Q and capital is
fixed at K, the equation for the production function
M O R E T H A N O N E VA R I A B L E I N P U T I N T H E S H O R T R U N
FIGURE 7.15 Short-Run Cost
Minimization with Two Variable Inputs and
One Fixed Input
To produce Q0 units of output, the costminimizing input combination occurs at point
A, where the Q0 isoquant is tangent to an
isocost line. Points E and F do not minimize
cost because the firm can lower cost from TC1
to TC0 by moving to input combination A.
M, quantity of materials per year
When the firm has more than one variable input, the analysis of cost minimization in
the short run is very similar to the long-run analysis. To illustrate, suppose that the
firm uses three inputs: labor L, capital K, and raw materials M. The firm’s production
function is f (L, K, M ). The prices of these inputs are denoted by w, r, and m, respectively. Again suppose that the firm’s capital is fixed at K. The firm’s short-run costminimization problem is to choose quantities of labor and materials that minimize
total cost, wL ⫹ mM ⫹ rK, given that the firm wants to produce an output level Q0.
Figure 7.15 analyzes this short-run cost-minimization problem graphically, by
plotting the two variable inputs against each other (L on the horizontal axis and M on
Slope of isocost line = –
TC1
m
TC0
m
E
A
F
Q0 isoquant
L, labor services per year
w
m
7 . 4 S H O RT- RU N C O S T M I N I M I Z AT I O N
275
the vertical axis). The figure shows two isocost lines and the isoquant corresponding to
output Q0. If the cost-minimization problem has an interior solution, the cost-minimizing
input combination will be at the point where an isocost line is tangent to the isoquant
(point A in the figure). At this tangency point, we have MRTSL, M MPL ⲐMPM wⲐm,
or, rearranging terms, MPL Ⲑw MPM Ⲑm. Thus, just as in the long run [see equation (7.2)],
the firm minimizes its total costs by equating the marginal product per dollar that it
spends on the variable inputs it uses in positive amounts. Learning-By-Doing Exercise 7.6
shows how to find the cost-minimizing combinations of inputs when the level of one
input is fixed and the levels of two other inputs are variable.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 7 . 6
D
Short-Run Cost Minimization with Two Variable Inputs
Suppose that a firm’s production function
is given by Q ⫽ 1L ⫹ 1K ⫹ 1M. For this production
function, the marginal products of labor, capital, and
materials are MPL ⫽ 1/(21L), MPK ⫽ 1/(21K ), and
MPM ⫽ 1/(21M ). The input prices of labor, capital,
and materials are w 1, r 1, and m 1, respectively.
Problem
(a) Given that the firm wants to produce 12 units of output, what is the solution to the firm’s long-run costminimization problem?
(b) Given that the firm wants to produce 12 units of
output, what is the solution to the firm’s short-run costminimization problem when K 4?
(c) Given that the firm wants to produce 12 units of output, what is the solution to the firm’s short-run costminimization problem when K 4 and L 9?
Solution
(a) Here we have two tangency conditions and the requirement that L, K, and M produce 12 units of output:
MPL
1
⫽ 1K⫽L
MPK
1
MPL
1
⫽ 1M⫽L
MPM
1
12 1L ⫹ 1K ⫹ 1M
This is a system of three equations in three unknowns.
The solution to this system gives us the long-run costminimizing input combination for producing 12 units of
output: L K M 16.
(b) With K fixed at 4 units, the firm must choose an
optimum combination of the variable inputs, labor and
materials. We thus have a tangency condition and the
requirement that L and M produce 12 units of output
when K ⫽ 4.
MPL
1
⫽ 1M⫽L
MPM
1
12 ⫽ 1L ⫹ 14 ⫹ 1M
This is a system of two equations in two unknowns, L
and M. The solution gives us the short-run costminimizing input combination for producing 12 units of
output, when K is fixed at 4 units: L ⫽ 25 and M ⫽ 25.
(c) With K fixed at 4 units and L fixed at 9 units, we do not
have a tangency condition to determine the short-run
cost-minimizing level of M because M is the only variable
factor of production. Instead, we can simply use the production function to find the quantity of materials M
needed to produce 12 units of output when L ⫽ 9 and
K ⫽ 4: 12 ⫽ 19 ⫹ 14 ⫹ 1M, which implies M ⫽ 49.
This is the short-run cost-minimizing quantity of materials to produce 12 units of output when L ⫽ 9 and K ⫽ 4.
The following table summarizes the results of this
exercise. In addition to showing the solutions to the costminimization problem, it also presents the firm’s minimized
total cost: the total cost incurred when the firm utilizes the
cost-minimizing input combination. (Recall that total cost
is simply wL ⫹ rK ⫹ mM.) Notice that the minimized cost
is lowest in the long run, next lowest in the short run with
one fixed input, and highest when the firm has two fixed
inputs. This shows that the more flexibility the firm has to
adjust its inputs, the more it can lower its costs.
Similar Problems: 7.29, 7.30
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Quantity of
Labor, L
Quantity of
Capital, K
Quantity of
Materials, M
Minimized
Total Cost
Long-run cost minimization
for Q 12
16 units
16 units
16 units
$48
Short-run cost minimization
for Q 12 when K 4
25 units
4 units
25 units
$54
Short-run cost minimization
for Q 12 when K 4
and L 9
9 units
4 units
49 units
$62
CHAPTER SUMMARY
• The opportunity cost of a decision is the payoff associated with the best of the alternatives that are not chosen.
• Opportunity costs are forward looking. When evaluating the opportunity cost of a particular decision, you
need to identify the value of the alternatives that the
decision forecloses in the future.
• From a firm’s perspective, the opportunity cost of
using the productive services of an input is the current
market price of the input.
• Explicit costs involve a direct monetary outlay.
Implicit costs do not involve an outlay of cash.
• Accounting costs include explicit costs only.
Economic costs include explicit and implicit costs.
• Sunk costs are costs that have already been incurred
and cannot be recovered. Nonsunk cost are costs that
can be avoided if certain choices are made.
• The long run is the period of time that is long
enough for the firm to vary the quantities of all its inputs. The short run is the period of time in which at least
one of the firm’s input quantities cannot be changed.
• An isocost line shows all combinations of inputs that
entail the same total cost. When graphed with quantity
of labor on the horizontal axis and quantity of capital on
the vertical axis, the slope of an isocost line is minus the
ratio of the price of labor to the price of capital.
• At an interior solution to the long-run cost-minimization problem, the firm adjusts input quantities so
that the marginal rate of technical substitution equals
the ratio of the input prices. Equivalently, the ratio of
the marginal product of one input to its price equals the
corresponding ratio for the other inputs.
• At corner point solutions to the cost-minimization
problem, the ratios of marginal products to input prices
may not be equal.
• An increase in the price of an input causes the costminimizing quantity of that input to go down or stay the
same. It can never cause the cost-minimizing quantity to
go up.
• An increase in the quantity of output will cause the
cost-minimizing quantity of an input to go up if the input
is a normal input and will cause the cost-minimizing quantity of the input to go down if the input is an inferior input.
• The expansion path shows how the cost-minimizing
quantity of inputs varies as quantity of output changes.
• An input demand curve shows how the costminimizing quantity of the input varies with its input price.
• The price elasticity of demand for an input is the
percentage change in the cost-minimizing quantity of
that input with respect to a 1 percent change in its price.
• When the elasticity of substitution between inputs is
small, the price elasticity of demand for each input is
also small. When the elasticity of substitution is large, so
is the price elasticity of demand.
• In the short run, at least one input is fixed. Variable
costs are output sensitive—they vary as output varies.
Fixed costs are output insensitive—they remain the
same for all positive levels of output.
• All variable costs are nonsunk. Fixed costs can be
sunk (unavoidable) or nonsunk (avoidable) if the firm
produces no output.
• The short-run cost-minimization problem involves a
choice of inputs when at least one input quantity is held
fixed.
277
PROBLEMS
REVIEW QUESTIONS
1. A biotechnology firm purchased an inventory of test
tubes at a price of $0.50 per tube at some point in the
past. It plans to use these tubes to clone snake cells.
Explain why the opportunity cost of using these test
tubes might not equal the price at which they were
acquired.
6. Explain why, at an interior optimal solution to the
firm’s cost-minimization problem, the additional output
that the firm gets from a dollar spent on labor equals the
additional output from a dollar spent on capital. Why
would this condition not necessarily hold at a corner
point optimal solution?
2. You decide to start a business that provides computer
consulting advice for students in your residence hall.
What would be an example of an explicit cost you would
incur in operating this business? What would be an example of an implicit cost you would incur in operating
this business?
7. What is the difference between the expansion path and
the input demand curve?
3. Why does the “sunkness” or “nonsunkness” of a cost
depend on the decision being made?
4. How does an increase in the price of an input affect
the slope of an isocost line?
5. Could the solution to the firm’s cost-minimization
problem ever occur off the isoquant representing the required level of output?
8. In Chapter 5 you learned that, under certain conditions, a good could be a Giffen good: An increase in the
price of the good could lead to an increase, rather than a
decrease, in the quantity demanded. In the theory of cost
minimization, however, we learned that, an increase in
the price of an input will never lead to an increase in the
quantity of the input used. Explain why there cannot be
“Giffen inputs.”
9. For a given quantity of output, under what conditions
would the short-run quantity demanded for a variable
input (such as labor) equal the quantity demanded in the
long run?
PROBLEMS
7.1. A computer-products retailer purchases laser
printers from a manufacturer at a price of $500 per
printer. During the year the retailer will try to sell the
printers at a price higher than $500 but may not be able
to sell all of the printers. At the end of the year, the manufacturer will pay the retailer 30 percent of the original
price for any unsold laser printers. No one other than the
manufacturer would be willing to buy these unsold printers at the end of the year.
a) At the beginning of the year, before the retailer has
purchased any printers, what is the opportunity cost of
laser printers?
b) After the retailer has purchased the laser printers, what is
the opportunity cost associated with selling a laser printer to
a prospective customer? (Assume that if this customer does
not buy the printer, it will be unsold at the end of the year.)
c) Suppose that at the end of the year, the retailer still has
a large inventory of unsold printers. The retailer has set
a retail price of $1,200 per printer. A new line of printers
is due out soon, and it is unlikely that many more old
printers will be sold at this price. The marketing manager of the retail chain argues that the chain should cut
the retail price by $1,000 and sell the laser printers at
$200 each. The general manager of the chain strongly
disagrees, pointing out that at $200 each, the retailer
would “lose” $300 on each printer it sells. Is the general
manager’s argument correct?
7.2. A grocery shop is owned by Mr. Moore and has the
following statement of revenues and costs:
Revenues
$250,000
Supplies
$25,000
Electricity
$6,000
Employee salaries
$75,000
Mr. Moore’s salary
$80,000
Mr. Moore always has the option of closing down his
shop and renting out the land for $100,000. Also, Mr.
Moore himself has job offers at a local supermarket at a
salary of $95,000 and at a nearby restaurant at $65,000.
He can only work one job, though. What are the shop’s
accounting costs? What are Mr. Moore’s economic costs?
Should Mr. Moore shut down his shop?
7.3. Last year the accounting ledger for an owner of a
small drug store showed the following information about
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her annual receipts and expenditures. She lives in a taxfree country (so don’t worry about taxes).
Revenues
Wages paid to hired labor
(other than herself )
Utilities (fuel, telephone, water)
$1,000,000
$300,000
$20,000
Purchases of drugs and
other supplies for the store
$500,000
Wages paid to herself
$100,000
She pays a competitive wage rate to her workers, and the
utilities and drugs and other supplies are all obtained at
market prices. She already owns the building, so she has
no cash outlay for its use. If she were to close the business, she could avoid all of her expenses, and, of course,
would have no revenue. However, she could rent out her
building for $200,000. She could also work elsewhere
herself. Her two employment alternatives include working at another drug store, earning wages of $100,000,
or working as a freelance consultant, earning $80,000.
Determine her accounting profit and her economic
profit if she stays in the drug store business. If the two are
different, explain the difference between the two values
you have calculated.
7.4. A consulting firm has just finished a study for a
manufacturer of wine. It has determined that an additional man-hour of labor would increase wine output by
1,000 gallons per day. Adding another machine-hour of
fermentation capacity would increase output by 200 gallons per day. The price of a man-hour of labor is $10 per
hour. The price of a machine-hour of fermentation
capacity is $0.25 per hour. Is there a way for the wine
manufacturer to lower its total costs of production and
yet keep its output constant? If so, what is it?
7.5. A firm uses two inputs, capital and labor, to produce
output. Its production function exhibits a diminishing
marginal rate of technical substitution.
a) If the price of capital and labor services both increase
by the same percentage amount (e.g., 20 percent), what
will happen to the cost-minimizing input quantities for a
given output level?
b) If the price of capital increases by 20 percent while the
price of labor increases by 10 percent, what will happen
to the cost-minimizing input quantities for a given output level?
7.6. A farmer uses three inputs to produce vegetables:
land, capital, and labor. The production function for
the farm exhibits diminishing marginal rate of technical
substitution.
a) In the short run the amount of land is fixed. Suppose
the prices of capital and labor both increase by 5 percent.
What happens to the cost-minimizing quantities of labor
and capital for a given output level? Remember that there
are three inputs, one of which is fixed.
b) Suppose only the cost of labor goes up by 5 percent.
What happens to the cost-minimizing quantity of labor
and capital in the short run.
7.7. The text discussed the expansion path as a graph
that shows the cost-minimizing input quantities as output
changes, holding fixed the prices of inputs. What the text
didn’t say is that there is a different expansion path for
each pair of input prices the firm might face. In other
words, how the inputs vary with output depends, in part,
on the input prices. Consider, now, the expansion paths
associated with two distinct pairs of input prices, (w1, r1)
and (w2, r2). Assume that at both pairs of input prices, we
have an interior solution to the cost-minimization problem for any positive level of output. Also assume that the
firm’s isoquants have no kinks in them and that they exhibit diminishing marginal rate of technical substitution.
Could these expansion paths ever cross each other at a
point other than the origin (L 0, K 0)?
7.8. Suppose the production of airframes is character1
1
ized by a CES production function: Q ⫽ (L2 ⫹ K 2 )2.
The marginal products for this production function are
1
1
1
1
1
1
MPL ⫽ (L2 ⫹ K 2 )L⫺2 and MPK (L2 ⫹ K 2)K ⫺2. Suppose
that the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of
labor and capital for an airframe manufacturer that wants to
produce 121,000 airframes.
7.9. Suppose the production of airframes is characterized by a Cobb–Douglas production function: Q LK.
The marginal products for this production function are
MPL K and MPK L. Suppose the price of labor is
$10 per unit and the price of capital is $1 per unit. Find
the cost-minimizing combination of labor and capital if
the manufacturer wants to produce 121,000 airframes.
7.10. The processing of payroll for the 10,000 workers
in a large firm can either be done using 1 hour of computer time (denoted by K ) and no clerks or with 10 hours
of clerical time (denoted by L) and no computer time.
Computers and clerks are perfect substitutes; for example, the firm could also process its payroll using 1/2 hour
of computer time and 5 hours of clerical time.
a) Sketch the isoquant that shows all combinations of
clerical time and computer time that allows the firm to
process the payroll for 10,000 workers.
b) Suppose computer time costs $5 per hour and clerical
time costs $7.50 per hour. What are the cost-minimizing
choices of L and K ? What is the minimized total cost of
processing the payroll?
c) Suppose the price of clerical time remains at $7.50 per
hour. How high would the price of an hour of computer
PROBLEMS
time have to be before the firm would find it worthwhile
to use only clerks to process the payroll?
7.11. A firm produces an output with the production
function Q KL, where Q is the number of units of output per hour when the firm uses K machines and hires L
workers each hour. The marginal products for this production function are MPK L and MPL K. The factor
price of K is 4 and the factor price of L is 2. The firm is
currently using K 16 and just enough L to produce Q
32. How much could the firm save if it were to adjust K
and L to produce 32 units in the least costly way possible?
7.12. A firm operates with the production function
Q K 2L. Q is the number of units of output per day when
the firm rents K units of capital and employs L workers
each day. The marginal product of capital is 2KL, and the
marginal product of labor is K 2. The manager has been
given a production target: Produce 8,000 units per day.
She knows that the daily rental price of capital is $400 per
unit. The wage rate paid to each worker is $200 day.
a) Currently, the firm employs 80 workers per day. What
is the firm’s daily total cost if it rents just enough capital
to produce at its target?
b) Compare the marginal product per dollar spent on K
and on L when the firm operates at the input choice in
part (a). What does this suggest about the way the firm
might change its choice of K and L if it wants to reduce
the total cost in meeting its target?
c) In the long run, how much K and L should the firm
choose if it wants to minimize the cost of producing
8,000 units of output day? What will the total daily cost
of production be?
7.13. Consider the production function Q LK, with
marginal products MPL K and MPK L. Suppose that
the price of labor equals w and the price of capital equals r.
Derive expressions for the input demand curves.
7.14. A cost-minimizing firm’s production function is
given by Q LK, where MPL K and MPK L. The
price of labor services is w and the price of capital services
is r. Suppose you know that when w $4 and r $2, the
firm’s total cost is $160. You are also told that when input
prices change such that the wage rate is 8 times the rental
rate, the firm adjusts its input combination but leaves total
output unchanged. What would the cost-minimizing
input combination be after the price changes?
7.15. Ajax, Inc., assembles gadgets. It can make each
gadget either by hand or with a special gadget-making
machine. Each gadget can be assembled in 15 minutes by
a worker or in 5 minutes by the machine. The firm can
also assemble some of the gadgets by hand and some with
machines. Both types of work are perfect substitutes, and
they are the only inputs necessary to produce the gadgets.
279
a) It costs the firm $30 per hour to use the machine and
$10 per hour to hire a worker. The firm wants to produce
120 gadgets. What are the cost-minimizing input quantities? Illustrate your answer with a clearly labeled graph.
b) What are the cost-minimizing input quantities if it
costs the firm $20 per hour to use the machine, and $10
per hour to hire a worker? Illustrate your answer with a
graph.
c) Write down the equation of the firm’s production
function for the firm. Let G be the number of gadgets assembled, M the number of hours the machines are used,
and L the number of hours of labor.
7.16. A construction company has two types of employees: skilled and unskilled. A skilled employee can build
1 yard of a brick wall in one hour. An unskilled employee
needs twice as much time to build the same wall. The
hourly wage of a skilled employee is $15. The hourly
wage of an unskilled employee is $8.
a) Write down a production function with labor. The inputs are the number of hours of skilled workers, LS, the
number of hours worked by unskilled employees, LU, and
the output is the number of yards of brick wall, Q.
b) The firm needs to build 100 yards of a wall. Sketch the
isoquant that shows all combinations of skilled and unskilled labor that result in building 100 yards of the wall.
c) What is the cost-minimizing way to build 100 yards of
a wall? Illustrate your answer on the graph in part (b).
7.17. A paint manufacturing company has a production
function Q ⫽ K ⫹ 1L. For this production function
MPK 1 and MPL ⫽ 1/(21L). The firm faces a price of
labor w that equals $1 per unit and a price of capital services r that equals $50 per unit.
a) Verify that the firm’s cost-minimizing input combination to produce Q 10 involves no use of capital.
b) What must the price of capital fall to in order for the
firm to use a positive amount of capital, keeping Q at 10
and w at 1?
c) What must Q increase to for the firm to use a positive
amount of capital, keeping w at 1 and r at 50?
7.18. A researcher claims to have estimated input demand curves in an industry in which the production technology involves two inputs, capital and labor. The input
demand curves he claims to have estimated are L wr 2Q
and K w2rQ. Are these valid input demand curves? In
other words, could they have come from a firm that minimizes its costs?
7.19. A manufacturing firm’s production function is Q
KL ⫹ K ⫹ L. For this production function, MPL ⫽ K ⫹ 1
and MPK ⫽ L ⫹ 1. Suppose that the price r of capital
services is equal to 1, and let w denote the price of labor
services. If the firm is required to produce 5 units of
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output, for what values of w would a cost-minimizing
firm use
a) only labor?
b) only capital?
c) both labor and capital?
7.20. Suppose a production function is given by Q
min(L, K)—that is, the inputs are perfect complements.
Draw a graph of the demand curve for labor when the
firm wants to produce 10 units of output (Q 10).
7.21. A firm’s production function is Q min(K, 2L),
where Q is the number of units of output produced using
K units of capital and L units of labor. The factor prices
are w 4 (for labor) and r 1 (for capital). On an optimal choice diagram with L on the horizontal axis and K
on the vertical axis, draw the isoquant for Q 12, indicate the optimal choices of K and L on that isoquant, and
calculate the total cost.
7.22. Suppose a production function is given by Q
K ⫹ L—that is, the inputs are perfect substitutes. For this
production function, MPL ⫽ 1 and MPK ⫽ 1. Draw a
graph of the demand curve for labor when the firm wants
to produce 10 units of output and the price of capital
services is $1 per unit (Q ⫽ 10 and r ⫽ 1).
7.23. Suppose a production function is given by Q ⫽
10K ⫹ 2L. The factor price of labor is 1. Draw the
demand curve for capital when the firm is required to
produce Q ⫽ 80.
7.24. Consider the production function Q K ⫹ 1L.
For this production function, MPL 1/(21L) and
MPK 1. Derive the input demand curves for L and K,
as a function of the input prices w (price of labor services)
and r (price of capital services). Show that at an interior
optimum (with K 0 and L 0) the amount of L demanded does not depend on Q. What does this imply
about the expansion path?
7.25. A firm has the production function Q LK. For
this production function, MPL K and MPK L. The
firm initially faces input prices w $1 and r $1 and is
required to produce Q 100 units. Later the price of
labor w goes up to $4. Find the optimal input combinations for each set of prices and use these to calculate the
firm’s price elasticity of demand for labor over this range
of prices.
7.26. A bicycle is assembled out of a bicycle frame and
two wheels.
a) Write down a production function of a firm that produces bicycles out of frames and wheels. No assembly is
required by the firm, so labor is not an input in this
case. Sketch the isoquant that shows all combinations of
frames and wheels that result in producing 100 bicycles.
b) Suppose that initially the price of a frame is $100 and
the price of a wheel is $50. On the graph you drew for
part (a), show the choices of frames and wheels that
minimize the cost of producing 100 bicycles, and draw
the isocost line through the optimal basket. Then repeat
the exercise if the price of a frame rises to $200, while the
price of a wheel remains $50.
7.27. Suppose that1 the firm’s production function is
given by Q ⫽ 10KL3. The firm’s capital is fixed at K.
What amount of labor will the firm hire to solve its
short-run cost-minimization problem?
7.28. A plant’s production function is Q ⫽ 2KL ⫹ K.
For this production function, MPK ⫽ 2L ⫹ 1 and MPL ⫽
2K. The price of labor services w is $4 and of capital services r is $5 per unit.
a) In the short run, the plant’s capital is fixed at K ⫽ 9.
Find the amount of labor it must employ to produce Q ⫽
45 units of output.
b) How much money is the firm sacrificing by not having
the ability to choose its level of capital optimally?
7.29. Suppose that the firm uses three inputs to produce its output: capital K, labor L, and materials M. The
1 1
1
firm’s production function is given by Q ⫽ K 3L3M 3 . For
this production function, the marginal products of
1
2 1
capital, labor, and materials are MPK ⫽ 31K⫺3L3M 3,
2
1
2
1
1 1
MPL ⫽ 13K 3L⫺3M 3, and MPM ⫽ 13K 3L3M ⫺3 . The prices
of capital, labor, and materials are r ⫽ 1, w ⫽ 1, and m ⫽ 1,
respectively.
a) What is the solution to the firm’s long-run costminimization problem given that the firm wants to
produce Q units of output?
b) What is the solution to the firm’s short-run costminimization problem when the firm wants to produce
Q units of output and capital is fixed at K ?
c) When Q ⫽ 4, the long-run cost-minimizing quantity
of capital is 4. If capital is fixed at K ⫽ 4 in the short run,
show that the short-run and long-run cost-minimizing
quantities of labor and materials are the same.
7.30. Consider the production function in LearningBy-Doing Exercise 7.6: Q ⫽ 1L ⫹ 1K ⫹ 1M. For
this production function, the marginal products of labor,
capital, and materials are MPL ⫽ 1/(21L), MPK ⫽
1/(2/ 1K), and MPM ⫽ 1/(21M). Suppose that the
input prices of labor, capital, and materials are w ⫽ 1, r ⫽
1, and m ⫽ 1, respectively.
a) Given that the firm wants to produce Q units of output, what is the solution to the firm’s long-run costminimization problem?
b) Given that the firm wants to produce Q units of
output, what is the solution to the firm’s short-run costminimization problem when K ⫽ 4? Will the firm want
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A P P E N D I X : A DVA N C E D TO P I C S I N C O S T M I N I M I Z AT I O N
to use positive quantities of labor and materials for all
levels of Q?
(c) Given that the firm wants to produce 12 units of
output, what is the solution to the firm’s short-run costminimization problem when K 4 and L 9? Will the
firm want to use a positive quantity of materials for all
levels of Q?
b) Suppose the price of computer time ( pC) is 10 and
the wage rate for a manual draftsman ( pD) is 5. The
firm has to produce 15 blueprints. What are the costminimizing choices of C and D? On a graph with C on
the horizontal axis and D on the vertical axis, illustrate
your answer showing the 15-blueprint isoquant and
isocost lines.
7.31. Acme, Inc., has just completed a study of its
production process for gadgets. It uses labor and capital to produce gadgets. It has determined that 1 more
unit of labor would increase output by 200 gadgets.
However, an additional unit of capital would increase
output by 150 gadgets. If the current price of capital is
$10 and the current price of labor is $25, is the firm
employing the optimal input bundle for its current output?
Why or why not? If not, which input’s usage should be
increased?
7.34. This problem will enable you to apply a revealed
preference argument to see if a firm is minimizing the
total cost of production. The firm produces output with
a technology characterized by a diminishing marginal
rate of technical substitution of labor for capital. It is required to produce a specified amount of output, which
does not change in this problem. When faced with input
prices w1 and r1, the firm chooses the basket of inputs at
point A on the following graph, and it incurs the total
cost on the isocost line IC1. When the factor prices
change to w2 and r2 the firm’s choice of inputs is at basket B, on isocost line IC2. Basket A lies on the intersection of the two isocost lines. Are these choices consistent
with cost-minimizing behavior?
7.32. A firm operates with a technology that is characterized by a diminishing marginal rate of technical
substitution of labor for capital. It is currently producing 32 units of output using 4 units of capital and
5 units of labor. At that operating point the marginal
product of labor is 4 and the marginal product of capital is 2. The rental price of a unit of capital is 2 when
the wage rate is 1. Is the firm minimizing its total
long-run cost of producing the 32 units of output? If
so, how do you know? If not, show why not and indicate whether the firm should be using (i) more capital
and less labor, or (ii) less capital and more labor to
produce an output of 32.
K
IC1
A
7.33. Suppose that in a given production process a
blueprint (B) can be produced using either an hour of
computer time (C) or 4 hours of a manual draftsman’s
time (D). (You may assume C and D are perfect substitutes. Thus, for example, the firm could also produce a
blueprint using 0.5 hour of C and 2 hours of D.)
a) Write down the production function corresponding to
this process (i.e., express B as a function of C and D).
B
A P P E N D I X : Advanced Topics in Cost Minimization
W H AT D E T E R M I N E S T H E P R I C E
O F C A P I TA L S E RV I C E S ?
In the Appendix to Chapter 4, we introduced basic concepts related to the time value
of money, in particular the concept of present value. We can use the concept of present value to explain the factors that determine the price per unit of capital services r.
Time value of money is relevant for determining the price of capital services because
the machines that provide capital services typically last for many years and thus provide services over a long period of time.
IC2
L
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C O S T S A N D C O S T M I N I M I Z AT I O N
The clearest way to explain how the price of capital services is determined is to
imagine that a firm obtains its capital services by renting machine-hours from another
firm. This sort of arrangement exists in the real world. For example, airlines often rent
airplanes from specialized leasing firms, and banks rent computer time from specialized data storage firms. We will imagine that the market for machine rentals is extremely competitive. Thus, firms that lease machine-hours compete with one another
for the business of firms that seek those services.
Suppose the machine that is being rented provides K machine-hours per year of
capital services. The machine itself costs A dollars to acquire. Thus, the acquisition
cost per machine hour is A/K, which we denote by a.
We further assume that the machine depreciates at a rate of d ⫻ 100 percent per
year. Thus, if d 0.05, the number of machine-hours that the machine is capable of
providing declines by 5 percent per year. If the machine can provide 100 machine
hours in its first year of life, then
• It would provide (1 ⫺ 0.05)100 ⫽ 95 machine-hours in its second year of life;
• It would provide (1 ⫺ 0.05)100 ⫺ 0.05(1 ⫺ 0.05)100 ⫽ (1 ⫺ 0.05)2100 ⫽ 90.25
machine-hours in its third year of life;
• It would provide (1 ⫺ 0.05)2100 ⫺ 0.05(1 ⫺ 0.05)2100 ⫽ (1 ⫺ 0.05)3100 ⫽
87.74 machine-hours in its fourth year of life;
and so on
Now, let r be the rental price charge by the owners of the machines for one
machine-hour of capital services. If owners of machines have a discount rate of i, the
net present value of the rental revenues to a machine owner would be:
r(1 ⫺ d )2K
r (1 ⫺ d)K
rK
⫹
⫹ p
⫹
(1 ⫹ i)
(1 ⫹ i)2
(1 ⫹ i )3
Though it is not obvious, we can use several steps of algebra to write the above expression as:
rK
i⫹d
Now, if the market for selling machine-hours is intensely competitive, the present
value of the revenues to the machine owner from renting the machine would just
cover the cost of acquiring the machine, or
rK
⫽A
i⫹d
or equivalently
rK
⫽ aK
i⫹d
Rearranging this expression gives us the expression for the rental price of machinehours, r :
r ⫽ a(i ⫹ d )
This is sometimes referred to as the implicit rental rate for capital services.
A P P E N D I X : A DVA N C E D TO P I C S I N C O S T M I N I M I Z AT I O N
This analysis tells us that the price of capital services reflects three factors: the
acquisition cost a of the equipment providing the services; the discount rate i of the
owner of the machine; and the rate of depreciation d of the capital equipment.
The above analysis pertains to the case of capital services that are purchased by a
firm that does not own the capital equipment that provides those services. What if the
firm actually owns its own capital equipment? The analysis is unchanged. In this case,
the price of capital services r would be the opportunity cost of using the machine to
provide productive services within the firm and thereby foregoing the opportunity to
sell the capital services outside the firm. Thus, the opportunity cost of capital services
would be r a(i ⫹ d ).
S O LV I N G T H E C O S T- M I N I M I Z AT I O N P R O B L E M U S I N G
T H E M AT H E M AT I C S O F C O N S T R A I N E D O P T I M I Z AT I O N
In this section, we set up the long-run cost-minimization problem as a constrained optimization problem and solve it using Lagrange multipliers.
With two inputs, labor and capital, the cost-minimization problem can be stated as:
min wL ⫹ rK
(A7.1)
subject to: f (L, K) ⫽ Q
(A7.2)
(L,K)
We proceed by defining a Lagrangian function
¶(L, K, l) wL ⫹ rK ⫺ l冤 f (L, K ) ⫺ Q冥
where is a Lagrange multiplier. The conditions for an interior optimal solution
(L 0, K 0) to this problem are
0f (L, K )
0¶
01wl
0L
0L
(A7.3)
0f (L, K )
0¶
01rl
0K
0K
(A7.4)
0¶
0 1 f(L, K ) Q
0l
(A7.5)
Recall from Chapter 6 that
MPL
MPK
0f (L, K )
0L
0f (L, K )
0K
We can combine (A7.3) and (A7.4) to eliminate the Lagrange multiplier, so our firstorder conditions reduce to:
MPL
w
r
MPK
(A7.6)
f (L, K ) Q
(A7.7)
283
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C O S T S A N D C O S T M I N I M I Z AT I O N
Conditions (A7.6) and (A7.7) are two equations in two unknowns, L and K. They are
identical to the conditions that we derived for an interior solution to the costminimization problem using graphical arguments. The solution to these conditions is
found in the long-run input demand functions, L*(Q, w, r) and K *(Q, w, r).
For more on the use of Lagrange multipliers to solve problems of constrained optimization, see the Mathematical Appendix in this book.
D UA L I T Y: “ BAC K I N G O U T ” T H E P R O D U C T I O N
FUNCTION FROM THE INPUT DEMAND FUNCTIONS
duality The correspondence between the production function and the input
demand functions.
This chapter has shown how we can start with a production function and derive the
input demand functions. But we can also reverse directions: If we start with input demand functions, we can characterize the properties of a production function and
sometimes even write down the equation of the production function. This is because
of duality, which refers to the correspondence between the production function and
the input demand functions.
We will illustrate duality by backing out the production function from the input
demand curves that we derived in Learning-By-Doing Exercise 7.4. We use that example because we already know what the underlying production function is, and we
can thus confirm whether the production function we derive is correct. We will proceed in three steps.
• Step 1. Start with the labor demand function and solve for w in terms of Q, r,
and L:
L
Q
r
50 A w
wa
Q 2
br
50L
• Step 2. Substitute the solution for w into the capital demand function
K (QⲐ50) 1(wⲐr):
Q ( 50L )2r 2
b
K
a
r
50
Q
which simplifies to K
1
Q2
.
2500 L
1
1
• Step 3. Solve this expression for Q in terms of L and K: Q 50K 2L2.
If you go back to Learning-By-Doing Exercise 7.4, you will see that this is indeed the
production function from which we derived the input demand functions.
You might wonder why duality is important. Why would we care about deriving production functions from input demand functions? We will discuss the significance of duality in Chapter 8, after we have introduced the concept of a long-run total cost function.
8
COST CURVES
8.1
L O N G - RU N C O S T C U RV E S
The Long-Run Total Cost of
Urban Transit Systems
APPLICATION 8.2 The Costs of Higher Education
APPLICATION 8.3 Hospitals Are Businesses Too
APPLICATION 8.4 Estimates of the Output
Elasticity of Total Cost in the Electric Utility
and Computer Industries
APPLICATION 8.1
8.2
S H O RT- RU N C O S T C U RV E S
APPLICATION 8.5
Tracking Railroad Costs
APPLICATION 8.6
Economies of Scope
8.3
S P E C I A L TO P I C S I N C O S T
for the Swoosh
Experience Curves
in Emissions Control
APPLICATION 8.7
8.4
E S T I M AT I N G C O S T F U N C T I O N S
Estimating Economies of Scale
in Payment Processing Services
APPLICATION 8.8
APPENDIX
S H E P H A R D ’ S L E M M A A N D D UA L I T Y
How Can HiSense Get a Handle on Costs?
Beginning in the 1990s and continuing in the 2000s, the Chinese economy underwent an unprecedented
boom. As part of that boom, enterprises such as HiSense Company grew rapidly. At one point in the
mid-1990s, HiSense, China’s largest producer of flat-panel television sets, increased its sales at a rate of
50 percent per year. Its goal was to transform itself from a sleepy domestic producer of television sets into
a consumer electronics and appliances giant whose brand name was recognized around the world. By 2010
HiSense seemed well on its way toward achieving this goal. In addition to selling televisions, HiSense was
285
one of China’s leading producers of personal computers, mobile phones, refrigerators, and air conditioners.
It had sales, manufacturing, and research operations all over the world, including television factories in
South Africa and Hungary and R&D centers in the United States (Chicago) and Belgium. In 2008, HiSense
took an important step in building global brand recognition by signing a sponsorship deal to name a stadium in Melbourne Park, the annual site of the Australian Open tennis tournament.
Of vital concern to HiSense and the thousands of other Chinese enterprises that were plotting similar growth strategies in the mid-2000s was how production costs would change as the volume of output
increased. There is little doubt that HiSense’s total production costs would go up as it produced more
televisions. But how fast would they go up? HiSense’s executives hoped that as it produced more televisions, the cost of each television set would go down; that is, its unit costs would fall as its annual rate of
output went up.
HiSense’s executives also needed to know how input prices would affect its production costs. For example,
demand for flat-panel television sets in China has been growing rapidly. Television producers like HiSense
were hoping that prices of key inputs in the process of assembling flat-panel television sets, such as liquid
crystal displays, would remain low so that the growth in demand remained profitable. As another example,
in the mid-1990s HiSense competed with other large Chinese television manufacturers to acquire the
production facilities of smaller television makers. This competition bid up the price of capital. HiSense had
to reckon with the impact of this price increase on its total production costs.
This chapter picks up where Chapter 7 left off: with the comparative statics of the cost-minimization
problem. The cost-minimization problem—both in the long run and the short run—gives rise to total,
average, and marginal cost curves. This chapter studies these curves.
CHAPTER PREVIEW
After reading and studying this
chapter, you will be able to:
• Describe and graph a long-run
total cost curve.
• Determine the long-run total
cost curve from a production
function.
• Demonstrate how the graph of a
long-run total cost curve changes
when an input price changes.
• Derive a long-run average cost
curve and a long-run marginal
cost curve from the long-run
total cost curve.
• Explain the difference between
average cost and marginal cost.
286
287
8 . 1 L O N G - RU N C O S T C U RV E S
• Distinguish between economies of scale and diseconomies of scale.
• Describe and a graph a short-run total cost curve.
• Determine the short-run total cost curve from a production function.
• Illustrate graphically the relationship between a short-run total cost curve and a long-run total cost curve.
• Derive a short-run average cost curve and a short-run marginal cost curve from a short-run total cost curve.
• Explain and distinguish between the concepts of short-run average cost, short-run marginal cost,
average variable cost, and average fixed cost.
• Explain the meaning of economies of scope.
• Discuss how a learning curve illustrates economies of experience.
• Identify several common functional forms used to estimate total cost functions.
L O N G - R U N TOTA L C O S T C U RV E
8.1
In Chapter 7, we studied the firm’s long-run cost-minimization problem and saw how
the cost-minimizing combination of labor and capital depended on the quantity of output Q and the prices of labor and capital, w and r. Figure 8.1(a) shows how the optimal
LONG-RUN
C O S T C U RV E S
K, capital services per year
TC2
r
TC1
r
K2
B
A
K1
2 million TVs per year
1 million TVs per year
0
TC1
w
TC2
w
L, labor services per year
(a)
Minimized total cost,
dollars per year
L1 L2
TC(Q)
B
TC2 = wL2 + rK2
A
TC1 = wL1 + rK1
0
(b)
1 million
Q, TVs per year
2 million
FIGURE 8.1 Cost Minimization
and the Long-Run Total Cost Curve
for a Producer of Television Sets
The quantity of output increases
from 1 million to 2 million television sets per year, with the prices
of labor w and capital r held constant. The comparative statics
analysis in panel (a) shows how
the cost-minimizing input combination moves from point A to
point B, with the minimized total
cost increasing from TC1 to TC2.
Panel (b) shows the long-run total
cost curve TC(Q), which represents
the relationship between output
and minimized total cost.
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CHAPTER 8
long-run total cost
curve A curve that
shows how total cost varies
with output, holding input
prices fixed and choosing
all inputs to minimize cost.
S
E
C O S T C U RV E S
input combination for a television manufacturer changes as we vary output, holding
input prices fixed. For example, when the firm produces 1 million televisions per year,
the cost-minimizing input combination occurs at point A, with L1 units of labor and K1
units of capital. At this input combination, the firm is on an isocost line corresponding
to TC1 dollars of total cost, where TC1 wL1 ⫹ rK1. TC1 is thus the minimized total
cost when the firm produces 1 million units of output. When the firm increases output
from 1 million to 2 million televisions per year, its isocost line shifts to the northeast,
and its cost-minimizing input combination moves to point B, with L2 units of labor and
K2 units of capital. Thus, its minimized total cost goes up (i.e., TC2 TC1). It cannot
be otherwise, because if the firm could decrease total cost by producing more output, it
couldn’t have been using a cost-minimizing combination of inputs in the first place.
Figure 8.1(b) shows the long-run total cost curve, denoted by TC(Q). The longrun total cost curve shows how minimized total cost varies with output, holding input
prices fixed and selecting inputs to minimize cost. Because the cost-minimizing input
combination moves us to higher isocost lines, the long-run total cost curve must be
increasing in Q. We also know that when Q 0, long-run total cost is 0. This is because, in the long run, the firm is free to vary all its inputs, and if it produces a zero
quantity, the cost-minimizing input combination is zero labor and zero capital. Thus,
comparative statics analysis of the cost-minimization problem implies that the long-run
total cost curve must be increasing in Q and must equal 0 when Q 0.
L E A R N I N G - B Y- D O I N G E X E R C I S E 8 . 1
D
Finding the Long-Run Total Cost Curve from a Production Function
Let’s return again to the production function Q 50 1LK that we introduced in Learning-ByDoing Exercise 7.2.
(b) What is the graph of the long-run total cost curve
when w 25 and r 100?
Solution
Problem
(a) In Learning-By-Doing Exercise 7.4, we saw that
the following equations describe the cost-minimizing
quantities of labor and capital: L (Q/50) 1r/w and
(a) How does minimized total cost depend on the output Q
and the input prices w and r for this production function?
TC, dollars per year
TC(Q) = 2Q
FIGURE 8.2
Long-Run Total Cost Curve
The graph of the long-run total cost curve
TC(Q) 2Q is a straight line.
$4 million
$2 million
0
1 million
2 million
Q, units per year
K ⫽ (Q/50) 1w/r. To find the minimized total cost, we
calculate the total cost the firm incurs when it uses this
cost-minimizing input combination:
TC(Q) ⫽ wL ⫹ rK ⫽ w
8 . 1 L O N G - RU N C O S T C U RV E S
Q r
Q w
⫹r
50 A w
50 A r
289
(b) If we substitute w ⫽ 25 and r ⫽ 100 into this equation
for the total cost curve, we get TC(Q) ⫽ 2Q. Figure 8.2
shows that the graph of this long-run total cost curve is
a straight line.
Similar Problems: 8.5, 8.11, 8.12, 8.13, 8.14, 8.17
Q
Q
1wr
1wr ⫹
1wr ⫽
Q
50
50
25
H O W D O E S T H E L O N G - R U N TOTA L C O S T C U RV E
SHIFT WHEN INPUT PRICES CHANGE?
What Happens When Just One Input Price Changes?
K, capital services per year
In the chapter introduction, we discussed how HiSense faced the prospect of higher
prices for certain inputs, such as capital. To illustrate how an increase in an input price
affects a firm’s total cost curve, let’s return to the cost-minimization problem for our
hypothetical television producer. Figure 8.3 shows what happens when the price of
capital increases, holding output and the price of labor constant. Suppose that at the
initial situation, the optimal input combination for an annual output of 1 million television sets occurs at point A on isocost line C1, where the minimized total cost is $50
million per year. After the increase in the price of capital, the optimal input combination is at point B on isocost line C3, corresponding to a total cost that is greater than
$50 million. To see why, note that the $50 million isocost line at the new input prices
(C2) intersects the horizontal axis in the same place as the $50 million isocost line at
the old input prices. However, C2 is flatter than C1 because the price of capital has gone
up. Thus, the firm could not operate on isocost line C2 because it would be unable to
produce the desired quantity of 1 million television sets. Instead, the firm must operate
on an isocost line that is farther to the northeast (C3) and thus corresponds to a higher
C1 = $50 million isocost line
before the price of capital goes up
C2 = $50 million isocost line
after price of capital goes up
C3 = $60 million isocost line
after price of capital goes up
C1
C3
C2
A
B
FIGURE 8.3
1 million TVs per year
L, labor services per year
How a Change in the
Price of Capital Affects the Optimal
Input Combination and Long-Run Total
Cost for a Producer of Television Sets
The firm’s long-run total cost increases
after the price of capital increases. The
isocost line moves from C1 to C3 and
the cost-minimizing input combination
shifts from point A to point B.
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CHAPTER 8
C O S T C U RV E S
TC(Q)
after increase
FIGURE 8.4
How a
Change in the Price of
Capital Affects the LongRun Total Cost Curve for a
Producer of Television Sets
An increase in the price of
capital causes the long-run
total cost curve TC (Q) to rotate upward. Points A and B
correspond to the cost-minimizing input combinations
in Figure 8.3.
TC, dollars per year
in price of capital
TC(Q)
before increase
in price of capital
$60 million
$50 million
0
B
A
1 million
Q, TVs per year
level of cost ($60 million perhaps). Thus, holding output fixed, the minimized total
cost goes up when the price of an input goes up.1
This analysis then implies that an increase in the price of capital results in a new
total cost curve that lies above the original total cost curve at every Q 0 (at Q 0,
long-run total cost is still zero). Thus, as Figure 8.4 shows, an increase in an input
price rotates the long-run total cost curve upward.2
What Happens When All Input Prices Change Proportionately?
What if the price of capital and the price of labor both go up by the same percentage
amount, say, 10 percent? The answer is that a given percentage increase in both input
prices leaves the cost-minimizing input combination unchanged, while the total cost curve
shifts up by exactly the same percentage.
As shown in Figure 8.5(a), at the initial prices of labor w and capital r, the
cost-minimizing input combination is at point A. After both input prices increase by
10 percent, to 1.10w and 1.10r, the ideal combination is still at point A. The reason is
that the slope of the isocost line is unchanged by the price increase (⫺w/r ⫽
⫺1.10w/1.10r), so the point of tangency between the isocost line and the isoquant is
also unchanged.
Figure 8.5(b) shows that the 10 percent increase in input prices shifts the total
cost curve up by 10 percent. Before the price increase, total cost TCA ⫽ wL ⫹ rK; after
the price increase, total cost TCB ⫽ 1.10wL ⫹ 1.10r K. Thus, TCB ⫽ 1.10TCA (i.e., the
total cost increases by 10 percent for any combination of L and K ).
1
An analogous argument would show that minimized total cost goes down when the price of capital
goes down.
2
There is one case in which an increase in an input price would not affect the long-run total cost curve.
If the firm is initially at a corner point solution using a zero quantity of the input, an increase in the price
of the input will leave the firm’s cost-minimizing input combination—and thus its minimized total cost—
unchanged. In this case, the increase in the input price would not shift the long-run total cost curve.
K, capital services per year
8 . 1 L O N G - RU N C O S T C U RV E S
291
A
1 million units
per year
0
(a)
L, labor services per year
FIGURE 8.5
TC, dollars per year
TC(Q)B
TC(Q)A
B
TCB = 1.10TCA
TCA
A
0
Q, TVs per year
(b)
A P P L I C A T I O N
How a
Proportionate Change in
the Prices of All Inputs
Affects the Cost-Minimizing
Input Combination and the
Total Cost Curve
The price of each input increases by 10 percent. Panel
(a) shows that the costminimizing input combination remains the same
(at point A), because the
slope of the isocost line is
unchanged. Panel (b) shows
that the total cost curve
shifts up by the same
10 percent.
8.1
The Long-Run Total Cost of Urban
Transit Systems
Transit systems in large cities around the world rely on
a variety of different modes of transportation, including commuter trains, subways, and light rail. But in the
United States, the most common mode of urban transit
is the bus. A bus system is an interesting setting in which
to study the behavior of long-run total costs because
when input prices or output changes, bus systems can
adjust their input mixes without much difficulty over
the long run. Drivers and dispatchers can be hired or
laid off, fuel purchases can be adjusted, and even
busses can be bought or sold as circumstances dictate.
Michael Iacono estimated long-run total cost
curves for urban bus systems, using data on input
prices and total ridership from bus systems in 23
292
CHAPTER 8
C O S T C U RV E S
medium and large U.S. cities from 1996 to 2003.3 He
calculated long-run total cost as a function of output
and the prices of four inputs: labor, capital (busses),
fuel, and materials other than fuel and busses.
Output was measured several ways, including number
of miles driven and number of passengers. For our
purposes we will consider Q measured by number of
passengers per year.
Figure 8.6 illustrates an example of cost curves for
a typical urban transit system suggested by Iacono’s
estimates. Note that total cost increases with the
quantity of output, as the theory we just discussed
implies. Total cost also increases with the price of each
input (holding the prices of the other three inputs
constant). Thus, doubling the price of labor causes the
total cost curve to shift upward to TC(Q)L. The effect
of doubling the price of either capital, fuel, or materials is approximately the same and is thus shown by
the single shifted total cost curve TC(Q)K,F,M.
Why is the impact of doubling the price of labor
on long-run total cost greater than the impact of doubling either the price of capital or the price of fuel or
the price of materials? That is, why does TC(Q)L lie
above TC(Q)K,F,M? The reason is that for a typical bus
system, labor costs constituted approximately 50 percent of long-run total costs, while the costs of each of
the other three inputs constituted only about 16 percent of the total. As a result, long-run total cost is
more sensitive to changes in the price of labor than it
is to changes in the prices of the other inputs.
$200
TC(Q)L
How Changes
in Input Prices Affect the LongRun Total Cost Curve for an Urban
Transit System
Total cost TC (Q) is more sensitive
to the price of labor than to the
price of capital (buses), fuel, or
materials. Holding the prices of
other inputs constant, doubling
the price of labor shifts the cost
curve up to TC (Q)L. The effect of
doubling the price of either capital, fuel, or materials is approximately the same and is thus
shown by the single shifted total
cost curve TC (Q)K,F,M.
TC, millions of dollars
FIGURE 8.6
TC(Q)K,F,M
TC(Q)
$150
$100
$50
$0
0
10
20
30
40
50
60
Q, thousands of passengers per year
L O N G - R U N AV E R AG E A N D M A R G I N A L C O S T C U RV E S
What Are Long-Run Average and Marginal Costs?
long-run average cost
The firm’s total cost per
unit of output. It equals
long-run total cost divided
by total quantity.
long-run marginal cost
The rate at which long-run
total cost changes with
respect to change in output.
Two other types of cost play an important role in microeconomics: long-run average
cost and long-run marginal cost. Long-run average cost is the firm’s cost per unit of
output. It equals long-run total cost divided by Q: AC(Q) [TC(Q)]/Q.
Long-run marginal cost is the rate at which long-run total cost changes with respect to a change in output: MC(Q) (TC )/(Q). Thus, MC(Q) equals the slope of
TC(Q).
3
Michael Iacano, “Modeling Cost Structure of Public Transit Firms: Scale Economies and Alternate
Functional Forms,” Transportation Research Board Annual Meeting, Paper #09-3435, 2009.
8 . 1 L O N G - RU N C O S T C U RV E S
293
TC, dollars
TC(Q)
$1,500
C
A
Slope of line BAC = 10
Slope of ray 0A = 30
B
FIGURE 8.7
0
50
Q, units per year
(a)
AC, MC dollars per unit
MC(Q) = Slope of TC(Q)
$30
A′
$10
A′′
0
(b)
AC(Q) = Slope of ray
from 0 to TC(Q) curve
50
Q, units per year
Deriving LongRun Average and Marginal Cost
Curves from the Long-Run Total
Cost Curve
Panel (a) shows the firm’s longrun total cost curve TC(Q). Panel
(b) shows the long-run average
cost curve AC(Q) and the longrun marginal cost curve MC(Q),
both derived from TC(Q). At point
A in panel (a), when output is
50 units per year, average cost
slope of ray 0A $30 per unit;
marginal cost slope of line
BAC $10 per unit. In panel
(b), points A and A correspond
to point A in panel (a), illustrating the relationship between the
long-run total, average, and marginal cost curves.
Although long-run average and marginal cost are both derived from the firm’s
long-run total cost curve, the two costs are generally different, as illustrated in Figure 8.7. At any particular output level, the long-run average cost is equal to the slope
of a ray from the origin to the point on the long-run total cost curve corresponding
to that output, whereas the long-run marginal cost is equal to the slope of the longrun total cost curve itself at that point. Thus, at point A on the total cost curve TC(Q)
in Figure 8.7(a), where the firm’s output level is 50 units per year, the average cost is
equal to the slope of ray 0A, or $1500/50 units $30 per unit. By contrast, the marginal cost at point A is the slope of the line BAC (the line tangent to the total cost
curve at A); the slope of this line is 10, so the marginal cost when output is 50 units
per year is $10 per unit.
Figure 8.7(b) shows the long-run average cost curve AC(Q) and the long-run
marginal cost curve MC(Q) corresponding to the long-run total cost curve TC(Q)
in Figure 8.7(a). The average cost curve shows how the slope of rays such as 0A
changes as we move along TC(Q), whereas the marginal cost curve shows how
the slope of tangent lines such as BAC changes as we move along TC(Q). Thus, in
Figure 8.7(b), when the firm’s output equals 50 units per year, the average cost
is $30 per unit (point A) and the marginal cost is $10 per unit (point A),
corresponding to the slope of ray 0A and line BAC, respectively, at point A in
Figure 8.7(a).
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S
E
C O S T C U RV E S
L E A R N I N G - B Y- D O I N G E X E R C I S E 8 . 2
D
Deriving Long-Run Average and Marginal Cost Curves from a Long-Run
Total Cost Curve
In Learning-By-Doing Exercise 8.1 we derived the
equation for the long-run total cost curve for the production function Q 50 1LK when the price of labor
L is w 25 and the price of capital K is r 100:
TC(Q) 2Q.
Problem What are the long-run average and marginal
cost curves associated with this long-run total cost curve?
Solution
FIGURE 8.8 Long-Run Average
and Marginal Cost Curves for the
Production Function Q ⴝ 501LK
The long-run average and marginal
cost curves are identical horizontal
lines at $2 per unit when w 25
and r 100.
AC, MC, dollars per unit
Long-run average cost is AC(Q)
[TC(Q)]/Q 2Q/Q 2. Note that average cost does not
depend on Q. Its graph would be a horizontal line, as
Figure 8.8 shows.
Long-run marginal cost is the slope of the long-run
total cost curve. With TC(Q) 2Q, the slope of the
long-run total cost curve is 2, and thus MC(Q) 2.
Long-run marginal cost also does not depend on Q. Its
graph is the same horizontal line.
This exercise illustrates a general point. Whenever
the long-run total cost is a straight line (as in Figure 8.2),
long-run average and long-run marginal cost curves will
be the same and will be a horizontal line.
Similar Problems:
8.6, 8.7, 8.8
AC(Q) = MC(Q) = 2
$2
0
1 million
2 million
Q, units per year
Relationship between Long-Run Average and Marginal Cost Curves
As with other average and marginal concepts (e.g., average product versus marginal
product, discussed in Chapter 6), there is a systematic relationship between the longrun average and long-run marginal cost curves:
• If average cost is decreasing as quantity is increasing, then average cost is
greater than marginal cost: AC(Q) MC(Q).
• If average cost is increasing as quantity is increasing, then average cost is less
than marginal cost: AC(Q) MC(Q).
• If average cost is neither increasing nor decreasing as quantity is increasing, then
average cost is equal to marginal cost: AC(Q) MC(Q).
8 . 1 L O N G - RU N C O S T C U RV E S
295
MC(Q)
AC, MC, dollars per unit
AC(Q)
A
AC(Q) > MC(Q)
FIGURE 8.9
AC(Q) < MC(Q)
AC(Q) = MC(Q)
Q, units per year
Relationship between
the Long-Run Average and Marginal Cost
Curves
To the left of point A, average cost AC is
decreasing as quantity Q is increasing, so
AC(Q) MC(Q). To the right of point A,
AC is increasing as Q is increasing, so
AC(Q) MC(Q). At point A, AC is at a
minimum, neither increasing nor decreasing, so AC(Q) MC(Q).
Figure 8.9 illustrates this relationship.
As we discussed in Chapter 6, the relationship between marginal cost and average
cost is the same as the relationship between the marginal of anything and the average
of anything. For example, suppose that your microeconomics teacher has just finished
grading your most recent quiz. Your average score on all of the quizzes up to that
point was 92 percent, and your teacher tells you that based on your most recent quiz
your average has risen to 93 percent. What can you infer about the score on your most
recent quiz? Since your average has increased, the “marginal score” (your grade on the
most recent quiz) must be above your average. If your average had fallen to 91 percent,
it would have been because your most recent quiz score was below your average. If
your average had remained the same, the reason would have been that the score on
your most recent quiz was equal to your average.
A P P L I C A T I O N
8.2
The Costs of Higher Education
How big is your college or university? Is it a large
school, such as Ohio State, or a smaller one, such as
Northwestern? At which school is the cost per student
likely to be lower? Does university size affect the
long-run average and marginal cost of “producing”
education?
4
Rajindar and Manjulika Koshal have studied how
school size affects the average and marginal cost of
education.4 They collected data on the average cost
per student from 195 U.S. universities from 1990 to
1991 and estimated an average cost curve for these
universities.5 To control for differences in cost that
stem from differences among universities in terms of
their commitment to graduate programs, the Koshals
R. Koshal and M. Koshal, “Quality and Economies of Scale in Higher Education,” Applied Economics 27
(1995): 773–778.
5
To control for variations in cost that might be due to differences in academic quality, their analysis also
allowed average cost to depend on the student–faculty ratio and the academic reputation of the school, as
measured by factors such as average SAT scores of entering freshmen. In Figure 8.10, these variables are
assumed to be equal to their national averages.
296
CHAPTER 8
C O S T C U RV E S
FIGURE 8.10
The Long-Run
Average and Marginal Cost Curves for
Undergraduate Education at U.S.
Universities
The marginal cost of an additional student
is less than the average cost per student
until enrollment reaches about 30,000 students. Until that point, average cost per
student falls with the number of students.
Beyond that point, the marginal cost of
an additional student exceeds the average cost per student, and average cost
increases with the number of students.
AC, MC, dollars per student
MC
$50,000
$40,000
$30,000
$20,000
$10,000
0
estimated average cost curves for four groups of universities, primarily distinguished by the number of
Ph.Ds awarded per year and the amount of government funding for Ph.D. students these universities received. For simplicity, we discuss the cost curves for
the category that includes the 66 universities nationwide with the largest graduate programs (e.g., schools
like Harvard, Northwestern, and the University of
California at Berkeley).
Figure 8.10 shows the estimated average and
marginal cost curves for this category of schools. It
shows that the average cost per student declines until
enrollment reaches about 30,000 full-time undergraduate students (about the size of Indiana University, for
example). Because few universities are this large, the
Koshals’ research suggests that for most universities in
the United States with large graduate programs, the
marginal cost of an additional undergraduate student
is less than the average cost per student, and thus an
economies of scale
A
characteristic of production
in which average cost decreases as output goes up.
diseconomies of scale
A characteristic of production in which average cost
increases as output goes up.
AC
10
20
30
40
50
Q, thousands of full-time students
increase in the size of the undergraduate student
body would reduce the cost per student.
This finding seems to make sense. Think about
your university. It already has a library and buildings
for classrooms. It already has a president and a staff
to run the school. These costs will probably not go up
much if more students are added. Adding students is,
of course, not costless. For example, more classes
might have to be added. But it is not that difficult to
find people who are able and willing to teach university classes (e.g., graduate students). Until the point
is reached at which more dormitories or additional
classrooms are needed, the extra costs of more students are not likely to be that large. Thus, for the typical university, while the average cost per student
might be fairly high, the marginal cost of matriculating an additional student is often fairly low. If so,
average cost will decrease as the number of students
increases.
Economies and Diseconomies of Scale
The change in long-run average cost as output increases is the basis for two important concepts: economies of scale and diseconomies of scale. A firm enjoys economies
of scale in a situation where average cost goes down when output goes up. By
contrast, a firm suffers from diseconomies of scale in the opposite situation, where
average cost goes up when output goes up. The extent of economies of scale can affect
the structure of an industry. Economies of scale can also explain why some firms are
297
8 . 1 L O N G - RU N C O S T C U RV E S
AC, dollars per unit
AC(Q)
Q′
Q″
Q, units per year
FIGURE 8.11 Economies and
Diseconomies of Scale for a Typical
Real-World Average Cost Curve
There are economies of scale for outputs less than Q. Average costs are
flat between and Q and Q and there
are diseconomies of scale thereafter.
The output level Q is called the minimum efficient scale.
more profitable than others in the same industry. Claims of economies of scale are
often used to justify mergers between two firms producing the same product.6
Figure 8.11 illustrates economies and diseconomies of scale by showing a longrun average cost curve that many economists believe typifies many real-world production processes. For this average cost curve, there is an initial range of economies of
scale (0 to Q), followed by a range over which average cost is flat (Q to Q), and then
a range of diseconomies of scale (Q Q).
Economies of scale have various causes. They may result from the physical properties of processing units that give rise to increasing returns to scale in inputs.
Economies of scale can also arise due to specialization of labor. As the number of
workers increases with the output of the firm, workers can specialize on tasks, which
often increases their productivity. Specialization can also eliminate time-consuming
changeovers of workers and equipment. This, too, would increase worker productivity and lower unit costs.
Economies of scale may also result from the need to employ indivisible inputs.
An indivisible input is an input that is available only in a certain minimum size; its
quantity cannot be scaled down as the firm’s output goes to zero. An example of an
indivisible input is a high-speed packaging line for breakfast cereal. Even the smallest
such lines have huge capacity––14 million pounds of cereal per year. A firm that might
only want to produce 5 million pounds of cereal a year would still have to purchase
the services of this indivisible piece of equipment.
Indivisible inputs lead to decreasing average costs (at least over a certain range of
output) because when a firm purchases the services of an indivisible input, it can
“spread” the cost of the indivisible input over more units of output as output goes up.
For example, a firm that purchases the services of a minimum-scale packaging line to
6
See Chapter 4 of F. M. Scherer and D. Ross, Industrial Market Structure and Economic Performance
(Boston: Houghton Mifflin, 1990) for a detailed discussion of the implications of economies of scale for
market structure and firm performance.
indivisible input
An
input that is available only
in a certain minimum size.
Its quantity cannot be
scaled down as the firm’s
output goes to zero.
298
managerial diseconomies A situation in
which a given percentage
increase in output forces
the firm to increase its
spending on the services
of managers by more than
this percentage.
minimum efficient
scale The smallest quantity at which the long-run
average cost curve attains
its minimum point.
CHAPTER 8
C O S T C U RV E S
produce 5 million pounds of cereal per year will incur the same total cost on this input
when it increases production to 10 million pounds of cereal per year.7 This will drive
the firm’s average costs down.
The region of diseconomies of scale (e.g., the region where output is greater
than Q⬙ in Figure 8.11) is usually thought to occur because of managerial diseconomies. Managerial diseconomies arise when a given percentage increase in
output forces the firm to increase its spending on the services of managers by
more than this percentage. To see why managerial diseconomies of scale can arise,
imagine an enterprise whose success depends on the talents or insight of one key
individual (e.g., the entrepreneur who started the business). As the enterprise
grows, that key individual’s contribution to the business cannot be replicated by
any other single manager. The firm may have to employ so many additional
managers that total costs increase at a faster rate than output, which then pushes
average costs up.
The smallest quantity at which the long-run average cost curve attains its minimum point is called the minimum efficient scale, or MES (in Figure 8.11, the MES
occurs at output Q). The size of MES relative to the size of the market often indicates the significance of economies of scale in particular industries. The larger MES
is, in comparison to overall market sales, the greater the magnitude of economies of
scale. Table 8.1 shows MES as a percentage of total industry output for a selected
group of U.S. food and beverage industries.8 The industries with the largest
MES-market size ratios are breakfast cereal and cane sugar refining. These industries
have significant economies of scale. The industries with the lowest MES-market size
ratios are mineral water and bread. Economies of scale in manufacturing in these industries appear to be weak.
TABLE 8.1 MES as a Percentage of Industry Output for Selected U.S. Food
and Beverage Industries
Industry
Beet sugar
Cane sugar
Flour
Bread
Canned vegetables
Frozen food
Margarine
MES as % of Output
Industry
MES as % of Output
1.87
12.01
0.68
0.12
0.17
0.92
1.75
Breakfast cereal
Mineral water
Roasted coffee
Pet food
Baby food
Beer
9.47
0.08
5.82
3.02
2.59
1.37
Source: Table 4.2 in J. Sutton, Sunk Costs and Market Structure: Price Competition, Advertising,
and the Evolution of Concentration (Cambridge, MA: MIT Press, 1991).
7
Of course, it may spend more on other inputs, such as raw materials, that are not indivisible.
In this table, MES is measured as the capacity of the median plant in an industry. The median plant is
the plant whose capacity lies exactly in the middle of the range of capacities of plants in an industry. That
is, 50 percent of all plants in a particular industry have capacities that are smaller than the median plant
in that industry, and 50 percent have capacities that are larger. Estimates of MES based on the capacity of
the median plant correlate highly with “engineering estimates” of MES that are obtained by asking wellinformed manufacturing and engineering personnel to provide educated estimates of minimum efficient
scale plant sizes. Data on median plant size in U.S. industries are available from the U.S. Census of
Manufacturing.
8
8 . 1 L O N G - RU N C O S T C U RV E S
A P P L I C A T I O N
8.3
Hospitals Are Businesses Too
The business of health care seems always to be in the
news. By 2009, total spending on health care represented about 15 percent of GDP. Whether this high
level of spending reflects high levels of medical care,
or high costs, is a matter of great controversy. One of
the most interesting trends in health care over the
last two decades has been the consolidation of hospitals through mergers. For example, in the Chicago
area in the 1990s, Northwestern Memorial Hospital
merged with several suburban hospitals to form a
large multihospital system covering the North Side of
Chicago and the North Shore suburbs. Such mergers
often create controversy.
Proponents of hospital mergers argue that mergers
enable hospitals to achieve cost savings through economies of scale in “back-office” operations—activities
such as laundry, housekeeping, cafeterias, printing
and duplicating services, and data processing that do
not generate revenue for the hospital directly, but
that no hospital can function without. Opponents
argue that such cost savings are illusory and that
hospital mergers mainly reduce competition in local
hospital markets. The U.S. antitrust authorities have
blocked several hospital mergers on this basis.
David Dranove has studied the extent to which
back-office activities within a hospital are subject to
economies of scale.9 Figure 8.12 summarizes some of
his findings. The figure shows the long-run average
cost curves for three different activities: cafeterias,
printing and duplicating, and data processing.
Output is measured as the annual number of patients
who are discharged by the hospital. (For each activity,
average cost is normalized to equal an index of 1.0,
at an output of 10,000 patients per year.) These figures show that economies of scale vary from activity
to activity. Cafeterias are characterized by significant
economies of scale. For printing and duplicating, the
average cost curve is essentially flat. And for data
processing, diseconomies of scale arise at a fairly low
level of output. Overall, averaging the 14 back-office
1.50
1.40
AC index
1.30
AC data processing
1.20
1.10
AC printing and
duplicating
1.00
0.90
0.80
2,500
AC cafeterias
10,000
Output, patients per year
17,500
FIGURE 8.12 Average Cost Curves for Three “Back-office” Activities in a Hospital
Cafeterias exhibit significant economies of scale. Data processing exhibits diseconomies of
scale beyond an output of about 5,000 patients per year. And the average cost curve for
printing and duplicating is essentially flat (i.e., there are no significant economies or diseconomies of scale in this activity).
9
299
David Dranove, “Economies of Scale in Non-Revenue Producing Cost Centers: Implications for
Hospital Mergers,” Journal of Health Economics 17 (1998): 69–83.
300
CHAPTER 8
C O S T C U RV E S
activities that he studied, Dranove found that there
are economies of scale in these activities, but they are
largely exhausted at an output of about 7,500 patient
discharges per year. This would correspond to a hospital with 200 beds, which is medium-sized by today’s
standards.
Dranove’s analysis shows that a merger of two
large hospitals would be unlikely to achieve
economies of scale in back-office operations. Thus,
claims that hospital mergers generally reduce costs
per patient should be viewed with skepticism, unless
both merging hospitals are small.
Economies of Scale and Returns to Scale
Economies of scale and returns to scale are closely related, because the returns to scale
of the production function determine how long-run average cost varies with output.
Table 8.2 illustrates these relationships with respect to three production functions
where output Q is a function of a single input, quantity of labor L. The table shows
each production function and the corresponding labor requirements function (which
specifies the quantity of labor needed to produce a given quantity of output, as discussed in Chapter 6), as well as the expressions for total cost and long-run average cost
given a price of labor w.
The relationships illustrated in Table 8.2 between economies of scale and returns
to scale can be summarized as follows:
• If average cost decreases as output increases, we have economies of scale and
increasing returns to scale (e.g., production function Q L2 in Table 8.2).
• If average cost increases as output increases, we have diseconomies of scale and
decreasing returns to scale (e.g., production function Q 1L in Table 8.2).
• If average cost stays the same as output increases, we have neither economies nor
diseconomies of scale and constant returns to scale (e.g., production function Q L
in Table 8.2).
Measuring the Extent of Economies of Scale:
The Output Elasticity of Total Cost
In Chapter 2 you learned that elasticities of demand, such as the price elasticity of
demand or income elasticity of demand, tell us how sensitive demand is to the various
TABLE 8.2
Relationship between Economies of Scale and Returns to Scale
Production Function
2
Q=L
Labor requirements function
Long-run total cost
Long-run average cost
How does long-run average
cost vary with Q?
Economies/diseconomies
of scale?
Returns to scale
√
L = Q
√
TC = w Q
√
AC = w/ Q
Q=
√
L
L = Q2
TC = wQ2
AC = wQ
Q=L
L =Q
TC = wQ
AC = w
Decreasing
Increasing
Constant
Economies of scale
Diseconomies of scale
Neither
Increasing
Decreasing
Constant
8 . 1 L O N G - RU N C O S T C U RV E S
301
TABLE 8.3 Relationship between Output Elasticity of Total Cost and
Economies of Scale
Value of
T C ,Q
T C ,Q
T C ,Q
<1
>1
=1
TC,Q
MC Versus AC
How AC Varies as
Q Increases
Economies/
Diseconomies of Scale
Decreases
Increases
Constant
Economies of scale
Diseconomies of scale
Neither
MC < AC
MC > AC
MC = AC
factors that drive demand, such as price or income. We can also use elasticities to tell
us how sensitive total cost is to the factors that influence it. An important cost elasticity is the output elasticity of total cost, denoted by ⑀TC,Q. It is defined as the percentage change in total cost per 1 percent change in output:
⑀TC,Q
¢TC
TC
¢Q
Q
¢TC
¢Q
output elasticity of
total cost The percentage change in total cost
per 1 percent change in
output.
TC
Q
Since TC/Q marginal cost (MC ) and TC/Q average cost (AC ),
⑀TC,Q
MC
AC
Thus, the output elasticity of total cost is equal to the ratio of marginal to average
cost.
As we have noted (see page 294), the relationship between long-run average and
marginal cost corresponds with the way average cost AC varies with output quantity
Q. This means that output elasticity of total cost tells us the extent of economies of
scale, as shown in Table 8.3.
A P P L I C A T I O N
8.4
Estimates of the Output Elasticity
of Total Cost in the Electric Utility
and Computer Industries
Estimates of the output elasticity of total cost can be
used to characterize the degree of scale economies in
an industry. For example, a study by Russell Rhine estimated the output elasticity of total cost using data
from 83 privately owned U.S. electric power companies from 1991 to 1995.10 These companies generated
electricity primarily through the burning of fossil fuels
10
such as coal, but approximately 25 percent of the
total output was generated by nuclear power plants.
Rhine was interested in determining the extent of
long-run economies of scale in generating electricity.
Table 8.4 shows Rhine’s point estimates of the
output elasticity of long-run total cost for the electric utilities studied. All are below 1, but only slightly
so. This could indicate that there are long-run
economies of scale in power generation and that
the firms in Rhine’s sample were able to take advantage of them almost completely by operating close to
the minimum level of long-run average cost. Or it
Russell Rhine, “Economies of Scale and Optimal Capital in Nuclear and Fossil Fuel Electricity
Production,” Atlantic Economic Journal 29, no. 2 ( June 2001): 203–214.
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TABLE 8.4 Output Elasticity of Total Cost
in Electric Power Generation
⑀TC,Q
All utilities
Nuclear utilities
Non-nuclear utilities
Mean
Median
0.993
0.995
0.992
0.994
0.995
0.993
may indicate that for the utilities in Rhine’s sample,
power generation is characterized by constant returns
to scale, with flat (or nearly flat) long-run average
cost curves.
As another example, Hyunbae Chun and M. Ishaq
Nadiri used data from 1978–1999 to develop estimates
of the output elasticity of total cost for four computer
industries: electronic computers, computer storage devices, computer terminals, and computer peripheral
equipment.11 Table 8.5 reports these estimates.
For each industry, the estimate of the output
elasticity of total cost is less than 1. This indicates
that each of these industries is characterized by
economies of scale. Unlike the case of electric power
generation, however, the estimates are not particularly close to 1, indicating that firms in these industries are not fully exploiting all available scale
economies. In an industry such as electronic computers, which consists of multiple producers of products
such as personal computers, this is quite possible. No
one firm may have a sufficiently large share of the
market to take full advantage of economies of
scale.
TABLE 8.5 Output Elasticity of Total Cost
in Four Computer Industries
Industry
⑀TC,Q
Electronic computers
Computer storage devices
Computer terminals
Computer peripheral equipment
0.759
0.652
0.636
0.664
8.2
S H O R T- R U N TOTA L C O S T C U RV E
S H O R T- R U N
C O S T C U RV E S
The long-run total cost curve shows how the firm’s minimized total cost varies
with output when the firm is free to adjust all its inputs. The short-run total cost
curve STC(Q) tells us the minimized total cost of producing Q units of output
when at least one input is fixed at a particular level. In the following discussion we
assume that the amount of capital used by the firm is fixed at K. The short-run
total cost curve is the sum of two components: the total variable cost curve
TVC(Q) and the total fixed cost curve TFC—that is, STC(Q) ⫽ TVC(Q) ⫹
TFC. The total variable cost curve TVC(Q) is the sum of expenditures on variable inputs, such as labor and materials, at the short-run cost-minimizing input
combination. Total fixed cost is equal to the cost of the fixed capital services
(i.e., TFC ⫽ rK) and thus does not vary with output. Figure 8.13 shows a graph
of the short-run total cost curve, the total variable cost curve, and the total
fixed cost curve. Because total fixed cost is independent of output, its graph is a
horizontal line with the value rK. Thus, STC(Q) ⫽ TVC(Q) ⫹ rK, which means
that the vertical distance between STC(Q) and TVC(Q) is equal to rK at every
quantity Q.
short-run total cost
curve A curve that
shows the minimized total
cost of producing a given
quantity of output when at
least one input is fixed.
total variable cost
curve A curve that
shows the sum of expenditures on variable inputs, such
as labor and materials, at the
short-run cost-minimizing
input combination.
total fixed cost curve
A curve that shows the cost
of fixed inputs and does
not vary with output.
11
Hyunbae Chun, and M. Ishaq Nadiri, “Decomposing Productivity Growth in the U.S. Computer
Industry, Review of Economics and Statistics 90, no. 1 (February 2008): 174–180.
8 . 2 S H O RT- RU N C O S T C U RV E S
303
STC(Q)
TC, dollars per year
TVC(Q)
TFC
rK
0
S
E
Q, units per year
FIGURE 8.13 Short-Run Total Cost Curve
The short-run total cost curve STC(Q) is the
sum of the total variable cost curve TVC(Q)
and the total fixed cost curve TFC. Total fixed
cost is equal to the cost rK of the fixed
capital services.
L E A R N I N G - B Y- D O I N G E X E R C I S E 8 . 3
D
Deriving a Short-Run Total Cost Curve
Let us return to the production function in
Learning-By-Doing Exercises 7.2, 7.4, 7.5, and 8.1,
Q 50 1LK.
Problem What is the short-run total cost curve for
this production function when capital is fixed at a level
K and the input prices of labor and capital are w 25
and r 100, respectively?
Solution In Learning-By-Doing Exercise 7.5, we
derived the short-run cost-minimizing quantity of labor
when capital was fixed at K : L Q2/(2500 K ). We can
obtain the short-run total cost curve directly from this
solution: STC(Q) wL ⫹ rK ⫽ Q2/(100K ) ⫹ 100K. The
total variable and total fixed cost curves follow:
TVC(Q) Q2/(100K ) and TFC 100K.
Note that, holding Q constant, total variable cost is
decreasing in the quantity of capital K. The reason is
that, for a given amount of output, a firm that uses more
capital can reduce the amount of labor it employs. Since
TVC is the firm’s labor expense, it follows that TVC
should decrease in K.
Similar Problems:
8.20, 8.21
R E L AT I O N S H I P B E T W E E N T H E L O N G - R U N
A N D T H E S H O R T- R U N TOTA L C O S T C U RV E S
Consider again a firm that uses just two inputs, labor and capital. In the long run, the
firm can freely vary the quantity of both inputs, but in the short run the quantity of
capital is fixed. Thus, the firm is more constrained in the short run than in the long
run, so it makes sense that it will be able to achieve lower total costs in the long run.
Figure 8.14 shows a graphical analysis of the long-run and short-run costminimization problems for a producer of television sets in this situation. Initially, the
firm wants to produce 1 million television sets per year. In the long run, when it is free
to vary both capital and labor, it minimizes total cost by operating at point A, using
L1 units of labor and K1 units of capital.
CHAPTER 8
FIGURE 8.14 Total Costs
Are Generally Higher in the
Short Run than in the Long Run
Initially, the firm produces
1 million TVs per year and operates at point A, which minimizes
cost in both the long run and the
short run, if the firm’s usage of
capital is fixed at K1. If Q is increased to 2 million TVs per year,
and capital remains fixed at K1 in
the short run, the firm operates
at point B. But in the long run,
the firm operates at point C, on a
lower isocost line than point B.
K, capital services per year
304
C O S T C U RV E S
Expansion path
C
K2
A
B
K1
Q = 2 million TVs per year
Q = 1 million TVs per year
L1
0
L2
L3
L, labor services per year
Suppose the firm wants to increase its output to 2 million TVs per year and that,
in the short run, its usage of capital must remain fixed at K1. In that case, the firm
would operate at point B, using L3 units of labor and the same K1 units of capital. In
the long run, however, the firm could move along the expansion path and operate at
point C, using L2 units of labor and the same K2 units of capital. Since point B is on a
higher isocost line than point C, the short-run total cost is higher than the long-run
total cost when the firm is producing 2 million TVs per year.
When the firm is producing 1 million TVs per year, point A is cost-minimizing
in both the long run and the short run, if the short-run constraint is K1 units of capital. Figure 8.15 shows the firm’s corresponding long-run and short-run total cost
STC(Q) when K = K1
TC, dollars per year
TC(Q)
FIGURE 8.15 Relationship between
Short-Run and Long-Run Total Cost Curves
When the quantity of capital is fixed at K1,
STC(Q) is always above TC(Q), except at
point A. Point A solves both the long-run
and the short-run cost-minimization problem when the firm produces 1 million TVs
per year.
B
A
C
rK1
0
1 million
2 million
Q, TVs per year
305
8 . 2 S H O RT- RU N C O S T C U RV E S
curves TC(Q) and STC(Q). We see that STC(Q) always lies above TC(Q) (i.e., shortrun total cost is greater than long-run total cost) except at point A, where STC(Q) and
TC(Q) are equal.
S H O R T- R U N AV E R AG E A N D M A R G I N A L C O S T C U RV E S
Just as we can define long-run average and long-run marginal cost curves (see page 292),
we can also define the curves for short-run average cost (SAC) and short-run marginal cost (SMC): SAC(Q) [STC(Q)]/Q and SMC(Q) (STC)/(Q). Thus, just as
long-run marginal cost is equal to the slope of the long-run total cost curve, short-run
marginal cost is equal to the slope of the short-run total cost curve. ( Note that in
Figure 8.15 at point A, when output equals 1 million units per year, the slopes of the
long-run total cost and short-run total cost curves are equal. It therefore follows that
at this level of output, not only does STC TC, but SMC MC.)
In addition, just as we can break short-run total cost into two pieces (total variable cost and total fixed cost), we can break short-run average cost into two pieces:
average variable cost (AVC) and average fixed cost (AFC ): SAC AVC ⫹ AFC.
Average fixed cost is total fixed cost per unit of output (AFC ⫽ TFC/Q). Average variable cost is total variable cost per unit of output (AVC ⫽ TVC/Q).
Figure 8.16 illustrates typical graphs of the short-run marginal cost, short-run average
cost, average variable cost, and average fixed cost curves. We obtain the short-run average cost curve by “vertically summing” the average variable cost curve and the average
fixed cost curve.12 The average fixed cost curve decreases everywhere and approaches
the horizontal axis as Q becomes very large. This reflects the fact that as output increases, fixed capital costs are “spread out” over an increasingly large volume of output, driving fixed costs per unit downward toward zero. Because AFC becomes smaller
and smaller as Q increases, the AVC(Q) and SAC(Q) curves get closer and closer together. The short-run marginal cost curve SMC(Q) intersects the short-run average
SMC(Q)
short-run average cost
The firm’s total cost per
unit of output when it has
one or more fixed inputs.
short-run marginal
cost The slope of the
short-run total cost curve.
average variable cost
Total variable cost per unit
of output.
average fixed cost
Total fixed cost per unit of
output.
SAC(Q)
Cost per unit
AVC(Q)
A
B
AFC(Q)
Q, units per year
12
FIGURE 8.16 Short-Run Marginal and
Average Cost Curves
The short-run average cost curve SAC(Q) is
the vertical sum of the average variable cost
curve AVC(Q) and the average fixed cost
curve AFC(Q). The short-run marginal cost
curve SMC(Q) intersects SAC(Q) at point A
and AVC(Q) at point B, where each is at a
minimum.
Vertically summing means that, for any Q, we find the height of the SAC curve by adding together the
heights of the AVC and AFC curves at that quantity.
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C O S T C U RV E S
cost curve and the average variable cost curve at the minimum point of each curve.
This property mirrors the relationship between the long-run marginal and long-run
average cost curves (see page 294), again reflecting the relationship between the average and marginal measures of anything.
R E L AT I O N S H I P S B E T W E E N T H E L O N G - R U N A N D T H E
S H O R T- R U N AV E R AG E A N D M A R G I N A L C O S T C U RV E S
The Long-Run Average Cost Curve as an Envelope Curve
The long-run average cost curve forms a boundary (or envelope) around the set of
short-run average cost curves corresponding to different levels of output and fixed
input. Figure 8.17 illustrates this for a producer of television sets. The firm’s long-run
average cost curve AC(Q) is U-shaped, as are its short-run average cost curves
SAC1(Q), SAC2(Q), and SAC3(Q), which correspond to different levels of fixed capital
K1, K2, and K3 (where K1 ⬍ K2 ⬍ K3). (Moving to an increased level of fixed capital
might mean increasing the firm’s plant size or its degree of automation.)
The short-run average cost curve corresponding to any level of fixed capital lies
above the long-run curve except at the level of output for which the fixed capital is optimal (points A, B, and D in the figure). Thus, the firm would minimize its costs when
producing 1 million TVs if its level of fixed capital were K1, but if it expanded its output to 2 million or 3 million TVs, it would minimize costs if its level of fixed capital
were K2 or K3, respectively. (In practice, if K represents plant size, the firm’s high
short-run average cost of $110 to produce 2 million TVs using fixed capital K1 might
reflect reductions in the marginal product of labor resulting from crowding too many
workers into a small plant. To achieve the minimal average cost of $35, the firm would
have to increase its plant size to K2.)
Now observe the dark scalloped lower boundary of the short-run cost curves in
Figure 8.17, and imagine that the figure included more and more short-run curves.
$110
C
SAC1(Q), when K = K1
Cost, dollars per unit
FIGURE 8.17 The Long-Run
Average Cost Curve as an Envelope
Curve
The short-run average cost curves
SAC1(Q), SAC2(Q), and SAC3(Q), lie
above the long-run average cost
curve AC(Q) except at points A, B,
and D. This shows that short-run average cost is always greater than
long-run average cost except at the
level of output for which a plant size
(K1, K2, or K3) is optimal. Point C
shows where the firm would operate
in the short run if it produced 2 million TV sets per year with capital
remaining fixed at K1. If the figure
included progressively more shortrun curves, the dark scalloped lower
boundary of the short-run curves
would smooth out and ultimately
coincide with the long-run curve.
AC(Q)
SAC3(Q), when K = K3
SAC2(Q),
when K = K2
$60
$50
D
A
$35
B
1 million
2 million
Q, TVs per year
3 million
8 . 2 S H O RT- RU N C O S T C U RV E S
307
The dark boundary would become progressively smoother (i.e., with increasingly
many shallow scallops instead of a few deep scallops), and as the number of short-run
curves grew larger the dark curve would more and more closely approximate the longrun curve. Thus, you can think of the long-run curve as the lower envelope of an infinite number of short-run curves. That’s why the long-run average cost curve is
sometimes referred to as the envelope curve.
W H E N A R E L O N G - R U N A N D S H O R T- R U N AV E R AG E
A N D M A R G I N A L C O S T S E Q UA L , A N D W H E N
A R E T H E Y N OT ?
The curves shown in Figure 8.18 are the same as those in Figure 8.17, but with the
addition of the long-run marginal cost curve MC(Q) and the three short-run marginal
cost curves SMC1(Q), SMC2(Q), and SMC3(Q). Figure 8.18 shows the special relationships between the short-run average and marginal cost curves and the long-run average and marginal cost curves. As we have seen, if the firm is required to produce
1 million units, in the long run it would choose a plant size K1. Therefore, if the firm
has a fixed plant of size K1, the combination of inputs it would use to produce 1 million
units in the short run is the same as the combination it would choose in the long run.
At an output of 1 million units not only are SAC1(Q) and AC(Q) equal (at point A ),
but also SMC1(Q) and MC(Q) are equal (at point G).
Similar relationships hold at all levels of output. For example, if the firm has a
fixed plant of size K3, it can produce 3 million units as efficiently in the short run as it
can in the long run. Therefore SAC3(Q) and AC(Q) are equal (at point D), and
SMC3(Q) and MC(Q) are also equal (at point E ).
Figure 8.18 also illustrates another feature of short-run average cost curves that
you may find surprising. A short-run average cost curve does not generally reach its
For SAC1(Q) and SMC1(Q), K = K1
For SAC2(Q) and SMC2(Q), K = K2
For SAC3(Q) and SMC3(Q), K = K3
K1 < K2 < K3
SMC3(Q)
AC(Q)
E
Cost per unit
SMC2(Q)
SAC3(Q)
SAC1(Q) SMC1(Q)
C
SAC2(Q)
A
F
D
B
G
1 million
2 million = MES
Q, TVs per year
MC(Q)
3 million
FIGURE 8.18 The
Relationship between the
Long-Run Average and
Marginal Cost Curves and
the Short-Run Average
and Marginal Cost Curves
When the firm’s short-run
and long-run average
costs are equal, its shortrun and long-run marginal
costs must also be equal.
308
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C O S T C U RV E S
minimum at the output where short-run and long-run average costs are equal. For
example, at point A, SAC1(Q) and AC(Q) are equal, and they are both downward sloping. SAC1(Q) must be falling because SMC1(Q) lies below SAC1(Q). The minimum of
SAC1(Q) occurs at point C, where SMC1(Q) equals SAC1(Q). Similarly, at point D,
SAC3(Q) and AC(Q) are equal and have the same upward slope. SAC3(Q) must be rising because SMC3(Q) lies above SAC3(Q). The minimum of SAC3(Q) occurs at point F,
where SMC3(Q) equals SAC3(Q).
The figure also illustrates that it is possible for a short-run average cost curve to
reach its minimum at the output where short-run and long-run average costs are
equal. For example, at point B, SAC2(Q) and AC(Q) are equal, and they both achieve
a minimum. SAC2(Q) must have a slope of zero because SMC2(Q) passes through
SAC2(Q) at B.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 8 . 4
D
The Relationship between Short-Run and Long-Run Average Cost Curves
Let us return to the production function
in Learning-By-Doing Exercises 8.1, 8.2, and 8.3:
Q 50 1LK.
Problem What is the short-run average cost curve
for this production function for a fixed level of capital K
and input prices w 25 and r 100? Sketch a graph of
the short-run average cost curve for levels of capital
K 1, K 2, and K 4.
Solution We derived the short-run total cost curve
for this production function in Learning-By-Doing
Exercise 8.3: STC(Q) ⫽ Q2/(100K) ⫹ 100K. Thus, the
short-run average cost curve is SAC(Q) ⫽ Q/
(100K ) ⫹ 100K/Q. Figure 8.19 shows graphs of the
short-run average cost curve for K 1, K 2, and
K 4. It also shows the long-run average cost curve for
this production function (derived in Learning-By-Doing
Exercise 8.2). The short-run average cost curves are
U-shaped, while the long-run average cost curve (a horizontal line) is the lower envelope of the short-run average cost curves.
Similar Problems: 8.23, 8.27
SAC(Q), K = 1
FIGURE 8.19
LongRun and Short-Run
Average Cost Curves
The long-run average cost
curve AC(Q) is a horizontal line. It is the lower envelope of the short-run
average cost curves.
AC, dollars per unit
SAC(Q), K = 2
SAC(Q), K = 4
$2.5
$2
0
AC(Q)
100
200
400
600
Q, units per year
800
1000
309
8 . 2 S H O RT- RU N C O S T C U RV E S
A P P L I C A T I O N
8.5
Tracking Railroad Costs
In the period from 2003 to 2006, U.S. railroads faced
increasing complaints about speed of delivery. By
2006, the chairman of the U.S. Surface Transportation
Board (the body responsible for overseeing U.S. railroads) requested that each of the seven major U.S.
railroads submit a plan for how it intended to deal
with service bottlenecks. Part of the problem, according to industry observers, arose because the industry
downsized too much in the 1980s and 1990s, selling
or abandoning 55,000 miles of track.
Concerns over the quality of rail services and how
they relate to the amount of track a railroad employs
might make you wonder how a railroad’s costs depend
on these factors. Would a railroad’s total variable costs
decrease as it adds track? If so, at what rate? Would
faster service cause an increase or decrease in costs?
A study of railroad costs in the 1980s by Ronald
Braeutigam, Andrew Daughety, and Mark Turnquist
(hereafter BDT) provides some hints at the answers.13
BDT obtained data on the costs of shipment, input
prices, volume of output, and speed of service for a
large railroad. In their study, total variable cost is the
sum of the railroad’s monthly costs for labor, fuel,
maintenance, rail cars, locomotives, and supplies. You
should think of track miles as a fixed input, analogous
to capital in our previous discussion. A railroad cannot
instantly vary the quantity or quality of its track to
adjust to month-to-month variations in shipment
volumes, and thus must regard track as a fixed input.
Table 8.6 shows the impact on total variable costs
of a hypothetical 10 percent increase in traffic volume
(carloads of freight per month); the quantity of the
railroad’s track (in miles); speed of service (miles per
day of loaded cars); and the prices of fuel, labor and
equipment.14
Table 8.6 contains several interesting findings.
First, total variable cost increases with total output
and with input prices. This is consistent with the
theory you have been learning in this chapter and
13
TABLE 8.6 What Affects Total Variable Costs
for a Railroad?
A 10 Percent
Increase in . . .
Volume of output
Track mileage
Speed of service
Price of fuel
Price of labor
Price of equipment
Changes Total
Variable Cost by . . .
+3.98%
−2.71%
−0.66%
+1.90%
+5.25%
+2.85%
Source: Adapted from Table 1 of R. R. Braeutigam, A. F.
Daughety, and M. A. Turnquist, “A Firm-Specific Analysis of
Economies of Density in the U.S. Railroad Industry,’’ Journal
of Industrial Economics 33 (September 1984): 3–20. The
percentage changes in the various factors are changes away
from the average values of these factors over the period
studied by BDT.
Chapter 7. Second, total variable costs decrease as
the volume of the fixed input is increased (as discussed in Learning-By-Doing Exercise 8.3). Holding
volume of output and speed of service fixed, an increase in track mileage (or an increase in the quality
of track, holding mileage fixed) would be expected
to decrease the amount the railroad spends on variable inputs, such as labor and fuel. For example, with
more track (holding output and speed fixed), the
railroad would reduce the congestion of trains on its
mainlines and in its train yards. As a result, it would
probably need fewer dispatchers (i.e., less labor) to
control the movement of trains. Third, improvements
in average speed may also reduce costs. Although this
impact is not large, it does suggest that improvements in service might benefit not only the railroad’s
customers, but also the railroad itself through lower
variable costs. For this railroad, higher speeds might
reduce the use of labor (e.g., fewer train crews
would be needed to haul a given amount of freight)
and increase the fuel efficiency of the railroad’s
locomotives.
Ronald Braeutigam, Andrew Daughety, and Mark Turnquist, “A Firm-Specific Analysis of Economics
of Density in the U.S. Railroad Industry,” Journal of Industrial Economics 33 (September 1984): 3–20. The
identity of the railroad remained anonymous to ensure confidentiality of its data.
14
In this study, the railroad’s track mileage was adjusted to reflect changes in the quality of its track
over time.
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C O S T C U RV E S
SAC1: Track mileage 7.9 percent higher than average
AC, in units of minimum AC
SAC2: Track mileage 200 percent higher than average
AC(Q)
SAC1
SAC2
1.0
Observed average output level = 0.4
0
0.2
0.4
0.6
0.8
1.0 = MES
1.2
Q, in units of MES
FIGURE 8.20 Long-Run and Short-Run Average Cost Curves for a Railroad
The two short-run average cost curves SAC1 and SAC2 correspond to a different amount of track
(expressed in relation to the average amount of track observed in the data). The cost curves show
that with a cost-minimizing adjustment in amount of track, this railroad could decrease its unit
costs over a wide range of output above its current output level. As we have seen with other
such U-shaped cost curves, the long-run curve AC(Q) is the lower envelope of the short-run curves.
BDT also estimated the long-run total and average cost curves for this railroad by finding the track
mileage that, for each Q, minimized the sum of total
variable costs and total fixed cost, where total fixed
cost is the monthly opportunity cost to the firm’s
owners of a given amount of track mileage. Figure
8.20 shows the long-run average cost function estimated by BDT using this approach. It also shows two
short-run average cost curves, each corresponding to
a different level of track mileage. (Track mileage is
stated in relation to the average track mileage observed in BDT’s data.) The units of output in Figure 8.20
are expressed as a percentage of MES; the average
level of output produced by the railroad at the time
of study was about 40 percent of MES. This study thus
suggests that increases in traffic volume, accompanied
by cost-minimizing adjustments in track mileage,
would reduce this railroad’s average production costs
over a wide range of output.
8.3
ECONOMIES OF SCOPE
SPECIAL
TO P I C S
IN COST
This chapter has concentrated on cost curves for firms that produce just one product or service. In reality, though, many firms produce more than one product. For
a firm that produces two products, total costs would depend on the quantity Q1 of
the first product the firm makes and the quantity Q2 of the second product it makes.
We will use the expression TC(Q1, Q2) to denote how the firm’s costs vary with Q1
and Q2.
In some situations, efficiencies arise when a firm produces more than one product. That is, a two-product firm may be able to manufacture and market its products
8 . 3 S P E C I A L TO P I C S I N C O S T
at a lower total cost than two single-product firms. These efficiencies are called
economies of scope. Mathematically, economies of scope are present when:
TC(Q1, Q2) 6 TC(Q1, 0) ⫹ TC(0, Q2)
(8.1)
The zeros in the expressions on the right-hand side of equation (8.1) indicate that the
single-product firms produce positive amounts of one good but none of the other.
These expressions are sometimes called the stand-alone costs of producing goods 1
and 2.
Intuitively, the existence of economies of scope tells us that “variety” is more efficient than “specialization,” which we can see mathematically by representing equation (8.1) as follows: TC(Q1, Q2) ⫺ TC(Q1, 0) ⬍ TC(0, Q2) ⫺ TC(0, 0). This is equivalent to equation (8.1) because TC (0, 0) ⫽ 0; that is the total cost of producing zero
quantities of both products is zero. The left-hand side of this equation is the additional
cost of producing Q2 units of product 2 when the firm is already producing Q1 units of
product 1. The right-hand side of this equation is the additional cost of producing Q2
when the firm does not produce Q1. Economies of scope exist if it is less costly for a firm
to add a product to its product line given that it already produces another product.
Economies of scope would exist, for example, if it were less costly for Coca-Cola to
add a cherry-flavored soft drink to its product line than it would be for a new company starting from scratch.
Why would economies of scope arise? An important reason is a firm’s ability to
use a common input to make and sell more than one product. For example, BSkyB,
the British satellite television company, can use the same satellite to broadcast a news
channel, several movie channels, several sports channels, and several general entertainment channels.15 Companies specializing in the broadcast of a single channel
would each need to have a satellite orbiting the Earth. BSkyB’s channels save hundreds
of millions of dollars as compared to stand-alone channels by sharing a common satellite. Another example is Eurotunnel, the 31-mile tunnel that runs underneath the
English Channel between Calais, France, and Dover, Great Britain. The Eurotunnel
accommodates both highway and rail traffic. Two separate tunnels, one for highway
traffic and one for rail traffic, would have been more expensive to construct and operate than a single tunnel that accommodates both forms of traffic.
A P P L I C A T I O N
economies of scope
A production characteristic
in which the total cost of
producing given quantities
of two goods in the same
firm is less than the total
cost of producing those
quantities in two singleproduct firms.
stand-alone cost The
cost of producing a good in
a single-product firm.
8.6
Economies of Scope for the Swoosh
An important source of economies of scope is marketing. A company with a well-established brand name
in one product line can sometimes introduce additional products at a lower cost than a stand-alone
company would be able to do. This is because when
consumers are unsure about a product’s quality, they
often make inferences about its quality from the
15
311
BSkyB is a subsidiary of Rupert Murdoch’s News Corporation.
product’s brand name. This can give a firm with an
established brand reputation an advantage in
introducing new products, as it would not have to
spend as much on advertising as a firm without the
established reputation. This is an example of
economies of scope.
A company with an extraordinary brand reputation is Nike. Nike’s “swoosh,” the symbol that appears on its athletic shoes and sports apparel, is one
312
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C O S T C U RV E S
of the most recognizable marketing symbols of the
modern age. Nike’s swoosh is so recognizable that
Nike can run television commercials that never mention its name and be confident that consumers will
know whose products are being advertised. To support its brand Nike spends approximately 12 percent
of revenue on marketing every year. That includes
advertising, but also endorsement fees that it pays
to sports leagues, teams such as the Spanish soccer
team F.C. Barcelona, and individual athletes such as
LeBron James. Nike also pays substantial amounts
to become the official supplier of major events
such as the 2008 Beijing Summer Olympics or the
European Football (soccer) Championships. For example, in 2006 the company reported that it owed
$2.25 billion in endorsement fees to various athletes
and organizations.
Nike originally extended its brand very successfully from athletic shoes into apparel. This was so successful that Nike has been the global market share
leader in both categories for many years. In the late
1990s, Nike turned its attention to the sports equipment market, introducing products such as hockey
sticks and golf balls. While no one can deny Nike’s
past success in the athletic shoe and sports apparel
markets, producing a high-quality hockey stick or an
innovative golf ball has little in common with making
sneakers or jogging clothes. It therefore seems unlikely that Nike could attain economies of scope in
manufacturing or product design.
Instead, Nike hoped to achieve economies of
scope in marketing, based on its strong brand reputation, close ties to sports equipment retailers, and special relationships with professional athletes such as
Tiger Woods. Nike’s plan was to develop sports equipment that it could claim was innovative, and then use
its established brand reputation and ties with retail
trade to convince consumers that its products were
superior to existing products. Nike would then be
able to introduce its new products at far lower costs
than a stand-alone company would incur to introduce
otherwise identical products.
Economies of scope in marketing can be powerful, but they also have their limits. A strong brand
reputation can induce consumers to try a product
once, but if it does not perform as expected or if its
quality is inferior, it may be difficult to penetrate the
market or get repeat business. Nike’s initial forays
into the sports equipment market illustrate this risk.
Its lines of roller skates and ice skates both suffered
quality problems when first introduced.
Celebrity endorsements can be a powerful way
to try to extend economies of scope, but they too
have their risks. In 2007, Atlanta Falcons quarterback
Michael Vick was prosecuted for running a dogfighting ring in his home. At the time he endorsed
Nike products. Nike immediately suspended its contract with him and pulled all products related to Vick
from store shelves. In late 2009, Nike signed a new
endorsement contract with Vick after he had served
18 months in prison and then returned to playing in
the National Football League. In late 2009, Tiger
Woods’s personal life erupted in scandal, after it became known that he had had several extramarital
affairs. Woods took a leave of absence from the
Professional Golf Association tour during the early
part of 2010. Many of the companies whose products
Woods had endorsed dropped his contract in the
wake of the scandal. Nike was one of the few that
announced it would continue to work with Woods,
stating that “Tiger has been part of Nike for more
than a decade. He is the best golfer in the world and
one of the greatest athletes of his era. We look forward
to his return to golf.”
As of 2010, Nike has yet to attain its desired
dominance of the sports equipment business.
Approximately 50 percent of its revenues come from
footwear and 30 percent from apparel. Equipment
accounts for only 6 percent of revenue. Still, Nike’s
performance is impressive. The sporting equipment
market has historically been highly fragmented, and
no one firm has ever done what Nike aspires to do:
provide products over the entire category, from athletic shoes to ice skates, from golf balls to soccer balls.
That Nike has done as well as it has in the product
categories it has entered is no doubt a testimony to
the impressive array of sports stars that use Nike’s
products. At the same time, that success transcends
the stars who endorse Nike’s products and reflects
more broadly the economies of scope that Nike has
been able to attain in marketing: the power of the
“swoosh.”
313
8 . 3 S P E C I A L TO P I C S I N C O S T
E C O N O M I E S O F E X P E R I E N C E : T H E E X P E R I E N C E C U RV E
Learning-by-Doing and the Experience Curve
Economies of scale refer to the cost advantages that flow from producing a larger output at a given point in time. Economies of experience refer to cost advantages that
result from accumulated experience over an extended period of time, or from learningby-doing, as it is sometimes called. This is the reason we gave that title to the exercises
in this book—they are designed to help you learn microeconomics by doing microeconomics problems.
Economies of experience arise for several reasons. Workers often improve their
performance of specific tasks by performing them over and over again. Engineers
often perfect product designs as they accumulate know-how about the manufacturing
process. Firms often become more adept at handling and processing materials as they
deepen their production experience. The benefits of learning are usually greater labor
productivity (more output per unit of labor input), fewer defects, and higher material
yields (more output per unit of raw material input).
Economies of experience are described by the experience curve, a relationship
between average variable cost and cumulative production volume.16 A firm’s cumulative production volume at any given time is the total amount of output that it has produced over the history of the product until that time. For example, if Boeing’s output
of a type of jet aircraft was 30 in 2001, 45 in 2002, 50 in 2003, 70 in 2004, and 60 in
2005, its cumulative output as of the beginning of 2006 would be 30 ⫹ 45 ⫹ 50 ⫹
70 ⫹ 60, or 255 aircraft. A typical relationship between average variable cost and cumulative output is AVC(N) ⫽ AN B, where AVC is the average variable cost of production and N denotes cumulative production volume. In this formulation, A and B are
constants, where A 0 and B is a negative number between ⫺1 and 0. The constant A
represents the average variable cost of the first unit produced, and B represents the
experience elasticity: the percentage change in average variable cost for every 1 percent increase in cumulative volume.
The magnitude of cost reductions that are achieved through experience is often expressed in terms of the slope of the experience curve,17 which tells us how much average variable costs go down as a percentage of an initial level when cumulative output
doubles.18 For example, if doubling a firm’s cumulative output of semiconductors results in average variable cost falling from $10 per megabyte to $8.50 per megabyte, we
would say that the slope of the experience curve for semiconductors is 85 percent, since
average variable costs fell to 85 percent of their initial level. In terms of an equation,
slope of experience curve ⫽
AVC(2N )
AVC(N )
The slope and the experience elasticity are systematically related. If the experience
elasticity is equal to B, the slope equals 2B. Figure 8.21 shows experience curves with
three different slopes: 90 percent, 80 percent, and 70 percent. The smaller the slope,
the “steeper” the experience curve (i.e., the more rapidly average variable costs fall
as the firm accumulates experience). Note, though, that all three curves eventually
16
The experience curve is also known as the learning curve.
The slope of the experience curve is also known as the progress ratio.
18
Note that the term slope as used here is not the usual notion of the slope of a straight line.
17
economies of experience Cost advantages
that result from accumulated experience, or as it is
sometimes called, learningby-doing.
experience curve
A
relationship between average variable cost and cumulative production volume. It
is used to describe the
economies of experience.
experience elasticity
The percentage change in
average variable cost for
every 1 percent increase in
cumulative volume.
slope of the experience curve How much
average variable costs go
down, as a percentage of
an initial level, when cumulative output doubles.
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C O S T C U RV E S
$1
FIGURE 8.21
AVC, dollars per unit
Experience Curves
with Different Slopes
The smaller the slope, the
“steeper” the experience
curve, and the more rapidly average variable costs
fall as cumulative output
goes up. No matter what
the slope, though, once
cumulative experience
becomes sufficiently large
(e.g., N 40), additional
increments to experience
do not lower average
variable costs by much.
$0.8
$0.6
Slope = 90%
$0.4
Slope = 80%
$0.2
Slope = 70%
1
10
20
30
40
50
N, cumulative output
flatten out. For example, beyond a volume of N 40, increments in cumulative experience have a small impact on average variable costs, no matter what the slope of the
experience curve is. At this point, most of the economies of experience are exhausted.
Experience curve slopes have been estimated for many different products. The
median slope appears to be about 80 percent, implying that for the typical firm, each
doubling of cumulative output reduces average variable costs to 80 percent of what
they were before. Slopes vary from firm to firm and industry to industry, however, so
that the slope enjoyed by any one firm for any given production process generally falls
between 70 and 90 percent and may be as low as 60 percent or as high as 100 percent
(i.e., no economies of experience).
A P P L I C A T I O N
8.7
Experience Curves in Emissions
Control
There are many examples of experience curves in actual production technologies. The manufacture of
products such as semiconductor, commercial and military airframes, and merchant vessels have been
shown to benefit from economies of experience.
But economies of experience also show up in
other, perhaps less obvious, settings. One example is
in electric power plant emissions control technologies. Edward Rubin, Sonia Yeh, David Hounshell, and
19
Margaret Taylor have estimated experience curves
for two pollution control technologies widely used in
the electric utility industry: flue gas desulphurization,
which is used to reduce sulphur dioxide (SO2) emissions, and selective catalytic reduction systems, which
is used to reduce nitrogen oxide (NOx) emissions.19
The study relied on 30 years of data on output and
emissions for both technologies to measure the impact of cumulative experience on the cost of operating and maintaining each of these emissions control
systems. The slope of the experience curve for flue
gas desulphurization systems was estimated to be
89 percent, while the slope for selective catalytic
Edward, Rubin, Sonia Yeh, David Hounshell, and Margaret Taylor, “Experience Curves for Power Plant
Emissions Control Technologies,” International Journal of Energy Technology and Policy 2(1–2) (2004): 52–69.
315
8 . 4 E S T I M AT I N G C O S T F U N C T I O N S
reduction systems was estimated to be 88 percent. To
put these estimates in perspective, cumulative experience with flue gas desulphurization technology increased by a factor of approximately 5 between 1983
and 1996. This corresponds to about 2.3 “doublings”
of cumulative experience.20 Given an experience curve
slope of 89 percent, this resulted in operating and
maintenance costs in 1996 equal to about 76 percent
of operating and maintenance expenses in 1983.21
This result has an important implication for public policy. Governments worldwide are currently debating various policies, such as cap & trade and carbon taxes, to deal with greenhouse gas emissions.
One approach to reducing greenhouse gas emissions
is CO2 capture-and-sequestration, a technology that
has many technological similarities to the systems
studied in this paper and that may, therefore, benefit
from economies of experience. If economies of experience are ignored, estimates of the costs of reducing
greenhouse gas emissions through capture-andsequestration technologies may be overstated, and
the benefits of early adoption of these technologies—which can be thought of as an investment in
the development of economies of experience—will be
understated. Therefore, in setting climate-change
policy, it is not only important to consider the current
costs of employing an emissions control technology,
but the impact of cumulative experience on what
those costs are likely to be in the future.
Economies of Experience versus Economies of Scale
Economies of experience differ from economies of scale. Economies of scale refer to
the ability to perform activities at a lower unit cost when those activities are performed on a larger scale at a given point in time. Economies of experience refer to reductions in unit costs due to accumulating experience over time. Economies of scale
may be substantial even when economies of experience are minimal. This is likely to
be the case in mature, capital-intensive production processes, such as aluminum can
manufacturing. Likewise, economies of experience may be substantial even when
economies of scale are minimal, as in such complex labor-intensive activities as the
production of handmade watches.
Firms that do not correctly distinguish between economies of scale and experience might draw incorrect inferences about the benefits of size in a market. For example, if a firm has low average costs because of economies of scale, reductions in the
current volume of production will increase unit costs. If the low average costs are the
result of cumulative experience, the firm may be able to cut back current production
volumes without raising its average costs.
S
uppose you wanted to estimate how the total costs for a television producer varied
with the quantity of its output or the magnitude of its input prices. To do this, you
might want to estimate what economists call a total cost function. A total cost function is a mathematical relationship that shows how total costs vary with the factors
that influence total costs. These factors are sometimes called cost drivers. We’ve
spent much of this chapter analyzing two key cost drivers: input prices and scale (volume of output). Our discussion in the previous section suggests two other factors that
could also be cost drivers: scope (variety of goods produced by the firm) and cumulative experience.
20
To find the number of “doublings” of experience that 5 represents, we solve the equation 2x 5, which
gives us x 2.3.
21
We get this by noting that (0.89)2.3 0.76.
total cost function
A
mathematical relationship
that shows how total costs
vary with the factors that
influence total costs, including the quantity of output
and the prices of inputs.
8.4
E S T I M AT I N G
COST
FUNCTIONS
cost driver
A factor
that influences or “drives”
total or average costs.
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C O S T C U RV E S
When estimating cost functions, economists first gather data from a cross section of firms or plants at a particular point in time. A cross section of television producers would consist of a sample of manufacturers or manufacturing facilities in a
particular year, such as 2010. For each observation in your cross section, you would
need information about total costs and cost drivers. The set of cost drivers that you
include in your analysis is usually specific to what you are studying. In television
manufacturing, scale, cumulative experience, labor wages, materials prices, and costs
of capital would probably be important drivers for explaining the behavior of average costs in the long run.
Having gathered data on total costs and cost drivers, you would then use statistical techniques to construct an estimated total cost function. The most common technique used by economists is multiple regression. The basic idea behind this technique
is to find the function that best fits our available data.
C O N S TA N T E L A S T I C I T Y C O S T F U N C T I O N
constant elasticity
cost function A cost
function that specifies constant elasticities of total
cost with respect to output
and input prices.
An important issue when you use multiple regression to estimate a cost function is
choosing the functional form that relates the dependent variable of interest—in this
case, total cost—to the independent variables of interest, such as output and input
prices. One commonly used functional form is the constant elasticity cost function,
which specifies a multiplicative relationship between total cost, output, and input
prices. For a production process that involves two inputs, capital and labor, the constant elasticity long-run total cost function is TC aQbwcrd, where a, b, c, and d are
positive constants. It is common to convert this into a linear relationship using logarithms: log TC log a ⫹ b log Q ⫹ c log w ⫹ d log r. With the function in this form,
the positive constants a, b, c, and d can be estimated using multiple regression.
A useful feature of the constant elasticity specification is that the constant b is the
output elasticity of total cost, discussed earlier. Analogously, the constants c and d are
the elasticities of long-run total cost with respect to the prices of labor and capital, respectively. These elasticities must be positive since, as we saw earlier, an increase in an
input price will increase long-run total cost. We also learned earlier that a given percentage increase in w and r would have to increase long-run total cost by the same
percentage amount. This implies that the constants c and d must add up to 1 (i.e., c ⫹
d ⫽ 1) for the estimated long-run total cost function to be consistent with long-run
cost minimization. This restriction can be readily incorporated into the multiple regression analysis.
TRANSLOG COST FUNCTION
translog cost function
A cost function that postulates a quadratic relationship between the log of
total cost and the logs of
input prices and output.
The constant elasticity cost function does not allow for the possibility of average costs
that first decrease and then increase as Q increases (i.e., economies of scale, followed
by diseconomies of scale). The translog cost function, which postulates a quadratic
relationship between the log of total cost and the logs of input prices and output, does
allow for this possibility. The equation of the translog cost function is
log TC ⫽ b0 ⫹ b1 log Q ⫹ b2 log w ⫹ b3 log r ⫹ b4(log Q)2
⫹ b5(log w)2 ⫹ b6(log r)2 ⫹ b7(log w)(log r)
⫹ b8(log w)(log Q) ⫹ b9(log r)(log Q)
8 . 4 E S T I M AT I N G C O S T F U N C T I O N S
317
This formidable-looking expression turns out to have many useful properties. For
one thing, it is often a good approximation of the cost functions that come from just
about any production function. Thus, if (as is often the case) we don’t know the
exact functional form of the production function, the translog might be a good
choice for the functional form of the cost function. In addition, the average cost
function can be U-shaped. Thus, it allows for both economies of scale and diseconomies of scale. For instance, the short-run average cost curves in Figure 8.20
(Application 8.5) were estimated as translog functions. Note, too, that if b4 b5
b6 b7 b8 b9 0, the translog cost function reduces to the constant elasticity
cost function. Thus, the constant elasticity cost function is a special case of the
translog cost function.
A P P L I C A T I O N
8.8
Estimating Economies of Scale
in Payment Processing Services
Whether it is the payment of bills online, transfer of
funds between bank accounts, or transfer of securities
from one party to another, paper transactions are
rapidly becoming replaced by digital transactions
over the Internet or private computer networks. Some
of the most important of these systems are operated
by the Federal Reserve. The Fedwire Funds system is
an electronic settlement system between banks. The
Fedwire Securities (formerly Book-Entry) system provides a similar service for transactions involving
stocks and bonds. In 2008, these programs combined
totaled over 150 million transactions valued at over
$1,100 trillion.
Another key part of the Fed’s payment systems is
the Automated Clearinghouse (ACH). Transactions
that take place through the ACH include direct deposits of paychecks, Social Security benefits, payments
to suppliers, direct debits of mortgages, and tax payments. In 2000, 4.8 billion transfers took place. That
number rose to over 18 billion in 2008, valued at over
$30 trillion.
The enormous scale of the Fed’s services raises
the question of whether payment processing services
22
are characterized by economies of scale. Robert
Adams, Paul Bauer, and Robin Sickles explored this
question by estimating a translog cost function for
each of these three services from 1990 to 2000.22 In
all three services there was clear evidence of
economies of scale. This implies that if the Federal
Reserve and its smaller competitors have the same
technology and face the same input prices, then the
Fed will have a lower average cost than its smaller
rivals. Put another way, in order for a smaller competitor to offset the Fed’s scale-based cost advantage,
the competitor would need to use superior technology (e.g., better software) or face more favorable
input prices.
The finding of economies of scale is interesting
for another reason, related to the way the Federal
Reserve prices its electronic transfer services. The U.S.
banks that purchase these services pay a price that
equals the Fed’s average costs. As the systems grow
over time, and assuming that factor prices remain
fixed, the price that the Federal Reserve charges for its
services should also decline. In fact, this is what happened during the 1990s. This suggests that the Fed’s
customers (U.S. banks)—and perhaps its customers’
customers (i.e., households that do business with
those banks)—benefited from the economies of scale
in payment processing services.
Robert Adams, Paul Bauer, and Robin Sickles, “Scale Economies, Scope Economies, and Technical
Change in Federal Reserve Payment Processing,” Journal of Money, Credit and Banking 36, no. 5 (October
2004): 943–958.
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C O S T C U RV E S
CHAPTER SUMMARY
• The long-run total cost curve shows how the
minimized level of total cost varies with the quantity of
output. (LBD Exercise 8.1)
• The short-run total cost curve tells us the minimized
total cost as a function of output, input prices, and the
level of the fixed input(s). (LBD Exercise 8.3)
• An increase in input prices rotates the long-run total
cost curve upward through the point Q 0.
• Short-run total cost is the sum of two components:
total variable cost and total fixed cost.
• Long-run average cost is the firm’s cost per unit of
output. It equals total cost divided by output. (LBD
Exercise 8.2)
• Long-run marginal cost is the rate of change of longrun total cost with respect to output. (LBD Exercise 8.2)
• Long-run marginal cost can be less than, greater
than, or equal to long-run average cost, depending on
whether long-run average cost decreases, increases, or
remains constant, respectively, as output increases.
• Economies of scale describe a situation in which
long-run average cost decreases as output increases.
Economies of scale arise because of the physical properties of processing units, specialization of labor, and indivisibilities of inputs.
• Diseconomies of scale describe a situation in which
long-run average cost increases as output increases. A key
source of diseconomies of scale is managerial diseconomies.
• The minimum efficient scale (MES) is the smallest
quantity at which the long-run average cost curve attains
its minimum.
• With economies of scale, there are increasing returns
to scale; with diseconomies of scale, there are decreasing
returns to scale; and with neither economies nor diseconomies of scale, there are constant returns to scale.
• The output elasticity of total cost measures the extent of economies of scale; it is the percentage change in
total cost per 1 percent change in output.
• Short-run total cost is always greater than long-run
total cost, except at the quantity of output for which the
level of fixed input is cost minimizing.
• Short-run average cost is the sum of average variable
cost and average fixed cost. Short-run marginal cost is
the rate of change of short-run total cost with respect to
output.
• The long-run average cost curve is the lower envelope of the short-run average cost curves. (LBD
Exercise 8.4)
• Economies of scope exist when it is less costly to produce given quantities of two products with one firm than
it is with two firms that each specialize in the production
of a single product.
• Economies of experience exist when average variable
cost decreases with cumulative production volume. The
experience curve tells us how average variable costs are
affected by changes in cumulative production volume.
The magnitude of this effect is often expressed in terms
of the slope of the experience curve.
• Cost drivers are factors such as output or the prices
of inputs that influence the level of costs.
• Two common functional forms that are used for realworld estimation of cost functions are the constant elasticity cost function and the translog cost function.
REVIEW QUESTIONS
1. What is the relationship between the solution to the
firm’s long-run cost-minimization problem and the longrun total cost curve?
2. Explain why an increase in the price of an input typically causes an increase in the long-run total cost of producing any particular level of output.
3. If the price of labor increases by 20 percent, but all
other input prices remain the same, would the long-run
total cost at a particular output level go up by more than
20 percent, less than 20 percent, or exactly 20 percent? If
the prices of all inputs went up by 20 percent, would
long-run total cost go up by more than 20 percent, less
than 20 percent, or exactly 20 percent?
4. How would an increase in the price of labor shift the
long-run average cost curve?
5. a) If the average cost curve is increasing, must the
marginal cost curve lie above the average cost curve?
Why or why not?
319
PROBLEMS
b) If the marginal cost curve is increasing, must the marginal cost curve lie above the average cost curve? Why or
why not?
6. Sketch the long-run marginal cost curve for the
“flat-bottomed” long-run average cost curve shown in
Figure 8.11.
7. Could the output elasticity of total cost ever be
negative?
curve be? What shape would the short-run average cost
curve be?
10. Suppose that the minimum level of short-run average cost was the same for every possible plant size.
What would that tell you about the shapes of the longrun average and long-run marginal cost curves?
8. Explain why the short-run marginal cost curve must
intersect the average variable cost curve at the minimum
point of the average variable cost curve.
11. What is the difference between economies of
scope and economies of scale? Is it possible for a twoproduct firm to enjoy economies of scope but not
economies of scale? Is it possible for a firm to have
economies of scale but not economies of scope?
9. Suppose the graph of the average variable cost curve
is flat. What shape would the short-run marginal cost
12. What is an experience curve? What is the difference
between economies of experience and economies of scale?
PROBLEMS
8.1. The following incomplete table shows a firm’s
various costs of producing up to 6 units of output. Fill
in as much of the table as possible. If you cannot determine the number in a box, explain why it is not possible to do so.
Q
TC
1
100
2
TVC
TFC
AC
MC
AVC
4
5
6
120
8.2. The following incomplete table shows a firm’s
various costs of producing up to 6 units of output. Fill
in as much of the table as possible. If you cannot determine the number in a box, explain why it is not possible to do so.
TVC
AFC
1
AC
MC
AVC
100
50
30
AC
MC
66
10
6
18
108
8.4. The following incomplete table shows a firm’s various costs of producing up to 6 units of output. Fill in as
much of the table as possible. If you cannot determine the
number in a box, explain why it is not possible to do so.
Q
TC
1
20
TVC
TFC
AC
MC
5
80
10
15
72
5
6
AVC
18
4
30
AVC
16
3
10
330
TFC
10
2
4
6
18
TVC
5
170
3
1
4
95
2
TC
3
20
TC
Q
2
160
3
Q
8.3. The following incomplete table shows a firm’s various costs of producing up to 6 units of output. Fill in as
much of the table as possible. If you cannot determine the
number in a box, explain why it is not possible to do so.
30
144
320
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C O S T C U RV E S
8.5. A firm produces a product with labor and capital,
and its production function is described by Q LK. The
marginal products associated with this production function are MPL K and MPK L. Suppose that the price
of labor equals 2 and the price of capital equals 1. Derive
the equations for the long-run total cost curve and the
long-run average cost curve.
8.6. A firm’s long-run total cost curve is TC(Q)
1000Q2. Derive the equation for the corresponding longrun average cost curve, AC(Q). Given the equation of the
long-run average cost curve, which of the following
statements is true?
a) The long-run marginal cost curve MC(Q) lies below
AC(Q) for all positive quantities Q.
b) The long-run marginal cost curve MC(Q) is the same
as the AC(Q) for all positive quantities Q.
c) The long-run marginal cost curve MC(Q) lies above
the AC(Q) for all positive quantities Q.
d) The long-run marginal cost curve MC(Q) lies below
AC(Q) for some positive quantities Q and above the
AC(Q) for some positive quantities Q.
1
8.7. A firm’s long-run total cost curve is TC(Q) ⫽ 1000Q2.
Derive the equation for the corresponding long-run average cost curve, AC(Q). Given the equation of the longrun average cost curve, which of the following statements
is true?
a) The long-run marginal cost curve MC(Q) lies below
AC(Q) for all positive quantities Q.
b) The long-run marginal cost curve MC(Q) is the same
as the AC(Q) for all positive quantities Q.
c) The long-run marginal cost curve MC(Q) lies above
the AC(Q) for all positive quantities Q.
d) The long-run marginal cost curve MC(Q) lies below
AC(Q) for some positive quantities Q and above the
AC(Q) for some positive quantities Q.
8.8. A firm’s long-run total cost curve is TC(Q)
1000Q ⫺ 30Q2 ⫹ Q3. Derive the expression for the corresponding long-run average cost curve and then sketch
it. At what quantity is minimum efficient scale?
8.9. A firm’s long-run total cost curve is TC(Q) ⫽ 40Q
⫺ 10Q2 ⫹ Q3, and its long-run marginal cost curve is
MC(Q) ⫽ 40 ⫺ 20Q ⫹ 3Q2. Over what range of output
does the production function exhibit economies of scale,
and over what range does it exhibit diseconomies of scale?
8.10. For each of the total cost functions, write the expressions for the total fixed cost, average variable cost,
and marginal cost (if not given), and draw the average
total cost and marginal cost curves.
a) TC(Q) ⫽ 10Q
b) TC(Q) ⫽ 160 ⫹ 10Q
c) TC(Q) ⫽ 10Q2, where MC(Q) ⫽ 20Q
d) TC(Q) ⫽ 10 1Q, where MC(Q) ⫽ 5/ 1Q
e) TC(Q) ⫽ 160 ⫹ 10Q2, where MC(Q) ⫽ 20Q
8.11. A firm produces a product with labor and capital
as inputs. The production function is described by Q ⫽
LK. The marginal products associated with this production function are MPL ⫽ K and MPK ⫽ L. Let w ⫽ 1 and
r ⫽ 1 be the prices of labor and capital, respectively.
a) Find the equation for the firm’s long-run total cost
curve as a function of quantity Q.
b) Solve the firm’s short-run cost-minimization problem
when capital is fixed at a quantity of 5 units (i.e., K ⫽ 5).
Derive the equation for the firm’s short-run total cost
curve as a function of quantity Q and graph it together
with the long-run total cost curve.
c) How do the graphs of the long-run and short-run total
cost curves change when w ⫽ 1 and r ⫽ 4?
d) How do the graphs of the long-run and short-run total
cost curves change when w ⫽ 4 and r ⫽ 1?
8.12. A firm produces a product with labor and capital.
Its production function is described by Q ⫽ min(L, K ). Let
w and r be the prices of labor and capital, respectively.
a) Find the equation for the firm’s long-run total cost
curve as a function of quantity Q and input prices, w
and r.
b) Find the solution to the firm’s short-run costminimization problem when capital is fixed at a quantity
of 5 units (i.e., K ⫽ 5). Derive the equation for the firm’s
short-run total cost curve as a function of quantity Q.
Graph this curve together with the long-run total cost
curve for w ⫽ 1 and r ⫽ 1.
c) How do the graphs of the long-run and short-run total
cost curves change when w ⫽ 1 and r ⫽ 2?
d) How do the graphs of the long-run and short-run total
cost curves change when w ⫽ 2 and r ⫽ 1?
8.13. A firm produces a product with labor and capital.
Its production function is described by Q ⫽ L ⫹ K. The
marginal products associated with this production function are MPL ⫽ 1 and MPK ⫽ 1. Let w ⫽ 1 and r ⫽ 1 be
the prices of labor and capital, respectively.
a) Find the equation for the firm’s long-run total cost
curve as a function of quantity Q when the prices of labor
and capital are w ⫽ 1 and r ⫽ 1.
b) Find the solution to the firm’s short-run costminimization problem when capital is fixed at a quantity
of 5 units (i.e., K ⫽ 5), and w ⫽ 1 and r ⫽ 1. Derive the
equation for the firm’s short-run total cost curve as a
function of quantity Q and graph it together with the
long-run total cost curve.
PROBLEMS
321
c) How do the graphs of the short-run and long-run total
cost curves change when w 1 and r 2?
d) How do the graphs of the short-run and long-run total
cost curves change when w 2 and r 1?
8.18. A firm has the linear production function Q ⫽
3L ⫹ 5K, with MPL ⫽ 3 and MPK ⫽ 5. Derive the expression for the 1ong-run total cost that the firm incurs,
as a function of Q and the factor prices, w and r.
8.14. Consider a production function of two inputs,
labor and capital, given by Q ⫽ ( 1L ⫹ 1K )2. The marginal products associated with this production function
are as follows:
8.19. A firm uses two inputs: labor and capital. The
price of labor is w and the price of capital is r. The
firm’s long-run
total cost is given by the equation
1 4
TC(Q) w 5r 5Q. Based on this equation, which change
would cause the greater upward rotation in the long-run
total cost curve: a 10 percent increase in w or a 10 percent increase in r? Based on your answer, is the firm’s
operation more capital intensive or more labor intensive? Explain your answer.
1
1
1
1
1
MPL ⫽ [ L2 ⫹ K 2 ] L⫺2
1
MPK ⫽ [ L2 ⫹ K 2 ] K ⫺ 2
Let w 2 and r 1.
a) Suppose the firm is required to produce Q units of output. Show how the cost-minimizing quantity of labor depends on the quantity Q. Show how the cost-minimizing
quantity of capital depends on the quantity Q.
b) Find the equation of the firm’s long-run total cost
curve.
c) Find the equation of the firm’s long-run average cost
curve.
d) Find the solution to the firm’s short-run costminimization problem when capital is fixed at a quantity
of 9 units (i.e., K ⫽ 9).
e) Find the short-run total cost curve, and graph it along
with the long-run total cost curve.
f ) Find the associated short-run average cost curve.
8.15. Tricycles must be produced with 3 wheels and 1
frame for each tricycle. Let Q be the number of tricycles,
W be the number of wheels, and F be the number of
frames. The price of a wheel is PW and the price of a
frame is PF.
a) What is the long-run total cost function for producing
tricycles, TC(Q, PW, PF)?
b) What is the production function for tricycles, Q(F, W )?
8.16. A hat manufacturing firm has the following production function with capital and labor being the inputs:
Q min(4L, 7K )—that is, it has a fixed-proportions production function. If w is the cost of a unit of labor and r
is the cost of a unit of capital, derive the firm’s long-run
total cost curve and average cost curve in terms of the
input prices and Q.
8.17. A packaging firm relies on the production function Q KL ⫹ K, with MPL ⫽ K and MPK ⫽ L ⫹ 1.
Assume that the firm’s optimal input combination is interior (it uses positive amounts of both inputs). Derive its
long-run total cost curve in terms of the input prices, w
and r. Verify that if the input prices double, then total
cost doubles as well.
8.20. When a firm uses K units of capital and L units of
labor, it can produce Q units of output with the production function Q K 1L. Each unit of capital costs 20,
and each unit of labor costs 25. The level of K is fixed
at 5 units.
a) Find the equation of the firm’s short-run total cost
curve.
b) On a graph, draw the firm’s short-run average cost.
8.21. When a firm uses K units of capital and L units of
labor, it can produce Q units of output with the production function Q 1L 1K. Each unit of capital costs
2, and each unit of labor costs 1.
a) The level of K is fixed at 16 units. Suppose Q 4.
What will the firm’s short-run total cost be? (Hint: How
much labor will the firm need?)
b) The level of K is fixed at 16 units. Suppose Q 4. Find
the equation of the firm’s short-run total cost curve.
8.22. Consider a production function of three inputs,
labor, capital, and materials, given by Q LKM. The
marginal products associated with this production function are as follows: MPL KM, MPK LM, and MPM
LK. Let w 5, r 1, and m 2, where m is the price
per unit of materials.
a) Suppose that the firm is required to produce Q units of
output. Show how the cost-minimizing quantity of labor
depends on the quantity Q. Show how the cost-minimizing
quantity of capital depends on the quantity Q. Show how
the cost-minimizing quantity of materials depends on the
quantity Q.
b) Find the equation of the firm’s long-run total cost
curve.
c) Find the equation of the firm’s long-run average cost
curve.
d) Suppose that the firm is required to produce Q units
of output, but that its capital is fixed at a quantity of
50 units (i.e., K 50). Show how the cost-minimizing
quantity of labor depends on the quantity Q. Show how
322
CHAPTER 8
C O S T C U RV E S
the cost-minimizing quantity of materials depends on the
quantity Q.
e) Find the equation of the short-run total cost curve
when capital is fixed at a quantity of 50 units (i.e.,
K ⫽ 50) and graph it along with the long-run total cost
curve.
f ) Find the equation of the associated short-run average
cost curve.
8.23. The production function Q KL ⫹ M has marginal products MPK ⫽ L, MPL ⫽ K, and MPM ⫽ 1. The
input prices of K, L, and M are 4, 16, and 1, respectively.
The firm is operating in the long run. What is the longrun total cost of producing 400 units of output?
8.24. The production function Q ⫽ KL ⫹ M has marginal products MPK ⫽ L, MPL ⫽ K, and MPM ⫽ 1. The
input prices of K, L, and M are 4, 16, and 1, respectively.
The firm is operating in the short run, with K fixed at
20 units. What is the short-run total cost of producing
400 units of output?
8.25. The production function Q ⫽ KL ⫹ M has marginal products MPK ⫽ L, MPL ⫽ K, and MPM ⫽ 1. The
input prices of K, L, and M are 4, 16, and 1, respectively.
The firm is operating in the short run, with K fixed at
20 units and M fixed at 40. What is the short-run total
cost of producing 400 units of output?
60
8.26. A short-run total cost curve is given by the equation STC(Q) ⫽ 1000 ⫹ 50Q2. Derive expressions for, and
then sketch, the corresponding short-run average cost,
average variable cost, and average fixed cost curves.
8.27. A producer of hard disk drives has a short-run
total cost curve given by STC(Q) ⫽ K ⫹ Q2/K. Within
the same set of axes, sketch a graph of the short-run average cost curves for three different plant sizes: K ⫽ 10,
K ⫽ 20, and K ⫽ 30. Based on this graph, what is the
shape of the long-run average cost curve?
8.28. Figure 8.18 shows that the short-run marginal
cost curve may lie above the long-run marginal cost
curve. Yet, in the long run, the quantities of all inputs are
variable, whereas in the short run, the quantities of just
some of the inputs are variable. Given that, why isn’t
short-run marginal cost less than long-run marginal cost
for all output levels?
8.29. The following diagram shows the long-run average and marginal cost curves for a firm. It also shows the
short-run marginal cost curve for two levels of fixed capital: K ⫽ 150 and K ⫽ 300. For each plant size, draw the
corresponding short-run average cost curve and explain
briefly why that curve should be where you drew it and
how it is consistent with the other curves.
SMC(Q),
K = 150
SMC(Q),
K = 300
MC(Q)
50
AC(Q)
AC(Q), MC(Q)
40
30
20
10
0
2
4
6
Q
8
10
A P P E N D I X : S H E P H A R D ’ S L E M M A A N D D UA L I T Y
8.30. Suppose that the total cost of providing satellite
television services is as follows:
TC(Q1, Q2) ⫽ e
0, if Q1 ⫽ 0 and Q2 ⫽ 0
1000 ⫹ 2Q1 ⫹ 3Q2, otherwise
where Q1 and Q2 are the number of households that subscribe to a sports and movie channel, respectively. Does
the provision of satellite television services exhibit
economies of scope?
8.31. A railroad has two types of services: freight service
and passenger service. The stand-alone cost for freight
service is TC1 500 ⫹ Q1, where Q1 equals the number
of ton-miles of freight hauled each day and TC1 is the
total cost in thousands of dollars per day. The standalone cost for passenger service is TC2 ⫽ 1000 ⫹ 2Q2,
where Q2 equals the number of passenger-miles per day
and TC2 is the total cost in thousands of dollars per day.
When a railroad offers both services jointly, its total is
TC(Q1, Q2) ⫽ 2000 ⫹ Q1 ⫹ 2Q2. Does the provision of
passenger and freight service exhibit economies of scope?
8.32. Suppose that the experience curve for the production of a certain type of semiconductor has a slope of
80 percent. Suppose over a five-year period that cumulative production experience increases by a factor of 8.
Input prices over this period did not change. At the beginning of the period, average variable cost was $10 per
unit. Assume that average variable cost is independent of
the level of output at any particular point in time. What
is your best estimate of average variable cost at the end
of this five-year period?
8.33. A railroad provides passenger and freight service.
The table shows the long-run total annual costs TC(F, P ),
where P measures the volume of passenger traffic and F the
volume of freight traffic. For example, TC(10,300) ⫽ 1,000.
Determine whether there are economies of scope for a railroad producing F ⫽ 10 and P ⫽ 300. Briefly explain.
A P P E N D I X:
Total Annual Costs for Freight and Passenger Service
F, Units of
Freight Service
0
10
P, Units of Passenger Service
0
300
Cost ⫽ 0
Cost ⫽ 400
Cost ⫽ 500
Cost ⫽ 1000
8.34. A researcher has claimed to have estimated a
long-run total cost function for the production of1 auto1
mobiles. His estimate is that TC(Q, w, r) ⫽ 100w⫺2 r 2Q3,
where w and r are the prices of labor and capital. Is
this a valid cost function—that is, is it consistent
with long-run cost minimization by the firm? Why or
why not?
8.35. A firm owns two production plants that make
widgets. The plants produce identical products, and each
plant (i) has a production function given by Qi ⫽ 1KiLi,
for i ⫽ 1, 2. The plants differ, however, in the amount of
capital equipment in place in the short run. In particular,
plant 1 has K1 ⫽ 25, whereas plant 2 has K2 ⫽ 100. Input
prices for K and L are w ⫽ r ⫽ 1.
a) Suppose the production manager is told to minimize
the short-run total cost of producing Q units of output.
While total output Q is exogenous, the manager can
choose how much to produce at plant 1 (Q1) and at plant
2 (Q2), as long as Q1 ⫹ Q2 ⫽ Q. What percentage of its
output should be produced at each plant?
b) When output is optimally allocated between the two
plants, calculate the firm’s short-run total, average, and
marginal cost curves. What is the marginal cost of the
100th widget? Of the 125th widget? The 200th widget?
c) How should the entrepreneur allocate widget production between the two plants in the long run? Find
the firm’s long-run total, average, and marginal cost
curves.
Shephard’s Lemma and Duality
W H AT I S S H E P H A R D ’ S L E M M A ?
Let’s compare our calculations in Learning-By-Doing Exercises 7.4 and 8.1. Both pertain to the production function Q ⫽ 50 1KL. Our input demand functions were
K *(Q, w, r) ⫽
Q w
50 A r
L*(Q, w, r) ⫽
Q
r
50 A w
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324
CHAPTER 8
C O S T C U RV E S
Our long-run total cost function was
TC(Q, w, r)
1wr
Q
25
How does the long-run total cost function vary with respect to the price of labor w,
holding Q and r fixed? The rate of change of long-run total cost with respect to the price of
labor is equal to the labor demand function:
0TC(Q, w, r)
Q
r
L *(Q, w, r)
0w
50 A w
(A8.1)
Similarly, the rate of change of long-run total cost with respect to the price of capital is equal
to the capital demand function:
0TC(Q, w, r)
Q
w
K *(Q, w, r)
0r
50 A r
Shephard’s Lemma
The relationship between
the long-run total cost
function and the input
demand functions: the rate
of change of the long-run
total cost function with
respect to an input price is
equal to the corresponding
input demand function.
(A8.2)
The relationships summarized in equations (A8.1) and (A8.2) are no coincidence. They reflect a general relationship between the long-run total cost function
and the input demand functions. This relationship is known as Shephard’s
Lemma, which states that the rate of change of the long-run total cost function with
respect to an input price is equal to the corresponding input demand function.23
Mathematically,
0TC(Q, w, r)
L *(Q, w, r)
0w
0TC(Q, w, r)
K *(Q, w, r)
0r
Shephard’s Lemma makes intuitive sense: If a firm experiences an increase in its
wage rate by $1 per hour, then its total costs should go up (approximately) by the $1
increase in wages multiplied by the amount of labor it is currently using; that is, the
rate of increase in total costs should be approximately equal to its labor demand function. We say “approximately” because if the firm minimizes its total costs, the increase
in w should cause the firm to decrease the quantity of labor and increase the quantity
of capital it uses. Shephard’s Lemma tells us that for small enough changes in w (i.e.,
w sufficiently close to 0), we can use the firm’s current usage of labor as a good approximation for how much a firm’s costs will rise.
23
Shephard’s Lemma also applies to the relationship between short-run total cost functions and the shortrun input demand functions. For that reason, we will generally not specify whether we are in the short
run or long run in the remainder of this section. However, to maintain a consistent notation, we will use
the “long-run” notation used in this chapter and Chapter 7.
A P P E N D I X : S H E P H A R D ’ S L E M M A A N D D UA L I T Y
D UA L I T Y
What is the significance of Shephard’s Lemma? It provides a key link between the
production function and the cost function, a link that in the appendix to Chapter 7 we
called duality. With respect to Shephard’s Lemma, duality works like this:
• Shephard’s Lemma tells us that if we know the total cost function, we can derive
the input demand functions.
• In turn, as we saw in the appendix to Chapter 7, if we know the input demand functions, we can infer properties of the production function from
which it was derived (and maybe even derive the equation of the production
function).
Thus, if we know the total cost function, we can always “characterize” the production function
from which it must have been derived. In this sense, the cost function is dual (i.e., linked)
to the production function. For any production function, there is a unique total cost
function that can be derived from it via the cost-minimization problem.
This is a valuable insight. Estimating a firm’s production function by statistical
methods is often difficult. For one thing, data on input prices and total costs are often
more readily available than data on the quantities of inputs. Researchers often take
advantage of Shephard’s Lemma in studies of economies of scale. They estimate cost
functions and then apply Shephard’s Lemma and the logic of duality to infer the
nature of returns to scale in the production function.
PROOF OF SHEPHARD’S LEMMA
For a fixed Q, let L0 and K0 be the cost-minimizing input combination for any arbitrary combination of input prices (w0, r0):
L0 ⫽ L *(Q, w0, r0)
K0 ⫽ K*(Q, w0, r0)
Now define a function of w and r, g(w, r):
g(w, r) TC(Q, w, r) ⫺ wL0 ⫺ rK 0
Since L0, K0 is the cost-minimizing input combination when w w0 and r r0, it
must be the case that
g(w0, r0) 0
(A8.3)
Moreover, since (L0, K 0) is a feasible (but possibly nonoptimal) input combination to
produce output Q at other input prices (w, r) besides (w0, r0), it must be the case that:
g(w, r) ⱕ 0
for
(w, r) ⫽ (w0, r0)
(A8.4)
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C O S T C U RV E S
Conditions (A8.3) and (A8.4) imply that the function g(w, r) attains its maximum when
w w0 and r r0. Hence, at these points, its partial derivatives with respect to w and
r must be zero:24
0g(w0, r0)
0w
0g(w0, r0)
0r
01
0TC(Q, w0, r0)
L0
0w
(A8.5)
01
0TC(Q, w0, r0)
K0
0r
(A8.6)
But since L0 L* (Q, w0, r0) and K0 K*(Q, w0, r0), (A8.5) and (A8.6) imply
0TC(Q, w0, r0)
L*(Q, w0, r0)
0w
(A8.7)
0TC(Q, w0, r0)
K *(Q, w0, r0)
0r
(A8.8)
Since (w0, r0) is an arbitrary combination of input prices, conditions (A8.7) and (A8.8)
hold for any pair of input prices, and this is exactly what we wanted to show to prove
Shephard’s Lemma.
24
For more on the use of partial derivatives to find the optimum of a function depending on more than
one variable, see the Mathematical Appendix in this book.
9
PERFECTLY
COMPETITIVE MARKETS
9.1
W H AT I S P E R F E C T C O M P E T I T I O N ?
APPLICATION 9.1
Perfectly Competitive Catfish
Farming
9.2
P R O F I T M A X I M I Z AT I O N
B Y A P R I C E - TA K I N G F I R M
APPLICATION 9.2
Wealth Creators and Wealth
Destroyers
9.3
H OW T H E M A R K E T P R I C E
I S D E T E R M I N E D : S H O RT- RU N
EQUILIBRIUM
APPLICATION 9.3
APPLICATION 9.4
Shutting Down an Oil Rig
How Much Corn at Which
Price?
APPLICATION 9.5
How Much Copper
at Which Price?
Growing Perfectly
Competitive Roses
APPLICATION 9.6
9.4
H OW T H E M A R K E T P R I C E
I S D E T E R M I N E D : L O N G - RU N
EQUILIBRIUM
APPLICATION 9.7
When the Supertanker
Market Sank
The U.S. Ethanol Industry
and the Price of Corn
APPLICATION 9.8
9.5
ECONOMIC RENT
AND PRODUCER SURPLUS
APPLICATION 9.9
Mining Copper for Profit
APPENDIX
P R O F I T M A X I M I Z AT I O N I M P L I E S
C O S T M I N I M I Z AT I O N
327
A Rose Is a Rose Is a Rose
Nevado Ecuador SA is a producer of fresh-cut roses located about 140 kilometers south of Quito, Ecuador.1
Ecuador’s warm days, cool nights, dry air, rich volcanic soil, and most of all, abundant and intense sunlight,
make it a near-perfect location for growing tall, bountiful roses. Perhaps not surprisingly, as the fresh-cut
rose market has globalized in the last two decades, the country of Ecuador has emerged as one of the
world’s leading suppliers of fresh-cut roses in the world. Of the nearly 1.5 billion roses bought annually by
U.S. households, nearly 400 million came from Ecuador, a quantity exceeded only by Colombia (which ships
about 900 million roses to the United States annually).
In an industry that has come under scrutiny from human rights activists who have called attention to
the use of child labor and the dangerous work conditions on rose plantations created by the heavy use of
pesticides, fungicides, and fertilizers, Nevado Equador stands out in sharp relief. The company—whose
slogan is “Roses with a conscience”—is known for its emphasis on environmental sustainability (e.g., it
forgoes the use of pesticides and uses organic rather than chemical fertilizers) and its humane treatment
of its workers (e.g., it provides educational loans and vocational training to its workers).
In the contemporary business world, the companies such as Starbucks and McDonald’s that are known
for their social responsibility are often quite large. And indeed, Nevado Ecuador is one the largest rose producers in Ecuador. Still, Nevado Ecuador is actually quite small in comparison to the overall size of the market.
Its 550 workers represent less than 10 percent of the workers employed in rose growing in Ecuador, and it is
but 1 of 400 or so rose growers operating in Ecuador. Since Ecuadorian rose growers compete with their
counterparts in Colombia, the United States, and other parts of the world, Ecuador Nevado is actually part
of a much larger pool of firms all producing fresh-cut roses. In the eyes of the typical consumer in, say, the
United States who purchases fresh-cut roses at his or her local flower shop, the specific grower is almost
certainly unknown and (notwithstanding Nevado Ecuador’s social responsibility) probably immaterial. In the
words of Gertrude Stein, from the perspective of the final consumer, “a rose is a rose is a rose.”
Given this reality, it is virtually certain that no single firm such as Nevado Ecuador can determine
the price of fresh cut roses on the world market. As a result, the key decision Nevado Ecuador faces is
1
This example draws from a number of sources: “Behind Roses’
Beauty, Poor and Ill Workers, New York Times (February 15, 2003),
http://www.nytimes.com/2003/02/13/us/behind-roses-beauty-poorand-ill-workers.html?scp=4&sq=roses⫹ecuador&st=nyt (accessed
December 18, 2009); Ross Wehner, “Deflowering Ecuador,” Mother
Jones ( January/February 2003), http://motherjones.com/politics/2002/
01/deflowering-ecuador (accessed December 18, 2008); “A Rose Is
[Not] a Rose,” Audubonmagazine.org ( January–February 2008),
http://www.audubonmagazine.org/(accessed December 18, 2009);
“Nevado Ecuador Launches Edible Culinary Rose,” Floriculture
International (December 8, 2009), http://www.floracultureinternational
.com/index.php?option=com_content&view=article&id=1531:nevadoecuador-launches-edible-culinary-roses&catid=52:business&Itemid=307
(accessed December 18, 2009); and the Nevado Ecuador company
website, http://www.nevadoecuador.com/en/index.html (accessed
December 18, 2009).
328
not what price to charge, but rather how many roses it should produce given the anticipated world
price for fresh-cut roses. That price is not determined by a single firm; rather it emerges out of the
interactions of hundreds of firms.
Nevado Ecuador is an example of a firm operating in a perfectly competitive market. A perfectly
competitive market consists of firms that produce identical products that sell at the same price. Each firm’s
volume of output is so small in comparison to overall market demand that no single firm has an impact on
the market price.
Perfect competition is worth studying for two reasons. First, a number of important real-world
markets—including most agricultural products, many minerals (e.g., copper and gold), metal fabrication,
commodity semiconductors, and oil tanker shipping—are like the fresh-cut rose industry: They consist of
many small firms, each producing nearly identical products, each with approximately equal access to the
resources needed to participate in the industry. The theory of perfect competition developed in this
chapter will help us understand the determination of prices and the dynamics of entry and exit in these
markets. Second, the theory of perfect competition forms an important foundation for the rest of microeconomics. Many of the key concepts that we develop in this chapter, such as the vital roles of marginal
revenue and marginal cost in output decisions, will apply when we study other market structures, such
as monopoly and oligopoly, in later chapters.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Describe the conditions that characterize a perfectly competitive market.
• Explain the difference between economic profit and accounting profit.
• Illustrate graphically the profit-maximization condition for a perfectly competitive firm.
• Derive a perfectly competitive firm’s short-run supply curve from the firm’s profit-maximization problem.
• Illustrate graphically an average nonsunk curve and explain how the presence of nonsunk fixed costs
affects a perfectly competitive firm’s short-run supply curve.
• Build up the short-run market supply curve from the short-run supply curves of individual firms.
• Perform comparative statics analysis of the short-run equilibrium in a perfectly competitive market.
• Indicate the difference between the short run and the long run.
• State the conditions for the long-run perfectly competitive equilibrium.
• Solve for the long-run equilibrium price, the equilibrium quantity demanded and supplied at the
market level, the quantity supplied by an individual firm in equilibrium, and the equilibrium number
of firms, given the market demand curve and the marginal and average cost curve for a typical firm.
• Show, using graphs, how the long-run market supply curve is determined in a constant-cost industry, an
increasing cost industry, and a decreasing cost industry.
• Explain what economic rent is and show graphically how it could arise in a perfectly competitive
equilibrium.
329
330
CHAPTER 9
P E R F E C T LY C O M P E T I T I V E M A R K E T S
• Define and compute producer surplus for a price-taking firm.
• Calculate producer surplus for the entire market in a short-run equilibrium and a long-run equilibrium.
• Explain the difference between economic profit, producer surplus, and economic rent.
9.1
W H AT I S
PERFECT
COMPETITION?
fragmented industry
An industry that consists of
many small buyers and
sellers; one of the characteristics of a perfectly
competitive industry.
undifferentiated
products Products that
consumers perceive as
being identical; one of the
characteristics of a perfectly
competitive industry.
perfect information
about prices Full
awareness by consumers of
the prices charged by all
sellers in the market; one of
the characteristics of a perfectly competitive industry.
equal access to
resources A condition
in which all firms—those
currently in the industry,
as well as prospective
entrants—have access to
the same technology and
inputs; one of the characteristics of a perfectly
competitive industry.
price taker A seller or a
buyer that takes the price
of the product as given
when making an output
decision (seller) or a purchase decision (buyer).
T
he market for fresh-cut roses is an example of a perfectly competitive market, and
Nevado Ecuador is an example of a perfectly competitive firm. But what is it, exactly,
that makes a market perfectly competitive? And what, if anything, is special about a
perfectly competitive firm?
Perfectly competitive markets have four characteristics:
1. The industry is fragmented. It consists of many buyers and sellers. Each
buyer’s purchases are so small that they have an imperceptible effect on market
price. Each seller’s output is so small in comparison to market demand that it
has an imperceptible impact on the market price. In addition, each seller’s input
purchases are so small that they have an imperceptible impact on input prices.
The market for fresh-cut roses is an excellent example of a fragmented market.
Even the largest producers, such as Nevado Ecuador, are very small in comparison to the overall scale of the market. Buyers that purchase fresh-cut roses from
the producers—wholesalers, brokers, and florists—are also small and numerous.
2. Firms produce undifferentiated products. That is, consumers perceive the
products to be identical no matter who produces them. When you buy fresh
roses from a local flower shop, it probably does not matter to you that they
were produced by Nevado Ecuador or one of its competitors. And because this
is true for you, it is also true for the flower shops and the wholesalers who buy
the roses directly from the growers. If the final consumer sees no difference in
the roses grown by the different growers, then florists and wholesalers don’t
care who they buy roses from either, as long as they get the best price. Roses
are thus an example of an undifferentiated product.
3. Consumers have perfect information about prices all sellers in the market
charge. This is certainly true in the rose market. The wholesalers and florists
that buy roses from the growers are keenly aware of the prevailing prices. These
consumers need to be deeply knowledgeable about prices because the price is
the main thing they care about when deciding which growers to buy roses from.
4. The industry is characterized by equal access to resources. All firms—those
currently in the industry, as well as prospective entrants—have access to the
same technology and inputs. Firms can hire inputs, such as labor, capital, and
materials, as they need them, and they can release them from their employment
when they do not need them. This characteristic is generally true of the freshcut rose industry: the technology for growing roses is well understood, and the
key inputs necessary to operate a rose-growing firm (land, greenhouses, rose
bushes, and labor) are readily available in well-functioning markets.
These characteristics have three implications for how perfectly competitive
markets work:
• The first characteristic—the market is fragmented—implies that sellers and
buyers act as price takers. That is, a firm takes the market price of the product
331
9 . 1 W H AT I S P E R F E C T C O M P E T I T I O N ?
as given when making an output decision, and a buyer takes the market price as
given when making purchase decisions. This characteristic also implies that a
firm takes input prices as fixed when making decisions about input quantities.2
• The second and third characteristics—firms produce undifferentiated products
and consumers have perfect information about prices—implies a law of one
price: Transactions between buyers and sellers occur at a single market price.
Because the products of all firms are perceived to be identical and the prices of
all sellers are known, a consumer will purchase at the lowest price available in
the market. No sales can be made at any higher price.
• The fourth characteristic—equal access to resources—implies that the industry
is characterized by free entry. That is, if it is profitable for new firms to enter
the industry, they will eventually do so. Free entry does not mean that a new
firm incurs no cost when it enters the industry, but rather that it has access to
the same technology and inputs that existing firms have.
In this chapter, we will develop a theory of perfect competition that includes each
of these three implications: price-taking behavior by firms, a common market price
charged by each firm in the industry, and free entry. To keep the development of this
theory manageable, we will organize our study of perfect competition in three steps:
law of one price In
a perfectly competitive
industry, the occurrence of
all transactions between
buyers and sellers at a single,
common market price.
free entry
Characteristic of an industry in which any potential
entrant has access to the
same technology and inputs
that existing firms have.
1. In the next section, we study profit maximization by a price-taking firm.
2. Then, we will study how the common market price is determined when the industry consists of a fixed number of firms (a number that is assumed to be large,
as in the case of the rose industry, which consists of hundreds of firms). This is
called the analysis of the short-run equilibrium of a perfectly competitive market.
3. Finally, we will study how the market price is affected by free entry. This is called
the analysis of the long-run equilibrium of a perfectly competitive market.
Once we have gone through all of these steps, we will have built a coherent theory
of perfect competition. In Chapter 10, we will then employ this theory to explore how
perfectly competitive markets facilitate the allocation of resources and the creation of
economic value.
A P P L I C A T I O N
9.1
Perfectly Competitive Catfish
Farming
Production of fresh-cut roses is a good example of a
perfectly competitive market. Another good example
is catfish farming. It may seem strange to characterize
production of catfish as farming rather than fishing,
but farming is a good description of the process.
Catfish are raised in ponds that range in size between
2
10 and 15 acres. Farmers harvest catfish by the use of
seine nets that capture the fish. The nets are then
hoisted by crane and placed on trucks with specially
designed hauling tanks.
Catfish farming is big business in the United
States. In 2005, U.S. catfish farmers had sales of over
$460 million and employed more than 10,000 people.
Catfish farming in the United States accounts for
more than one-third of the sales revenues from all
U.S. “aquacultural” products. The geographic locus
This is the assumption that we maintained throughout our analysis of input choices and cost functions
in Chapters 7 and 8.
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P E R F E C T LY C O M P E T I T I V E M A R K E T S
of the catfish farming industry is the Deep South,
with most catfish farms located in one of four states:
Mississippi, Louisiana, Alabama, and Arkansas.
Catfish farming satisfies all the conditions of perfect
competition:
• The industry is highly fragmented. In the state of
Arkansas alone, there are over 120 catfish farmers,
and in Mississippi there are over 350; in the United
States as a whole, there are over 1,000 catfish farms.3
• Catfish farmers produce what amounts to an
undifferentiated product. The catfish produced by
any one farm are, in the eyes of the ultimate consumers, a perfect substitute for the catfish
produced by any other farm.
• Catfish farmers sell their products to processing
plants or directly to retailers such as supermarkets or
restaurants. Sellers and buyers are well aware of prevailing prices, and information about prices is easy to
get. For example, the U.S. Department of Agriculture
publishes monthly reports on catfish prices.
• Finally, the technology of catfish farming is well understood and easily accessible, and the financial requirements to enter the industry are not onerous. For
example, state agricultural extension services
publish manuals to provide guidance to would-be
catfish farmers about how to set up catfish farms.
The minimum efficient scale of catfish farm is estimated to require about 80 to 100 acres of ponds.
The upfront investment in capital required to
build a catfish farm of this scale is estimated to be
9.2
PROFIT
MAXIMIZATION
BY A PRICETA K I N G F I R M
between $400,000 and $500,000. Though not trivial,
this is approximately what it would cost to purchase a medium-size home in a large metropolitan
area in the United States. Thus, the financial
requirements needed to set up a catfish farm are
potentially within reach of many individuals.4
Perfectly competitive markets are not easy businesses in
which to prosper. As we will see, when opportunities for
profit arise in such industries, entry of new participants
typically occurs. This has recently occurred in the catfish
farming industry. In recent years, catfish exports by both
China and Vietnam have surged. Between 2004 and
2008, Vietnam more than tripled its already substantial
exports of catfish.5 The U.S. catfish farming industry has
also been hurt by rising input prices. A key input in the
production of catfish is feed made from corn and
soybeans. As we documented in the introduction to
Chapter 2, corn prices in the United States have increased since 2006, driven to a significant degree by the
increased demand for corn from producers of ethanol.
The implication for the catfish industry has been an
increase in catfish feed prices by about 33 percent.
Perhaps not surprisingly, U.S. catfish producers
have attempted to cope with their travails through the
political system. In 2002, the U.S. Congress enacted
labeling regulations that require Vietnamese catfish
to be labeled under different names (e.g., pengasius). In
addition, as a result of an antidumping suit, imported
catfish from Vietnam have been subject to tariffs. We
will study the impact of government interventions on
competitive markets in Chapter 10.
We begin our analysis of perfect competition by studying decision making by a
price-taking firm that maximizes economic profit. To do this, though, we need to explore briefly what we mean by economic profit.
3
U.S. Department of Agriculture, Table 8, Census of Aquaculture, 2005, http://www.agcensus.usda.gov/
Publications/2002/Aquaculture/aquacen2005_08.pdf (accessed December 21, 2009).
4
“Catfish Farming in Kentucky,” Aquaculture Program, Kentucky State University, http://www
.ksuaquaculture.org/PDFs/Publications/Catfish.pdf (accessed December 21, 2009).
5
Anson, Adam, “The Changing Shape of U.S. Farm-Raised Catfish, TheFishSite.com, http://www
.thefishsite.com/articles/744/the-changing-shape-of-us-farmraised-catfish (accessed December 21, 2009).
9 . 2 P R O F I T M A X I M I Z AT I O N B Y A P R I C E - TA K I N G F I R M
333
E C O N O M I C P R O F I T V E R S U S AC C O U N T I N G P R O F I T
In Chapter 7, we distinguished between economic cost and accounting cost. Economic
cost measures the opportunity cost of the resources that the firm uses to produce and
sell its products, whereas accounting cost measures the historical expenses the firm incurred to produce and sell its output.
We will now make a similar distinction between economic profit and accounting profit:
economic profit ⫽ sales revenue ⫺ economic costs
accounting profit ⫽ sales revenue ⫺ accounting costs
That is, economic profit is the difference between a firm’s sales revenue and the totality of its economic costs, including all relevant opportunity costs. To illustrate, consider a small consulting firm operated by its owner. In 2010, the firm earned revenues
of $1 million and incurred expenses on supplies and hired labor of $850,000. The
owner’s best outside employment opportunity would have been to work for another
firm for $200,000 a year. The firm’s accounting profit is $1,000,000 $850,000 ⫽
$150,000. The firm’s economic profit deducts the opportunity cost of the owner’s
labor and is thus $1,000,000 $850,000 $200,000 ⫽ ⫺50,000. The fact that this
firm earns a negative economic profit of $50,000 means that the owner made $50,000
less in income by operating this business than he could have made by taking advantage of the best outside alternative. We might say that the business “destroyed”
$50,000 of the owner’s wealth: By operating his own business, the owner earned
$50,000 less income than he might have otherwise.
We use similar logic to account for the cost of the funds that a firm receives from
its owners in order to finance the acquisition of its capital assets (e.g., buildings, machines, and computers). To illustrate, let’s return to the example of our small consulting firm, but let’s modify the story. Suppose that the firm is owned by an investor
who is not involved in the day-to-day management of the firm (thus, we do not need
to worry about the opportunity cost of the owner’s time). The owner invested $2 million of her savings to finance the acquisition of the assets that were needed to start
the business (e.g., an office building, computers, telephones, fax machines, and so
forth). Suppose that the owner’s best alternative use of these funds would have been
to invest them in a portfolio of stocks and bonds yielding an annual return of 10 percent, or $200,000 per year. The owner invested her money in the consulting business
in the hope that the company’s annual accounting profit would be at least $200,000
per year. If the consulting firm delivers an accounting profit that is less than
$200,000, the firm will have a negative economic profit. Supposing (as before) that
the firm’s revenues are $1 million per year and its supply and labor expenses are
$850,000, the firm’s accounting profit is $150,000 per year, but its economic profit is
$1,000,000 ⫺ $850,000 ⫺ $200,000 ⫽ ⫺$50,000. This negative economic profit signals
that the business is not delivering financial returns commensurate with the returns that
the owner of the firm could have earned had she devoted her financial resources to their
best alternative use. By contrast, if the consulting firm’s accounting profit had exceeded
the minimum return of $200,000 demanded by the owner, the firm would have had a
positive economic profit, signaling that the business was delivering financial returns that
exceed those that the owner could have earned in her best alternative investment.
Whenever we discuss profit maximization, we are talking about economic profit
maximization. Economic profit is the appropriate objective for a firm that is acting on
its owners’ behalf, whether it be Nevado Ecuador, Coca-Cola, or Microsoft.
economic profit The
difference between a firm’s
sales revenue and the
totality of its economic
costs, including all relevant
opportunity costs.
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A P P L I C A T I O N
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9.2
Wealth Creators and Wealth
Destroyers
TABLE 9.1
Top Wealth Creators, 2009
Company
One widely used measure of economic profit is
economic value added (EVA), a term coined by the
financial consulting firm Stern Stewart. To compute
EVA, Stern Stewart starts with a company’s accounting
profit and deducts the minimum return on invested
capital demanded by the firm’s investors. A company
with a positive EVA has delivered a return on invested
capital that exceeds the minimum return demanded
by investors. A company with negative EVA, by contrast,
has failed to deliver the minimum return demanded by
investors. A firm that consistently delivers a positive
EVA over time creates wealth for its owners: The market
value of the firm, as reflected in its share price, will
exceed the investments made in the firm’s assets. By
contrast, a firm that consistently delivers a negative
EVA over time destroys the wealth of its owners: The
market value of the firm will be less than the investment
cost of its assets.
Stern Stewart regularly tracks EVA for a broad
range of firms around the world through its subsidiary,
EVA Dimensions. Tables 9.1 and 9.2 show EVA for a few
well-known U.S. firms in September 2009.6 ExxonMobil
had a positive economic profit of more than $24.4 billion over the prior 12 months. Microsoft had an economic profit of $11.6 billion and, in fact, has recorded a
positive economic profit every year since 1986.
By contrast, some well-known firms had negative
economic profit. For example, Ford Motor had a negative economic profit of over $10.4 billion. Its returns
clearly did not measure up to the minimum return
demanded by its owners, and as a result, it destroyed
economic value added
A widely used measure of
economic profit, equal to
the company’s accounting
profit minus the minimum
return on invested capital
demanded by the firm’s
investors.
EVA (millions)
ExxonMobil
Microsoft
Chevron
Altria Group
Wal-Mart
General Electric
Johnson & Johnson
Merck
Philip Morris
Apple
TABLE 9.2
24,444
11,589
10,419
9,094
7,650
7,544
5,395
5,294
4,791
4,612
Top Wealth Destroyers, 2009
Company
Morgan Stanley
JDS Uniphase
Freeport-McMoran
Sprint Nextel
Verizon
Devon Energy
Ford Motor
AT&T
Citigroup
AIG
EVA (millions)
⫺5,746
⫺6,781
⫺7,030
⫺7,114
⫺7,744
⫺8,370
⫺10,474
⫺11,386
⫺27,787
⫺50,066
shareholder wealth. Many of the worst-performing
firms in 2009 were banks or insurance companies
(with AIG—insurance company American International
Group—at the bottom) because of the financial crisis
in 2008–2009.
T H E P R O F I T- M A X I M I Z I N G O U T P U T C H O I C E
F O R A P R I C E - TA K I N G F I R M
We can now study the problem of a price-taking firm that seeks to maximize its economic
profit. Assuming that the firm produces and sells a quantity of output Q, its economic
profit (denoted by 7) is ⫽ TR(Q) ⫺ TC(Q), where TR(Q) is the total revenue derived
6
We thank Bennett Stewart and Ling Yang of Stern Stewart subsidiary EVA Dimensions for providing
the data.
7
Economists commonly use the Greek letter to denote profit. In this book, does not refer to the number 3.14 used in geometry.
9 . 2 P R O F I T M A X I M I Z AT I O N B Y A P R I C E - TA K I N G F I R M
TABLE 9.3
Total Revenue, Cost, and Profit for a Price-taking Rose Producer
(thousands of
roses per month)
TR (Q)
(thousands of
$ per month)
TC (Q)
(thousands of
$ per month)
(thousands of
$ per month)
0
60
120
180
240
300
360
420
0
60
120
180
240
300
360
420
0
95
140
155
170
210
300
460
0
35
20
25
70
90
60
40
Q
from selling the quantity Q and TC(Q) is the total economic cost of producing the quantity Q. Total revenue equals the market price P multiplied by the quantity of output Q
produced by the firm: TR(Q) ⫽ P ⫻ Q. Total cost TC(Q) is the total cost curve discussed
in Chapter 8; it tells us the total cost of producing Q units of output.
Because the firm is a price taker, it perceives that its volume decision has a negligible impact on market price. Thus, it takes the market price P as given. Its goal is to
choose a quantity of output Q to maximize its total profit.
To illustrate the firm’s problem, suppose that a rose grower anticipates that the
market price for fresh-cut roses will be P ⫽ $1.00 per rose. Table 9.3 shows total revenue,
total cost, and profit for various output levels, and Figure 9.1(a) graphs these numbers.
Figure 9.1(a) shows that profit is maximized at Q ⫽ 300 (i.e., 300,000 roses per
month). It also shows that the graph of total revenue is a straight line with a slope of 1.
Thus, as we increase Q, the firm’s total revenue goes up at a constant rate equal to the
market price, $1.00.
For any firm (price taker or not), the rate at which total revenue changes with
respect to a change in output is called marginal revenue (MR). It is defined by
⌬TR/⌬Q. For a price-taking firm, each additional unit sold increases total revenue by
an amount equal to the market price—that is, ⌬TR/⌬Q ⫽ P. Thus, for a price-taking
firm, marginal revenue is equal to the market price, or MR ⫽ P.
As we learned in Chapter 8, marginal cost (MC), the rate at which cost changes with
respect to a change in output, can be defined similarly to marginal revenue: MC ⫽
⌬TC/⌬Q. Figure 9.1 shows that for quantities between Q ⫽ 60 and the profit-maximizing
quantity Q ⫽ 300, producing more roses increases profit. Increasing the quantity in this
range increases total revenue faster than total cost: ⌬TR/⌬Q ⬎ ⌬TC/⌬Q, or P ⬎ MC.
When P ⬎ MC, each time the rose producer increases its output by one rose, its profit
goes up by P ⫺ MC, the difference between the marginal revenue and the marginal cost
of that extra rose.
Figure 9.1 shows that for quantities greater than Q ⫽ 300, producing fewer roses
increases profit. Decreasing quantity in this range decreases total cost faster than it
decreases total revenue—that is, marginal revenue is less than marginal cost, or P ⬍ MC.
When P ⬍ MC, each time the producer reduces its output by one rose, its profit goes
up by MC ⫺ P, the difference between the marginal cost and the marginal revenue of
that extra rose.8
8
335
Or, equivalently, each extra rose produced decreases profit by P ⫺ MC.
marginal revenue The
rate at which total revenue
changes with respect to
output.
CHAPTER 9
Total revenue, total cost, and total profit
(thousands of dollars per month)
336
P E R F E C T LY C O M P E T I T I V E M A R K E T S
TR
$300
TC
210
90
Total profit π
60
300
Quantity (thousands of roses per month)
(a)
FIGURE 9.1
Profit Maximization by a Price-Taking Firm
Panel (a) shows that the firm’s
profit is maximized when
Q ⫽ 300,000 roses per year.
Panel (b) shows that at this point
marginal cost is MC ⫽ P. Marginal cost also equals price when
Q ⫽ 60,000 roses per year, but
this point is a profit minimum.
Price (dollars per rose)
MC
$1
MR = P
0
60
(b)
300
Quantity (thousands of roses per month)
If the producer can increase its profit when either P ⬎ MC or P ⬍ MC, quantities at which these inequalities hold cannot maximize its profit. It must be the case,
then, that at the profit-maximizing output,
P ⫽ MC
(9.1)
Equation (9.1) tells us that a price-taking firm maximizes its profit when it produces a
quantity Q* at which the marginal cost equals the market price.
Figure 9.1(b) illustrates this condition. The rose grower’s marginal revenue curve
is a horizontal line at the market price of $1.00. The profit-maximizing quantity
occurs at Q ⫽ 300, where this MR curve intersects the MC curve. This tells us that when
the rose grower faces a market price of $1.00 per fresh-cut rose, its profit-maximizing
decision is to produce and sell 300,000 fresh-cut roses per month.
Figure 9.1(b) also illustrates that there is another quantity, Q ⫽ 60, at which
MR ⫽ MC. The difference between Q ⫽ 60 and Q ⫽ 300 is that at Q ⫽ 300, the marginal cost curve is rising, while at Q ⫽ 60 the marginal cost curve is falling. Is Q ⫽ 60
also a profit-maximizing quantity? The answer is no. Figure 9.1(a) shows us that Q ⫽ 60
represents the point at which profit is minimized rather than maximized. This shows
that there are two profit-maximization conditions for a price-taking firm:
• P ⫽ MC.
• MC must be increasing.
337
9 . 3 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : S H O RT- RU N E Q U I L I B R I U M
If either of these conditions does not hold, the firm cannot be maximizing its profit.
It would be able to increase profit by either increasing or decreasing its output.
T
he previous section showed that a price-taking firm such as Nevado Ecuador would
maximize its profit by producing an output level at which the market price equals marginal cost. But how does the market price get determined in the first place? In this
section, we study how the market price is determined in the short run. The short run is
the period of time in which (1) the number of firms in the industry is fixed and (2) at
least one input, such as the plant size (i.e., quantity of capital or land) of each firm, is
fixed. For example, in the market for fresh-cut roses, short-run swings in the market
price from one month to the next are determined by the interaction of a fixed number of firms (several hundred very small firms), each of which operates with a fixed
amount of land, a fixed quantity of greenhouses, and a fixed quantity of rose bushes.
With land, greenhouses, and rose plants fixed, rose producers control their output
through pinching and pruning decisions, as well as through the amounts of fertilizer
and pesticide they apply to the rose plants. These decisions determine how many
fresh-cut rose stems will be available to meet demand throughout the year.
We will see that the profit-maximizing output decisions of individual producers
such as Nevado Ecuador will give rise to short-run supply curves for these firms. If we
then add together the short-run supply curves for all of the producers currently in the
industry, we will obtain a market supply curve. The market price is then determined
by the interaction of this market supply curve and the market demand curve.
9.3
HOW THE
MARKET
PRICE IS
DETERMINED:
S H O R T- R U N
EQUILIBRIUM
T H E P R I C E - TA K I N G F I R M ’ S S H O R T- R U N
COST STRUCTURE
Our goal in the next several sections is to learn how to construct an individual firm’s
short-run supply curve. To do this, we need to explore the cost structure of a typical
firm in the industry.
The firm’s short-run total cost of producing a quantity of output Q is
STC(Q) ⫽ e
SFC ⫹ NSFC ⫹ TVC(Q),
SFC, when Q ⫽ 0
when Q 7 0
This equation identifies three categories of costs for this firm.
• TVC(Q) represents total variable costs. These are output-sensitive costs—that is,
they go up or down as the firm increases or decreases its output. Total variable
costs include materials costs and the costs of certain kinds of labor (e.g., factory
labor). Total variable costs are zero if the firm produces zero output and thus
are examples of nonsunk costs. If a rose producer decided to shut down its rose
growing operations, it would avoid the need to spend money on fertilizer and
pesticide. These costs would thus be nonsunk.
• SFC represents the firm’s sunk fixed costs. A sunk fixed cost is a fixed cost that
a firm cannot avoid if it temporarily suspends operations and produces zero output. For this reason, sunk fixed costs are often also called unavoidable costs. For
example, suppose that a rose grower has signed a long-term lease (e.g., for five
sunk fixed cost A fixed
cost that the firm cannot
avoid if it shuts down and
produces zero output.
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P E R F E C T LY C O M P E T I T I V E M A R K E T S
9.3
Shutting Down an Oil Rig9
Whether a fixed cost is sunk or nonsunk (i.e., not
avoidable or avoidable) often depends on how long
the firm anticipates it will suspend operations and
produce zero output. To illustrate, consider the offshore
oil drilling business, which consists of numerous independent contractors who are hired by large petroleum
companies to drill for oil in the open seas. These contractors operate offshore oil rigs, large platforms that
are transported out to sea and used to drill for oil.
Generally speaking, a given offshore oil platform
is allocated a fixed number of wells that may operate
at a time. The company cannot drill a new well if all
allocated wells are producing at an economic rate (as
determined by the government). Once a well begins
to operate below that rate, the oil company can drill
a new well for the platform, if it has the resources and
decides that it would be profitable to do so.
A rig consists of a crew of managers (e.g., the rig
boss), engineers, marine personnel, and workers who
conduct the drilling operations and maintain the rig
(e.g., drillers, crane operators, mechanics, and electricians). From the perspective of a drilling contractor,
the quantity of output can be measured by the number of wells drilled within a particular period of time.
The most significant variable costs of operating the
rig include drilling supplies, such as drill bits, and fuel.
A rig’s fixed costs include maintenance, food, medical
care, insurance, and the wages of its crew. The crew
costs are fixed because a contractor typically commits
to hiring a crew for a particular period of time, and
thus its labor cost does not vary with the number of
wells drilled within that time period.
There are three ways that a contractor can idle its
rig and produce zero output:
Hot Stacking: A “hot-stacked” rig is taken out of
service temporarily (perhaps for a few weeks),
but remains fully staffed and ready on short
notice to begin drilling again. By hot stacking a
rig, the contractor avoids its variable costs, but
all other costs continue to be incurred. When a
rig is hot-stacked, all fixed costs are sunk.
Warm Stacking: A “warm-stacked” rig is taken
out of service temporarily, but typically for a
9
longer period of time than a hot-stacked rig
(perhaps for a few months). By warm stacking a
rig, the contractor avoids all of the costs that are
avoided by a hot-stacked rig, and it also avoids
some maintenance expenses and some labor
costs (since some workers may be laid off). When
a rig is warm-stacked, some fixed costs are sunk,
while others are nonsunk.
Cold Stacking: A “cold-stacked” rig is taken out
of service for a significant period of time. The
rig’s crew is laid off, and its doors are welded
shut. When a rig is cold-stacked, all fixed costs
are avoided except for insurance. Insurance
would thus be a sunk fixed cost, while all other
fixed costs (maintenance, food, medical supplies,
and crew costs) would be nonsunk.
Consider a typical oil platform in the Gulf of
Mexico, which has eight well slots (the identity of the
rig and company are confidential). For most of 2008 and
2009 all eight wells were producing at economic rates,
and this was not anticipated to change in the near future.
The recession added more uncertainty, since the price of
oil was expected to be lower than it otherwise would
be. For these reasons, no drilling for new wells was
anticipated for the foreseeable future, and so the rig
associated with that platform was cold-stacked. The
crew that had occupied the rig was sent to another platform to drill. (Note that the company therefore did not
need additional employees for the other platform, so
labor costs for the rig were nonsunk in this case.) If the
company anticipated that one or more slots would soon
need to be drilled, it would have the contractor warmstack or hot-stack the rig, depending on how soon it
expected that the drilling would need to commence.
Oil rigs are quite expensive. For example, in 2009
the cost of operating a rig was approximately $250,000
per day. When hot-stacked, the rig costs about $150,000
per day, while it costs about $40,000 per day if warmstacked. Thus there are substantial nonsunk fixed costs
even in the short run.
In thinking through which fixed costs are sunk (unavoidable) and which are nonsunk (avoidable), keep in
mind how temporary the firm’s shutdown decision is. The
longer the firm plans to produce zero output, the larger
will be the proportion of fixed costs that are avoidable.
We thank Jason Sheridan for sharing his expertise with offshore oil rigs in preparing this application.
This application also relies on information presented in K. Corts, “The Offshore Oil Drilling Industry,”
Harvard Business School Case 9-799-11.
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9 . 3 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : S H O RT- RU N E Q U I L I B R I U M
years) to rent land on which to grow roses and that the lease prevents it from
subletting the land to anyone else. The lease cost is fixed because it does not
vary with the quantity of roses that the firm produces. It is output insensitive. It
is also sunk because the firm cannot avoid the rental payments, even by producing zero output.10
• NSFC represents the firm’s nonsunk fixed costs. A nonsunk fixed cost is a fixed
cost that must be incurred if the firm is to produce any output, but it does not
have to be incurred if the firm produces no output. Nonsunk fixed costs, as well
as variable costs, are also often called avoidable costs. For a rose grower, an example
of a nonsunk fixed cost would be the cost of heating the greenhouses. Because
greenhouses must be maintained at a constant temperature whether the firm
grows 10 or 10,000 roses within the greenhouses, so the cost of heating the
greenhouses is fixed (i.e., it is insensitive to the number of rose stems produced).
But the heating costs are nonsunk because they can be avoided if the grower
chooses to produce no roses in the greenhouses.
nonsunk fixed cost
A fixed cost that must be
incurred for a firm to
produce any output but
that does not have to be
incurred if the firm produces
no output.
The firm’s total fixed (or output-insensitive) cost, TFC, is thus given by TFC ⫽
NSFC ⫹ SFC. If NSFC ⫽ 0, there are no fixed costs that are nonsunk. In that case,
TFC ⫽ SFC. This is the case that we consider in the next section.
S H O R T- R U N S U P P LY C U RV E F O R A P R I C E - TA K I N G
FIRM WHEN ALL FIXED COSTS ARE SUNK
In this section, we derive the supply curve for a price-taking firm in the easiest case,
when all fixed costs are sunk—that is, NFSC ⫽ 0 and thus TFC ⫽ SFC. Figure 9.2 depicts the short-run marginal cost curve, SMC, short-run average cost curve, SAC, and
average variable cost curve, AVC, for such a firm in the fresh-cut rose industry.
Consider three possible market prices for fresh-cut roses: $0.25 per rose, $0.30
per rose, and $0.35 per rose. If we apply the P ⫽ MC profit-maximization condition
from the previous section, the firm’s profit-maximizing output level when the price is
$0.25 is 50,000 roses per month (point A in Figure 9.2). Similarly, when the market
price is $0.30 and $0.35 per rose, the profit-maximizing output levels are 55,000 and
60,000 roses per month (points B and C, respectively). Each of these quantities represents a point at which the firm’s short-run marginal cost SMC equals the relevant market price P, or P ⫽ SMC.
The firm’s short-run supply curve tells us how its profit-maximizing output
decision changes as the market price changes. Graphically, for the prices $0.25, $0.30,
and $0.35, the firm’s short-run supply curve coincides with the short-run marginal
cost curve SMC. Thus, points A, B, and C are all on the firm’s short-run supply curve.
However, the firm’s short-run marginal cost curve and the firm’s short-run supply curve do not necessarily coincide at all possible prices. To see why, suppose the
price of roses is $0.05. To maximize its profits at this price, the firm would produce at
the point at which price equals marginal cost, an output of 25,000 roses per month.
But at this price, the firm would earn a loss: It would incur its total fixed cost TFC,
and, on top of that, it would lose the difference between the price of $0.05 and the average variable cost, AVC25, on each of the 25,000 roses it produces. That is, the firm’s
total loss would be TFC plus 25,000(AVC25 ⫺ 0.05) (the shaded region in Figure 9.2).
10
Of course, the firm eventually avoids having to make payments on the lease, but not because it decides
to shut down its operations today. Rather, the lease payments will go away once the five-year term of the
lease expires.
short-run supply curve
The supply curve that shows
how the firm’s profitmaximizing output decision
changes as the market price
changes, assuming that the
firm cannot adjust all of its
inputs (e.g., quantity of
capital or land).
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AVC
SMC
$0.35
C
B
0.30
Price (dollars per rose)
SAC
A
0.25
SAC40
0.18
AVC25
AVC40
0.10
PS = minimum AVC
0.05
0
25
33
40 50 55 60
Quantity (thousands of roses per month)
FIGURE 9.2
Short-Run Supply Curve for a Price-Taking Firm Whose Fixed Costs Are
All Sunk
The firm’s short-run supply curve is the portion of its short-run marginal cost (SMC ) above
the minimum level of average variable cost, denoted by PS. This is the firm’s shutdown
price. For prices below the shutdown price, the firm supplies zero output, and its supply
curve is a vertical line coinciding with the vertical axis.
If the firm did not produce, its loss would only be its (sunk) total fixed cost TFC. At a
price of $0.05, then, the firm cuts its loss by not producing.
More generally, the firm is better off cutting its losses by temporarily shutting
down if the market price P is less than the average variable cost AVC(Q*) at the output level Q* at which P equals short-run marginal cost, or P ⬍ AVC(Q*).
We can now draw the firm’s short-run supply curve. We have seen that
• A profit-maximizing price-taking firm, if it produces positive output, produces
where P ⫽ SMC and SMC slopes upward.
• A profit-maximizing price-taking firm never produces where P ⬍ AVC.
Thus, the firm would never produce on the portion of the SMC curve where SMC ⬍
AVC. This is the portion below the minimum level of the AVC curve. It then follows
that if price is below the minimum level of AVC, the firm will produce Q ⫽ 0.
In light of this, the firm’s supply curve has two parts:
shutdown price The
price below which a firm
supplies zero output in the
short run.
• If the market price is less than the minimum level of AVC—a level we denote by
PS in Figure 9.2—the firm will supply zero output (i.e., Q ⫽ 0). In Figure 9.2,
PS is $0.10 per rose. As Figure 9.2 shows, this portion of the firm’s supply curve
is a vertical “spike” that coincides with the vertical axis. We call PS the firm’s
shutdown price, the price below which it produces a quantity of zero in the
short run.
9 . 3 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : S H O RT- RU N E Q U I L I B R I U M
341
• If the market price is greater than PS, the firm will produce a positive amount of
output, and its short-run supply curve will coincide with its short-run marginal
cost curve. (If the market price is equal to PS, the firm is indifferent between
shutting down and producing 33,000 roses. In either case, it incurs a loss equal
to its sunk fixed costs.)
This analysis implies that perfectly competitive firms might operate during periods
in which they earn negative economic profit. For example, Figure 9.2 shows that when
the price is $0.18 per rose, the firm produces 40,000 roses per month. It earns a loss because at this level of output, the price $0.18 is less than the short-run average cost corresponding to 40,000 roses per month, SAC40. However, because the price of $0.18 exceeds
the average variable cost at 40,000 roses per month, AVC40, the firm’s total revenue
exceeds its total variable cost. Thus, by continuing to produce, the firm offsets some of
the loss it would incur if it produced nothing. Of course, if the rose grower expects the
price of $0.18 per rose to persist, then given enough time, it would reduce its plant size
(i.e., devote less land to growing roses), or it might even exit the industry altogether.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 9 . 1
D
Deriving the Short-Run Supply Curve for a Price-Taking Firm
Suppose that a firm has a short-run total
cost curve given by STC 100 ⫹ 20Q ⫹ Q2, where the
total fixed cost is 100 and the total variable cost is 20Q ⫹
Q2. The corresponding short-run marginal cost curve is
SMC ⫽ 20 ⫹ 2Q. All of the fixed cost is sunk.
Problem
(a) What is the equation for average variable cost (AVC)?
(b) What is the minimum level of average variable cost?
(c) What is the firm’s short-run supply curve?
Solution
(a) As we saw in Chapter 8, average variable cost is total
variable cost divided by output. Thus, AVC ⫽ (20Q ⫹ Q2)/
Q ⫽ 20 ⫹ Q.
equal—in this case, where 20 ⫹ Q ⫽ 20 ⫹ 2Q, or Q ⫽ 0.
If we substitute Q ⫽ 0 into the equation of the AVC
curve 20 ⫹ Q, we find that the minimum level of AVC
equals 20.
(c) For prices below 20 (the minimum level of average
variable cost), the firm will not produce. For prices
above 20, we can find the supply curve by equating price
to marginal cost and solving for Q: P ⫽ 20 ⫹ 2Q, or Q ⫽
⫺10 ⫹ P/2. The firm’s short-run supply curve, which we
denote by s(P ), is thus:
s (P) ⫽ •
0, when P 6 20
⫺10 ⫹
Similar Problems:
1
P, when P 20
2
9.8, 9.9, 9.10
(b) We know that the minimum level of average variable
cost occurs at the point at which AVC and SMC are
S H O R T- R U N S U P P LY C U RV E F O R A P R I C E - TA K I N G
FIRM WHEN SOME FIXED COSTS ARE SUNK AND
SOME ARE NONSUNK
Let’s now consider the possibility that the firm has some nonsunk fixed costs. That is,
TFC SFC ⫹ NSFC, where NSFC ⬎ 0. As before, the firm maximizes its profit by
equating price to marginal cost. However, the rule that defines when the firm produces
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SMC
SAC
ANSC
Price (dollars per rose)
AVC
ANSC35
PS = minimum ANSC
$0.15
0
35
Quantity (thousands of roses per month)
FIGURE 9.3
Short-Run Supply Curve for a Firm with Some Nonsunk Fixed Costs
The shutdown price PS is the minimum level of average nonsunk cost. The firm’s supply curve
coincides with the short-run marginal cost curve SMC for prices above PS. For prices below PS, it
is a vertical spike that coincides with the vertical axis.
average nonsunk cost
The sum of average variable
cost and average nonsunk
fixed cost.
zero, as opposed to positive, output is different from the case discussed in the previous
section.
To show why, we first need to define a new cost curve. The firm’s average nonsunk
cost, ANSC, is equal to the sum of its average variable cost and its average nonsunk
fixed cost: ANSC ⫽ AVC ⫹ NSFC/Q.
Figure 9.3 shows that the average nonsunk cost curve is U-shaped and lies between the short-run average cost curve SAC and the average variable cost curve AVC.
At its minimum point, SMC ⫽ ANSC. In this sense, the ANSC curve behaves much
like the SAC curve.
To illustrate how we modify the price-taking firm’s shutdown rule when it has
nonsunk fixed costs, suppose, as shown in Figure 9.3, that the price of roses is $0.15.
If the firm maximized its profits at this price, it would produce at the point at which
price equals marginal cost, an output of 35,000 roses per month. But at this price,
the firm would earn a loss: it would incur its sunk fixed cost SFC, and, on top of that,
for every rose it produced, it would lose the difference between the price of $0.15
and its average nonsunk costs, ANSC35. By contrast, if the firm did not produce, its
loss would only be its sunk fixed cost SFC. That is, by temporarily shutting down,
the firm would avoid both its variable costs and its nonsunk fixed costs. At a price
of $0.15, then, the firm cuts its loss by not producing. By doing so, it avoids an additional loss of 35,000 (ANSC35 ⫺ $0.15) (represented by the shaded region in
Figure 9.3).
More generally, the firm is better off cutting its short-run losses by not producing if the market price P is less than the average nonsunk cost ANSC(Q* ) at the output Q* at which P equals short-run marginal cost, P ⬍ ANSC(Q*).
9 . 3 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : S H O RT- RU N E Q U I L I B R I U M
343
We can now draw the firm’s short-run supply curve. We have seen that
• A profit-maximizing price-taking firm, if it produces positive output, produces
where P ⫽ SMC and SMC slopes upward.
• A profit-maximizing price-taking firm with nonsunk fixed costs would never
produce where P ⬍ ANSC.
Thus, the firm would never produce on the portion of the SMC curve where
SMC ⬍ ANSC. This is the portion below the minimum level of the ANSC curve. It
then follows that if price is below the minimum level of ANSC—denoted by PS in
Figure 9.3—the firm will produce Q ⫽ 0.
Figure 9.3 shows the short-run supply curve for a rose-growing firm when there
are nonsunk fixed costs. It is a vertical spike for prices below the minimum level of
average nonsunk cost, and it coincides with the short-run marginal cost curve for
prices above this level.
The concept of average nonsunk cost is sufficiently flexible that we can identify
the firm’s supply curve and shutdown price for three special cases:
• All fixed costs are sunk. This is the case we studied in the previous section. When
all fixed costs are sunk, ANSC ⫽ AVC, and our shutdown rule, P ⬍ ANSC, becomes P ⬍ AVC. The firm’s short-run supply curve is thus the portion of SMC
above the minimum point of the average variable cost curve.
• All fixed costs are nonsunk. In this case, ANSC ⫽ SAC.11 Our shutdown rule,
P ⬍ ANSC, now becomes P ⬍ SAC. When all fixed costs are nonsunk, the
firm’s short-run supply curve is the portion of SMC above the minimum point
of the short-run average cost curve.
• Some fixed costs are sunk and some are nonsunk. This is the case we studied in this
section. As we have seen, the firm’s short-run supply curve is the portion of
SMC above the minimum point of the average nonsunk cost curve. As Figure 9.3
shows, the shutdown price PS when some, but not all, fixed costs are sunk is
above the minimum level of AVC but below the minimum level of SAC.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 9 . 2
D
Deriving the Short-Run Supply Curve for a Price-Taking Firm
with Some Nonsunk Fixed Costs
As in Learning-By-Doing Exercise 9.1, suppose that a
firm’s short-run total cost curve is STC ⫽ 100 ⫹ 20Q ⫹
Q2. The corresponding short-run marginal cost curve is
SMC ⫽ 20 ⫹ 2Q.
Problem
(a) Suppose that SFC ⫽ 36, while NSFC ⫽ 64. What is
the firm’s average nonsunk cost curve?
(b) What is the minimum level of average nonsunk cost?
(c) What is the firm’s short-run supply curve?
Solution
(a) The average nonsunk cost curve is ANSC ⫽ AVC ⫹
NSFC/Q ⫽ 20 ⫹ Q ⫹ 64/Q.
(b) As Figure 9.4 shows, the average nonsunk cost curve
ANSC reaches its minimum when average nonsunk cost
equals short-run marginal cost: 20 ⫹ 2Q ⫽ 20 ⫹
Q ⫹ 64/Q. Solving for Q, we find that Q ⫽ 8. Thus, the
average nonsunk cost curve attains its minimum value at
Q ⫽ 8. Substituting Q ⫽ 8 back into the equation for the
average nonsunk cost curve will tell us the minimum level
of average nonsunk cost: ANSC ⫽ 20 ⫹ 8 ⫹ 64/8 ⫽ 36.
11
This is because SFC ⫽ 0, and thus TNSC ⫽ TVC ⫹ TFC. As a result ANSC ⫽ (TVC ⫹ TFC )/Q, which
equals SAC.
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Price (dollars per unit)
SMC
SAC
ANSC
AVC
$40
36
20
0
8
Quantity (units per year)
FIGURE 9.4
Short-Run Supply Curve for a Price-Taking Firm with Some Nonsunk
Fixed Costs
The firm’s shutdown price is the minimum level of average nonsunk cost, or $36. The firm’s
supply curve coincides with the short-run marginal cost curve SMC for prices above $36,
and it is a vertical spike for prices below $36. For prices between $36 and $40, the firm
produces but earns negative economic profit.
Thus, as Figure 9.4 shows, the minimum level of average nonsunk cost is $36 per unit.
(c) As Figure 9.4 shows, for prices below the minimum
level of ANSC (i.e., for P ⬍ 36), the firm does not produce.
For prices above this level, the firm’s profit-maximizing
quantity is given by equating price to marginal cost—
that is, P ⫽ 20 2Q, or Q ⫽ ⫺10 ⫹ P/2. The firm’s
short-run supply curve s(P ) is thus:
s (P) ⫽ •
when P 6 36
1
⫺10 ⫹ P, when P ⱖ 36
2
0,
When the market price is between 36 and 40, the
firm will continue to produce in the short run, even
though its economic profit is negative. Its losses from
operating will be less than its losses if it shuts down.
Similar Problems:
A P P L I C A T I O N
9.4
How Much Corn at Which Price?12
Agricultural markets are often cited as the classic
example of perfect competition. An individual farmer’s
output of a product, such as corn, soybeans, or cotton,
is small in comparison to the overall market for such
products. Therefore, it is reasonable to view an individual farm as a price taker in the markets in which it
participates.
12
9.11, 9.12, 9.13
Figure 9.5 illustrates a supply curve for a typical
Iowa corn farmer. The figure shows the farmer’s
short-run marginal cost curve, as well as its short-run
average cost curve and its average variable cost curve.
Economist Daniel Suits constructed these cost curves
based on data collected by the U.S. Department of
Agriculture.13
If we assume that all fixed costs are sunk, the
farmer would not supply corn at prices below the
This example draws from D. B. Suits, “Agriculture,” Chapter 1 in The Structure of American Industry,
9th ed., in W. Adams and J. W. Brock, eds. (Englewood Cliffs, NJ: Prentice Hall, 1995).
13
Updated to 1991 dollars.
9 . 3 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : S H O RT- RU N E Q U I L I B R I U M
345
SMC
Price (dollars per bushel)
$3.50
SAC
2.56
AVC
Farmer’s loss
when production = 46,000 bushels
FIGURE 9.5
1.75
1.36
0
36
46 50 52
Quantity (thousands of bushels per year)
minimum level of average variable cost. In Figure 9.5,
the minimum level of average variable cost occurs at
about $1.36 per bushel. Thus, at prices below $1.36,
the farmer’s supply curve is a vertical spike. For prices
above $1.36 per bushel, the supply curve coincides
with the short-run marginal cost curve. This curve rises
rapidly. For example, at an output of 52,000 bushels,
short-run marginal cost is about $3.50 per bushel. At
this output, the farm is close to the effective capacity
of its land, and the incremental cost of additional
bushels of corn is very high.
When the price of corn is greater than $1.36, the
farm may produce even though economic profit
Supply Curve
for a Typical Iowa Corn Farmer
in 1991
Short-run marginal cost (SMC )
is constant at $1.36 until output
of about 36,000 bushels and
increases sharply thereafter. The
farmer’s supply curve coincides
with the short-run marginal cost
curve for prices above $1.36,
and is a vertical spike for prices
below $1.36.
might be negative. For example, at a price of $1.75,
the profit-maximizing output for the farm would be
46,000 bushels. The difference between price and
average cost at this point is about $0.81, so the farm
would lose about $37,260 for the year by producing
corn at this price (represented by the shaded region
in Figure 9.5). Nevertheless, the farmer is better off
producing 46,000 bushels of corn than producing
nothing. If the farm produced nothing, it would earn
a loss equal to its annual fixed cost of about $47,250.
The farm cuts its annual loss by $9,990 by producing
the profit-maximizing quantity rather than shutting
down.
S H O R T- R U N M A R K E T S U P P LY C U RV E
Having derived the short-run supply curve for an individual price-taking firm, let’s now
see how to go from the firm’s supply curve to the supply curve for the entire industry.
Because the number of producers in the industry is fixed in the short run, market
supply at any price is equal to the sum of the quantities that each established firm supplies at that price. To illustrate, suppose that the market for fresh-cut roses consists of
the two types of firms illustrated in Figure 9.6(a): 100 firms of type 1, each with a
short-run supply curve ss1, and 100 firms of type 2, each with a short-run supply curve
ss2. A type 1 firm has a shutdown price of $0.20 per rose, while a type 2 firm has a shutdown price of $0.40 per rose. Table 9.4 shows the quantity of roses produced by each
type of firm and the quantity produced by the total market, when the price per rose is
$0.10, $0.30, $0.40, and $0.50.
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ss2
ss1
Price (dollars per rose)
Price (dollars per rose)
SS
$0.50
0.40
0.30
$0.50
0.40
0.30
0.20
0.20
0.10
0.10
0
10
20
30
Quantity (thousands of roses per month)
(a)
1
2
4
Quantity (millions of roses per month)
(b)
FIGURE 9.6
Short-Run Market Supply Curve
Panel (a) shows the short-run supply curves for two types of firms. ss1 is the short-run supply
curve for a firm with a shutdown price of $0.20 per rose; ss2 is the short-run supply curve for
a firm with a shutdown price of $0.40 per rose. Panel (b) shows the short-run market supply
curve SS, which is the horizontal sum of the supply curves in panel (a). At prices between
$0.20 and $0.40 per rose, the market supply curve is 100 times the quantity given by ss1 because the firms represented by ss2 do not produce any output at prices below $0.40 per rose.
At prices below $0.20 per rose, SS is a vertical spike because neither type of firm supplies
output at prices below $0.20.
short-run market
supply curve The supply
curve that shows the quantity supplied in the aggregate
by all firms in the market for
each possible market price
when the number of firms in
the industry is fixed.
Figure 9.6(b) shows the short-run market supply curve SS. The short-run
market supply curve is derived by horizontally summing the supply curves of the
individual firms. The short-run market supply curve tells us the quantity supplied
in the aggregate by all firms in the market. Note that while the scales of the vertical
axes of the two parts of Figure 9.6 are the same, the scales of the horizontal axes differ because total market output is much larger than the output of any individual
firm.
TABLE 9.4
Short-Run Market Supply of Roses
Quantity of Roses Produced by
Price
per Rose
$0.10
$0.30
$0.40
$0.50
Type 1 Firms
100
100
100
100
0 0
10,000
20,000
30,000
1,000,000
2,000,000
3,000,000
Type 2 Firms
100
100
100
100
0 0
0 0
0 0
10,000
1,000,000
Total Market
0
1,000,000
2,000,000
4,000,000
9 . 3 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : S H O RT- RU N E Q U I L I B R I U M
A P P L I C A T I O N
347
9.5
How Much Copper at Which Price?
Copper is produced all over the world. In the year
2000, there were more than 70 copper mines worldwide, operated by 29 different companies. Analysts
following the copper industry collect detailed data on
the production capacities and costs of production of
these mines. It is reasonable to view copper producers
as price-taking firms because each one is small in comparison to the scale of that market. Given this, we can
describe their behavior with supply curves. Figure 9.7
shows supply curves for an individual copper mine
(the Bingham Canyon mine), for all producers in the
United States, and for the overall world market.14
The curve for the Bingham Canyon mine (located
in Utah and owned by copper producer Rio Tinto) is
Supply curve
for Bingham
100¢
Canyon
U.S. supply curve
mine
rather flat when the price of copper is 47 cents per
pound, but then rises sharply as the price rises. At a
price of 70 cents per pound (the price that prevailed
in early 1999), the Bingham Canyon mine would operate at full capacity, producing 285 kilotons of copper
per year (point A on its supply curve). Beyond that
point, marginal costs rise rapidly and the supply curve
becomes almost vertical.
The U.S. supply curve in Figure 9.7 is the horizontal sum of the supply curves of all 17 U.S. copper
mines15 (including Rio Tinto and its Bingham Canyon
mine). The upward slope of this curve tells us that
different mines have different marginal costs of production. The lower the price, the fewer the number of
mines that would supply copper (e.g., at a price below
45 cents per pound, only four U.S. mines would produce
World supply curve
Price (cents per pound)
80
70
A
B
C
60
FIGURE 9.7
47
40
20
0 285
1320 2000
4000
6000
Quantity (kilotons per year)
14
8000 8518
10000
Supply
Curves for Copper in 2000
The supply curves for the
Bingham Canyon mine, for
all 17 U.S. mines, and for all
70 mines worldwide become
nearly vertical after the mines
reach full production capacity.
The U.S. and world supply
curves slope upward because
some mines don’t supply copper or don’t operate at full
capacity when the price of
copper is too low.
We constructed these curves using data from the Mine Cost Data Exchange (www.minecost.com), a firm
that specializes in the analysis of mining operations in a variety of mineral industries, including copper.
15
Strictly speaking, the horizontal summation of the supply curves of individual mines with different vertical intercepts will result in a supply curve that has kinks in it, like the curve in Figure 9.6 (b). But when
we add together so many supply curves (17 of them), this kinked curve will very nearly be smooth. The
U.S. supply curve shown here is the best smooth approximation to the kinked curve that results from
summing the supply curves of the 17 U.S. mines. Likewise, the world supply curve is a smooth approximation of the kinked supply curve that results from summing the supply curve of all 70 mines worldwide.
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any copper, and Bingham Canyon would not be one
of them). At a price of 70 cents per pound, U.S. producers would supply a total of about 1,320 kilotons of
copper per year (point B on the U.S. supply curve).
This is less than the total capacity of U.S. mines of
about 1,560 kilotons per year, indicating that at this
price some mines might not supply any copper or
might not operate at full capacity. Beyond 1,560 kilotons per year, marginal costs rise rapidly and the U.S.
supply curve becomes almost vertical.
The world supply curve in Figure 9.7 is the horizontal sum of the supply curves of all 70 copper mines
worldwide. Like the U.S. supply curve, this curve is also
upward sloping because different mines have different
marginal costs. At a price of 70 cents per pound, world
copper production would be about 8,518 kilotons
per year (point C on the supply curve). Again like U.S.
production, world production at this price is less than
world capacity, which is nearly 9,000 kilotons of copper
per year. Beyond this level, the world supply curve also
becomes almost vertical.
The fact that the three supply curves in Figure 9.7
become almost vertical after certain points indicates
that, in the short run, the supply of copper cannot be
easily expanded beyond current capacity levels, neither
at individual mines (like Bingham Canyon) nor at U.S.
or world mines considered together.
Because each firm’s supply curve coincides with its marginal cost curve (over the range
of prices for which the firm is willing to produce positive output), the market supply curve
tells us the marginal cost of producing the last unit supplied in the market. For example,
in Figure 9.6, when the quantity of roses supplied in the market is 4 million, the marginal
cost of supplying the four-millionth rose is $0.50. This must be the case because, as we
have seen, profit-maximizing behavior induces each rose producer to expand production
to the point at which its marginal cost of the last unit produced equals the market price.
The process of obtaining the market supply curve by summing the individual firm
supply curves is subject to one important qualification: This approach is valid only if
the prices that firms pay for their inputs are constant as the market output varies. The
assumption that input prices are constant may be valid in many markets. For example,
if the industry’s demand for the services of unskilled labor is but a small fraction of the
overall demand for unskilled labor throughout the economy, then changes in industry
output would have a negligible effect on the wage rate for unskilled workers.
However, in some markets the prices of certain inputs might vary as market output
changes. For example, suppose that an industry employs a kind of skilled labor that no
other industry employs. As the quantity supplied increases in response to a higher price,
the industry’s demand for skilled labor would rise, possibly leading to a higher wage rate.
If so, each producer’s marginal cost curve would shift upward. The higher marginal cost
would mean that a producer in this industry would supply less output at any market price
than it would have if the wage rate of skilled labor had not increased. This implies that
the market supply for this product would be less responsive to a change in the price of
this product than it would be if the wage rate for skilled workers were constant.
We will further discuss the effects of changing input prices on market supply in
the section that deals with long-run market supply curves. In what follows, unless
otherwise explicitly stated, we will assume that input prices do not change as industry
output varies in the short run.
S H O R T- R U N P E R F E C T LY C O M P E T I T I V E E Q U I L I B R I U M
short-run perfectly
competitive equilibrium
The market price and quantity at which quantity demanded equals quantity
supplied in the short run.
We can now explore how market price is determined in a competitive market. A
short-run perfectly competitive equilibrium occurs when the quantity demanded
by consumers equals the total quantity supplied by all the firms in the market—that
is, at a point where the market demand curve and the market supply curve intersect.
Figure 9.8(b) shows the market demand curve D and the short-run market supply
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9 . 3 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : S H O RT- RU N E Q U I L I B R I U M
SS
Price (dollars per unit)
Price (dollars per unit)
SMC
P*
PS
0
Q*
P*
D
0
Quantity (units per year)
(a) Typical firm
100Q *
Quantity (units per year)
(b) Market
FIGURE 9.8
Short-Run Equilibrium
The short-run equilibrium price is P*, the price at which market supply equals market demand.
Panel (a) shows that a typical firm produces Q*, where short-run marginal cost equals price.
Panel (b) shows that total quantity supplied and demanded at P* is equal to 100Q*.
curve SS in an industry that consists of 100 identical producers. The equilibrium price
is P *, where quantity supplied is equal to quantity demanded. Figure 9.8(a) shows that
a typical firm will produce output Q*, at which its marginal cost equals the market
price P *. Since there are 100 firms, each supplying Q* units of output, market supply
(which equals market demand at the price P *) must equal 100Q*.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 9 . 3
D
Short-Run Market Equilibrium
A market consists of 300 identical firms, and
the market demand curve is given by D(P) 60 P.
Each firm has a short-run total cost curve STC 0.1
150 Q2, and all fixed costs are sunk. The corresponding
short-run marginal cost curve is SMC 300Q, and the
corresponding average variable cost curve is AVC
150Q. The minimum level of AVC is 0; thus, a firm will
continue to produce as long as price is positive. (You can
verify this by sketching the SMC and AVC curves.)
Solution Each firm’s profit-maximizing quantity is
given by equating marginal cost and price: 300Q P.
Thus the supply curve s(P) of an individual firm is s(P)
P/300.
Since the 300 firms in this market are all identical,
short-run market supply equals 300s(P). The short-run
equilibrium occurs where market supply equals market
demand, or 300(P/300) 60 P. Solving for P, we
find that the equilibrium price is P $30 per unit.
Problem What is the short-run equilibrium price in
Similar Problems: 9.10, 9.11, 9.12, 9.13, 9.14,
this market?
9.15, 9.16, 9.18, 9.19
C O M PA R AT I V E S TAT I C S A N A LYS I S
O F T H E S H O R T- R U N E Q U I L I B R I U M
The competitive equilibrium shown in Figure 9.8(b) should look familiar. We introduced it in Chapter 1 and studied it extensively in Chapter 2. As in those chapters, it
is useful to perform comparative statics analysis on the competitive equilibrium so that
we can better understand the factors that determine the market equilibrium price.
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Price (dollars per unit)
SS0
FIGURE 9.9
Comparative Statics
Analysis: Increase in the Number of Firms
An increase in the number of firms shifts
the short-run supply curve rightward, from
SS0 to SS1. The quantity supplied at any price
goes up. The rightward shift drives the
equilibrium price down and the equilibrium
quantity up.
SS1
$10
Equilibrium
price falls
D
Equilibrium
quantity rises
0
Quantity (units per year)
Figure 9.9 shows one example of a comparative statics analysis: what happens
when the number of firms in the market goes up. Adding more firms moves the shortrun market supply curve rightward, from SS0 to SS1, which means that at any given
market price, such as $10 per unit, the quantity supplied goes up. Thus, as a result of
the increase in the number of firms, the price falls and the equilibrium quantity rises.
Figure 9.10 shows another comparative statics analysis: what happens when the
market demand increases from D to D⬘. As a result of the increase in market demand,
the equilibrium price and quantity both go up.
Market supply curve
New
price
Initial
price
D
New
price
Initial
price
D′
Quantity (units per year)
(a) Effect of shift in demand:
Supply is relatively elastic
Price (dollars per unit)
Price (dollars per unit)
Market supply curve
D
D′
Quantity (units per year)
(b) Effect of shift in demand:
Supply is relatively inelastic
FIGURE 9.10 The Impact of a Shift in Demand on Price Depends on the Price Elasticity
of Supply
In panel (a), supply is relatively elastic, and a shift in demand has a modest impact on price. In
panel (b), supply is relatively inelastic, and the identical shift in demand has a more dramatic
impact on the equilibrium price.
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9 . 3 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : S H O RT- RU N E Q U I L I B R I U M
Figure 9.10 also shows that the price elasticity of supply is an important determinant of the extent to which the equilibrium price fluctuates in response to a shift in
demand. Comparing panel (a) to panel (b) shows that a given shift in demand in a
market with relatively inelastic supply will have a more dramatic impact on the market
price than the same shift in demand in a market with relatively elastic supply. The
boom-and-bust cycles experienced in industries such as oil tankers can be explained,
at least in part, by the inelasticity of short-run market supply.16
A P P L I C A T I O N
9.6
Growing Perfectly Competitive Roses
Figure 9.11 shows wholesale prices and quantities of
long-stem red roses in the United States in 1991, 1992,
and 1993 in four distinct one-month periods: May,
August, November, and the last two weeks of
January and first two weeks of February.17 These are
the prices that rose growers faced as they contemplated supply decisions during the early 1990s.
$0.55
Price (dollars per rose)
SS
January/
February
1991–1993
0.22
May 1991–1993
DJ F
August
and November
1991–1993
DAN
0
DM
4.5
8.9
Quantity (millions of roses per month)
FIGURE 9.11 The Short-Run Supply Curve for Roses
DAN is the demand curve for August and November; DM is the demand curve for May; and DJF
is the demand curve for the January–February period just before Valentine’s Day. The short-run
supply curve SS is flat (perfectly elastic) for quantities up to about 4.5 million roses per month
and increases (slopes up) thereafter.
16
We discuss the example of oil tankers in detail in the next section, on long-run competitive equilibrium.
The data are derived from Tables 12 and 17 of “Fresh Cut Roses from Colombia and Ecuador,”
Publication 2766, International Trade Commission (March 1994). Figure 9.11 shows a weighted average
of prices of U.S. and Colombian growers. These prices have been adjusted for decreases in the value of
the Colombian peso relative to the U.S. dollar and to reflect the normal “quality premium” that U.S. roses
commanded vis-à-vis Colombian roses during 1991–1993. The reference cited above reports quarterly
quantities. The monthly quantities in Figure 9.11 are estimated based on the seasonal pattern of roses
imported from Colombia.
17
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Monthly demand in the U.S. rose market varies
in a predictable way. It is lowest from July through
December because gifts of roses are not customary
for any holidays during this period. It is highest during the last two weeks of January and the first two
weeks of February because of Valentine’s Day. Finally,
it is in-between from April through June because of
Mother’s Day (mid-May) and because May and June
are the busiest months for weddings. In Figure 9.11,
DAN is the demand curve for August and November,
in the period when demand is lowest; DJF is the
demand curve for the end-of-January–beginningof-February period, when demand is highest; and DM
is the demand curve for May, in the period when
demand is in-between.
Supply conditions were stable during 1991–1993,
so we can use back-of-the-envelope techniques to
identify the short-run market supply curve for freshcut roses—that is, we can use the shifts in demand
over the year to trace out the supply curve. As shown
in Figure 9.11, the supply curve was perfectly elastic at
a price of about $0.22 per rose for quantities up to
about 4.5 million roses per month. In other words, at
9.4
HOW THE
MARKET
PRICE IS
DETERMINED:
LONG-RUN
EQUILIBRIUM
that price, rose growers were willing to supply any
quantity up to that amount. But an increase in price
was needed to induce growers to supply the additional
quantity demanded during the month before
Valentine’s Day.
In particular, the price and quantity during the
month before Valentine’s Day were (on average)
$0.55 and 8.9 million roses per month, respectively.
We estimate the slope of the supply curve over the
range between 4.5 and 8.9 million roses per month as
(8.9 ⫺ 4.5)
¢Qs
⫽
⫽ 0.1333
¢P
(55 ⫺ 22)
That is, supply increases at a rate of 0.1333 million
roses for every 1 cent increase in price. We can use
this calculation to determine the price elasticity of
supply for fresh-cut roses in the month before
Valentine’s Day: ⑀Qs,P ⫽ 0.1333 ⫻ (55/8.9) ⫽ 0.82. That
is, the supply of roses around Valentine’s Day increases
at a rate of 0.82 percent for every 1 percent increase
in price. The short-run market supply of roses is thus
relatively inelastic.
I
n the short run, firms operate within a given plant size, and the number of firms in
the industry does not change. As a result, at the short-run perfectly competitive equilibrium, firms might earn positive or negative economic profits. By contrast, in the
long run, established firms can adjust their plant sizes and can even leave the industry
altogether. In addition, new firms can enter the industry. In the long run, these forces
drive a firm’s economic profits to zero.
L O N G - R U N O U T P U T A N D P L A N T- S I Z E A D J U S T M E N T S
B Y E S TA B L I S H E D F I R M S
In the long run, an established firm can adjust both its plant size and its rate of output
to maximize its profit. Thus, as the firm looks out over the long-run horizon and contemplates the possible output levels it might produce, it should evaluate the cost of
those outputs using its long-run cost functions.
To illustrate, Figure 9.12 shows a rose producer that faces a price of $0.40 per rose.
With its current plant size—its current stock of rose bushes, land, and greenhouses—
the firm’s short-run marginal and average cost curves are SMC0 and SAC0, respectively.
Its short-run profit-maximizing output is 18,000 roses per month. At this quantity and
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9 . 4 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : L O N G - RU N E Q U I L I B R I U M
SMC1 SAC1
SMC0
MC
SAC0
AC
Price (dollars per rose)
$0.40
MR = P
0.22
0.16
0
18
75
Quantity (thousands of roses per month)
FIGURE 9.12 Long-Run Output and Plant Size Adjustment by a Price-Taking Firm
A rose grower expects that the market price will be $0.40 per rose. At its existing plant size,
represented by short-run marginal and average cost curves SMC0 and SAC0, the grower’s profitmaximizing output is 18,000 roses per month. To maximize profit over the long run, the
grower would increase output to 75,000 roses per month, the quantity at which the price P
equals the long-run marginal cost MC. To do so, the grower would expand its plant size to the
cost-minimizing level represented by curves SMC1 and SAC1. (The long-run average cost curve
AC is shown to facilitate comparison with Figure 9.13.)
the price of $0.40, the firm earns a positive economic profit because the price exceeds
the firm’s short-run average cost of about $0.22 per rose.
In the long run, however, the grower can increase its profits by expanding its plant
size and harvesting more roses within this expanded plant size. Figure 9.12 shows the
long-run profit-maximizing output for a rose grower that expects the market price to
be $0.40 per rose.18 The profit-maximizing quantity (75,000 roses per month) is the
point at which long-run marginal cost equals the market price (MC ⫽ P, as shown in
Figure 9.12). To produce this quantity, the firm utilizes a plant size that is cost minimizing for this output level.
T H E F I R M ’ S L O N G - R U N S U P P LY C U RV E
The preceding analysis suggests that a firm’s long-run supply curve is its long-run
marginal cost curve. This is almost correct. For prices above the minimum level of
long-run average cost ($0.20 per rose, as shown in Figure 9.13), the firm’s long-run
18
This analysis assumes that the rose grower faces an unchanging market price over time. In reality, the
market price for roses might fluctuate, in which case the rose grower’s long-run profit-maximizing problem
is more complex. The analysis of this more complex problem is beyond the scope of the text.
CHAPTER 9
P E R F E C T LY C O M P E T I T I V E M A R K E T S
MC
AC
$0.40
Price (dollars per rose)
354
0.30
0.20
0.10
0
18
50
75
Quantity (thousands of roses per month)
FIGURE 9.13 The Firm’s Long-Run Supply Curve
For prices greater than the minimum level of long-run average cost (about $0.20 here),
the firm’s long-run supply curve coincides with its long-run marginal cost curve. For prices
below the minimum level of long-run average cost, the firm’s supply curve is a vertical
spike that coincides with the vertical axis.
supply curve coincides with its long-run marginal cost curve. For prices below the
minimum long-run average cost, however, a firm would produce no output, and its
long-run supply curve would be a vertical spike that coincides with the vertical axis
(representing zero output). The reason for this is that at market prices below the minimum long-run average cost, the firm would earn negative economic profit, even after
making all available adjustments in its input mix to minimize total costs. If the firm
anticipated that the market price would remain at such a level for the foreseeable future, its best course of action would be to exit the industry.
The logic underlying the construction of the firm’s long-run supply curve is analogous to the logic we used to construct the firm’s short-run supply curve. In both
cases, we considered the relationship between price and marginal cost to determine
the optimal level of output if indeed the firm produced positive output. And in both
cases, we asked whether the firm would be better off not producing in light of the
costs it avoids if it does not produce. The difference is that in the long run, all costs
are avoidable (i.e., they are nonsunk), whereas in the short run, some costs might not
be avoidable (i.e., they are sunk) if the firm produces a quantity of zero.
FREE ENTRY AND LONG-RUN PERFECTLY
COMPETITIVE EQUILIBRIUM
In our analysis of short-run perfectly competitive equilibrium, we assumed that the
number of firms in the industry was fixed. But in the long run, new firms can enter
the industry. A firm will enter the industry if, given the market price, it can earn positive economic profits and thereby create wealth for its owners.
9 . 4 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : L O N G - RU N E Q U I L I B R I U M
355
A long-run perfectly competitive equilibrium occurs at a price at which supply
equals demand and firms have no incentive to enter or exit the industry. More specifically, a long-run perfectly competitive equilibrium is characterized by a market price
P*, a number of identical firms n*, and a quantity of output Q* per firm that satisfies
three conditions:
long-run perfectly
competitive
equilibrium The market
1. Each firm maximizes its long-run profit with respect to output and plant size. Given
the price P*, each active firm chooses a level of output that maximizes its profit
and selects a plant size that minimizes the cost of producing that output. This
condition implies that a firm’s long-run marginal cost equals the market price,
or P* ⫽ MC(Q*).
2. Each firm’s economic profit is zero. Given the price P*, a prospective entrant cannot
earn positive economic profit by entering this industry. Moreover, an active firm
cannot earn negative economic profit by participating in this industry. This
condition implies that a firm’s long-run average cost equals the market price,
or P * ⫽ AC(Q*).
3. Market demand equals market supply. At the price P*, market demand equals market
supply, given the number of firms n* and individual firm supply decisions Q*. This
implies that D(P *) ⫽ n*Q*, or equivalently, n* ⫽ D(P *)/Q*.
price and quantity at which
supply equals demand,
established firms have no
incentive to exit the industry,
and prospective firms have
no incentive to enter the
industry.
Figure 9.14 shows these conditions graphically. (The numbers in the figure correspond to Learning-By-Doing Exercise 9.4.) Because the equilibrium price simultaneously equals long-run marginal cost and long-run average cost, each firm produces
at the bottom of its long-run average cost curve. If the minimum of the average cost
occurs at a single level of output such as Q* in Figure 9.14, the firm produces at minimum efficient scale. The condition that supply equals demand then implies that the
equilibrium number of firms equals market demand divided by minimum efficient
scale output.
Price (dollars per unit)
Price (dollars per unit)
SMC MC SAC AC
P* = $15
Q* = 50
0
Quantity (thousands of units per year)
(a) Typical firm
$15
D(P )
D(P*) = 10
0
Quantity (millions of units per year)
(b) Market
FIGURE 9.14 Long-Run Equilibrium in a Perfectly Competitive Market
The long-run equilibrium price P* equals the minimum level of long-run average cost ($15 per
unit). Each firm produces a quantity Q* equal to its minimum efficient scale (50,000 units). The
equilibrium quantity demanded is 10 million units. The equilibrium number of firms is this
amount divided by the output per firm of 50,000 (n* ⫽ D(P *)ⲐQ* ⫽ 10,000,000Ⲑ50,000 ⫽ 200).
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E
P E R F E C T LY C O M P E T I T I V E M A R K E T S
L E A R N I N G - B Y- D O I N G E X E R C I S E 9 . 4
D
Calculating a Long-Run Equilibrium
Problem In this market, all firms and
potential entrants are identical. Each has a long-run
average cost curve AC(Q) 40 Q 0.01Q2 and a corresponding long-run marginal cost curve MC(Q) 40
2Q 0.03Q2 where Q is thousands of units per year.
The market demand curve is D(P) 25,000 1,000P,
where D(P) is also measured in thousands of units. Find
the long-run equilibrium quantity per firm, price, and
number of firms.
Solution Let asterisks denote equilibrium values.
The long-run competitive equilibrium satisfies the following three equations.
P* MC(Q*) 40 2Q* 0.03(Q*) 2
( profit maximization)
By combining the first two equations, we can solve for
the quantity per firm, Q*: 40 ⫺ 2Q* 0.03(Q*)2 40
Q* 0.01 (Q*)2, or Q* 50. Thus, each firm in equilibrium produces 50,000 units per year. By substituting
Q* 50 back into the average cost function, we can
solve for the equilibrium price, P*: P* 40 50
0.01(50)2 15. The equilibrium price of $15 per unit
corresponds to each firm’s minimum level of average
cost. By substituting P* into the demand function, we
can find the equilibrium market demand: 25,000
1,000(15) 10,000, or 10 million units per year. The
equilibrium number of firms is equilibrium market demand divided by minimum efficient scale: 10,000,000Ⲑ
50,000 200 firms.
Similar Problems: 9.23, 9.24, 9.25
P* AC(Q*) 40 Q* 0.01(Q*) 2 (zero profit)
n*
D(P*)
25,000 1,000P*
Q*
Q*
(supply equals demand)
L O N G - R U N M A R K E T S U P P LY C U RV E
long-run market
supply curve A curve
that shows the total quantity of output that will be
supplied in the market at
various prices, assuming
that all long-run adjustments (plant size, new
entry) take place.
In our analysis of the short-run competitive equilibrium, we depicted the equilibrium
price by the intersection of the market demand curve and the short-run market supply curve. In this section, we will see that the long-run equilibrium can be depicted in
a similar way: by the intersection of the market demand curve and the long-run market
supply curve. (In this section we will make the same assumption that we made when
obtaining the short-run market supply curve—namely, that changes in industry output
do not affect input prices. In the next section, we will see how to obtain the long-run
market supply curve when this assumption doesn’t hold.)
The long-run market supply curve tells us the total quantity of output that will be
supplied in the market at various prices, assuming that all long-run adjustments take
place (such as adjustments in plant size and new firms entering the market). However,
we cannot obtain the long-run market supply curve in the same way we obtained the
short-run curve, by horizontally summing the individual firm supply curves. The reason
is that, in the long-run as opposed to the short run, market supply can vary as firms
enter or exit the market; thus, there is no fixed set of individual firm supply curves that
we can sum together.
Figure 9.15 shows how to construct a long-run market supply curve. Initially, the
market is in long-run equilibrium at a price of $15. At this price, each of the 200 identical firms produces at its minimum efficient scale of 50,000 units per year, so market
supply is 10 million units per year (the quantity demanded is also 10 million units per
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9 . 4 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : L O N G - RU N E Q U I L I B R I U M
SAC
B
$23
15
A
AC
Price (dollars per unit)
Price (dollars per unit)
SMC
SS0
SS1
LS
$15
D1
D0
50 52
0
0
Quantity (thousands of units per year)
10 10.4
200 firms
(a) Typical firm
18
360 firms
Quantity (millions of units per year)
(b) Market
FIGURE 9.15 Long-Run Market Supply Curve
Initially, the industry is in long-run equilibrium at a price of $15 per unit. Each of the 200 identical firms in the market produces its minimum efficient scale output of 50,000 units per year,
as indicated by point A in panel (a); thus, total market supply is 10 million units per year
(50,000 ⫻ 200 ⫽ 10 million), at the intersection of the initial demand curve D0 and the longrun supply curve LS in panel (b). If demand then shifts rightward from D0 to D1, the short-run
equilibrium price is $23, where the short-run supply curve SS0 intersects D1. In the short run,
each firm is at point B in panel (a), supplying 52,000 units per year and earning a positive economic profit equal to the area of the shaded region. The opportunity to earn a profit induces
new entry, which shifts the short-run supply curve rightward, until it reaches SS1. At this new
long-run equilibrium, the industry now has 360 firms, each firm is again supplying 50,000 units
per year, and the equilibrium price is again $15 per unit. Thus, the long-run supply curve LS is a
horizontal line at $15—in the long run, all market supply occurs at this price.
year, of course, because the market is in equilibrium). Point A in Figure 9.15(a) represents the position of a typical firm at this long-run equilibrium.
Now suppose that market demand shifts from D0 to D1, as shown in Figure 9.15(b).
Also suppose that this demand shift is expected to persist, so the market will reach a new
long-run equilibrium.
In the short run, with 200 firms in the market, equilibrium occurs at a price of
$23, with each firm maximizing profit by producing 52,000 units per year and with
total market supply and demand at 200 ⫻ 52,000 ⫽ 10.4 million units per year. For
the individual firm, this situation is represented by point B in Figure 9.15(a); for the
market, it is represented by the intersection of the short-run supply curve SS0 and the
new demand curve D1 in Figure 9.15(b).
At a price of $23, each of the 200 firms in the market earns a positive economic
profit equal to the area of the shaded rectangle in Figure 9.15(a). The availability
of an economic profit attracts new firms into the market, shifting the short-run
supply curve rightward. Entry of new firms continues until the short-run supply
curve has shifted to SS1 and the price has fallen back to $15 per unit, as represented
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by the intersection of SS1 and D1 in Figure 9.15(b). At this point, 160 new firms have
entered the industry, and each firm (new and old) maximizes its profit by producing
at its minimum efficient scale of 50,000 units per year. Once price falls to $15, there
is no incentive for additional entry or exit because each firm earns zero economic
profit. Moreover, the market clears because market demand at $15 equals the total
market supply of 360 ⫻ 50,000 ⫽ 18 million units per year.
This analysis shows that, in a perfectly competitive market that is initially in
long-run equilibrium at a price P, additional market demand will be satisfied in the
long run by the entry of new firms. Although the equilibrium price may increase in
the short run, in the long run this process of new entry will drive the equilibrium
price back down to its original level. Thus, the long-run market supply curve will be
a horizontal line corresponding to the long-run equilibrium price P. In Figure 9.15(b),
LS is the long-run market supply curve corresponding to the long-run equilibrium
price of $15.
C O N S TA N T- C O S T, I N C R E A S I N G - C O S T,
AND DECREASING-COST INDUSTRIES
Constant-Cost Industry
constant-cost industry
An industry in which the
increase or decrease of
industry output does not
affect the prices of inputs.
When constructing the long-run supply curve in the previous section, we assumed
that the expansion of industry output that occurs as a result of new entry does not affect
the prices of inputs (e.g., labor, raw materials, capital) used by firms in the industry. As
a result, when new firms enter the industry, the cost curves of incumbent producers
do not shift. This assumption holds when an industry’s demand for an input is a small
part of the total demand for that input. In this case, increases or decreases in the industry’s use of that input would not affect its market price. For example, firms in the
rose industry use a significant amount of natural gas, distillates, and other fuels to heat
greenhouses. But many other industries also use these fuels. Because of this, an increase or a decrease in the amount of rose production—and a corresponding increase
in the demand for heating fuels by rose growers—would be unlikely to have much impact on overall demand for heating fuels and would probably not significantly change
the free-market prices of such fuels.
When changes in industry output have no effect on input prices, we have a
constant-cost industry, like the industry depicted in Figure 9.15. (“Constant cost” is
not the same as “constant returns to scale,” which, as you learned in Chapter 8, implies
a horizontal long-run average cost function. Figure 9.15 shows that we can have a
constant-cost industry even though firms do not have constant returns to scale.
Conversely, firms in an industry can have constant returns to scale, but the industry
need not be constant cost.)
increasing-cost industry
An industry in which
increases in industry output
increase the prices of inputs.
industry-specific inputs
Scarce inputs that are used
only by firms in a particular
industry and not by other
industries in the economy.
Increasing-Cost Industry
When an expansion of industry output increases the price of an input, we have an
increasing-cost industry. An industry is likely to be increasing cost if firms use
industry-specific inputs—scarce inputs that only firms in that industry use. For example,
rose producers typically employ a master grower who is responsible for planting rose
bushes, determining fertilizer and pesticide levels, scheduling harvesting, and creating
hybrids. Good master growers are hard to find, and those with a track record of success
are highly sought after.
9 . 4 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : L O N G - RU N E Q U I L I B R I U M
A P P L I C A T I O N
9.7
When the Supertanker Market Sank19
Supertankers are enormous ships that transport crude
oil around the world. The tanker business has been
called the “world’s largest poker game,” a reference
not only to the high risks and large stakes involved in
entering the business—a single tanker can cost more
than $100 million—but also to the colorful figures,
such as Aristotle Onassis and Sir Y. K. Pao, who amassed
fortunes by owning tankers.
No episode underscores how quickly fortunes in
the tanker business can shift than the collapse of the
supertanker market in the 1970s. Figure 9.16 shows the
spot price for supertanker services—the price to charter
a supertanker for a single voyage—between 1973 and
1976.20 In September 1973, the spot rate for supertanker voyages averaged W205. But then the price tum-
bled to under W50 by the end of the year, well under
the level (approximately W80) that would allow supertankers to earn a positive economic profit. Thereafter,
despite some fluctuations, the price continued to fall,
until it reached a fairly stable but abysmally low rate in
the range of W20–W30 during 1975 and 1976.
What happened? The demand for tanker services
depends on the world demand for oil and on the distance between producers and consumers of oil. In the
1960s and early 1970s, the demand for oil grew briskly,
and more oil came from the Middle East. Oil sales from
the Persian Gulf grew at close to 10 percent each year
in the early 1970s, and most industry observers expected
that growth to continue. Demand growth for oil, and
thus for tankers, was especially strong in the first nine
months of 1973. This accounted for the big increase in
the spot price for tankers during the summer of 1973.
Spot price of tanker services (Worldscale)
250
Sept-73
200
150
Positive economic profit
100
80
50
0
Negative economic profit
Jan-73
Jan-74
Jan-75
Jan-76
Month and year
FIGURE 9.16 Spot Price to Charter a Supertanker, January 1973–March 1976
Source: Table 2, p. 14, of Market Conditions and Tanker Economics (London: H. P. Drewry, 1976).
19
359
This example draws from a variety of sources, including “The Oil Tanker Shipping Industry,” Harvard
Business School Case 9-379-086; “The Oil Tanker Shipping Industry in 1983,” Harvard Business School
Case 9-384-034; R. Thomas, “Perfect Competition among Supertankers: Free Enterprise’s Greatest
Mistake,” Chapter 14 in Microeconomic Applications (Cincinnati, OH: South-Western, 1981); and Market
Conditions and Tanker Economics (London: H. P. Drewry, 1976).
20
This price is measured in units called Worldscale (abbreviated W), a price index for tanker services based
on a standard-sized ship operating under standard conditions.
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Figure 9.17 depicts this increase in price as a short-run
equilibrium response to a shift in demand, with the
industry operating on the short-run supply curve SS0.
In the late 1960s and early 1970s, expectations of
high prices for tankers led owners to invest in new
tanker capacity. By 1973, just six years after the first
supertanker was launched, there were nearly 400 supertankers worldwide, and 500 more were on order. Had
the demand side of the market unfolded as expected,
this increase in tanker capacity would have driven the
market price toward the long-run equilibrium price P* at
which supertankers earn zero economic profit (indicated
by the long-run supply curve LS in Figure 9.17).
But demand conditions did not unfold as expected.
In October 1973, war broke out between Israel and the
Arab states, and shortly thereafter, the Organization of
Petroleum Exporting Countries (OPEC) imposed an oil
embargo on the United States. Oil prices skyrocketed,
and OPEC exports to the United States dropped substantially. Oil tankers, whose services had been desperately needed in September 1973, floated empty in
December 1973. Figure 9.17 depicts this as a leftward
shift in demand, from demand curve D0 to D1. Given the
supply curve SS0, the price of tanker services fell far
below the long-run equilibrium level P*.
The increase in price in 1973 and the subsequent
drop in price later that year were especially dramatic
because the short-run supply of supertankers is quite
inelastic. Tanker operators have limited options for adjusting output in the short run: They can steam their
tankers faster or slower to increase or decrease supply,
but such tactics have only a modest effect. Operators can
also deactivate tankers, either by “mothballing” them
with the option of activating them later, or selling them
for scrap. Mothballing is costly, and sale for scrap is irreversible, so neither is done unless low prices are expected to persist. Moreover, supertankers have no alternative uses. In particular, an owner cannot easily convert
a tanker from shipping oil to, say, shipping grain. All of
this implies that short-run supply curves, such as SS0, are
nearly vertical over a wide range of prices.
The oil embargo eventually ended, but the demand
for tanker services remained low throughout 1974 and
1975. Prices of OPEC-produced oil stayed high, and demand fell as Western nations, such as the United States,
cut back their oil consumption. Oil tankers last for a
long time (typically 20 years), so it takes capacity a long
time to leave the industry. In fact, in 1974 and 1975, the
short-run supply curve actually shifted rightward, to SS1
in Figure 9.17, as new supertankers that were ordered
in the early 1970s were commissioned for service. For
example, in 1974 worldwide tanker capacity increased
18 percent despite record-low prices for tanker services.
This accentuated the fall in price.
Eventually, tanker supply did adjust. In 1977 and
1978, over 20 million tons worth of tanker capacity was
sold for scrap. In addition, almost half of the orders for
unfinished tankers were canceled, costing owners millions of dollars in lost down payments and cancellation
fees. The decrease in tanker capacity, coupled with a
gradual increase in demand for oil, caused tanker
prices to creep upward in the late 1970s. Still, it took
more than 10 years for the industry to recover from the
collapse in prices that began in the autumn of 1973.
FIGURE 9.17 The Collapse of the Oil Tanker
Market, 1973–1975
In the early fall of 1973, the demand curve for tanker
services was D0, the short-run supply curve was SS0, and
the price was at the level marked by the intersection of
these two curves (well above the long-run equilibrium
price at the level of the long-run supply curve LS). Then
the demand curve shifted leftward to D1, and the price
fell to the level marked by the intersection of D1 and
SS0. Subsequently, the short-run supply curve shifted
rightward to SS1, and by 1975 the price had fallen even
further, to the level of the intersection of D1 and SS1.
Price (Worldscale)
SS0 SS
1
Sept.
1973
P*
LS
Dec.
1973
1975
D1
D0
0
Quantity (millions of deadweight tons per year)
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9 . 4 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : L O N G - RU N E Q U I L I B R I U M
Price (dollars per unit)
SMC1
SMC0
$23
20
AC1
AC0
LS
$23
20
B
15
SS0 SS
1
A
15
D1
D0
0
50 52
Quantity (thousands of units per year)
(a) Typical firm
0
10 10.4 14
200 firms
280 firms
Quantity (millions of units per year)
(b) Market
FIGURE 9.18 Long-Run Industry Supply Curve in an Increasing-Cost Industry
Initially, the industry is in long-run equilibrium at a price of $15 per unit. Each of the 200 identical firms in the market produces its minimum efficient scale output of 50,000 units per year,
as indicated by point A in panel (a); thus, total market supply is 10 million units per year
(50,000 ⫻ 200 ⫽ 10 million), at the intersection of the initial demand curve D0 and the longrun supply curve LS in panel (b). If demand then shifts rightward from D0 to D1, the short-run
equilibrium price is $23, where the short-run supply curve SS0 intersects D1. In the short run,
each firm is at point B in panel (a), supplying 52,000 units per year and earning a positive
economic profit. The opportunity to earn a profit induces new entry, which shifts the short-run
supply curve rightward, until it reaches SS1. As new firms enter, the prices of industry-specific
inputs go up, shifting the long-run and short-run cost curves upward, as shown in panel (a)—in
particular, the minimum level of long-run average cost increases from $15 to $20. At the new
long-run equilibrium, the industry now has 280 firms, each firm is again supplying 50,000 units
per year, and the equilibrium price is $20 per unit. Thus, the long-run supply curve LS is upward
sloping.
Figure 9.18 illustrates the equilibrium adjustment process in an increasing-cost
industry, based on the same initial scenario as in Figure 9.15. At an initial long-run
equilibrium price of $15, the 200 identical firms in the industry each produce
50,000 units per year [each is at the position marked by point A in Figure 9.18(a)].
Suppose the market demand shifts rightward, from demand curve D0 to D1 in
Figure 9.18(b). Initially, assuming no entry by new firms and no change in input
prices, the short-run supply curve is SS0. The equilibrium price would be $23, at the
intersection of D1 and the initial short-run supply curve SS0. At that price, firms can
earn a positive economic profit, which attracts new entrants and thus shifts the
short-run supply curve rightward. So far, all this parallels the situation depicted in
Figure 9.15.
But now, as industry output increases through new entry, the prices of industryspecific inputs (such as master growers) begin to rise (e.g., as new entrants seek to lure
master growers away from their current employers by offering them higher salaries).
The increase in input prices causes each firm’s long-run and short-run cost functions
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to shift upward, as shown in Figure 9.18(a).21 [Figure 9.18(a) depicts an upward shift
that leaves each firm’s minimum efficient scale unchanged at 50,000 units per year, as
indicated by point B, but in general a firm’s minimum efficient scale could also change
as input prices change.] The new short-run market supply curve SS1 is drawn with the
number of firms in the industry after all new entry has occurred (280 firms) and with
input prices at their new (higher) levels. The new equilibrium price is $20, and the
quantity exchanged in the market is 14 million units per year. While the short-run
supply curves are each drawn for a given number of firms and given input prices, the
long-run supply curve LS takes into account both entry by new firms and changes in
input prices.
The adjustment process stops when price falls to a point at which firms earn zero
profits. This occurs at a price of $20, where the new short-run supply curve SS1 intersects the new demand curve D1. That price equals the minimum level of the new
long-run average cost curve AC1 that results from the increase in input prices.
Industry output expands from 10 million to 14 million units per year. Since each firm
produces output of 50,000 units, the equilibrium number of firms is now
14,000,000/50,000 ⫽ 280. Thus, an additional 80 firms have entered the industry.
The long-run market supply curve in an increasing-cost industry is upward sloping, like curve LS in Figure 9.18(b). The upward-sloping market supply curve tells us
that increases in price are needed to elicit additional industry output in the long run.
The increases in price compensate for the increases in the minimum level of long-run
average cost that are driven by the increase in industry output and the resulting increase in input prices.
Decreasing-Cost Industry
decreasing-cost
industry An industry in
which increases in industry
output decrease the prices
of some or all inputs.
In some situations, an increase in industry output can lead to a decrease in the price
of an input. We then have a decreasing-cost industry. To illustrate, suppose an industry relies heavily on a special kind of computer chip as an input. The industry may
be able to acquire computer chips more inexpensively as the industry’s demand for
chips rises, perhaps because manufacturers of computer chips can employ costreducing techniques of production at higher volumes. In a decreasing-cost industry,
each firm’s average and marginal cost curves may fall, not because the firms produce
with economies of scale, but because input prices fall when the industry produces
more.
Figure 9.19 illustrates that the long-run supply curve LS is downward sloping
in a decreasing-cost industry. At an initial long-run equilibrium price of $15, the
200 identical firms in the industry each produces 50,000 units per year [each is at
the position marked by point A in Figure 9.19(a)]. Initially, assuming no entry by
new firms and no change in input prices, the short-run supply curve is SS0. If
the market demand shifts rightward, from demand curve D0 to D1 in Figure
9.19(b), the equilibrium price in the short run would be $23, at the intersection of
D1 and the initial short-run supply curve SS0. At that price, firms can earn a positive
21
For the case of a rose-growing firm that employs a single master grower, the salary of the master grower
would be a fixed cost. An increase in the salaries of master growers would thus affect the AC curve but not
the SMC curve. Figure 9.18(a) shows the case of an increase in the price of an input that firms use in
variable amounts. Increases in the price of a variable input would shift the short-run marginal cost curve
from SMC0 to SMC1, as shown in the figure.
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9 . 4 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : L O N G - RU N E Q U I L I B R I U M
SMC0
Price (dollars per unit)
SMC1
AC0
AC1
SS0
SS1
$23
$23
A
15
12
B
15
12
LS
D0
50 52
0
Quantity (thousands of units per year)
(a) Typical firm
0
10 10.4
20
200 firms
400 firms
Quantity (millions of units per year)
(b) Market
FIGURE 9.19 Long-Run Industry Supply Curve in a Decreasing-Cost Industry
Initially, the market consists of 200 identical firms. In panel (a), point A is the position of a single
firm when the market is in long-run equilibrium at a price of $15 per unit, with the firm
producing 50,000 units per year and with total market supply at 10 million units per year. After
demand increases (and input prices decrease), each firm operates at point B when the market
reaches long-run equilibrium at a price of $12 per unit.
In panel (b), LS is the long-run market supply curve. The initial equilibrium is at the intersection of LS and the initial demand curve D0. The increase in demand shifts the demand curve
from D0 to D1. Initially, when there are 200 firms paying the initial input prices, the short-run
supply curve is SS0. After 200 additional firms enter the market and input prices decrease,
the short-run supply curve has shifted to SS1. In the long run, the equilibrium price will be
$12 (following the decrease in input prices), at the intersection of LS and the new demand
curve D1.
economic profit, and entry would occur. So far, all this parallels the situation depicted in Figures 9.15 and 9.18.
However, as industry output increases through new entry, the prices of industryspecific inputs (such as computer chips) begin to fall, causing each firm’s long-run and
short-run cost curves to shift downward, as shown in Figure 9.19(a). (As before, this
example assumes that the shift from AC0 to AC1 leaves each firm’s minimum efficient
scale unchanged at 50,000 units per year, as indicated by point B.) The new market
short-run supply curve SS1 is drawn with the 400 firms in the industry after entry has
occurred and with input prices at their new (lower) levels. The new equilibrium price
is $12, and the quantity exchanged in the market is 20 million units per year. The
long-run supply curve LS is drawn taking into account both entry by new firms and
changes in input prices; it is downward sloping because producers face lower input
prices when the market produces larger quantities.
D1
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9.8
The U.S. Ethanol Industry and
the Price of Corn
The ethanol industry in the United States provides an
excellent example of an increasing-cost industry.
Ethanol (or ethyl alcohol, CH3CH2OH) is a colorless,
flammable liquid that is used in a variety of applications including alcoholic beverages, solvents, scents,
and fuel. Ethanol is produced through a process of
fermentation of sugar found in grains such as corn,
maize, or sorghum or other crops such as sugar cane.
When people refer generically to “alcohol,” they are
usually referring to ethyl alcohol. In Brazil, the second
largest ethanol producer in the world after the United
States, ethanol is manufactured using sugar cane as
feedstock. Ethanol can also be produced, through
somewhat more difficult processes, from trees, grasses,
crop residues, algae, or even old newspapers. In the
United States, though, most ethanol for fuel is made
from corn.
The ethanol industry in the United States has long
been supported by the U.S. government. For example,
the United States imposes tariffs on foreign ethanol
produced from sugar cane. The tariffs effectively block
Brazilian producers of sugar-based ethanol (whose
average production costs tend to be lower than cornbased producers in the United States) from competing
in the U.S. market. As another example, when motor
fuel producers blend ethanol with gasoline to enhance
octane (to reduce engine knock and increase engine
power), they are eligible for a tax refund of $0.45 for
every gallon of ethanol that is blended with gasoline.22
Producers of E85, an alternative fuel that consists of 85
percent ethanol and 15 percent gasoline, also receive
the tax credit. This tax credit is a subsidy to motor fuel
producers that purchase ethanol and has the effect of
increasing the demand for ethanol.
Demand for ethanol in the United States began to
increase in the mid-2000s as the price of oil rose steadily.
22
The rise in the price of oil, and the attendant increase in
the price of gasoline, made ethanol-based alternative
fuels such as E85 more attractive to U.S. motorists. The
U.S. government reinforced the increase in the demand
for ethanol through a number of important changes
in policy. In 2005, the federal government withdrew
liability protection from motor fuel producers who used
a compound called Methyl Tertiary Butyl Ether (MTBE)
to enhance octane ratings. MTBE had been linked to
cancer, and beginning in the early 2000s, many U.S.
states banned its use. With MTBE either banned or
more costly because of the withdrawal of liability protection, motor fuel producers switched from MTBE to
ethanol. In addition, in 2005 and 2007, the Congress
passed energy bills that included rules mandating the
use of certain quantities of biofuels, including ethanol.
For example, the Energy Independence and Security
Act of 2007 requires usage of 20.5 billion gallons of
biofuel annually by 2015 and 36 billion gallons by
2022, of which 15 billion gallons can be ethanol. These
mandates also increased the demand for ethanol.
By the mid-2000s, demand in the ethanol market
was surging. The model of perfect competition suggests that in the short run this should have led to
increases in prices and producer profits. The price of
ethanol, which had been about $1 per gallon in 2005,
increased by a factor of 4 during 2006.23 As prices
rose, so did the profits of existing producers. Accounts
of the industry in the press spoke about the “biofuels
boom.”24
Booms in perfectly competitive industries typically attract the entry of new capacity, which is exactly what happened in the ethanol industry. As the
top panel of Figure 9.20 shows, significant amounts
of new capacity entered the industry after 2005. For
example, the number of U.S. ethanol plants at the
beginning of 2005 was 81; by 2009, there were 190
ethanol plants nationwide.25 In 2007 alone, more
The tax credit was $0.51 per gallon until passage of the 2008 Farm Bill.
“Corn Farmers Smile as the Price of Ethanol Rises but Experts on Food Prices Worry,” New York
Times ( January 16, 2006), Section A, p. 13; “U.S. Ethanol Ends Pivotal Year Amid Uncertainty: Rising
Production Threatens Margins,” Platts Oilgram Price Report 85, no. 1 ( January 2, 2007): 1.
24
See, for example, “Biofuels Boom,” CQ Researcher 16, no. 34 (September 29, 2006).
25
“Ethanol’s Boom Stalling as Glut Depresses Prices,” New York Times (September 30, 2007).
23
U.S. ethanol plants and new plant construction: 1999–2009
180
80
Number of U.S. ethanol plants
160
70
Number of ethanol
plants under construction
140
60
120
50
100
80
40
Number of ethanol plants
30
60
40
20
20
10
0
1999
0
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
365
Number of U.S. ethanol plants under construction
9 . 4 H OW T H E M A R K E T P R I C E I S D E T E R M I N E D : L O N G - RU N E Q U I L I B R I U M
Year
U.S. ethanol production capacity: 1999–2009
10,000
8,000
6,000
4,000
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
01/01/2009
0
1999
01/01/2008
2,000
01/01/2007
U.S. ethanol production capacity
(millions of gallons per year)
12,000
Year
U.S. price of corn: 1999–2009
5.00
4.00
3.00
2.00
1.00
Date
01/01/2006
01/01/2005
01/01/2004
01/01/2003
01/01/2002
01/01/2001
01/01/2000
0
01/01/1999
U.S. price of corn (dollars per bushel)
$6.00
FIGURE 9.20 Ethanol
Plants, Production Capacity,
and the Price of Corn in the
United States, 1999–2009
The upper panel shows the
number of ethanol plants in
the United States as of
January of each year. It also
shows the number of new
plants under construction.
The middle panel shows the
total amount of U.S. ethanol
production capacity as of
January of each year. The
bottom panel shows the
price of corn in the United
States as of January of each
year. Source: Ethanol plant,
plant construction, and production capacity data come
from the website of the
Renewable Fuel Association
http://www.ethanolrfa.org/
industry/statistics/#C (accessed
December 26, 2009).
Data on corn prices comes
from U.S. Department of
Agriculture Economic
Research Service, Feed
Grains Database, U.S.
Department of Agriculture,
http://www.ers.usda.gov/data/
feedgrains/ (accessed July 9,
2009).
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than 75 ethanol plants were under construction in the
United States, a number that exceeded the population
of active plants before 2004. Total ethanol production
capacity, shown in the middle panel of Figure 9.20,
rose from about 3,650 million gallons per year in
January 2005 to about 10,570 million gallons per year
four years later.26
In a constant-cost industry, the expansion of
industry capacity into the industry does not affect
input prices, but in an increasing-cost industry, the
entry of new capacity causes the prices of one or
more inputs to increase. The ethanol industry is an
increasing-cost industry. As noted, corn is the feedstock used to produce ethanol in the United States.
In the late 2000s, ethanol alone accounted for between 15 and 20 percent of overall corn demand in
the United States. Perhaps not surprisingly, as the
ethanol industry expanded during the boom of the
late-2000s, the price of corn increased. As shown in
the bottom panel of Figure 9.20, at the beginning of
2005, the price of corn in the United States was
about $2 per bushel, the norm for the industry in the
1990s and early 2000s. Four years later, the price had
doubled to $4 per bushel (and indeed the price averaged more than $5 per bushel during much of 2008).
Not all of the increase in the price of corn was due
to ethanol, but a significant fraction was. The U.S.
Congressional Budget Office estimates that between
28 and 47 percent of the increase in the price of corn
between April 2007 and April 2008 was due to increased demand for corn due to increases in ethanol
production.27
As a perfectly competitive market moves toward
a new long-run equilibrium, the entry of new capacity begins to slow down and eventually stop.
Producers that enjoyed high profits during the boom
begin to feel squeezed as new entry drives down
the price of the product and—in an increasing–cost
industry—drives up the prices of scarce inputs. As 2009
came to end, this was the saga of the U.S. ethanol
industry. A story in the Minneapolis Star-Tribune titled
“Ethanol Boom Goes Bust” epitomized much of the
press coverage of the industry during 2009.28 As the
top panel of Figure 9.20 shows, the rate of new construction of new ethanol plants decreased sharply. Still,
despite the “bust,” the price of ethanol in the United
States in 2009 was over $2 per gallon, more than twice
as high as it was in 2005.29 This is consistent with the
theory of long-run equilibrium in an increasing-cost industry: A rightward shift in market demand will move
the market along its long-run supply curve to a new
long-run equilibrium at a higher price.
W H AT D O E S P E R F E C T C O M P E T I T I O N T E AC H U S ?
In this section, we have studied how free entry affects the long-run equilibrium price
in a perfectly competitive market. In doing so, we have seen a key implication of the
theory of perfect competition: Free entry will eventually drive economic profit to
zero. This is one of the most important ideas in microeconomics. It tells us that when
profit opportunities are freely available to all firms, economic profits will not last.
This confirms the conventional business wisdom: “If anyone can do it, you can’t make
money at it.” The lesson of the theory of perfect competition for managers is that if
you base your firm’s strategy on skills that can easily be imitated or resources that can
easily be acquired, you put yourself at risk from the forces that are highlighted by the
theory of perfect competition. In the long run, your economic profit will be competed
away.
26
Data on the number of ethanol plants and total production capacity come from the website of the
Renewable Fuels Association, http://www.ethanolrfa.org/industry/statistics/#C (accessed December 26, 2009).
27
“The Impact of Ethanol Use on Food Prices and Greenhouse-Gas Emissions,” Congressional Budget
Office (April 2009).
28
“Ethanol Boom Goes Bust,” StarTribune.com (November 29, 2009), http://www.startribune.com/
politics/state/78108802.html (accessed December 24, 2009).
29
Current ethanol price data are available at EthanolMarket.com, http://www.ethanolmarket.com /(accessed
December 24, 2009).
367
9.5 ECONOMIC RENT AND PRODUCER SURPLUS
I
n the preceding sections, we studied how price-taking firms adjust their production
decisions in light of the market price. We also explored how the market price is determined. We now explore how firms and input owners (e.g., providers of labor services or owners of land or capital) profit from their activities in perfectly competitive
markets. We will introduce two concepts to describe the profitability of firms and
input owners in perfectly competitive markets: economic rent and producer surplus.
9.5
ECONOMIC
RENT AND
PRODUCER
SURPLUS
ECONOMIC RENT
In the theory we have developed so far, we have assumed that all firms that operate in
a perfectly competitive market have access to identical resources. This was reflected
in our assumption that all active firms and potential entrants had the same long-run
cost curves.
But in many industries some firms gain access to extraordinarily productive
resources, while others do not. For example, in the rose industry, several thousand
individuals might be good enough to be master growers, but only a handful are truly
extraordinary master growers. The rose producers lucky enough to hire this handful
will be more productive than firms that hire the merely good growers.
Economic rent measures the economic surplus that is attributable to an extraordinarily productive input whose supply is limited. Specifically, economic rent is equal
to the difference between the maximum amount a firm is willing to pay for the services of the input and the input’s reservation value. The input’s reservation value, in
turn, is the return that the input owner could get by deploying the input in its best
alternative use outside the industry. Putting the pieces of this definition together, we
thus have: economic rent ⫽ A ⫺ B, where
A ⫽ maximum amount firm is willing to pay for services of input
B ⫽ return that input owner gets by deploying the input in its best
alternative use outside the industry
To illustrate this definition, suppose that the maximum amount that a rose firm
would be willing to pay to hire an extraordinary master grower—the A term in our
definition of economic rent—is equal to $105,000.30 Suppose further that the grower’s
best available employment opportunity outside the rose industry is to work as a
grower in the tulip industry for an annual salary of $70,000. This is the B term in our
definition. The economic rent attributable to the extraordinary master grower is thus
$105,000 ⫺ $70,000 ⫽ $35,000 per year.
Economic rent is frequently confused with economic profit. These concepts are
related but distinct. To illustrate the difference, let’s develop our rose-growing example
further. Suppose that every rose-producing firm needs one and only one master
grower. Also suppose that there are two types of master growers: extraordinary and
run of the mill. There are a limited number—let’s say 20—of the former, but a virtually
unlimited supply of the latter. Imagine that the reservation value of either type of master
grower is $70,000 per year, and for now, let’s suppose that all master growers are paid
an annual salary that equals this reservation value.
30
Later in this section, we will see how we would determine this maximum willingness to pay.
economic rent The
economic return that is attributable to extraordinarily
productive inputs whose
supply is scarce.
reservation value The
return that the owner of an
input could get by deploying the input in its best
alternative use outside the
industry.
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P E R F E C T LY C O M P E T I T I V E M A R K E T S
Price (dollars per rose)
AC *
MC′
MC AC
AC′
$0.25
0.20
D
0
700
Quantity
(thousands of roses per year)
(a) Rose firm with extraordinary
master grower
0
600
Quantity
(thousands of roses per year)
(b) Rose firm with run-of-the-mill
master grower
0
134
Quantity
(millions of roses per year)
(c) Market for roses
FIGURE 9.21 Economic Rent
Panels (a) and (b): When all master growers are paid the same annual salary, a rose firm with
an extraordinary master grower has a lower marginal cost curve than a firm with a run-of-themill master grower (MC⬘ versus MC ) and a lower average cost curve (AC⬘ versus AC ). In this
case, at the equilibrium price of $0.25 per rose, the economic rent of an extraordinary master
grower [equal to the area of the shaded region in panel (a)] is entirely captured as economic
profit by the firm that employs him or her. But if firms must compete for extraordinary master
growers and if their salary is bid up to the maximum of $105,000 annually that firms would be
willing to pay, the cost curve of a firm with an extraordinary master grower shifts upward to
AC*, the same as the cost curve AC of a firm with a run-of-the-mill master grower. At that
point, each extraordinary master grower captures all the economic rent he or she generates,
and the firm’s economic profit drops to zero.
Panel (c) shows the market demand curve and the total quantity of roses produced at the
equilibrium price.
An extraordinary master grower can grow more roses with the same inputs (labor,
capital, land, materials) than a run-of-the-mill master grower. Thus, as Figure 9.21
shows, when all master growers are paid the same annual salary of $70,000, a rose firm
that employs an extraordinary master grower has lower average and marginal cost
curves than a firm that employs a run-of-the-mill master grower [AC⬘ and MC⬘ in
panel (a) versus AC and MC in panel (b)]. Note that the average cost curves, AC and
AC⬘, are the sum of two parts: the cost per unit for all of the expenses incurred by a
rose firm other than the salary of the master grower (e.g., labor, materials, land, capital)
and the master grower’s salary per unit of output, which equals $70,000 divided by the
number of roses produced. It is the “other expenses” that the firm economizes on if it
employs an extraordinary master grower. Also note that because the master grower’s
salary is independent of the quantity of roses produced (i.e., the grower’s salary is a
fixed cost), the magnitude of the grower’s salary does not influence the position of a
rose firm’s marginal cost curve. The difference between MC and MC⬘ is attributable
solely to the extra productivity that a firm gains from hiring an extraordinary master
grower.
Figure 9.21 shows the market equilibrium when all master growers are paid the same
salary. A firm with a run-of-the-mill master grower produces 600,000 roses per year, its
minimum efficient scale [panel (b)]. A firm with an extraordinary master grower produces
9.5 ECONOMIC RENT AND PRODUCER SURPLUS
700,000 roses per year, the point at which its marginal cost curve MC⬘ intersects the equilibrium market price of $0.25 per rose [panel (a)]. Total market demand for roses at $0.25
is 134 million roses [panel (c)]. Of that, 20 ⫻ 700,000 ⫽ 14 million roses are supplied by
the 20 firms that hire the 20 extraordinary master growers; the remaining 120 million
roses are supplied by firms with run-of-the-mill master growers. Notice from Figure 9.21(a)
that when a firm hires an extraordinary master grower at a salary of $70,000, its average
cost is equal to $0.20 per rose. By contrast, a firm that hires a run-of-the-mill master
grower at the same $70,000 annual salary has an average cost equal to the equilibrium
price of $0.25 per rose. Thus, by employing an extraordinary master grower, a rose firm
attains a cost savings of $0.05 per rose produced.
Now, let’s identify the economic rent generated by an extraordinary master grower.
In light of our definition above, we must first ask: What is the maximum salary that a
firm would be willing to pay to hire an extraordinary master grower? The most that a
firm would be willing to pay an extraordinary master grower would be the salary—call
it S *—that would make the firm’s economic profit equal to zero. At any higher salary,
the firm would be better off dropping out of the industry. From Figure 9.21, we can
see that paying this maximum salary of S* would have to push a firm’s average cost
upward, from AC⬘ to AC*, so that at a quantity of 700,000, average cost would just
equal the market price of $0.25 per rose.31 That is, a salary of S* rather than $70,000
is just enough to offset the $0.05 per rose cost advantage created by the extraordinary
grower’s talent. The upward shift in the average cost curve is equal to the difference
between the salary per unit at S*, S*/ 700,000, and the salary per unit at $70,000, or
70,000/ 700,000, and this upward shift must be exactly equal to $0.05. Thus:
S*Ⲑ700,000 ⫺ 70,000Ⲑ700,000 ⫽ 0.05, or S* ⫽ $105,000. That is, the highest salary a
rose firm would be willing to pay an extraordinary master grower is $105,000 per year.
The economic rent is the difference between this maximum willingness to pay and a
master grower’s reservation value of $70,000: economic rent ⫽ $105,000 ⫺ $70,000 ⫽
$35,000. Notice that this economic rent of $35,000 corresponds to the shaded region
in Figure 9.21(a).32
Now let’s compute a rose firm’s economic profit. Firms with run-of-the-mill master
growers earn zero economic profit. By contrast, the 20 firms with the extraordinary
master growers earn positive economic profit equal to their $0.05 per rose cost advantage times the number of roses they produce. This product also equals the area of the
shaded region in Figure 9.21(a). When an extraordinary master grower is paid the
same as a run-of-the-mill master grower, economic profit equals economic rent. That
is, each of the 20 firms that employs an extraordinary master grower captures all of
the economic rent for itself as positive economic profit. An extraordinary grower, by
contrast, captures none of the economic rent that his or her talent generates. This is
clearly a great outcome for a firm that is lucky enough to hire an extraordinary master
grower at a salary of $70,000 per year.
But suppose that rose firms had to compete to hire the extraordinary master
growers. This would be a market not unlike the market for free agents in major league
baseball or professional basketball. The competition among rose firms to hire the best
master growers would bid up the salaries of the extraordinary ones. If competition is
31
Remember, the magnitude of the grower’s salary does not affect the position of the rose firm’s marginal
cost curve, so a firm that hires an extraordinary master grower would still produce 700,000 roses per year,
the point at which its (unshifting) MC⬘ curve equals the market price of $0.25.
32
This is because the area of this region ⫽ (0.25 ⫺ 0.20) ⫻ 700,000 ⫽ $35,000.
369
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P E R F E C T LY C O M P E T I T I V E M A R K E T S
TABLE 9.5
Relationship between Economic Rent and Economic Profit
Master Grower’s
Annual Salary
$70,000
Between $70,000
and $105,000
$105,000
Economic Rent
Generated by
Extraordinary
Master Grower
$35,000
$35,000
$35,000
“Salary Premium”
(part of economic
rent captured by
extraordinary
master grower)
Economic Profit
(part of economic
rent captured by
firm that employs
extraordinary
master grower)
$0
Between $0
and $35,000
$35,000
$35,000
Between $35,000
and $0
$0
sufficiently intense, the salaries of extraordinary master growers would be bid up to
$105,000, the maximum a firm would be willing to pay. Firms with such master growers would then, in fact, operate on long-run average cost curve AC* in Figure 9.21(a).33
In a long-run equilibrium, these firms, like their run-of-the-mill counterparts, earn
zero economic profit. The cost advantage gained by employing an extra productive
master grower is just offset by the higher salary that must be paid to lure the grower
from other rose firms that also want to employ his or her services. The economic rent
of the scarce input is still the area of the shaded region. In this case, though, the rent
is captured by an extraordinary master grower as a “salary premium” above the reservation value of $70,000, rather than by rose firms as positive economic profit.
In general, the salary of an extraordinary master grower could fall anywhere
between $70,000 per year and $105,000. Depending on this salary, the economic profit
of a rose firm that hires an extraordinary master grower would range between $35,000
and $0. Table 9.5 illustrates this point. The table shows that the economic rent is a pie,
or a surplus that gets divided between firms and input owners. The economic rent is
always $35,000, but economic profit depends on how the “rent pie” gets divided.
The division of the economic rent between firms and master growers ultimately
depends on resource mobility. If master growers can easily move from firm to firm, we
would expect intense bidding for their services and master grower salaries close to firms’
maximum willingness to pay of $105,000. In this case, the economic profits of rose growers are dissipated through competition in the market to hire master growers ( just as the
profits of baseball teams are dissipated as they compete for talented free agents). If, by contrast, master growers cannot easily move from firm to firm, or if a master grower’s extraordinary talent is specialized to a particular firm (i.e., the master grower is extraordinary for
one particular firm but run-of-the-mill for all others), master grower salaries might not be
bid up. If not, the economic rents would be captured by firms as positive economic profits.
PRODUCER SURPLUS
In Chapter 5, we introduced the concept of consumer surplus, a monetary measure of
the net benefit enjoyed by price-taking consumers from being able to purchase a
product at the going market price. In Chapter 5, we saw that consumer surplus was
the area between the demand curve and the market price.
33
Recall that the marginal cost curves would be unaffected since a master grower’s salary is a fixed cost.
9.5 ECONOMIC RENT AND PRODUCER SURPLUS
In this section we show that there is an analogous concept for price-taking firms:
producer surplus. Producer surplus is the difference between the amount that a firm
actually receives from selling a good in the marketplace and the minimum amount the
firm must receive in order to be willing to supply the good in the marketplace. Just as
consumer surplus provides a measure of the net benefit enjoyed by price-taking consumers, producer surplus provides a measure of the net benefit enjoyed by price-taking
firms from supplying a product at a given market price.
371
producer surplus A
measure of the monetary
benefit that producers
derive from producing a
good at a particular price.
Producer Surplus for an Individual Firm
To illustrate the producer surplus for an individual firm, let us begin with a simple
example. Suppose that a shipbuilder can either build one ship in the upcoming year
or no ships at all. The firm would be willing to supply this ship as long it receives
at least $50 million, the additional cost that the firm incurs if it builds the ship (or
equivalently, the cost that it avoids if it does not build the ship). If the market price
for ships of this type is $75 million, the firm would be willing to supply a ship. By
doing so, it receives $75 million in additional revenue, while incurring $50 million
in additional cost, thus increasing its total profit. The firm’s producer surplus would
be $75 million ⫺ $50 million ⫽ $25 million. Notice that producer surplus is simply
the difference between the firm’s total revenue and its total nonsunk (i.e., avoidable)
cost.
Of course, as we have seen throughout this chapter, firms typically would be
willing to supply more than one unit. For example, suppose that our shipbuilder could
potentially build as many as four ships during a particular year. The firm’s supply
curve S is shown in Figure 9.22. It shows that the firm must receive at least $50 million per ship in order to be willing to supply the first ship. The lowest price at which
it would be willing to supply a second ship would be $60 million. The minimum price
at which it would supply a third ship would be $70 million, and the minimum price at
Price (millions of $ per unit)
S
$80
75
70
Producer surplus
60
50
1
2
3
Quantity (ships per year)
4
FIGURE 9.22 Producer Surplus for a
Shipbuilder
The supply curve S shows that the firm must receive
at least $50 million per ship in order to be willing
to supply one ship. To be willing to supply two
ships, the firm must receive at least $60 million per
ship. To be willing to supply three ships, the firm
must receive at least $70 million per ship, and to
supply four ships, the firm must receive at least $80
million per ship. If the market price of ships is $75
million per ship, the shipbuilder would supply three
ships. The shipbuilder’s producer surplus is $45 million, the area of the shaded region between the
market price and the supply curve.
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P E R F E C T LY C O M P E T I T I V E M A R K E T S
which it would supply a fourth ship would be $80 million. As in our initial example,
the minimum price at which the shipbuilder would be willing to supply ships reflects
the additional cost of producing a ship. The shipbuilder requires a higher price in
order to supply the second ship because if it builds two ships in the upcoming year
rather than one, it must utilize an older portion of its shipyard with less modern
equipment (which in turn makes its workers less productive). The shipbuilder requires
a higher price still in order to be willing to supply the third and fourth ships for the
same reason.
Suppose that the market price of ships is $75 million per ship. At this price, the
shipbuilder’s supply curve indicates that it would supply three ships in the upcoming
year. What is the shipbuilder’s producer surplus? To find out, you would add the surpluses of each of the ships built. The producer surplus of the first ship is (as before)
$25 million: the market price of $75 million minus the avoidable cost of $50 million
of building that ship. The producer surplus of the second ship is $75 million minus
$60 million, or $15 million, while the producer surplus of the third ship is $75 million minus $70 million, or $5 million. The shipbuilder’s producer surplus is thus $25
million ⫹ $15 million ⫹ $5 million ⫽ $45 million, the difference between the shipbuilder’s total revenue and its total nonsunk cost.
As Figure 9.22 shows, the shipbuilder’s producer surplus is the area between the
firm’s supply curve and the market price. In this example, the firm’s supply curve was
a series of “steps,” which makes it easy to see the producer surplus of each unit produced. However, the concept of producer surplus readily applies to the case in which
a firm has a smooth supply curve.
Figure 9.23 shows the producer surplus for a firm that faces a marginal cost curve
MC and an average nonsunk cost curve ANSC. For this firm, the supply curve is a vertical spike 0E up to the shutdown price of $2 per unit. Above this price, it is the solid
portion of MC. When the market price is $3.50 per unit, the firm supplies 125 units.
The firm’s producer surplus when the market price is $3.50 is the area between the supply curve and the market price, or the area of region FBCE. This area is the sum of two
parts: rectangle FACE and triangle ABC. Rectangle FACE is the difference between
total revenue and the total nonsunk cost of the first 100 units supplied. It thus represents the producer surplus of these 100 units. Triangle ABC is the difference between
the additional revenue and the additional cost if the firm expands output from 100 units
to 125 units. It thus represents the producer surplus of the last 25 units supplied. For
each additional unit of output in this range, the firm’s profit goes up by the difference
between the price and the marginal cost MC of that additional unit, and so area ABC
is the additional profit due to increasing output from 100 to 125 units. As before, the
overall producer surplus at a market price of $3.50 (area FBCE ) equals the difference
between the firm’s total revenue and its total nonsunk cost when it supplies 125 units.
In the short run, when some of the firm’s fixed costs might be sunk, a firm’s producer surplus and its economic profit are not equal, but differ by the extent of the
firm’s sunk costs—in particular, economic profit equals total revenue minus total
costs, while producer surplus equals total revenue minus total nonsunk cost. However,
in the long run, when all costs are nonsunk (i.e., avoidable), producer surplus and economic profit are the same.
Notice that in both cases the difference in producer surplus at one market price and
producer surplus at another price is equal to the difference in the firm’s economic profits
at these two prices (because fixed costs do not change). Thus, for example, in Figure 9.23,
area P1P2GH is the increase in economic profit as well as the increase in producer surplus
that the firm enjoys when the price increases from P1 to P2.
9.5 ECONOMIC RENT AND PRODUCER SURPLUS
MC
P2
P1
Price (dollars per unit)
ANSC
G
H
F
$3.50
2.00
E
A
B
PS = minimum ANSC
C
Area P1P2GH = Increase in “firm’s” economic
profit when price increases from P1 to P2
Area FBCE = Producer surplus at a price of $3.50
0
100
125
Quantity (units per year)
FIGURE 9.23 Producer Surplus for a Price-Taking Firm
The producer surplus at price $3.50 is equal to the area between the price and the supply curve,
area FBCE. This area is equal to the difference between the firm’s total revenue and its total
nonsunk cost when it produces 125 units of output. The change in producer surplus when the
market price moves from P1 to P2 is equal to the area of P1P2GH. This is the change in the firm’s
economic profit that results when the market price increases from P1 to P2.
Producer Surplus for the Entire Market: Short Run
In the short run, the number of producers in the industry is fixed, and the market
supply curve is the horizontal sum of the supply curves of the individual producers. Because of this, the area between the short-run market supply curve and the
market price is the sum of the producer surpluses of the individual firms in the
market.
Figure 9.24 illustrates this for a market that consists of 1,000 identical firms,
each with a supply curve ss. The market supply curve SS in Figure 9.24(b) is the horizontal sum of these individual supply curves. The area between this supply curve and
the price—the producer surplus for the entire market—equals total market revenue
minus the total nonsunk costs of all firms in the industry. For example, when the
price is $10 per unit, each individual firm in Figure 9.24 produces 200 units per year
and has a producer surplus equal to area ABCD, which in this case equals $350.34
Total market supply at $10 is equal to 200,000 units per year, and the area between
the market supply curve and price, area EFGH, is equal to $350,000. This is the combined producer surplus of 1,000 individual firms, each with a producer surplus of
$350 ($350,000 ⫽ $350 ⫻ 1,000). The market-level producer surplus of $350,000 is
thus the difference between the total revenue of all 1,000 firms and their total nonsunk costs.
34
The area of ABCD equals (10 ⫺ 8) ⫻ 150 plus (1/2) ⫻ (10 ⫺ 8) ⫻ (200 ⫺ 150), which equals 350.
373
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CHAPTER 9
P E R F E C T LY C O M P E T I T I V E M A R K E T S
Price (dollars per unit)
MC
ss
$10
8
B
Firm’s
producer
surplus
SS
ANSC
C
E
A
D
ss
Market-level
producer
surplus
H
SS
150
0
G
F
200
150
0
200
Quantity (thousands of units per year)
(b) Market of 1000 firms
Quantity (units per year)
(a) Typical firm
FIGURE 9.24 Market-Level Producer Surplus: Number of Firms in the Industry Is Fixed
Panel (a): A typical firm has a supply curve ss. At a price of $10, a firm supplies 200 units, and
its producer surplus is area ABCD. This area equals $350. Panel (b): With 1,000 firms in the industry, the market supply curve is SS. At a price of $10, market supply is 200,000 units, and the
market-level producer surplus is area EFGH. This area equals $350,000.
A P P L I C A T I O N
9.9
Mining Copper for Profit
In the late 1990s, the world copper market was
rocked by declining demand and falling prices. We
can use the concept of producer surplus, along with
the world supply curve for copper that we presented
in Application 9.5, to illustrate the impact of falling
copper prices on industry producer surplus.
In early 1998, the price of copper was about
90 cents a pound. By early 1999, the price had fallen to
Price (cents per pound)
Bingham
Canyon
supply
curve
90¢
70
F
G
Change in individual
mine's producer surplus
E
H
World
supply
curve
90¢
70
40
40
20
20
0
100
200
274 283
Quantity (kilotons per year)
(a) Typical low-cost mine
(Bingham Canyon)
0
B
C
Change in market-level
producer surplus
D
A
2000
4000
6000
8518 9142
Quantity (kilotons per year)
(b) World market
FIGURE 9.25 Producer Surplus in the World Copper Market
Area ABCD in panel (b) shows the reduction in industrywide producer surplus when the price of
copper dropped from 90 cents per pound to 70 cents per pound. Area EFGH in panel (a) shows
the reduction in producer surplus for a particular mine, the Bingham Canyon mine in Utah, with
medium to low costs, that continues to produce at close to full capacity despite the drop in price.
9.5 ECONOMIC RENT AND PRODUCER SURPLUS
about 70 cents a pound, a drop of about 22 percent.
The resultant decrease in market-level producer surplus
was equal to the area of the shaded region ABCD in
Figure 9.25(b), roughly $3.5 billion. This is a significant
decrease. The producer surplus at a price of 90 cents
per pound—the area between the supply curve and a
price of 90 cents—is approximately $6.5 billion. The
22 percent drop in copper prices during 1999 reduced
industry producer surplus by more than 50 percent.
The reason for part of the drop in industry producer
surplus was that some high-cost mines that were profitable at a price of 90 cents were no longer profitable at
a price of 70 cents. These high-cost mines significantly
reduced their operations or shut down altogether. But
much of the drop in producer surplus was due to the fact
that many lower-cost mines—such as the Bingham
Canyon mine described in Application 9.5 and whose
supply curve is reproduced in closeup in Figure 9.25(a)—
continued to operate at near full capacity but at lower
profit margins. These mines were less profitable to operate when copper sold at 70 cents a pound than at
90 cents a pound, as indicated by the shaded region
EFGH. But their owners still earned higher profits by
keeping them open instead of shutting them down.
L E A R N I N G - B Y- D O I N G E X E R C I S E 9 . 5
S
E
375
D
Calculating Producer Surplus
Suppose that the market supply curve for
milk is given by Q 60P, where Q is the quantity of
milk sold per month (measured in thousands of gallons)
when the price is P dollars per gallon.
Problem
sold per month [Q 60(2.50) 150]. The producer
surplus is triangle A, the area between the supply curve
and the market price. This area equals (1/2)(2.50 0)
(150,000) 187,500. Producer surplus in this market is
thus $187,500 per month.
(b) By how much does producer surplus increase when
the price of milk increases from $2.50 to $4.00 per gallon?
(b) If the price increases from $2.50 to $4.00, the quantity supplied will increase to 240,000 gallons per month.
Producer surplus will increase by area B ($225,000) plus
area C ($67,500). Producer surplus in this market thus
increases by $292,500 per month.
Solution
Similar Problems: 9.30, 9.33, 9.34
(a) What is the producer surplus in this market when the
price of milk is $2.50 per gallon?
(a) Figure 9.26 shows the supply curve for milk. When
the price is $2.50 per gallon, 150,000 gallons of milk are
SMilk
Price ($ per gallon)
$4.00
B
C
2.50
A
0
150
240
Quantity (thousands of gallons per month)
FIGURE 9.26 Producer Surplus in the
Milk Market
The producer surplus when the price of milk
is $2.50 per gallon is the area of triangle A,
or $187,500. If the price increases from
$2.50 to $4.00, the increase in producer
surplus is the sum of area B ($225,000) and
area C ($67,500), or $292,500.
CHAPTER 9
P E R F E C T LY C O M P E T I T I V E M A R K E T S
Price (dollars per unit)
376
LS
P*
E
Economic
rent
F
D
0
Quantity (millions of units per year)
FIGURE 9.27
Producer Surplus at the Long-Run Equilibrium in an Increasing-Cost
Industry
At a long-run equilibrium price P*, each firm earns zero economic profit. The area between
the long-run industry supply curve LS and the equilibrium price, area FP*E, equals the
economic rent that goes to the inputs whose supply is scarce.
Producer Surplus for the Entire Market: Long Run
In a long-run equilibrium, a price-taking firm earns zero economic profit. Since a
firm’s producer surplus in the long run equals its economic profit, it follows that the
producer surplus for a perfectly competitive firm in a long-run equilibrium must equal
zero as well.
But Figure 9.27 shows that there is a positive area (FP*E) between the long-run
industry supply curve LS and the market equilibrium price. Since all firms earn zero
economic profit, area FP *E cannot represent the economic profit of the firms in the industry. What is it then?
Recall that when a perfectly competitive industry has an upward-sloping long-run
supply curve, it is because firms must compete for the services of a scarce input (e.g.,
extraordinary master growers in the rose industry). As we discussed in the previous
section on economic rent, the result of such competition is that the economic rents
are fully captured by the owners of the input. Thus, area FP *E is not the economic
profit of firms (which is equal to zero). Rather it is the economic rent that is captured
by owners of scarce industry-specific inputs. For example, if the market in Figure 9.27
is the rose market, then area FP*E is the salary earned by the extraordinary master
growers above and beyond the minimum salary that would be necessary to induce
them to supply their services to a rose firm.35
E C O N O M I C P R O F I T, P R O D U C E R S U R P L U S,
ECONOMIC RENT
We conclude this section with the following table, summarizing the relationship between
the three measures of performance that we have discussed in this chapter: economic
profit, producer surplus, and economic rent.
35
There is an area between a downward-sloping industry supply curve and the market price in a decreasingcost industry. To interpret what this area means would take us beyond the scope of this text, and so we will
not discuss it here.
C H A P T E R S U M M A RY
377
Long-Run Competitive
Equilibrium
Short Run
Economic profit for industry
Producer surplus for industry
⫽ total revenue ⫺ total cost
⫽ total revenue ⫺ total
nonsunk cost
⫽ total revenue ⫺ total cost ⫽ 0
⫽ total revenue ⫺ total cost ⫽ 0
Area between industry supply
curve and market price industry
⫽ producer surplus for industry
In a constant-cost industry, this area
equals zero.
In an increasing-cost industry, this
area is positive and equals the
economic rent captured by owners
of scarce industry-specific inputs.
CHAPTER SUMMARY
• Perfectly competitive markets have four characteristics: the industry is fragmented, firms produce undifferentiated products, consumers have perfect information
about prices, and all firms have equal access to resources.
These characteristics imply that firms act as price takers,
output sells at a single price, and the industry is characterized by free entry.
• Economic profit (not accounting profit) represents
the appropriate profit-maximization objective for a firm.
Economic profit is the difference between a firm’s sales
revenue and its total economic costs, including all relevant opportunity costs.
• Marginal revenue is the additional revenue a firm
generates by selling one additional unit or the revenue it
sacrifices by producing one fewer unit.
• A price-taking firm’s marginal revenue curve is a
horizontal line equal to market price.
• The short-run equilibrium price occurs at the point
where market demand equals short-run market supply.
(LBD Exercise 9.3)
• The price elasticity of supply measures the percentage change in quantity supplied for each percent change
in price.
• In the long run, perfectly competitive firms can adjust
their plant sizes and thus maximize profit by producing a
quantity at which long-run marginal cost equals price.
• In the long run, free entry drives the market price to
the minimum level of long-run average cost. If firms
have identical U-shaped long-run average cost curves,
each firm supplies a quantity equal to its minimum efficient scale. The equilibrium number of firms is such that
total market supply equals the quantity demanded at the
equilibrium price. (LBD Exercise 9.4)
• A price-taking firm maximizes its profit by producing
an output level at which marginal cost equals the market
price, and the marginal cost curve is upward sloping.
• In a constant-cost industry, the expansion of industry
output that occurs as firms enter the industry does not
affect market price. The long-run market supply curve is
horizontal.
• If all fixed costs are sunk, a perfectly competitive firm
will produce positive output in the short run only if the
market price for its output exceeds average variable cost.
The shutdown price—the price below which the firm
produces zero output—is the minimum level of average
variable cost. (LBD Exercise 9.1)
• In an increasing-cost industry, the expansion of industry output that occurs as firms enter the industry increases the prices of industry-specific inputs. The long-run
market supply curve is upward sloping. In a decreasing-cost
industry, the long-run market supply curve is downward
sloping.
• If some fixed costs are nonsunk, the firm produces
positive output only if price exceeds average nonsunk
costs. The shutdown price is the minimum level of average
nonsunk cost. (LBD Exercise 9.2)
• If input prices do not change as market output varies,
the short-run market supply is the sum of the short-run
supplies of individual firms.
• The economic rent attributable to a scarce input is
the difference between a firm’s maximum willingness
to pay for the input and the input’s reservation value.
When a firm captures the input’s economic rent, it
earns positive economic profits. Competition for the
scarce input, however, will dissipate these profits. In
this case, economic rent is positive while economic
profit is zero.
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P E R F E C T LY C O M P E T I T I V E M A R K E T S
• Producer surplus is the area between the supply
curve and the market price.
• For a firm with sunk fixed costs, producer surplus
differs from economic profit. In particular, producer
surplus equals the difference between total revenues and
total nonsunk costs, while economic profit equals the
difference between total revenues and total costs. If a
firm has no sunk fixed costs, producer surplus equals
economic profit.
• In the short run, the market-level producer surplus is
the area between the short-run supply curve and the
market price. It equals the sum of the producer surpluses
of individual firms in the market. (LBD Exercise 9.5)
• In an increasing-cost industry, the long-run industry
supply curve is upward sloping. The area between the
price and the long-run supply curve measures the economic rents of inputs that are in scarce supply and whose
price is bid up as more firms enter the industry.
REVIEW QUESTIONS
1. What is the difference between accounting profit and
economic profit? How could a firm earn positive accounting profit but negative economic profit?
2. Why is the marginal revenue of a perfectly competitive firm equal to the market price?
3. Would a perfectly competitive firm produce if price
were less than the minimum level of average variable
cost? Would it produce if price were less than the minimum level of short-run average cost?
4. What is the shutdown price when all fixed costs are
sunk? What is the shutdown price when all fixed costs are
nonsunk?
5. How does the price elasticity of supply affect changes
in the short-run equilibrium price that results from an
exogenous shift in the market demand curve?
6. Consider two perfectly competitive industries—
Industry 1 and Industry 2. Each faces identical demand
and cost conditions except that the minimum efficient
scale output in Industry 1 is twice that of Industry 2. In a
long-run perfectly competitive equilibrium, which industry
will have more firms?
7. What is economic rent? How does it differ from economic profit?
8. What is the producer surplus for an individual firm?
What is the producer surplus for a market when the
number of firms in the industry is fixed and input prices
do not vary as industry output changes? When is producer surplus equal to economic profit (for either a firm
or an industry)? When producer surplus and economic
profit are not equal, which is bigger?
9. In the long-run equilibrium in an increasing-cost industry, each firm earns zero economic profits. Yet there is
a positive area between the long-run industry supply
curve and the long-run equilibrium price. What does this
area represent?
10. Explain the difference between the following concepts: producer surplus, economic profit, and economic
rent.
PROBLEMS
9.1. The annual accounting statement of revenues and
costs for a local flower shop shows the following:
Revenues
Supplies
Employee salaries
$250,000
$ 25,000
$170,000
If the owners of the firm closed its operations, they could
rent out the land for $100,000. They would then avoid
incurring any of the expenses for employees and supplies.
Calculate the shop’s accounting profit and its economic
profit. Would the owners be better off operating the shop
or shutting it down? Explain.
9.2. Last year, the accounting ledger for an owner of a
small drug store showed the following information about
her annual receipts and expenditures (she lives in a taxfree country, so don’t worry about taxes):
Revenues
Wages paid to hired labor
(other than herself )
Utilities (fuel, telephone, water)
Purchases of drugs and other supplies
for the store
Wages paid to herself
$1,000,000
$ 300,000
$ 20,000
$ 500,000
$ 100,000
379
PROBLEMS
She pays a competitive wage rate to her workers,
and the utilities and drugs and other supplies are all
obtained at market prices. She already owns the building,
so she pays no money for its use. If she were to close the
business, she could avoid all of her expenses and, of
course, would have no revenue. However, she could rent
out her building for $200,000. She could also work elsewhere herself. Her two employment alternatives include
working as a lawyer, earning wages of $100,000, or working at a local restaurant, earning $20,000. Determine her
accounting profit and her economic profit if she stays in the
drug store business. If the two are different, explain the
difference.
9.3. A firm sells a product in a perfectly competitive
market, at a price of $50. The firm has a fixed cost of $30.
Fill in the following table and indicate the level of output
that maximizes profit. How would the profit-maximizing
choice of output change if the fixed cost increased from
$40 to $60? More generally, explain how the level of
fixed cost affects the choice of output.
Total
Total
Output Revenue Cost
(units) ($/unit) ($/unit)
0
Profit
($)
Marginal Marginal
Revenue Cost
($/unit) ($/unit)
0
1
50
2
20
3
30
4
42
5
54
6
70
9.4. A firm can sell its product at a price of $150 in a
perfectly competitive market. Below is an incomplete
table of a firm’s various costs of producing up to 6 units
of output. Fill in the remaining cells of the table, and
then calculate the profit the firm earns when it maximizes profit.
Q
TC
1
200
2
TVC
AFC
MC
100
3
20
4
240
5
6
AC
24
660
160
AVC
9.5. A competitive, profit-maximizing firm operates at
a point where its short-run average cost curve is upward
sloping. What does this imply about the firm’s economic
profits? Briefly explain.
9.6. A bicycle-repair shop charges the competitive
market price of $10 per bike repaired. The firm’s shortrun total cost is given by STC(Q) ⫽ Q2/2, and the associated marginal cost curve is SMC(Q) ⫽ Q.
a) What quantity should the firm produce if it wants to
maximize its profit?
b) Draw the shop’s total revenue and total cost curves, and
graph the total profit function on the same diagram. Using
your graph, state (approximately) the profit-maximizing
quantity in each case.
9.7. A producer operating in a perfectly competitive
market has chosen his output level to maximize profit. At
that output, his revenue and costs are as follows:
Revenue
Variable costs
Sunk fixed costs
Nonsunk fixed costs
$200
$120
$60
$40
Calculate his producer surplus and his profits. Which (if
either) of these should he use to determine whether he
should exit the market in the short run? Briefly explain.
9.8. Dave’s Fresh Catfish is a northern Mississippi farm
that operates in the perfectly competitive catfish farming
industry. Dave’s short-run total cost curve is STC(Q) ⫽
400 ⫹ 2Q ⫹ 0.5Q2, where Q is the number of catfish harvested per month. The corresponding short-run marginal
cost curve is SMC(Q) ⫽ 2 ⫹ Q. All of the fixed costs are
sunk.
a) What is the equation for the average variable cost
(AVC)?
b) What is the minimum level of average variable
costs?
c) What is Dave’s short-run supply curve?
9.9. Ron’s Window Washing Service is a small business
that operates in the perfectly competitive residential window washing industry in Evanston, Illinois. The short-run
total cost of production is STC(Q) ⫽ 40 ⫹ 10Q ⫹ 0.1Q2,
where Q is the number of windows washed per day. The
corresponding short-run marginal cost function is
SMC(Q) ⫽ 10 ⫹ 0.2Q. The prevailing market price is
$20 per window.
a) How many windows should Ron wash to maximize
profit?
b) What is Ron’s maximum daily profit?
c) Graph SMC, SAC, and the profit-maximizing quantity.
On this graph, indicate the maximum daily profit.
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d) What is Ron’s short-run supply curve, assuming that
all of the $40 per day fixed costs are sunk?
e) What is Ron’s short-run supply curve, assuming that if
he produces zero output, he can rent or sell his fixed
assets and therefore avoid all his fixed costs?
9.10. The bolt-making industry currently consists of 20
producers, all of whom operate with the identical short-run
total cost curve STC(Q) ⫽ 16 ⫹ Q2, where Q is the annual
output of a firm. The corresponding short-run marginal
cost curve is SMC(Q) ⫽ 2Q. The market demand curve for
bolts is D(P) ⫽ 110 ⫺ P, where P is the market price.
a) Assuming that all of each firm’s $16 fixed cost is sunk,
what is a firm’s short-run supply curve?
b) What is the short-run market supply curve?
c) Determine the short-run equilibrium price and quantity in this industry.
9.11. Newsprint (the paper used for newspapers) is
produced in a perfectly competitive market. Each identical firm has a total variable cost TVC(Q) ⫽ 40Q ⫹ 0.5Q2,
with an associated marginal cost curve SMC(Q) ⫽ 40 ⫹
Q. A firm’s fixed cost is entirely nonsunk and equal to 50.
a) Calculate the price below which the firm will not produce any output in the short run.
b) Assume that there are 12 identical firms in this industry. Currently, the market demand for newsprint is
D(P) ⫽ 360 ⫺ 2P, where D(P) is the quantity consumed
in the market when the price is P. What is the short-run
equilibrium price?
9.12. The oil drilling industry consists of 60 producers,
all of whom have an identical short-run total cost curve,
STC(Q) ⫽ 64 ⫹ 2Q2, where Q is the monthly output of a
firm and $64 is the monthly fixed cost. The corresponding short-run marginal cost curve is SMC(Q) ⫽ 4Q.
Assume that $32 of the firm’s monthly $64 fixed cost can
be avoided if the firm produces zero output in a month.
The market demand curve for oil drilling services is
D(P) ⫽ 400 ⫺ 5P, where D(P) is monthly demand at
price P. Find the market supply curve in this market, and
determine the short-run equilibrium price.
9.13. There are currently 10 identical firms in the perfectly competitive gadget manufacturing industry. Each
firm operates in the short run with a total fixed cost of F
and total variable cost of 2Q2, where Q is the number of
gadgets produced by each firm. The marginal cost for
each firm is MC ⫽ 4Q. Each firm also has nonsunk fixed
costs of 128. Each firm would just break even (earn zero
economic profit) if the market price were 40. (Note: The
equilibrium price is not necessarily 40 when there are
10 firms in the market.)
The market demand for gadgets is QM ⫽ 180 ⫺ 2.5P,
where QM is the amount purchased in the entire market.
a) How large are the total fixed costs for each firm?
Explain.
b) What would be the shutdown price for each firm?
Explain.
c) Draw a graph of the short-run supply schedule for this
firm. Label it clearly.
d) What is the equilibrium price when there are 10 firms
currently in the market?
e) With the cost structure assumed for each firm in this
problem, how many firms would be in the market at an
equilibrium in which every firm’s economic profits are zero?
9.14. A perfectly competitive industry consists of two
types of firms: 100 firms of type A and 30 firms of type B.
Each type A firm has a short-run supply curve sA(P) ⫽ 2P.
Each type B firm has a short-run supply curve sB(P) ⫽
10P. The market demand curve is D(P) ⫽ 5000 ⫺ 500P.
What is the short-run equilibrium price in this market?
At this price, how much does each type A firm produce,
and how much does each type B firm produce?
9.15. A market contains a group of identical price-taking
firms. Each firm has a marginal cost curve SMC(Q) ⫽
2Q, where Q is the annual output of each firm. A study
reveals that each firm will produce if the price exceeds
$20 per unit and will shut down if the price is less than
$20 per unit. The market demand curve for the industry
is D(P) ⫽ 240 ⫺ P/2, where P is the market price. At the
equilibrium market price, each firm produces 20 units.
What is the equilibrium market price, and how many
firms are in this industry?
9.16. The wood-pallet market contains many identical
firms, each with the short-run total cost function
STC(Q) ⫽ 400 ⫹ 5Q ⫹ Q2, where Q is the firm’s annual
output (and all of the firm’s $400 fixed cost is sunk). The
corresponding marginal cost function is SMC(Q) ⫽ 5 ⫹
2Q. The market demand curve for this industry is D(P) ⫽
262.5 ⫺ P/2, where P is the market price. Each firm in
the industry is currently earning zero economic profit.
How many firms are in this industry, and what is the
market equilibrium price?
9.17. Suppose a competitive, profit-maximizing firm
operates at a point where its short-run average cost curve
is upward sloping. What does this imply about the firm’s
economic profits? If the profit-maximizing firm operates
at a point where its short-run average cost curve is downward sloping, what does this imply about the firm’s economic profits?
9.18. A firm in a competitive industry produces its
output in two plants. Its total cost of producing Q1 units
from the first plant is TC1 ⫽ (Q1)2, and the marginal cost
at this plant is MC1 ⫽ 2Q1. The firm’s total cost of producing Q2 units from the second plant is TC2 ⫽ 2(Q2)2;
PROBLEMS
the marginal cost at this plant is MC2 ⫽ 4Q2. The price
in the market is P. What fraction of the firm’s total supply
will be produced at plant 2?
9.19. A competitive industry consists of six type A
firms and four type B firms.
Each firm of type A operates with the supply curve:
Supply
QA
⫽ e
⫺ 10 ⫹ P,
0,
when P 7 10
when P ⱕ 10
Each firm of type B operates with the supply curve:
Supply
QB
⫽ 2P,
for P ⱖ 0.
Demand
a) Suppose the market demand is QMarket ⫽ 108 ⫺ 10P.
At the market equilibrium, which firms are producing,
and what is the equilibrium price?
Demand
b) Suppose the market demand is QMarket ⫽ 228 ⫺ 10P.
At the market equilibrium, which firms are producing,
and what is the equilibrium price?
9.20.
A firm’s short-run supply curve is given by
s(P ) ⫽ e
0, if P 6 10
3P ⫺ 30, if P ⱖ 10
What is the equation of the firm’s marginal cost curve
SMC(Q)?
9.21. Consider a point on a supply curve where price
and quantity are positive. Determine the numerical value
of the price elasticity of supply at that point when the
supply curve is
a) vertical at a positive quantity
b) horizontal at a positive price
c) a straight line through the origin, with a positive slope
9.22. During the week of February 9–15, 2001, the U.S.
rose market cleared at a price of $1.00 per stem, and 4 million stems were sold that week. During the week of
June 5–11, 2001, the U.S. rose market cleared at a price of
$0.20 per stem, and 3.8 million stems were sold that week.
From this information, what would you conclude about
the price elasticity of supply in the U.S. rose market?
9.23. The global cobalt mining industry is perfectly
competitive. Each existing firm and every potential entrant faces an identical U-shaped average cost curve. The
minimum level of average cost is $5 per ton and occurs
when a firm produces 2 million tons of cobalt per year.
The market demand curve for cobalt is D(P) ⫽ 205 ⫺ P,
where D(P) is the demand for cobalt in millions of tons
per year when the market price is P dollars per ton. What
is the long-run equilibrium price for cobalt? How much
cobalt does each producer make at this equilibrium
price? How many active cobalt producers will be in the
market?
381
9.24. The global propylene industry is perfectly competitive, and each producer has the long-run marginal cost
function MC(Q) ⫽ 40 ⫺ 12Q ⫹ Q2. The corresponding
long-run average cost function is AC(Q) ⫽ 40 ⫺ 6Q ⫹
Q2/3. The market demand curve for propylene is D(P) ⫽
2200 ⫺ 100P. What is the long-run equilibrium price in
this industry, and at this price, how much would an individual firm produce? How many active producers are in the
propylene market in a long-run competitive equilibrium?
9.25. The raspberry growing industry in the United
States is perfectly competitive, and each producer has a
long-run marginal cost curve given by MC(Q) ⫽ 20 ⫹
2Q. The corresponding long-run average cost function is
given by AC(Q) ⫽ 20 ⫹ Q ⫹ 144
Q . The market demand
curve is D(P) ⫽ 2,488 ⫺ 2P. What is the long-run equilibrium price in this industry, and at this price, how much
would an individual firm produce? How many active producers are in the raspberry growing industry in a long-run
competitive equilibrium?
9.26. Suppose that the world market for calcium is perfectly competitive and that, as a first approximation, all
existing producers and potential entrants are identical.
Consider the following information about the price of
calcium:
• Between 1990 and 1995, the market price was stable
at about $2 per pound.
• In the first three months of 1996, the market price
doubled, reaching a high of $4 per pound, where it remained for the rest of 1996.
• Throughout 1997 and 1998, the market price of
calcium declined, eventually reaching $2 per pound by
the end of 1998.
• Between 1998 and 2002, the market price was stable
at about $2 per pound.
Assuming that the technology for producing calcium did
not change between 1990 and 2002 and that input prices
faced by calcium producers have remained constant,
what explains the pattern of prices that prevailed between
1990 and 2002? Is it likely that there are more producers
of calcium in 2002 than there were in 1990? Fewer? the
same number? Explain your answer.
9.27. It is 2017, and you work for a prestigious management consultant firm whose client is a large agribusiness company that is considering acquiring an ownership
stake in several U.S. yellow perch farming operations.
(The yellow perch is a fresh fish found in the United
States and raised commercially for sale as food.) As a
member of the consulting team working on this project,
you have been assigned the task of understanding why
the U.S. farm-raised perch industry has evolved as it has
over the last six years.
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Between 2010 and 2013, the farm-raised yellow
perch market was stable. However, in 2013 an unexpected exogenous shock occurred that affected prices
and quantities in the market. You don’t know much
about the details of the industry, and since the industry
is not covered extensively in the press, it is hard to find
articles on the Web about what happened to the industry. From talking to the client, you learn that the shock
might have had something to do with either a change in
the market demand for yellow perch or a change in the
price of corn (which affects the price of perch feed). But
you do not know for sure, nor do you know whether the
shock was a permanent change or merely a temporary
one. However, you do have data (obtained from the
client), shown in the accompanying table, on yellow
perch prices, market demand, quantity supplied, and the
number of producers. The data pertain to 2010–2013,
2014 (within one year of the shock), and 2016 (three
years after the shock). You also know (from the client)
that yellow perch farms are virtually identical, with
U-shaped long-run average cost curves. You also learn
from the client that the minimum efficient scale of a typical yellow perch farm occurs at a rate of production of
about 1,000 pounds per month (and this is unaffected by
changes in the prices of key inputs such as feed or labor).
a) Based on the data in the table, what type of shock
most likely explains the evolution of the yellow perch
farming industry from 2010–2013 to 2016?
2014: within
6 months of
the shock
2010–2013
$3.00 per pound
100,000 pounds
per month
1,000 pounds
per month
100
Market price of yellow perch
Total quantity yellow perch
demanded in the United States
Quantity of yellow perch supplied
by a typical yellow perch farm
Number of active yellow perch farms
b) How would your answer change if the number of
active yellow perch farms in 2016 was 100?
$4.00 per pound
120,000 pounds
per month
1,200 pounds
per month
100
$3.00 per pound
150,000 pounds
1,000 pounds
per month
150
c) How would your answer change if the data in the
table looked like this?
2010–2013
$3.00 per pound
100,000 pounds
per month
1,000 pounds
per month
100
Market price of yellow perch
Total quantity yellow perch
demanded in the United States
Quantity of yellow perch supplied
by a typical yellow perch farm
Number of active yellow perch farms
2016: 3 years
after shock
9.28. The long-run total cost function for producers of
mineral water is TC(Q) ⫽ cQ, where Q is the output of an
individual firm expressed as thousands of liters per year.
The market demand curve is D(P) ⫽ a ⫺ bP. Find the
long-run equilibrium price and quantity in terms of a, b,
and c. Can you determine the equilibrium number of
firms? If so, what is it? If not, why not?
2014: within
6 months of
the shock
2016: 3 years
after shock
$3.50 per pound
90,000 pounds
$4.00 per pound
80,000 pounds
900 pounds
per month
100
1,000 pounds
per month
80
9.29. Support or refute the following: “In the long run
the firm’s producer surplus and profits will be equal.”
9.30. Each firm in the perfectly competitive widget industry produces with the levels of marginal cost (MC)
and total variable cost (TVC) at various levels of output Q
shown in the following table. Each firm has a total fixed
cost of 64 and a sunk fixed cost of 48.
Q
1
2
3
4
5
6
7
8
9
10
11
12
MC
4
6
8
10
12
14
16
18
20
22
24
26
TVC
3
8
15
24
35
48
63
80
99
120
143
168
PROBLEMS
a) Draw a clearly labeled graph of the short-run supply
schedule for this firm. Be sure to indicate the shutdown
price for each firm and to explain your reasoning for the
shape of the supply curve.
b) What is each firm’s producer surplus when the market
price is 16?
c) What is the breakeven price for each firm?
9.31. In a constant-cost industry in which firms have
U-shaped average cost curves, the long-run market supply
curve is a horizontal line. This market supply curve is not
the horizontal sum of individual firms’ long-run supply
curves. In this respect, the long-run market supply curve
differs from the short-run market supply curve, which, in
a constant-cost industry, will equal the horizontal sum of
individual firms’ short-run supply curves. Why does the
derivation of the long-run market supply curve differ
from the derivation of the short-run market supply curve?
9.32. The long-run average cost for production of
hard-disk drives is given by AC(Q) ⫽ 1wr(120 ⫺
20Q ⫹ Q2 ), where Q is the annual output of a firm, w is
the wage rate for skilled assembly labor, and r is the price
of capital services. The corresponding long-run marginal
cost curve is MC(Q) ⫽ 1wr(120 ⫺ 40Q ⫹ 3Q2 ). The
demand for labor for an individual firm is
L(Q, w, r) ⫽
1r(120Q ⫺ 20Q2 ⫹ Q3 )
21w
The price of capital services is fixed at r ⫽ 1.
a) In a long-run competitive equilibrium, how much output will each firm produce?
b) In a long-run competitive equilibrium, what will be
the market price? Note that your answer will be expressed as a function of w.
c) In a long-run competitive equilibrium, how much
skilled labor will each firm demand? Again, your answer
will be in terms of w.
d) Suppose that the market demand curve is given by
D(P) ⫽ 10,000/P. What is the market equilibrium quantity as a function of w?
e) What is the long-run equilibrium number of firms as
a function of w?
f ) Using your answers to parts (c) and (e), determine the
overall demand for skilled labor in this industry as a function of w.
g) Suppose that the supply curve for the skilled labor
used in this industry is ⌫(w) ⫽ 50w. At what value of w
does the supply of skilled labor equal the demand for
skilled labor?
h) Using your answer from part (g), go back through parts
(b), (d), and (e) to determine the long-run equilibrium price,
market demand, and number of firms in this industry.
383
i) Repeat the analysis in this problem, now assuming that
the market demand curve is given by D(P) ⫽ 20,000/P.
9.33. A price-taking firm’s supply curve is s(P) ⫽ 10P.
What is the producer surplus for this firm if the market
price is $20? By how much does producer surplus change
when the market price increases from $20 to $21?
9.34. The semiconductor market consists of 100
identical firms, each with a short-run marginal cost
curve SMC(Q) ⫽ 4Q. The equilibrium price in the
market is $200. Assuming that all of the firm’s fixed
costs are sunk, what is the producer surplus of an individual firm and what is the overall producer surplus for
the market?
9.35. Consider an industry in which chief executive
officers (CEOs) run firms. There are two types of CEOs:
exceptional and average. There is a fixed supply of 100
exceptional CEOs and an unlimited supply of average
CEOs. Any individual capable of being a CEO in this
industry is willing to work for a salary of $144,000 per year.
The long-run total cost of a firm that hires an exceptional
CEO at this salary is
TCE (Q) ⫽ e
144 ⫹ 21 Q2, if Q 7 0
0, if Q ⫽ 0
where Q is annual output in thousands of units and total
cost is expressed in thousands of dollars per year.
The corresponding long-run marginal cost curve is
MCE (Q) ⫽ Q, where marginal cost is expressed as dollars
per unit. The long-run total cost for a firm that hires an
average CEO for $144,000 per year is TCA(Q) ⫽ 144 ⫹
Q2. The corresponding marginal cost curve is MCA(Q) ⫽
2Q. The market demand curve in this market is D(P) ⫽
7,200 ⫺ 100P, where P is the market price and D(P) is
the market quantity, expressed in thousands of units
per year.
a) What is the minimum efficient scale for a firm run by
an average CEO? What is the minimum level of longrun average cost for such a firm?
b) What is the long-run equilibrium price in this industry,
assuming that it consists of firms with both exceptional
and average CEOs?
c) At this price, how much output will a firm with an
average CEO produce? How much output will a firm
with an exceptional CEO produce?
d) At this price, how much output will be demanded?
e) Using your answers to parts (c) and (d), determine how
many firms with average CEOs will be in this industry at
a long-run equilibrium.
f ) What is the economic rent attributable to an exceptional CEO?
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CHAPTER 9
P E R F E C T LY C O M P E T I T I V E M A R K E T S
g) If firms with exceptional CEOs hire them at the reservation wage of $144,000 per year, how much economic
profit do these firms make?
APPENDIX:
h) Assuming that firms bid against each other for the services of exceptional CEOs, what would you expect their
salaries to be in a long-run competitive equilibrium?
Profit Maximization Implies Cost Minimization
In Chapters 7 and 8, we studied decision making by firms that chose an input combination to minimize the total cost of producing a given level of output. In this chapter,
we studied the output choice of a price-taking firm seeking to maximize profit. How
are these analyses related?
Intimately. In particular, profit-maximizing output choice implies cost-minimizing
input choices, or in short, profit maximization implies cost minimization. To develop this
point, note that we could study the profit-maximization problem of a price-taking
firm in two ways:
• The input choice method: We could view the firm as choosing inputs (e.g., quantities
of capital and labor) to maximize profits, recognizing that these input choices
determine the firm’s output through the production function.
• The output choice method: We could view the firm as first choosing output and then
choosing input quantities to minimize total costs, given the selected output level.
We used the output choice method in this chapter. To persuade you that profit maximization implies cost minimization, we will show you here that the input choice
method implies that a profit-maximizing firm must produce its output with a costminimizing input combination. That, in turn, implies that the output choice method
and the input choice method, though analytically different, are equivalent approaches
to analyzing the behavior of a profit-maximizing firm.
Suppose that a firm uses two inputs, capital and labor. Input prices are w and r, respectively. The firm’s production function is Q ⫽ f (L, K ). This firm is a price taker in
the output and input markets (i.e., it takes as given the market price P and the input
prices w and r). The firm chooses quantities of its inputs, L and K, recognizing that
output is determined through the production function f (L, K ). We can thus state the
firm’s profit-maximization problem this way:
max p(L, K ) ⫽ Pf (L, K ) ⫺ wL ⫺ rK
(L,K )
The term Pf (L, K ) is the firm’s total revenue (i.e., market price multiplied by the volume
of output). The last two terms are the total labor costs and total capital costs, respectively.
The expression (L, K ) denotes the firm’s total profit as a function of its choices of labor
and capital.
Profit maximization implies two conditions:
0f
0p
w
⫽P
⫺w⫽01P⫽
0L
0L
MPL
(A9.1)
0f
0p
r
⫽P
⫺r⫽01P⫽
0K
0K
MPK
(A9.2)
A P P E N D I X : P R O F I T M A X I M I Z AT I O N I M P L I E S C O S T M I N I M I Z AT I O N
In writing these expressions, we have used the notation for marginal product that we
introduced in Chapter 6 and used frequently in Chapter 7.
These two conditions say that a profit-maximizing firm will choose its inputs so
that (1) the additional output that the firm gets from every additional dollar spent on
labor (i.e., MPL /w) equals the reciprocal of market price and (2) the additional output
that the firm gets from every additional dollar spent on capital (i.e., MPK /r) also
equals the reciprocal of market price. This implies that, given the profit-maximizing
input choices,
MPL
MPK
⫽
w
r
(A9.3)
But this is the condition for cost minimization derived in Chapter 7. Thus, of the
many input combinations that the firm might use to produce its output, condition
(A9.3) tells us that the profit-maximizing firm employs the cost-minimizing one.
Thus, profit maximization implies cost minimization.
385
10
COMPETITIVE
MARKETS:
APPLICATIONS
10.1
T H E I N V I S I B L E H A N D, E X C I S E TA X E S,
AND SUBSIDIES
APPLICATION 10.1
Gallons and Dollars: Gasoline
Taxes
10.2
PRICE CEILINGS AND FLOORS
Who Gets the Housing
with Rent Controls?
APPLICATION 10.3 Scalping Super Bowl Tickets
on the Internet
APPLICATION 10.4 Ceilings and Shortages: Food
in Venezuela
APPLICATION 10.2
10.3
P R O D U C T I O N Q U OTA S
APPLICATION 10.5
Quotas for Taxicabs
APPLICATION 10.6
A Bailout of the King
10.4
P R I C E S U P P O RT S I N T H E
AG R I C U LT U R A L S E C TO R
of Cheeses
10.5
I M P O RT Q U OTA S A N D TA R I F F S
APPLICATION 10.7
Sweet Deal: The U.S. Sugar
Quota Program
APPLICATION 10.8
APPLICATION 10.9
Dumping
Tariffs, Tires, and Trade Wars
Is Support a Good Thing?
Price and income support programs are commonplace in the world. In the United States, major agricultural
programs have been around since the 1930s. Government expenditures on these programs have ranged in the
billions of dollars annually, especially prior to 1996, when Congress passed a major farm bill that eliminated or
386
reduced many of the program benefits.1 Historically, Congress has required the Department of Agriculture
to support the prices of about 20 commodities, including sugar (sugar cane and beets), cotton, rice, feed
grains (including corn, barley, oats, rye, and sorghum), peanuts, wheat, tobacco, milk, soybeans, and various
types of oil seeds (such as sunflower seeds, and mustard seeds). During the fiscal years between 1983 and
1992, government expenditures on agricultural programs like the ones described here were more than
$140 billion. The most recent farm bill, the Food, Conservation, and Energy Act of 2008, builds on earlier
legislation to provide an array of programs that support the prices of agricultural products and increase the
income of America’s farmers.
Price support programs can take many forms. For example, under “acreage limitation programs”
wheat or feed grain farmers agree to restrict the number of acres they plant. In exchange, the government
gives the farmers an option to sell their crops to the government at a guaranteed price. Farmers are not
required to sell their crops to the government and would not do so if the market price exceeds the guaranteed price. But a farmer will take the option to sell to the government if the market price is lower than
the guaranteed price. Further, because an acreage limitation program reduces the amount of the crop on
the market, the market price is higher than it otherwise would be.
Other programs have supported prices for other commodities. For example, the government has supported the price of peanuts by establishing “poundage quotas,” limiting the quantity of edible peanuts
that a farmer could sell. For many years domestic sugar producers have relied on restrictive import quotas
to raise sugar prices in the United States. The government has also supported tobacco prices by restricting
production to certain farms and by limiting the amounts that those farms could produce.
Since there are many small consumers and producers of agricultural commodities, agricultural markets are
often good examples of perfectly competitive markets. Absent price supports, the forces of supply and demand
would lead to a competitive equilibrium and an economically efficient allocation of agricultural resources.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Analyze the consequences of many forms of government intervention in perfectly competitive markets,
including the impositions of excise taxes, subsidies to producers, price ceilings, price floors, production
quotas, and import tariffs and quotas.
• Explain how government intervention creates deadweight losses in perfectly competitive markets as
economic resources are reallocated.
• Show how intervention affects the distribution of
income and the net benefits to consumers and producers, typically making some people better off
while leaving others worse off.
• Employ economic analysis to understand the forces
and issues underlying public policy discussions about
government intervention in many kinds of competitive markets.
1
Some farm program benefits were restored or increased in a farm bill passed by Congress in 2002.
387
388
CHAPTER 10
10.1
Before we turn to the analysis of specific government interventions, it is important to
THE INVISIBLE
H A N D, E X C I S E
TA X E S, A N D
SUBSIDIES
partial equilibrium
analysis An analysis
that studies the determination of equilibrium price
and output in a single market, taking as given the
prices in all other markets.
general equilibrium
analysis An analysis
that determines the equilibrium prices and quantities
in more than one market
simultaneously.
externality The effect
that an action of any decision maker has on the wellbeing of other consumers
or producers, beyond the
effects transmitted by
changes in prices.
C O M P E T I T I V E M A R K E T S : A P P L I C AT I O N S
preview how we will be conducting our analysis. In this chapter, we will use a partial
equilibrium approach, usually focusing on only a single market. For example, we may
examine the effect of rent controls on the market for housing. A partial equilibrium
approach will not allow us to ask how rent controls affect prices in other markets, including the market for housing that is not rented and the markets for furniture, automobiles, and computers. To examine how a change in one market affects all markets
simultaneously, we would need to employ a general equilibrium model. A general
equilibrium analysis determines the equilibrium prices and quantities in all markets
simultaneously. We will introduce you to this more complex form of analysis in
Chapter 16. The conclusions we draw from a partial equilibrium analysis may not
always be the same as those found with a general equilibrium approach. Nevertheless,
a partial equilibrium framework can often be used to gain important insights about the
primary effects of government intervention.
In this chapter we examine markets that would be perfectly competitive absent
government intervention. As we observed in Chapter 9, in a competitive market all
producers and consumers are fragmented; that is, they are so small in the market that
they behave as price takers. If decision makers have the ability to influence the price in
the market, we cannot use supply and demand analysis. Instead, we would need to apply
an appropriate model of market power, such as the ones discussed in Chapters 11–14.
As we also learned in Chapter 9, in a perfectly competitive market consumers
have perfect information about the nature of the product being provided, as well as
the price of the product. Sometimes governments intervene in markets because consumers are unable to gather enough information about the products in the market.
For example, the health care sector would seem to have a competitive structure, with
many providers and consumers of health care services. Yet health care products, including medication and medical procedures, can be so complex that the average consumer finds it difficult to make informed choices. Government intervention in this
sector is often designed to protect consumers in such a complicated market.
Furthermore, in perfectly competitive markets there are no externalities.
Externalities are present in a market if the actions of either consumers or producers
lead to costs or benefits that are not reflected in the price of the product in that market. For example, a production externality will be present if a producer pollutes the
environment. Pollution creates a social cost that might be ignored by a producer
absent government intervention. A consumption externality exists when the action of
an individual consumer imposes costs on, or leads to benefits for, other consumers.
For example, zoning ordinances in housing markets are often intended to ensure that
consumers of housing do not undertake activities that reduce the value of property
owned by others in a neighborhood. In this chapter we do not consider the effects of
externalities; instead, we will address them in Chapter 17.
Finally, throughout this chapter we use consumer surplus to measure how much
better off or worse off a consumer is when intervention affects the price in the market. As we showed in Chapter 5, when income effects are negligible (as they typically
would be for goods that represent a small fraction of a consumer’s budget), changes in
consumer surplus will often serve as a good measure of the impact of price changes on
the well-being of consumers. However, we also saw in Chapter 5 that consumer surplus
may not always be a good way to measure the impact of a price change on a consumer.
For goods with large income effects it may be important to measure the effects of
price changes on consumers by examining compensating or equivalent variations instead of using changes in consumer surplus.
1 0 . 1 THE INVISIBLE H A N D, E X C I S E TA X E S, A N D S U B S I D I E S
389
THE INVISIBLE HAND
One of the remarkable features of a perfectly competitive market is this: In equilibrium, a competitive market allocates resources efficiently. Figure 10.1 illustrates this
point. In a competitive equilibrium, the market price is $8, with 6 million units per
year exchanged in the market (point R). The sum of consumer and producer surplus
will be VRW, the area below the demand curve D and above the supply curve S, or $54
million per year.
Why is it economically efficient for the market to produce 6 million units? Let’s
answer this question by asking why it is not efficient to produce some other level of
output. For example, why is it not efficient for the market to produce only 4 million
units? The demand curve tells us that there is a consumer who is willing to pay $12
for the 4 millionth unit. Yet the supply curve reveals that it only costs society $6 to
produce that unit. (Remember, the supply curve indicates the marginal cost of producing the next unit in the market.) Thus, total surplus would be increased by $6 (i.e.,
$12 $6) if the 4 millionth unit is produced. When the demand curve lies above the
supply curve, total surplus will increase if another unit is produced. If output is expanded from 4 to 6 million units, total surplus will increase by area RNT, or $6 million.
Is it efficient for the market to produce 7 million units? The demand curve indicates that the consumer of the last unit is willing to pay $6. But the supply curve shows
that it costs an extra $9 to produce that unit. Thus, total surplus would be decreased by
$3 (i.e., $6 $9) if the 7 millionth unit is produced. When the demand curve lies
below the supply curve, total surplus can be increased by cutting back the quantity of
the good produced. If output is cut back from 7 to 6 million units, total surplus will
increase by area RUZ, or $1.5 million.
To sum up, any production level other than 6 million units per year will lead to a
total surplus that is less than $54 million. It follows that the efficient (total surplusmaximizing) level of output is the one determined by the intersection of the supply
and demand curves, that is, the perfectly competitive equilibrium!
$20
V
Price (dollars per unit)
Consumer surplus
Producer surplus
N
$12
$9
$8
S
R
A
$6
$2
U
Z
T
W
D
4
6
7
Quantity (millions of units per year)
10
FIGURE 10.1 Economic Efficiency in a Competitive
Market
In a competitive equilibrium the market price is $8 per
unit and the quantity exchanged is 6 million units.
Consumer surplus is area AVR ($36 million), and producer
surplus is area AWR ($18 million). The supply curve indicates that the marginal cost of producing the 6 millionth
unit is $8. The market is allocating resources efficiently
because every consumer willing to pay at least the marginal cost of $8 is receiving the good, and every producer
who wants to supply the good at that price is doing so.
The sum of consumer and producer surplus ($54 million)
is as large as it can be given the supply and demand
curves.
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C O M P E T I T I V E M A R K E T S : A P P L I C AT I O N S
This brings us to a second major lesson. In a perfectly competitive market, each
producer acts in its own self-interest, deciding whether to be in the market and, if so,
how much to produce to maximize its own producer surplus. Further, each consumer
also acts in his or her own self-interest, maximizing utility to determine how many
units of the good to buy. There is no omniscient social planner telling producers and
consumers how to behave so that the efficient level of output is produced. Nevertheless,
the output produced in a perfectly competitive market is the one that maximizes net economic
benefits (as measured by the sum of the surpluses). As Adam Smith described it in his
classic treatise in 1776 (An Inquiry into the Nature and Causes of the Wealth of Nations),
it is as though there is an “Invisible Hand” guiding a competitive market to the efficient level of production and consumption.2
E X C I S E TA X E S
An excise tax is a tax on a specific commodity, such as gasoline, alcohol, tobacco, or
airline tickets. Economists often use a partial equilibrium model to study the effects
of an excise tax on a competitive market. For example, we might ask how a gasoline
tax will affect the price consumers pay for gasoline, as well as the price producers receive. A partial equilibrium analysis of the gasoline market will treat the prices of
other goods (such as automobiles, tires, and even ice cream) as constant. However, if
a gasoline tax is imposed, the prices of other goods may change, and the partial equilibrium framework will not capture the effects of those changes.
When there is no tax, the equilibrium in a competitive market will be like the one
depicted in Figure 10.1. Since the market clears in equilibrium, the quantity supplied
(Q s ) equals the quantity demanded (Q d ). In Figure 10.1 we observe that in equilibrium
Q s Q d 6 million units. With no tax, the price that consumers pay (call this P d )
equals the price producers receive (P s ). In the equilibrium illustrated in the figure,
P s P d $8 per unit.
Suppose the government imposes an excise tax of $6 per unit. The tax creates a
“tax wedge” between the price consumers pay for the good and the price that sellers
receive. One way to think about this wedge is to imagine a seller has the “administrative responsibility” to collect the tax. (This is how most excise taxes actually work in
practice.) If buyers are charged a market price of, say, $10 per unit, the seller immediately transfers $6 per unit to the government and pockets the remaining $4 per unit
as revenue. More generally, the price P s that a seller receives will be $6 less than the
price P d that a buyer pays, P s P d 6, or equivalently, P d P s 6. This relationship holds for a tax of any amount: With a tax of T per unit (T $6 in this example),
P d P s T.
In a market with an upward-sloping supply curve and a downward-sloping demand curve, the effects of an excise tax are as follows:
• The market will underproduce relative to the efficient level (i.e., the amount
that would be supplied with no tax).
• Consumer surplus will be lower than with no tax.
2
Adam Smith, An Inquiry into the Nature and Causes of the Wealth of Nations, printed for W. Strahan
and T. Cadell, London, 1776.
1 0 . 1 THE INVISIBLE H A N D, E X C I S E TA X E S, A N D S U B S I D I E S
$20
391
V
Price (dollars per unit)
S + $6
M
$12
$8
R
A
$6
N
$4
$2
S
J
E
$10
F
W
D
2
4 5
6
7
Quantity (millions of units per year)
10
FIGURE 10.2 Equilibrium with an Excise Tax
If the government imposes an excise tax of $6 per unit,
the curve labeled S $6 shows what quantity producers
will offer for sale when the price charged to consumers
covers the marginal production cost plus the tax. The intersection of the demand curve D and the S $6 curve
determines the equilibrium quantity, 4 million units.
Consumers pay $12 per unit (point M), the government
collects the $6 tax on each unit sold, and producers receive
a price of $6 (point N).
• Producer surplus will be lower than with no tax.
• The impact on the government budget will be positive because tax receipts are
collected. The tax receipts are part of the net benefit to society because they
will be distributed to people in the economy.
• The tax receipts will be less than the decrease in consumer and producer surplus.
Thus, the tax will cause a reduction in net economic benefits (a deadweight
loss—see discussion below).
One way to see the effect of the tax is to draw a new curve that adds the amount
of the tax vertically to the supply curve—for example, the curve labeled S $6 in
Figure 10.2. We shift the supply curve upward vertically by $6 because the impact of
the excise tax is “as if ” every seller’s marginal cost has increased by $6 per unit. This
new “as if ” supply curve tells us how much producers will offer for sale when the price
charged to consumers covers the marginal cost of production on the actual supply
curve plus the $6 tax. For example, if the price including tax is $10, producers offer
2 million units for sale (point E in Figure 10.2). When consumers pay a market price
of $10 per unit, producers receive only $4 after the tax is deducted from the sales price.
Point F on the actual supply curve S indicates that 2 million units will be offered for
sale when the producer receives the net after-tax price of $4.
Figure 10.2 indicates that the market will not clear if consumers pay a price P d
$10. At that price consumers want to buy 5 million units ( point J ), but producers want
to sell only 2 million units (point E). There would be an excess demand of 3 million
units (the horizontal distance between points E and J ).
The equilibrium with the tax is determined at the intersection of the demand
curve and the “as if ” supply curve, S $6 (point M), where the market-clearing
quantity is 4 million units and consumers pay P d $12. The government collects its
$6 tax on each unit produced, and producers receive a price P s $6 (point N ).
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CHAPTER 10
C O M P E T I T I V E M A R K E T S : A P P L I C AT I O N S
$20
Area
Price (dollars per unit)
S + $6
A
P d = $12
S
B
C
$8
Size (dollars/year)
$16 million
8 million
8 million
4 million
2 million
8 million
8 million
E
F
G
P s = $6
A
B
C
E
F
G
H
H
$2
D
4
6
Quantity (millions of units per year)
With No Tax
10
With Tax
Impact of Tax
Consumer surplus
A+B+C+E
($36 million)
A ($16 million)
–B – C – E
(–$20 million)
Producer surplus
F + G + H ($18 million)
H ($8 million)
–F – G (–$10 million)
Government receipts from tax
zero
B + C + G ($24 million)
B + C + G ($24 million)
Net benefits (consumer surplus +
producer surplus + government
receipts)
A+B+C+E+
F+G+H
($54 million)
A+B+C+G+H
($48 million)
–E – F
(–$6 million)
Deadweight loss
zero
E + F ($6 million)
E + F ($6 million)
FIGURE 10.3 Impact of a $6 Excise Tax
With no tax, the sum of consumer and producer surplus is $54 million, the maximum net benefit
possible in this market. The excise tax of $6 reduces consumer surplus by $20 million, reduces
producer surplus by $10 million, generates government tax receipts of $24 million, and
reduces the net benefit by $6 million (the deadweight loss).
deadweight loss A
reduction in net economic
benefits resulting from an
inefficient allocation of
resources.
Now we can compare the equilibria with and without the excise tax,3 using
Figure 10.3 to calculate the consumer surplus, producer surplus, government receipts
from the tax, net economic benefits, and deadweight loss (the potential net economic
benefit that no one captures when the tax is imposed—neither producers, nor consumers, nor the government).
With no tax, consumer surplus is the area below the demand curve D and above
the price consumers pay ($8) (consumer surplus areas A B C E $36 million
3
The comparison of the market with and without the tax is an exercise in comparative statics, as described
in Chapter 1. The exogenous variable is the size of the tax, which changes from zero to $6 per unit. We
can ask how various endogenous variables (such as the quantity exchanged, the price producers receive,
and the price consumers pay) change as the size of the tax varies.
393
1 0 . 1 THE INVISIBLE H A N D, E X C I S E TA X E S, A N D S U B S I D I E S
per year). Producer surplus is the area above the actual supply curve S and below the
price producers receive (also $8) (producer surplus areas F G H $18 million
per year). There are no tax receipts, so the net economic benefit is $54 million per
year (consumer surplus producer surplus), and there is no deadweight loss.
With the tax, consumer surplus is the area below the demand curve and above the
price consumers pay (P d $12) (consumer surplus area A $16 million per year).
What about producer surplus? The producer surplus on a unit sold is equal to the difference between the net after-tax price that sellers receive (P s $6) and the marginal
cost of that unit. Because it is the actual supply curve S that shows the relationship between the net after-tax price and the quantity supplied, we compute the producer surplus as the area above the actual supply curve S and below the $6 net after-tax price
that producers receive (P s ) (producer surplus area H $8 million per year). Tax receipts are the number of units sold (4 million) times the tax per unit ($6) (tax receipts
the rectangle consisting of areas B C G $24 million per year). The net economic benefit is $48 million per year (consumer surplus producer surplus tax
receipts), so the deadweight loss is $6 million per year (net economic benefit with no
tax net economic benefit with tax $54 million $48 million).
The deadweight loss of $6 million arises because the tax reduces consumer surplus by $20 million and producer surplus by $10 million (equals $30 million total),
while generating tax receipts of only $24 million ($24 million $30 million $6
million). In Figure 10.3, the deadweight loss is the sum of areas E ($4 million per year)
and F ($2 million per year), both of which were part of the net benefit with no tax.
Area E was part of consumer surplus and area F was part of producer surplus, and both
of these benefits disappeared because the tax caused consumers to reduce their purchases
and producers to reduce their output, from 6 million units to 4 million units.
The potential net economic benefit is constant and is equal to the sum of consumer surplus, producer surplus, tax receipts, and deadweight loss (in this case, $54
million). The actual net economic benefit, however, decreases by an amount equal to
the deadweight loss. All this is shown in the following table:
Consumer
Surplus
Producer
Surplus
Tax
Receipts
Deadweight
Loss
Net Economic
Benefit
With No Tax
$36 million
$18 million
0
0
Potential: $54 million
Actual: $54 million
With Tax
$16 million
$8 million
$24 million
$6 million
Potential: $54 million
Actual: $48 million
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 0 . 1
D
Impact of an Excise Tax
In this exercise we determine the equilibrium prices and quantities in Figure 10.3, using algebra.
The demand and supply curves in Figure 10.3 are as follows:
Q d 10 0.5P d
Qs e
2 P s, when P s 2
0, when P s 6 2
where Q d is the quantity demanded when the price
consumers pay is P d, and Q s is the quantity supplied
when the price producers receive is P s. The last line of
the supply equation indicates that nothing will be supplied if the price producers receive is less than $2 per
unit. Thus, for prices between zero and $2, the supply
curve lies on the vertical axis.
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Problem
(a) With no tax, what are the equilibrium price and
quantity?
(b) Suppose the government imposes an excise tax of
$6 per unit. What will the new equilibrium quantity
be? What price will buyers pay? What price will sellers
receive?
Solution
(a) With no tax, two conditions must be satisfied:
(i) P d ⫽ P s (there is no tax wedge). Since there is only
one price in the market, let’s call it P *.
(ii) Also, the market clears, so that Q d ⫽ Q s.
Together these conditions require that 10 ⫺ 0.5P* ⫽
⫺2 ⫹ P*, so the equilibrium price P* ⫽ $8 per unit. The
equilibrium quantity can be found by substituting P* ⫽
$8 into either the supply or demand equation. If we use
the demand equation, we find that the equilibrium
quantity Q d ⫽ 10 ⫺ 0.5(8) ⫽ 6 million units.
(b) With a $6 excise tax, there are two conditions that
must be satisfied:
(i) P d ⫽ P s ⫹ 6: there is a tax wedge between the
market price P d consumers pay and the net after-tax
price P s that sellers receive.
(ii) Also, the market clears, so that Q d ⫽ Q s, or
10 ⫺ 0.5P d ⫽ ⫺2 ⫹ P s.
Thus 10 ⫺ 0.5(P s ⫹ 6) ⫽ ⫺2 ⫹ P s, so the price producers receive P s ⫽ $6 per unit. The price consumers
pay P d ⫽ P s ⫹ $6 ⫽ $12 per unit. The equilibrium
quantity can be found by substituting P d ⫽ $12 into the
demand equation: Q d ⫽ 10 ⫺ 0.5P d ⫽ 10 ⫺ 0.5(12) ⫽
4 million units. (Alternatively, we could have substituted
P s ⫽ $6 into the supply equation.)
Similar Problems:
10.2, 10.6, 10.10, 10.17, 10.21
I N C I D E N C E O F A TA X
incidence of a tax A
measure of the effect of a
tax on the prices consumers
pay and sellers receive in a
market.
In a market with an upward-sloping supply curve and a downward-sloping demand curve,
an excise tax will increase the market price that consumers pay but will decrease the net
after-tax price that sellers receive. Which price will change more as a result of the tax: the
market price paid by buyers or the net, after-tax price received by sellers? In LearningBy-Doing Exercise 10.1, the price consumers pay increases by $4 (rising from $8 to $12).
The price producers receive falls by $2 (decreasing from $8 to $6). The incidence of
a tax is the effect that the tax has on the prices consumers pay and sellers receive in a
market. The incidence, or burden, of the tax is shared by both consumers and producers (in Learning-By-Doing Exercise 10.1, the larger share is borne by consumers).
The incidence of a tax depends on the shapes of the supply and demand curves.
Figure 10.4 illustrates two cases. In both cases the equilibrium price with no tax is $30
per unit. However, the effects of a tax of $10 are quite different in the two markets.
In Case 1 the demand curve is relatively inelastic, and the supply curve is quite
elastic. The tax increases the amount consumers pay by $8 and reduces the amount
producers receive by $2. The price change resulting from the tax is larger for consumers because demand is comparatively inelastic.
In Case 2 the supply curve is relatively inelastic, while the demand curve is comparatively elastic. Therefore, the tax has a larger impact on producers, decreasing the
price they receive by $8, while increasing the price consumers pay by only $2.
As shown in these two cases, a tax will have a larger impact on consumers if demand
is less elastic than supply at the competitive equilibrium, and a larger impact on producers if the reverse is true. At least for small price changes, it is reasonable to assume that
the demand and supply curves have approximately constant own-price elasticities, ⑀Q d,P
and ⑀Q s,P, which means we can summarize the quantitative relationship between the incidence of a tax and the price elasticities of supply and demand as follows:
⑀Q s,P
¢P d
⑀Q d,P
¢P s
(10.1)
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1 0 . 1 THE INVISIBLE H A N D, E X C I S E TA X E S, A N D S U B S I D I E S
S
$38
$30
$28
Case 1:
Demand is relatively
inelastic compared
with supply
S + $10
S
Price (dollars per unit)
Price (dollars per unit)
S + $10
$32
$30
$22
D
Case 2:
Supply is relatively
inelastic compared
with demand
D
(a)
Quantity with
no tax
Quantity with tax
Quantity
(b)
Quantity with
no tax
Quantity with tax
FIGURE 10.4 Incidence of a Tax
In Case 1, where the demand curve is relatively inelastic, the incidence of the $10 tax is
borne primarily by consumers. In Case 2, where the supply curve is relatively inelastic, the
incidence of the tax is borne primarily by producers.
Equation (10.1) tells us that the impact of the price change on consumers and producers will be equal when the absolute values of the price elasticities are the same (remember that the price elasticity of demand is negative and the price elasticity of supply
is positive).4 For example, if ⑀Q d,P ⫽ ⫺0.5 and ⑀Qs,P ⫽ ⫹0.5, then ⌬P d/⌬P s 1. In
other words, if a tax of $1 were imposed, the price consumers pay would rise by $0.50,
while the price producers receive would fall by $0.50.
Now suppose that supply is relatively elastic compared with demand (e.g.,
⑀Q d,P ⫽ ⫺0.5 and ⑀Q s,P ⫽ 2.0). Then ¢P d/ ¢P s 4. In this case, the increase in the
price consumers pay will be four times as much as the decrease in the price producers
receive. Thus, if an excise tax of $1 were imposed, the price consumers pay would rise
by $0.80, while the price producers receive would fall by $0.20. The incidence of the
tax is therefore primarily borne by consumers.
Equation (10.1) explains much about the impact of federal and state taxes on
many markets. For example, the demands for goods such as alcohol and tobacco are
quite inelastic, while their supply curves are comparatively elastic. Thus, the incidence
of an excise tax falls more on consumers in these markets than on producers.
4
To see why equation (10.1) is true, consider the effect of a small tax in a market. Suppose that the
equilibrium price and quantity in the market with no tax are, respectively, P * and Q*. For a
small tax, ⑀Q d,P ⫽ (¢Q/Q*)/(¢P d/P*), which can be written as ¢Q/Q* ⫽ (¢P d/P*)⑀Q d,P. Similarly,
⑀Q s,P ⫽ (¢Q/Q*)/(¢P s/P*), which means that ¢Q/Q* ⫽ (¢P s/P*)⑀Q s,P. Because the market will clear,
a tax will reduce the quantity demanded and supplied by the same amount (¢Q/Q*). This requires that
(¢P d/P*)⑀Q d,P ⫽ (¢P s/P*)⑀Q s,P, which can be simplified to equation (10.1).
Quantity
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C O M P E T I T I V E M A R K E T S : A P P L I C AT I O N S
10.1
Gallons and Dollars: Gasoline Taxes
In 2009, about 140 billion gallons of gasoline were
purchased annually in the United States. Consumer
prices for gasoline fluctuate a great deal over time
and vary by region, but the average price consumers
paid at the pump (P d) was about $2.65 per gallon.
Taxes on gasoline are often imposed at the federal
level, but also by state and local governments. Thus,
the taxes vary by region. In 2009 the federal tax was
18.4 cents per gallon, and the average state and local
tax was just under 22 cents per gallon. Thus, the total
tax per gallon averaged about 40 cents per gallon.
In a “back-of-the-envelope” exercise, let’s assume
that the tax on gasoline (T ) was $0.40 per gallon. This
means that the price producers received (P s ) was
about $2.25 per gallon. Studies have shown that in
the intermediate run (say, two to five years) the ownprice elasticities of demand and supply are about
⑀Qd,P ⫽ ⫺0.5 and ⑀Qs,P ⫽ ⫹0.4. Using the information
about the current equilibrium, let’s examine two
questions:
1. What quantities and prices would we anticipate
if the taxes were removed?
FIGURE 10.5
Effects of a Gasoline Tax
With an excise tax of $0.40 per gallon,
consumers pay about $2.65 per gallon (at
point R), and producers receive about $2.25
per gallon (at point W ). If there were no
tax, the equilibrium price would be about
$2.46 per gallon (at point E). The incidence
of the tax is shared nearly equally by consumers and producers.
Price (dollars per gallon)
2. By how much do gasoline tax revenues rise for
each one-cent increase in the gasoline tax?
In this application we assume that the demand
and supply curves are both linear and that the elasticities are correct at the equilibrium with the excise tax
of $0.40 per gallon. Let’s begin by determining the
equation of the demand curve, which must pass
through point R in Figure 10.5, where the price is
$2.65 and the quantity (measured in billions of gallons)
is 140. If the demand curve is linear, it has the form:
Qd a bP d
(10.2)
Using the data, let us find the constants a and
b in equation (10.2). By definition, the own-price
elasticity of demand is ⑀Qd,P ⫽ (¢Q/ ¢P)(P d/Qd ). In the
linear demand curve, ¢Q/ ¢P ⫽ ⫺b. Thus, 0.5
b(2.65/140), or b 26.42. Now we know that Qd
a 26.42P d. We can calculate a by using the price and
quantity data at point R. Thus, 140 a 26.42(2.65),
so a 210. The equation of the demand curve is
Q d 210 26.43P d.
The equation of a linear supply curve is:
Q s e fP s
(10.3)
Now let us find e and f. By definition, the ownprice elasticity of supply is ⑀Qs,P ⫽ (¢Q/ ¢P)(P s/Qs). In
equation (10.3), ¢Q/ ¢P f. Thus, at point W in
Figure 10.5, 0.4 f(2.25/140), or f 24.89. Therefore,
Q s e 24.89P s.
S
R
$2.65
E
$2.46
$2.25
Tax of $0.40
per gallon
W
D
140 145
Quantity (billions of gallons of gasoline per year)
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1 0 . 1 THE INVISIBLE H A N D, E X C I S E TA X E S, A N D S U B S I D I E S
We can calculate e by using the price and quantity data at point W. Thus, 140 e 24.89(2.25), so
e 84. Thus, the equation of the supply curve is Q s
84 24.89P s.
The supply and demand curves are drawn in
Figure 10.5. If there were no taxes, the equilibrium
would be at point E, where the equilibrium price
P* P s P d (there is no tax wedge). Since the market clears (Q s Q d ), we know that 210 26.43P*
84 24.89P*, so the equilibrium price is P* ⬇ 2.46. With
no tax, about 145 billion gallons of gas would be sold.
The incidence of the current tax (T $0.40 per
gallon) is almost evenly shared by consumers and
producers. This is not surprising because the elasticities of supply and demand are about the same.
With the tax, consumers pay $2.65 instead of $2.46
per gallon, while producers receive $2.25 instead
of $2.46.
We can repeat Learning-By-Doing Exercise 10.1
to find how different levels of the gasoline tax will
affect the quantity sold, the prices paid by consumers
and received by producers, and the revenues from
gasoline taxes. The following table shows the results
of this exercise (the calculations are not shown, but
you should be able to do them yourself) for taxes
varying between zero and $0.60 per gallon.
Tax per Gallon
Quantity (billions of
gallons per)
Price Producers
Receive (P s )
Price Consumers
Pay(P d )
Tax Revenues (billions of
dollars per year)
$0.00
$0.10
$0.20
$0.30
$0.40
$0.50
$0.60
145.1
143.8
142.6
141.3
140.0
138.7
137.4
$2.46
$2.40
$2.35
$2.30
$2.25
$2.20
$2.15
$2.46
$2.50
$2.55
$2.60
$2.65
$2.70
$2.75
$ 0.00
$14.38
$28.51
$42.38
$56.00
$69.36
$82.46
The table indicates that revenues from gasoline
taxes will increase by about $13.36 billion per year
(from $56 billion to about $69.36 billion) if the gasoline tax is raised from its current level of $0.40 per gallon to $0.50 per gallon. Thus, at least near the current
equilibrium, the tax receipts rise about $1.3 billion for
each cent of increase in the tax.
While this example helps us to understand the effects of gasoline taxes, we must remember that a
number of strong assumptions may limit the usefulness
of the model, especially if we try to use it to predict
the effects of very large tax changes. First, the supply
and demand curves are assumed to be linear, even for
large variations in price. While linear approximations
are often quite good for relatively small movements
around the current equilibrium, they may not be accurate for large movements. Second, large changes in
gasoline taxes may have significant effects on prices
in other markets. To study how other markets are affected by changes in the gasoline tax, we would need
to do more than a partial equilibrium analysis of a single market.
SUBSIDIES
Instead of taxing a market, a government might decide to subsidize it. We can think
of a subsidy as a negative tax: buyers pay the market price P d, and the government then
pays each seller a subsidy of $T per unit on top of this price so that the after-subsidy
price received by a seller, P s, is equal to P d T. As you might suspect, many of the
effects of a subsidy are the opposite of the effects of a tax.
• The market will overproduce relative to the efficient level (i.e., the amount that
would be supplied with no subsidy).
• Consumer surplus will be higher than with no subsidy.
• Producer surplus will be higher than with no subsidy.
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• The impact on the government budget will be negative. Government expenditures
on the subsidy constitute a negative net economic benefit since the money to pay
for the subsidy must be collected elsewhere in the economy.
• Government expenditures on the subsidy will be larger than the increase in
consumer and producer surplus. Thus, there will be a deadweight loss from
overproduction.
Figure 10.6 shows how a subsidy of $3 per unit affects the same market depicted
in Figure 10.1. In Figure 10.6, the curve labeled S $3 subtracts the amount of the
subsidy vertically from the supply curve. We shift the supply curve downward vertically
Price (dollars per unit)
$20
A
P s = $9
P * = $8
S
C
B
E
P d = $6
G K
J
S – $3
Subsidy of $3
per unit
produced
F
2
D
Q* = 6 Q1 = 7
Quantity (millions of units per year)
With No Subsidy
10
With Subsidy
Impact of Subsidy
Consumer surplus
A+B
($36 million)
A+B+E+G+K
($49 million)
E+G+K
($13 million)
Producer surplus
E+F
($18 million)
B+C+E+F
($24.5 million)
B+C
($6.5 million)
Impact on government
budget
zero
–B – C – E – G – K – J
(–$21 million)
(–$21 million)
( $54 million)
A+B+E+F–J
($52.5 million)
–J
(– $1.5 million)
zero
J ($1.5 million)
Net benefits
(consumer surplus +
producer surplus –
government expenditures)
Deadweight loss
A+B+E+F
–B – C – E – G – K – J
FIGURE 10.6 Impact of a $3 Subsidy
With no subsidy, the sum of consumer and producer surplus is $54 million, the maximum net
benefit possible in the market. The subsidy increases consumer surplus by $13 million, increases
producer surplus by $6.5 million, has a negative impact of $21 million on the government
budget, and reduces the net benefit by $1.5 million (the deadweight loss).
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1 0 . 1 THE INVISIBLE H A N D, E X C I S E TA X E S, A N D S U B S I D I E S
by $3 because the impact of the subsidy is “as if ” every seller’s marginal cost has decreased by $3 per unit. The “as if ” supply curve S $3 tells us how much producers
will offer for sale when the price received by producers includes the price consumers
pay plus the subsidy.
With no subsidy, equilibrium occurs at the point where the demand curve D and
the supply curve S intersect. At this point, P d P s $8, and the market-clearing
quantity is Q* 6 million units per year. With the subsidy, the equilibrium quantity
is Q1 7 million units per year where the demand curve and the “as if ” supply curve
S $3 intersect. At this quantity, P d $6 and P s $9 (i.e., P d plus the $3 subsidy).
Now we can compare the equilibria with and without the subsidy, using Figure 10.6
to calculate the consumer surplus, producer surplus, impact on government budget,
net economic benefits, and deadweight loss.
With no subsidy, consumer surplus is the area below the demand curve and above
the price consumers pay ($8) (consumer surplus areas A B $36 million per
year). Producer surplus is the area above the supply curve and below the price producers receive (also $8) (producer surplus areas E F $18 million per year). There
are no government expenditures, so the net economic benefit is $54 million per year
(consumer surplus producer surplus), and there is no deadweight loss.
With the subsidy, consumer surplus is the area below the demand curve and above
the price consumers pay (P d $6) (consumer surplus areas A B E G K
$49 million per year). Producer surplus is the area above the actual supply curve S and
below the after-subsidy price producers receive (P s $9) (producer surplus areas
B C E F $24.5 million per year). Government expenditures are the number of
units sold (7 million) times the subsidy per unit ($3). (Government expenditures the
rectangle consisting of areas B C E G K J $21 million per year; note
that, in the table within Figure 10.6, this is represented as a negative benefit because
it must be financed by taxes collected elsewhere in the economy.) The net economic
benefit is $52.5 million per year (consumer surplus producer surplus government
expenditures), so the deadweight loss is $1.5 million per year. (Net economic benefit
with no subsidy net economic benefit with subsidy $54 million $52.5 million.)
The deadweight loss of $1.5 million (area J ) arises because the subsidy increases
consumer surplus by $13 million and producer surplus by $6.5 million (equals $19.5
million total), while necessitating government expenditures of $21 million ($19.5 million $21 million $1.5 million). Another way of looking at this is to say that the
deadweight loss arises because the quantity produced rises from 6 million units with
no subsidy to 7 million units with the subsidy. Over that range of output, the supply
curve lies above the demand curve, so net benefits are reduced as each of these units
is produced. Thus net economic benefits are reduced because the subsidy causes the
market to overproduce relative to the efficient level of production.
Similar to the case with an excise tax, the potential net economic benefit is constant
and is equal to the sum of consumer surplus, producer surplus, the impact on the government budget, and deadweight loss, while the actual net economic benefit decreases
by an amount equal to the deadweight loss. All this is shown in the following table:
Consumer
Surplus
Producer
Surplus
Impact on Government
Budget
Deadweight
Loss
Net Economic
Benefit
With No
Subsidy
$36 million
$18 million
0
0
Potential: $54 million
Actual: $54 million
With
Subsidy
$49 million
$24.5 million
$21 million
$1.5 million
Potential: $54 million
Actual: $52.5 million
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E
C O M P E T I T I V E M A R K E T S : A P P L I C AT I O N S
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 0 . 2
D
Impact of a Subsidy
As in Learning-By-Doing Exercise 10.1,
the demand and supply curves are
Q d 10 0.5P d
2 P s, when P s 2
Qs e
0, when P s 6 2
where Q d is the quantity demanded when the price
consumers pay is P d, and Q s is the quantity supplied
when the price producers receive is P s.
Problem
Suppose the government provides a subsidy of $3 per
unit. Find the equilibrium quantity, the price buyers pay,
and the price sellers receive.
(a) There is a subsidy wedge of $3 that makes the
after-subsidy price received by sellers $3 more than
the market price received by buyers: P s P d 3, or
equivalently, P d P s 3.
(b) Also, the market clears, so that Q d Q s, or 10
0.5P d 2 P s.
Thus, 10 0.5(P s 3) 2 P s, so producers
receive a price of P s $9. The equilibrium price consumers pay is P d P s $3 $6 per unit. The equilibrium quantity can be found by substituting P d $6 into
the demand equation: Q d 10 0.5P d 10 0.5(6)
7 million units. (Alternatively, we could have substituted
P s $9 into the supply equation.)
Similar Problems:
10.17, 10.18
Solution
With a $3 subsidy, two conditions must be satisfied in
equilibrium:
10.2
PRICE
CEILINGS
AND FLOORS
S
ometimes a government may impose a price ceiling in a market, such as a maximum
allowable price for food or gasoline. Rent controls provide another common example
of a price ceiling because they specify maximum prices that landlords may charge tenants. Price ceilings will affect the distribution of income and economic efficiency
when they hold the price for a good or service below the level that would be observed
in equilibrium without the ceiling.
In other cases policy makers may impose a floor on the price allowed in a market.
For example, many governments have enacted laws that specify a minimum wage that
must be paid to workers. Legislative bodies often set other kinds of price floors, such
as usury laws (laws that set a minimum interest rate that can be charged for loans).
Price floors are designed to hold the price for a good or service above the level that
would be observed in equilibrium without the floor.
In contrast to the outcomes we observed with excise taxes and subsidies, markets
do not clear with price ceilings and floors. This means that we will need to think carefully about the way the goods or services are allocated as we analyze the effects of price
ceilings and floors on the distribution of income and economic efficiency.
PRICE CEILINGS
If the price ceiling is below the equilibrium price in a market with an upward-sloping
supply curve and a downward-sloping demand curve, the ceiling will have the following effects:
• The market will not clear. There will be an excess demand for the good.
• The market will underproduce relative to the efficient level (i.e., the amount
that would be supplied in an unregulated market).
10.2 PRICE CEILINGS AND FLOORS
• Producer surplus will be lower than with no price ceiling.
• Some (but not all) of the lost producer surplus will be transferred to consumers.
• Because there is excess demand with a price ceiling, the size of the consumer surplus will depend on which of the consumers who want the good are able to purchase it. Consumer surplus may either increase or decrease with a price ceiling.
• There will be a deadweight loss.
Let’s examine the effects of a price ceiling in the form of rent controls. For
decades rent controls have been in force in many cities around the world. Rent controls are legally imposed ceilings on the rents that landlords may charge their tenants.
They often originated as temporary ceilings imposed in the inflationary time of war,
as was the case in London and Paris during World War I, in New York during World
War II, and in Boston and several nearby suburbs during the Vietnam conflict in the
late 1960s and early 1970s.
In 1971 President Richard Nixon imposed wage and price controls throughout
the United States, freezing all rents. After the federal controls expired, many city governments continued to place ceilings on rents. In 1997 William Tucker noted,
“During the 1970s it appeared that rent control might be the wave of the future. . . .
By the mid-1980s, more than 200 separate municipalities nationwide, encompassing
about 20 percent of the nation’s population, were living under rent control. However,
this proved to be the high tide of the movement. As inflationary pressures eased, the
agitation for rent control subsided.”5
Figure 10.7 illustrates the supply and demand curves in the market for a particular type of housing, such as the market for studio apartments in New York City. For
various rental prices the supply curve S shows how many units landlords would be
willing to make available, and the demand curve D indicates how many units consumers would like to rent.
With no rent control, equilibrium occurs at the point where the demand curve
and the supply curve intersect (point V ). At this point, the equilibrium price is P*
$1,600 per month and the market-clearing quantity is Q* 80,000 housing units.
Every consumer willing to pay the equilibrium price (consumers between points Y and
V on the demand curve) will find housing, and every landlord willing to supply housing units at that price will serve the market.
Suppose the government imposes rent controls by setting a maximum rental price
of $1,000 per month. At that price, the market will not clear. Landlords will be willing to supply 50,000 housing units (point W ), while consumers will want to rent
140,000 units (point X ). Thus, rent control has reduced the supply by 30,000 units
(80,000 50,000) and increased the demand by 60,000 units (140,000 80,000), resulting in an excess demand of 90,000 units (30,000 60,000). (Excess demand in the
housing market is commonly referred to as a housing shortage.)
Now we can use Figure 10.7 to calculate the consumer surplus, producer surplus,
net economic benefits, and deadweight loss, with and without rent control.
With no rent control, consumer surplus is the area below the demand curve and
above the price consumers pay ($1,600) (consumer surplus areas A B E ).
Producer surplus is the area above the supply curve and below the price producers receive (also $1,600) (producer surplus areas C F G). The net economic benefit is
5
William Tucker, “How Rent Control Drives Out Affordable Housing,” Cato Policy Analysis, paper
no. 274 (Washington, DC: The Cato Institute, May 21, 1997).
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Y
Rental price (dollars per month)
Y
S
A
B
E
P * = $1,600
PR = $1,000
C
F
G
W
S
A
U
U
B
V
X
D
Z
E
C
F
G
W
V
T
X
H
D
Z
Q s = 50 Q* = 80
Q d = 140
Q s = 50
80 90
Q d = 140
Housing shortage
Housing shortage
Quantity (thousands of housing units)
Quantity (thousands of housing units)
Case 1: Maximum consumer surplus
Case 2: Minimum consumer surplus
With Rent Control
Free Market
(with no
rent control)
Case 1
(maximum
consumer
surplus)
Case 2
(minimum
consumer
surplus)
Impact of Rent Control
Case 1
(maximum
consumer
surplus)
Case 2
(minimum
consumer
surplus)
Consumer surplus
A+B+E
A+B+C
H
C–E
–A – B – E + H
Producer surplus
C+F+G
G
G
–C – F
–C – F
Net benefits
(consumer surplus +
producer surplus)
A+B+C+E+F+G
A+B+
C+G
H+G
–E – F
–A – B – C –
E–F+H
Deadweight loss
zero
E+F
A+B+C+
E+F–H
E+F
A+B+C+E+
F–H
FIGURE 10.7 Impact of Rent Controls
Rent controls require that landlords charge no more than $1,000 per month for housing
units that would rent for $1,600 without rent controls. The graph shows two cases (explained
below). In both cases, producer surplus is equal to area G. Case 1: If all 50,000 available housing units are rented by the consumers with the highest willingness to pay (those between
points Y and U on the demand curve D), consumer surplus under rent control is maximized,
net economic benefits are also maximized, and deadweight loss is minimized. Case 2: If all
50,000 available housing units are rented by the consumers with the lowest willingness to pay
(those between points T and X on the demand curve), consumer surplus under rent control is
minimized, net economic benefits are also minimized, and deadweight loss is maximized.
the sum of consumer surplus and producer surplus (net economic benefit areas A
B C E F G), and there is no deadweight loss.
With rent control, as you can see from Figure 10.7, we will consider two cases,
differing by which consumers actually rent the available housing units: Case 1 maximizes consumer surplus, while Case 2 minimizes consumer surplus. In both cases, the
10.2 PRICE CEILINGS AND FLOORS
A P P L I C A T I O N
403
10.2
Who Gets the Housing
with Rent Controls?
As Figure 10.7 illustrates, because the market does
not clear with rent controls, the consumers who most
value housing will not necessarily be the ones who
actually rent the available units. In Case 1 the consumers who are lucky enough to find housing are
those who value it most (the consumers between
points Y and U on the demand curve). However, Case
2 illustrates the other extreme possibility, with the
available housing instead allocated to the consumers
between points T and X; in this case none of the consumers who most value housing are able to rent it. In
an actual market the available housing might be allocated in many other possible ways, with some of it
rented by people who greatly value housing, and
some by consumers who value it less. What does empirical evidence tell us about the allocation of housing under rent controls?
Edward Glaeser and Erzo Luttmer have studied
the effects of rent controls in New York City, using
census data from 1990. (See E. Glaeser and E. Luttmer,
“The Misallocation of Housing under Rent Controls,”
The American Economic Review, September 2003).
Since rent controls at the time largely excluded build-
ings with fewer than five apartments, the authors
focused on buildings with at least five units.
The authors recognized two ways in which rent
controls might lead to a misallocation of housing.
First, “there is the possibility that apartments are allocated randomly or by some alternative queue-type
mechanism instead of by price. Second, rent control
creates an incentive for people to stay in the same
apartment instead of moving.” Overall, they found
that “approximately 20 percent of the apartments
are in the wrong hands.” These apartments are
rented to consumers who are not in the set of consumers with the highest value for housing (corresponding to the consumers between points Y and U
on the demand curve in Figure 10.7).
Glaeser and Luttmer observed, “Theorists have
long been aware that wage and price controls may
cause the misallocation of goods. However, this insight has, so far, both failed to create an empirical literature or even to penetrate into most economics
textbooks.” Their study examined one rent-controlled
city in one year, and the percentages might well vary
across time and over different cities. However, the
study does suggest that in analyzing the welfare effects of rent controls, it would not be a good idea to
assume that housing is always distributed to consumers who value it the most.
landlords serving the market are the ones between points Z and W on the supply curve,
and the producer surplus they receive is the area above that portion of the supply curve
and below the price they receive (P R $1,000) (producer surplus area G). Thus, with
rent control, producer surplus falls by an amount equal to areas C F. This decline in
producer surplus explains why landlords often strongly oppose rent controls.
Also in both cases, consumers who are lucky enough to get one of the 50,000
available units will pay only $1,000 per month instead of $1,600. The amount of income the producer collects for these units is reduced by area C.
To see how consumer surplus, net economic benefit, and deadweight loss are affected by rent control, we need to recognize that 140,000 consumers will want to rent
housing at $1,000 per month, but only 50,000 units will be available. We will find the
possible range of consumer surplus (i.e., maximum consumer surplus and minimum
consumer surplus) by assuming, in Case 1, that consumers with the highest willingness to pay rent all the available housing units, and in Case 2, that consumers with the
lowest willingness to pay rent all the available housing units.
• Case 1 (maximum consumer surplus). Consumers with the highest willingness to pay
rent all the available housing units (i.e., consumers between points Y and U on the
demand curve). Consumer surplus is the area below the portion of the demand
curve between points Y and U and above the price consumers pay (PR $1,000)
(consumer surplus areas A B C ); this is the maximum possible consumer
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10.3
Scalping Super Bowl Tickets
on the Internet
When the National Football League (NFL) sells tickets
to the Super Bowl, it establishes face values (the
prices printed on the tickets) that are far below the
market prices. The NFL understands that there will be
a large excess demand for tickets sold at face value. It
therefore accepts requests for tickets a year in advance of the event and then chooses the recipients of
the tickets in a random drawing.
In the month before Super Bowl XLIII in Tampa,
Florida, in 2009, tickets with a face value of $800 were
offered for sale on several Internet sites at prices
ranging from $2,350 to $3,750. In other Super Bowls,
markups have been even higher, with market prices as
much as 10 times the face value.
The winners of the random drawing are indeed
lucky. They can use the tickets themselves or resell the
tickets at a handsome profit. The existence of an easily
accessible, active resale market helps move the tickets
ultimately into the hands of people who most highly
value the opportunity to see the game in person.
Two types of transactions costs affect the possibility of resale. First, in some states resale (“scalping”) is
illegal. A law prohibiting resale is likely to be more effective when the penalty for a violation is high and
when the probability of being caught reselling is
high. Even though resale is illegal in many areas, it
may nevertheless be common where penalties are low
or there is little risk of being caught. Second, resellers
incur transactions costs in searching out supplies of
tickets and locating buyers.
In recent years the Internet has lowered both
types of transactions costs considerably. Buyers and
sellers can conduct business from the comfort of
home or the office. With a website, scalpers can
widely advertise tickets at a very low cost and with
less risk of being caught than would be the case if
the transactions took place in the shadow of the
stadium.
If resale involves low transactions costs, total surplus will be close to the maximum possible, as assumed
in Case 1 of Figure 10.7 in the discussion of price ceilings. Part of the surplus may go to middlemen
(scalpers and brokers) instead of the final holders of
the tickets, but the net benefits do not disappear
from the economy.
Of course, scalping typically involves a certain
amount of risk, including the possibility that the tickets are not as desirable as advertised or perhaps are
not valid at all. Those supporting laws against scalping often cite examples of fraud. If the original sellers of tickets or governing authorities are willing to
impose very strict conditions, it may be possible to reduce resale greatly. For example, the seller could put
the buyer’s picture on the ticket (as is often done with
monthly passes on urban transport systems) or write
the buyer’s name on the ticket and require the buyer
to produce a picture I.D. when she uses the ticket (as
airlines often do). However, these measures add significant costs to businesses and to law enforcement efforts and are often difficult to implement.
surplus with rent control. The net economic benefit consumer surplus
producer surplus areas A B C G. The deadweight loss net economic benefit with no rent control net economic benefit with rent control
(areas A B C E F G) (areas A B C G) areas E F. The
deadweight loss arises because rent control has reduced the available housing
supply by 30,000 units, so the consumer surplus represented by area E and the
producer surplus represented by area F have been lost to society.
• Case 2 (minimum consumer surplus). Consumers with the lowest willingness to pay
rent all the available housing units (i.e., consumers between points T and X on the
demand curve,6 which means that consumers between points Y and T on the demand curve will be unable to find housing, despite their willingness to pay more
6
We do not consider consumers to the right of point X on the demand curve because they would not be
willing to rent housing at $1,000 even if they could find it.
405
10.2 PRICE CEILINGS AND FLOORS
than $1,000 per month). Consumer surplus is the area below the portion of the
demand curve between points T and X and above the price consumers pay
(PR $1,000) (consumer surplus area H ); this is the minimum possible consumer surplus with rent control. The net economic benefit consumer surplus
producer surplus areas H G. The deadweight loss net economic benefit
with no rent control net economic benefit with rent control (areas A B
C E F G ) (areas H G ) areas A B C E F H. The
deadweight loss is larger than in Case 1 (by an amount equal to A B C H )
due to the inefficiency in the way in which available housing units are rationed
to consumers.
The two cases just considered define upper and lower limits on the consumer surplus and deadweight loss related to rent controls. The actual consumer surplus and
deadweight loss may be in between the levels in these two polar cases. To find the exact
amounts of consumer surplus and deadweight loss, we would need to know more about
how the available housing is actually allocated. Most textbooks depict the effects of a
price ceiling with a graph like the one in Case 1 of Figure 10.7, assuming that the good
ends up in the hands of consumers with the highest willingness to pay. This assumption is reasonable when consumers can easily resell the good to other consumers with
a higher willingness to pay, but as Application 10.2 suggests, it may not hold in practice, even though they might not be able to obtain the good when it is initially sold.
A P P L I C A T I O N
10.4
Ceilings and Shortages: Food
in Venezuela
7
Price
In 2003, the government of Venezuela imposed price
ceilings on various basic food items as a response to
inflation rates of 30 percent or more per year.7 Hugo
Chavez, Venezuela’s president, has strengthened the
price controls since then in an attempt to maintain
popularity with his primary electoral constituency,
poor citizens. By late 2009, roughly 400 food items had
mandated price ceilings.
Figure 10.8 illustrates the market for white rice in
Venezuela with a price ceiling PR below the price that
would prevail with no constraint, P*. At the price ceiling, the quantity supplied (Q S) will be below the quantity demanded (Q D ), creating a severe shortage of
rice. The deadweight loss caused by this regulation is
the area UVW.
Indeed, Venezuela has been plagued by sporadic
food shortages ever since the price controls were first
imposed. Consumers have had difficulty finding foods
at regulated prices and have often had to wait in long
Y
S
U
V
P*
PR
X
W
Qs
Q*
D
Qd
Shortage of white rice
Quantity of white rice
FIGURE 10.8 Price Ceilings for White Rice
in Venezuela
By law, the price of white rice was set to P R. The unregulated price would be P*. The ceiling induced a
shortage. The deadweight loss is at least as large as
the area bounded by the points UVW.
See, for example, “Venezuela’s Hugo Chavez Tightens State Control of Food Amid Rocketing Inflation
and Food Shortages,” Telegraph, March 4, 2009.
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lines to purchase those foods that were available.
Food companies have complained that the regulated
prices are below their average costs, so that they are
in danger of going out of business. For example, the
price of rice in 2009 was set at 2.15 bolivares per kilogram, while rice companies claim that their cost per
kilogram is about 4.41 bolivares.
In an attempt to avoid the price ceilings, food
companies have attempted to alter their products to
versions that are not regulated. For example, the
price of white rice is regulated, but the price of flavored rice is not. Rice companies altered their product
lines, moving away from white rice toward flavored,
so that they could raise prices. The government then
responded by imposing production quotas on many
food producers to force them to produce more of
foods with price ceilings. Rice companies are now required to have 80 percent of their production sold as
white rice. However, there were still shortages as
food companies limited total production (to Q S in
Figure 10.8). In 2009, the government seized control
of several food processing factories to force increases
in production, including a rice processing plant and
several coffee plants. The government is also contending with increased smuggling of low-priced food
across the border into Colombia.
Before leaving rent controls, we note that government attempts to regulate the
price of a commodity rarely work in a straightforward fashion. For example, when a
shortage develops in the rental market for housing, some landlords may demand key
money, or a fee—that is, an extra payment from a prospective renter—before agreeing
to lease an apartment. Although such payments are illegal, they are difficult to monitor, and renters who are willing to pay more than the rent-controlled price may willingly (though not happily) pay the key money. Landlords may also recognize that with
excess demand, they will be able to find renters even if they allow the quality of the
apartments to deteriorate. Rent control laws often attempt to specify that the quality
should be maintained, but it is quite difficult to write the laws to enforce this intent
effectively. Further, landlords may recognize that they would be better off in the long
run if they can convert apartments under rent control to other uses not subject to
price controls, such as condominiums or even parking lots. Critics of rent controls
often observe that the amounts of housing available have been reduced over time as
owners of controlled housing convert to alternative uses of land.8
We must remember that there are limitations in a partial equilibrium analysis of
the effect of a price ceiling, such as the one in Figure 10.7. If a rent control is imposed
in the market for studio apartments, people who cannot find a studio apartment will
seek another type of housing, such as a larger apartment, a condominium, or even a
house. This will affect the demand for other types of housing and thus the equilibrium
prices in those markets. As the prices of other types of housing change, the demand
for studio apartments may shift, with additional effects on the size of the shortage of
studio apartments, as well as on consumer and producer surplus and deadweight loss.
Calculating these additional effects is beyond the scope of a simple partial equilibrium
analysis, but you should recognize that they may be important.
The unintended consequences of price ceilings are present in many markets other
than housing. For example, in an effort to fight inflation in the 1970s, the Nixon administration imposed price ceilings on domestic suppliers of oil, creating a shortage
of domestic oil. The excess demand for oil led to increased imports of oil. When the
price controls were imposed in 1971, imports constituted only 25 percent of the nation’s
8
See, for example, Denton Marks, “The Effects of Partial-Coverage Rent Control on the Price and
Quantity of Rental Housing,” Journal of Urban Economics 16 (1984): 360–369.
10.2 PRICE CEILINGS AND FLOORS
407
supply. As time passed, the shortage grew substantially. By 1973, imports made up
nearly 33 percent of the total oil consumed in the United States. OPEC countries
recognized the growing dependence on imports in the United States, and they responded by quadrupling the price of imported oil. In the end the domestic price controls contributed to still higher inflation in the United States, working against their
original intent.9
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D
Impact of a Price Ceiling
As in the previous Learning-By-Doing
Exercises in this chapter, the demand and supply curves are
Q d 10 0.5P d
Qs e
s
s
2 P , when P 2
0, when P s 6 2
where Q d is the quantity demanded when the price consumers pay is P d, and Q s is the quantity supplied when
the price producers receive is P s.
Suppose the government imposes a price ceiling of
$6 in the market, as illustrated in Figure 10.9.
Problem
(a) What is the size of the shortage in the market with
the price ceiling? What is the producer surplus?
(b) What is the maximum consumer surplus, assuming
the good is purchased by consumers with the highest
willingness to pay? What is the net economic benefit?
What is the deadweight loss?
(c) What is the minimum consumer surplus, assuming
the good is purchased by consumers with the lowest
willingness to pay? What is the net economic benefit?
What is the deadweight loss?
Solution
(a) With the price ceiling, consumers demand 7 million
units (point X ), but producers supply only 4 million
units (point W ). Thus, the shortage (i.e., the excess demand) is 3 million units, equal to the horizontal distance
between points W and X.
9
Producer surplus is the area above the supply curve
S and below the price ceiling of $6. This is area SWZ
$8 million.
(b) If consumers with the highest willingness to pay
(those between points Y and T on the demand curve D)
purchase the 4 million units available, consumer surplus
will be the area below that portion of the demand curve
and above the price ceiling. This is area YTWS
$40 million.
The net economic benefit is the sum of consumer
surplus ($40 million) and producer surplus ($8 million)
$48 million.
The deadweight loss is the difference between the
net economic benefit with no price ceiling ($54 million)
and the net economic benefit with the price ceiling
($48 million) $6 million.
(c) If consumers with the lowest willingness to pay (those
between points U and X on the demand curve) purchase
the 4 million units available, consumer surplus will be
the area below that portion of the demand curve and
above the price ceiling. This is area URX 16 million.
The net economic benefit is the sum of consumer
surplus ($16 million) and producer surplus ($8 million)
$24 million.
The deadweight loss is the difference between the
net economic benefit with no price ceiling ($54 million)
and the net economic benefit with the price ceiling ($24
million) $30 million.
Similar Problems:
See George Horwich and David Weimer, “Oil Price Shocks, Market Response, and Contingency
Planning” (Washington, DC: American Enterprise Institute, 1984).
10.1, 10.12, 10.13
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Price (dollars per unit)
$20
C O M P E T I T I V E M A R K E T S : A P P L I C AT I O N S
Y
U
$14
T
$12
$8
Price
$6
ceiling
$2
Supply
V
A
X
S
R
W
Z
Demand
3 4
6 7
10
Quantity (millions of units per year)
With Price Ceiling
With No Price
Ceiling
With Maximum
Consumer Surplus
With Minimum
Consumer Surplus
Consumer surplus
area YAV = $36 million
area YTWS = $40 million
area URX = $16 million
Producer surplus
area AVZ = $18 million
area SWZ = $8 million
area SWZ = $8 million
Net benefits (consumer
surplus + producer
surplus)
$54 million
$48 million
$24 million
Deadweight loss
zero
$6 million
$30 million
FIGURE 10.9 Impact of a $6 Price Ceiling
With no price ceiling, the sum of consumer and producer surplus is $54 million, the maximum
net benefit possible in the market. With the price ceiling, producer surplus decreases by $10
million. When consumer surplus is maximized, consumer surplus increases by $4 million and
net benefit decreases by $6 million (the deadweight loss). When consumer surplus is minimized,
consumer surplus decreases by $20 million and net benefit decreases by $30 million (the
deadweight loss).
PRICE FLOORS
When the government imposes a price floor higher than the free-market price, we observe the following effects in a market with an upward-sloping supply curve and a
downward-sloping demand curve:
• The market will not clear. There will be an excess supply of the good or service
in the market.
• Consumers will buy less of the good than they would in a free market.
10.2 PRICE CEILINGS AND FLOORS
• Consumer surplus will be lower than with no price floor.
• Some (but not all) of the lost consumer surplus will be transferred to producers.
• Because there is excess supply with a price floor, the size of the producer surplus
will depend on which of the producers actually supply the good. Producer surplus
may either increase or decrease with a price floor.
• There will be a deadweight loss.
Let’s begin by studying the effects of a minimum wage law. There are many types
of labor in an economy. Some workers are unskilled, while others are highly skilled.
For most types of skilled labor, the minimum wage set by the government will be well
below the equilibrium wage rate in a free market. A minimum wage law will have no
effect in such a market. We therefore focus on the market for unskilled labor, where
the minimum wage requirement may be above the wage level in a free market. (In the
labor market, the producers are the workers who supply the labor, while the consumers are the employers who purchase the labor—i.e., hire the workers.)
Figure 10.10 illustrates the supply and demand curves in the market for unskilled
labor. The vertical axis shows the price of labor, that is, the hourly wage rate, w. The
horizontal axis measures the number of hours of labor, L. The supply curve S shows
how many hours workers will supply at any wage rate. The demand curve D indicates
how many hours of labor employers will hire.
With no minimum wage law, equilibrium occurs at the point where the demand
curve and the supply curve intersect (point V ). At this point, the equilibrium wage rate
is $5 per hour, and the market-clearing quantity of labor is 100 million hours per year.
Every worker willing to supply labor at the equilibrium wage rate (workers between
points Z and V on the supply curve) will find work, and every employer willing to pay
that rate (employers between points Y and V on the demand curve) will be able to hire
all the workers he wants.
Suppose the government enacts a minimum wage law requiring employers to pay
at least $6 per hour. At that wage rate, the labor market will not clear. Employers will
demand 80 million hours of labor (point R), but workers will want to supply 115 million hours (point T ). Thus, the minimum wage law has decreased the demand for
labor by 20 million hours (100 million 80 million) and has caused an excess labor
supply (unemployment) of 35 million hours (115 million 80 million, or the horizontal distance between points T and R). Unemployment measures more than just the
decrease in the demand for labor (20 million hours); rather, it measures the excess supply of labor (35 million hours).
Now we can use Figure 10.10 to calculate the consumer surplus, producer surplus, net economic benefits, and deadweight loss, with and without the minimum
wage law. (Note that Figure 10.10 is divided into two cases, as explained below.)
With no minimum wage, consumer surplus is the area below the demand curve
and above the equilibrium wage rate of $5 per hour. In Figure 10.10, this is areas A
B C E F. Producer surplus is the area above the supply curve and below the
equilibrium wage rate. In Figure 10.10, this is areas H I J. The net economic
benefit is the sum of consumer surplus and producer surplus. In Figure 10.10, this is
areas A B C E F H I J.
With the minimum wage, as you can see from Figure 10.10, we will consider two
cases, differing by which producers (i.e., workers) actually find jobs: Case 1 maximizes
producer surplus, while Case 2 minimizes producer surplus. In both cases, employers
are willing to hire labor up to point R on the demand curve, and the consumer surplus
they receive is the area below that portion of the demand curve and above the rate
409
410
w, wage rate (dollars per hour)
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S
Y
T
B R
C E F G
w = $5
J V
I
H
W
X
Z
wmin = $6
A
35
80
D
100 115
Excess labor supply
L, quantity of labor
(millions of hours per year)
With Minimum Wage
Free Market
(with no
minimum wage)
Case 1
(maximum
producer
surplus)
Case 2
(minimum
producer
surplus)
Impact of Minimum Wage
Case 1
Case 2
Consumer surplus
A+B+C+E+F
A+B
A+B
–C – E – F
–C – E – F
Producer surplus
H+I+J
C+E+H+I
E+F+G+I+J
C+E–J
E+F+G–H
Net benefits
(consumer surplus +
producer surplus)
A+B+C+E+
F+H+I+J
A+B+C+E+
H+I
A+B+E+F+
G+I+J
–F – J
–C – H + G
Deadweight loss
zero
F+J
C+H–G
F+J
C+H–G
FIGURE 10.10 Impact of Minimum Wage Law
A minimum wage law requires employers to pay at least $6 per hour, whereas in a free market
(i.e., with no minimum wage law) the equilibrium wage rate would be $5 per hour. The table
shows two cases (explained below). Consumer surplus is the same in both cases.
Case 1: If the most efficient workers get all the jobs (workers between points Z and W on
the supply curve S), producer surplus with the minimum wage is maximized, net economic
benefits are somewhat reduced, and there is some deadweight loss.
Case 2: If the least efficient workers get all the jobs (workers between points X and T on
the supply curve), producer surplus with the minimum wage is minimized, net economic
benefits are less than in Case 1, and the deadweight loss is greater than in Case 1.
they pay ($6). Thus, with the minimum wage, consumer surplus falls by an amount
equal to areas C E F. This decline in consumer surplus explains why businesses
often strongly lobby policy makers to keep the minimum wage from being raised.
Also in both cases, employers of the 80 million hours hired at the minimum wage
will pay $6 per hour instead of $5 per hour, thereby incurring an extra cost measured
by areas C E.
10.2 PRICE CEILINGS AND FLOORS
To see how producer surplus, net economic benefit, and deadweight loss are
affected by the minimum wage, we need to recognize that all the suppliers of labor
between points Z and T on the supply curve will want to work, but only some of them
will find jobs. We will determine the possible range of producer surplus (i.e., maximum
producer surplus and minimum producer surplus) by assuming, in Case 1, that the
most efficient workers find jobs, and in Case 2, that the least efficient workers find jobs.
• Case 1 (maximum producer surplus). The most efficient workers find jobs (i.e., workers between points Z and W on the supply curve; the other workers, those between points W and T, are unable to find jobs even though they are willing to
work at $6 per hour). Producer surplus is the area above the portion of the supply curve between points Z and W and below the wage rate ($6 per hour) (producer surplus areas C E H I ); this is the maximum possible producer
surplus with the minimum wage. The net economic benefit consumer surplus
producer surplus areas A B C E H I. The deadweight loss net
economic benefit with no minimum wage net economic benefit with the minimum wage (areas A B C E F H I J ) (areas A B C
E H I ) areas F J.
• Case 2 (minimum producer surplus). The least efficient workers find jobs (i.e., workers
between points X and T on the supply curve),10 which means that workers between points Z and X on the supply curve will be unable to find jobs, despite
their willingness to work at $6 per hour. Producer surplus is the area above the
portion of the supply curve between points X and T and below the wage rate
($6 per hour) (producer surplus areas E F G I J); this is the minimum
possible producer surplus with the minimum wage. The net economic benefit
consumer surplus producer surplus areas A B E F G I J.
The deadweight loss net economic benefit with no minimum wage net
economic benefit with the minimum wage (areas A B C E F H
I J ) (areas A B E F G I J ) areas C H G. The deadweight loss is larger than in Case 1 because producer surplus is smaller when
less efficient workers replace more efficient workers.
These two cases define upper and lower limits on the producer surplus and deadweight loss from a minimum wage law. The actual producer surplus and deadweight
loss typically falls in between the levels in these two polar cases, depending on which
workers find the available jobs.
Several simplifying assumptions are important in the analysis of minimum wage
laws. First, we assume that the quality of labor does not change as the minimum wage
rises. It is sometimes suggested that employers are able to hire better workers at
higher wages. If this is the case, the analysis would need to be modified to recognize
that the quality of labor changes as the wage rate rises. Also, a minimum wage law in
one market may affect wage rates in other markets, ultimately affecting the prices of
many goods and services. Finally, it is important to note that our discussion of the effects of a minimum wage law is a partial equilibrium analysis. To analyze the economywide impact of a minimum wage law, one would want to use a general equilibrium
analysis using tools like those presented in Chapter 16.
Empirical studies of the effects of minimum wages in some industries have suggested
that the effects of a minimum wage law may not be as predicted with the competitive
10
We do not consider workers to the right of point T on the supply curve because they would not be
willing to take jobs at a wage of $6 per hour.
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market analysis we have just presented. The competitive market model predicts that
an increase in the minimum wage law should lead to a decrease in employment in a
market with an upward-sloping supply curve and a downward-sloping demand curve
for labor. However, David Card and Alan Krueger examined the effect of an increase
in the minimum wage from $4.25 to $5.05 in New Jersey in 1992.11 Using data from
the fast-food industry, Card and Krueger found no indication that the increase in the
minimum wage led to any decrease in employment in the industry. The authors suggest that this industry may not have been perfectly competitive, perhaps because employers did not act as price takers in the labor market, or perhaps for other reasons.
A study of the effects of minimum wage laws in noncompetitive markets, as well as
the effects of a minimum wage law in one market on other markets, is beyond the scope
of the analysis here, but you should recognize that these complications may be important.
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E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 0 . 4
D
Impact of a Price Floor
As in the previous Learning-By-Doing
Exercises in this chapter, the demand and supply curves are
Q d 10 0.5P d
Qs e
2 P s, when P s 2
0, when P s 6 2
where Q d is the quantity demanded when the price consumers pay is P d, and Q s is the quantity supplied when
the price producers receive is P s.
Suppose the government sets a price floor of $12 in
the market, as illustrated in Figure 10.11.
Problem
(a) What is the size of the excess supply in the market
with the price floor? What is the consumer surplus?
(b) What is the maximum producer surplus, assuming
producers with the lowest costs sell the good? What is
the net economic benefit? What is the deadweight loss?
(c) What is the minimum producer surplus, assuming
producers with the highest costs sell the good? What is
the net economic benefit? What is the deadweight loss?
Solution
(a) With the price floor, consumers demand only 4 million units (point T ), but producers want to supply 10 million units (point N). Thus, the excess supply is 6 million
units, equal to the horizontal distance between points T
and N.
11
Consumer surplus is the area below the demand
curve D and above the price floor of $12. This is area
YTR $16 million.
(b) If the most efficient suppliers (those between points
Z and W on the supply curve S) produce the 4 million
units that consumers want, producer surplus will be the
area above that portion of the supply curve and below
the price floor. This is area RTWZ $32 million.
The net economic benefit is the sum of consumer
surplus ($16 million) and producer surplus ($32 million)
$48 million.
The deadweight loss is the difference between the
net economic benefit with no price ceiling ($54 million)
and the net economic benefit with the price ceiling ($48
million) $6 million.
(c) If the least efficient suppliers (those between points
V and N on the supply curve) produce the 4 million units
that consumers want, producer surplus will be the area
above that portion of the supply curve and below the
price floor. This is area MNV $8 million.
The net economic benefit is the sum of consumer
surplus ($16 million) and producer surplus ($8 million)
$24 million.
The deadweight loss is the difference between the
net economic benefit with no price floor ($54 million)
and the net economic benefit with the price floor ($24
million) $30 million.
Similar Problems:
10.1, 10.21
D. Card and Alan Krueger, “Minimum Wages and Employment: A Case Study of the Fast-Food
Industry in New Jersey and Pennsylvania,”American Economic Review 84, no. 4 (September 1994): 772.
413
1 0 . 3 P R O D U C T I O N Q U OTA S
Price (dollars per unit)
$20
Y
Excess supply
Price
floor
$12
$8
N
A
V
$6
$2
S
M
T
R
W
D
Z
4
6
10
Quantity (millions of units per year)
With Price Floor
With No Price Floor
With Maximum
With Minimum
Producer Surplus
Producer Surplus
Consumer surplus
area YVA = $36 million
area YTR = $16 million
area YTR = $16 million
Producer surplus
area AVZ = $18 million
area RTWZ = $32 million
area MNV = $8 million
Net benefits
(consumer surplus +
producer surplus)
$54 million
$48 million
$24 million
Deadweight loss
zero
$6 million
$30 million
FIGURE 10.11
Impact of a $12 Price Floor
With no price floor, the sum of consumer and producer surplus is $54 million, the maximum
net benefit possible in the market. With the price floor, consumer surplus decreases by $20
million. When producer surplus is maximized, producer surplus increases by $14 million and
net benefit decreases by $6 million (the deadweight loss). When producer surplus is minimized, producer surplus decreases by $10 million and net benefit decreases by $30 million
(the deadweight loss).
I
f the government wants to support the price at a level above the equilibrium price in
a free market, it may use a quota to restrict the quantity that producers can supply. A
quota is a limit on the number of producers in the market or on the amount that each
producer can sell.
Historically, quotas have been set in many agricultural markets. For example, the
government may limit the number of acres a farmer can plant. Quotas are used in
other industries, too. In many cities, governments limit the number of taxis that may
10.3
PRODUCTION
Q U OTA S
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be operated, often leading to fares higher than those that would be observed in unregulated markets.
When the government imposes a quota in a market with an upward-sloping
supply curve and a downward-sloping demand curve, we observe the following
effects:
• The market will not clear. There will be an excess supply of the good or service
in the market.
• Consumers will buy less of the good than they would in a free market.
• Consumer surplus will be lower than with no quota.
• Some (but not all) of the lost consumer surplus will be transferred to producers.
• Because there is excess supply with a quota, the size of the producer surplus will
depend on which of the producers actually supply the good. Producer surplus may
either increase or decrease with a quota.12
• There will be a deadweight loss.
Figure 10.12 illustrates the effects of a production quota of 4 million units, for the
same market depicted in Figure 10.6. (Figure 10.12 and the following discussion assume that the most efficient suppliers—those with the lowest costs—supply the 4 million units allowed by the quota.)
With no quota, equilibrium occurs at point G, where the demand curve D and the
supply curve S intersect. At this point, the equilibrium price is $8, and the marketclearing quantity is 6 million units per year.
Now we can compare the market with and without the quota, using Figure 10.12
to calculate the consumer surplus, producer surplus, net economic benefits, and deadweight loss.
With no quota, consumer surplus is the area below the demand curve and above
the price consumers pay ($8) (consumer surplus areas A B F $36 million per
year). Producer surplus is the area above the supply curve and below the price producers receive (also $8) (producer surplus areas C E $18 million per year). The
net economic benefit is $54 million per year (consumer surplus producer surplus),
and there is no deadweight loss.
With the quota, consumers will pay $12 per unit (point H ). Producers would
like to supply 10 million units at that price but are limited to the quota of 4 million
units, so there will be an excess supply of 6 million units. Consumer surplus is the
area below the demand curve and above the price consumers pay ($12) (consumer
surplus area F $16 million per year). Producer surplus is the area above the
supply curve (between points J and K, since we are assuming that the most efficient
suppliers produce all 4 million units) and below the price producers receive (also
$12) (producer surplus areas A E $32 million per year). The net economic
benefit is $48 million per year (consumer surplus producer surplus), so the deadweight loss is $6 million per year (net economic benefit with no quota net economic benefit with quota).
The reduction in consumer surplus occurs because the quota supports the
price at $12, well above the $8 equilibrium price in a competitive market. The size
12
If the most efficient producers serve the market, producer surplus will increase for some levels of the
quota. However, if the quota is too low (e.g., close to zero), producer surplus could actually decrease.
1 0 . 3 P R O D U C T I O N Q U OTA S
$20
Consumer surplus
Price (dollars per unit)
Quota
Deadweight loss
F
$8
C
$6
E
K
L
B
A
Producer
surplus
$2
S
H
$12
G
K
J
D
4
6
Quantity (millions of units per year)
With No Quota
10
With Quota
Impact of Quota
Consumer surplus
A+B+F
($36 million)
F
($16 million)
–A – B
(–$20 million)
Producer surplus
C+E
($18 million)
A+E
($32 million)
A– C
($14 million)
Net benefits
(consumer surplus + producer surplus)
A+B+C+E+F
($54 million)
A+E+F
($48 million)
–B – C
(–$6 million)
Deadweight loss
zero
B + C ($6 million)
B + C ($6 million)
FIGURE 10.12
Impact of a 4 Million Unit Production Quota
With no quota, the sum of consumer and producer surplus is $54 million, the maximum
net benefit possible in the market. The quota decreases consumer surplus by $20 million,
increases producer surplus by $14 million, and reduces the net benefit by $6 million (the
deadweight loss).
of the producer surplus depends on which suppliers are in the market. Because producers would like to supply 10 million units when the price is $12, there is no guarantee that the most efficient producers will supply the 4 million units allowed by
the quota. The 4 million units might be supplied by inefficient suppliers, such as
those located between points G and K on the supply curve. Then producer surplus
will be much lower (area L $8 million). Note that in this case, the quota leads to
a decrease in producer surplus, and the deadweight loss is $30 million (can you verify this?).
415
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C O M P E T I T I V E M A R K E T S : A P P L I C AT I O N S
10.5
Quotas for Taxicabs
The taxicab industry has the features of a competitive
market. There are many small consumers of taxi service, and if entry were unregulated there would also
be many firms providing service. However, in many cities
around the world, taxis are regulated. Sometimes
government control takes the form of direct price
regulation. More often cities restrict the number of
licenses authorizing a taxi to operate on the street.
Historically, the licenses have often been metallic objects (called medallions) issued by the government to
certify that the driver has permission to provide taxi
service. These days, a medallion is often just a paper
document.
It is not surprising that taxi fares are substantially higher in cities with quotas than in cities that
allow free entry, because the number of medallions
limits the supply of taxis. For example, in Washington,
D.C., it is quite easy to enter the market, and fares
are low, often half as high as they are in cities with
quotas.
There are usually active markets that enable the
owner of a medallion to sell it to other prospective
drivers. If you want to operate a taxi in a market with
a quota, you must buy or rent an existing medallion
from someone who has one. Because the quotas support the price above the equilibrium level, the medallions can be quite valuable. For example, in New York
City the average price for a taxi medallion was
$766,000 in August 2009.13
When medallions can be sold, a more efficient
supplier will be willing to pay more for a medallion
than a less efficient supplier. The suppliers of taxi service are likely to be those with the lowest costs. This
suggests that the deadweight loss from the quota system will be at the lower end of the theoretically possible range (e.g., if the supply and demand curves are
similar to those in Figure 10.12, the deadweight loss
should be close to the sum of areas B C ).
13
In recent years many cities have increased the
number of medallions, with the goal of making the
market more competitive. For example, in the early
1980s Chicago had a restrictive quota system with
only two major suppliers of taxi service (Yellow and
Checker). The number of medallions had been set at
4,600 in 1959 and not increased since that year. In
1987, Yellow and Checker owned 80 percent of
those licenses. In that year the city government initiated a program to increase the number of medallions gradually over time. In 2009, Chicago had
approximately 6,900 medallions, and the city announced that it would soon auction off an undisclosed number of new medallions. The last auction,
in 2006, had resulted in auction prices of $78,000 per
medallion.
The political reasons for the move toward competition are interesting. As the number of medallions increases, the value of medallions will fall.
Owners of medallions often form a powerful interest group, strenuously objecting to increasing the
number of medallions. However, there are also
strong interests in favor of entry. People with low
incomes frequently use taxi service, and they are
strongly in favor of the program to increase competition. Politicians understand that customers of taxi
service will benefit from lower fares, and these taxi
customers are voters. In the end, in Chicago the voters carried the day, initiating the move toward more
competition.
One might ask why Chicago did not deregulate
taxis all at once by simply eliminating the need for
medallions. Out of fairness to existing holders of
medallions, the government phased in increased
entry over time. Anyone who bought a medallion just
before the program of increased entry was announced
paid a handsome price for it. By phasing in the program over a number of years, the program allowed
existing holders to recover much of their investment
in medallions.
“Driver Competition Hot as NYC Taxi Medallions Hit $766,000,” USA Today (August 7, 2009).
417
1 0 . 4 P R I C E S U P P O RT S I N T H E AG R I C U LT U R A L S E C TO R
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 0 . 5
D
Comparing the Impact of an Excise Tax, a Price Floor,
and a Production Quota
Before going further, let’s compare three types of government intervention that lead consumers to pay a price
higher than the free-market price. Throughout this
chapter we have used the supply and demand curves in
Figure 10.1 to study the effects of government intervention. We have found that the price consumers pay will be
$12 per unit for each of the following forms of intervention:
• An excise tax of $6 (Learning-By-Doing
Exercise 10.1)
• A price floor of $12 (Learning-By-Doing
Exercise 10.4)
• A production quota of 4 million units (Figure 10.13)
To review and compare the results of these exercises, answer the following questions:
Problem
(a) How will consumer surplus differ in each of the
three cases?
Solution
(a) Since the price charged to consumers is $12 with
each type of intervention, consumer surplus is the same
in all three cases.
(b) Since the market clears with an excise tax, the suppliers in the market will be the efficient ones. The market
does not clear with a price floor or a quota, so inefficient
suppliers may serve the market. However, if the quota is
implemented with a certificate that authorizes production (as with taxi medallions in Application 10.5), and if
the certificates can be resold in a competitive market,
then we would expect the suppliers who ultimately acquire
the certificates to be efficient.
(c) Producers would prefer the price floor or the quota,
both of which may increase producer surplus. Producers
will least prefer the excise tax because it will reduce producer surplus.
(b) For which forms of intervention will we expect the
producers in the market to be the efficient suppliers (the
ones at the lower end of the supply curve)?
(d) Since the price and output levels are the same with
all three forms of intervention, the deadweight loss will
be smallest when there are efficient producers in the
market [and the conditions under which efficient producers will serve the market are summarized in part (b)].
(c) Which type (or types) of government intervention
might producers prefer?
Similar Problems:
10.1, 10.14, 10.15, 10.16, 10.19
(d) Which type (or types) of government intervention
lead to the lowest deadweight loss?
This exercise helps us appreciate why programs that have a common consequence
(here, the price consumers pay) may differ substantially in other ways. For example, a
higher consumer price does not necessarily mean that producers are better off or that
alternative programs are equally efficient. Furthermore, people who do not consume
the good may benefit if tax revenues collected in this market can be used to reduce tax
burdens elsewhere.
A
s noted in the opening of this chapter, price support programs are common in the
agricultural sector. These programs typically increase producer surplus for farmers. In
the United States, supports for products such as soybeans, corn, and peanuts often
hold prices above their free-market levels. Because price support programs are expensive to taxpayers, many governments have reduced such programs over the last
decade. However, many remain in place and sometimes enjoy a resurgence in years
when low prices threaten farming incomes.
10.4
PRICE
SUPPORTS
IN THE
AGRICULTURAL
S E C TO R
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In this section we discuss two price support programs that have been used
in the agricultural sector: acreage limitation programs and government purchase
programs.
AC R E AG E L I M I TAT I O N P R O G R A M S
With an acreage limitation program, the government gives farmers an incentive to
hold production below the free-market level by paying them not to plant. Figure 10.13
illustrates how such a program works, using supply and demand curves similar to
those in Figure 10.1. (We have labeled the horizontal axis in billions of bushels because agricultural support programs often involve billions of dollars instead of millions of dollars.) In equilibrium, the price is $8 per bushel, and farmers produce 6 billion bushels per year.
Suppose the government wants to support a price of $10 per bushel. Instead of
imposing a quota, it provides farmers with an incentive to reduce output to 5 billion
bushels, the level that would lead consumers to pay a price of $10. At a price of $10,
farmers would like to produce 8 billion bushels, which would create an excess supply
of 3 billion bushels. They would be willing to restrict production to 5 billion bushels
only if the government compensates them for not producing this additional 3 billion
bushels. The compensation farmers will require is equal to the producer surplus they
will forgo if they limit production to 5 billion bushels. This amount is equal to areas
B C G in Figure 10.13, or $4.5 billion.
The program decreases consumer surplus by $11 billion (areas A B) and increases producer surplus by $14 billion (areas A B G ). It costs the government
$4.5 billion (areas B C G). The net benefit to society is the sum of consumer surplus ($25 billion) and producer surplus ($32 billion), less the cost to the government
($4.5 billion), or $52.5 billion. The deadweight loss is $1.5 billion (areas B C ).
Since the program introduces a deadweight loss, one might ask why the government does not simply give farmers a cash transfer equal to their $14 billion producer
surplus gain under the acreage limitation program and then let the market function
without intervention to produce 6 billion bushels at a price of $8. This might seem
attractive because the deadweight loss would then be zero. The government would
collect the money to pay for the program from taxes imposed elsewhere. Although
such a program would be efficient, the public may find it more palatable to pay
farmers $4.5 billion to reduce output (and forgo a profit opportunity) than to give
farmers $14 billion to do nothing at all.14
G OV E R N M E N T P U R C H A S E P R O G R A M S
As an alternative to an acreage limitation program, the government can support a
price of $10 per bushel with a government purchase program. Figure 10.14 illustrates
how such a program might work still using the same supply and demand curves as in
Figure 10.13. At a price of $10 per bushel, farmers would like to produce 8 billion
bushels, but the market demand would be only 5 billion bushels. Thus, there would
be an excess supply of 3 billion bushels.
14
Of course, we must recognize that the government might create deadweight losses in other markets if it
imposed taxes to raise $14 billion to pay for the acreage limitation program.
419
1 0 . 4 P R I C E S U P P O RT S I N T H E AG R I C U LT U R A L S E C TO R
Price (dollars per bushel)
$20
F
S
Support
price
$10
A
$8
$7
B G
C
E
$2
D
5
6
8
10
Quantity (billions of bushels per year)
With No Program
With Acreage
Limitation Program
Consumer surplus
A+B+F
($36 billion)
F
($25 billion)
–A – B
(– $11 billion)
Producer surplus
C+E
($18 billion)
A+B+C+E+G
($32 billion)
A+B+G
($14 billion)
Impact on government budget
zero
–B – C – G
(−$4.5 billion)
–B – C – G
(–$4.5 billion)
Net benefits
(consumer surplus + producer surplus –
government expenditures)
A+B+C+E+F
($54 billion)
A+E+F
($52.5 billion)
–B – C
(–$1.5 billion)
Deadweight loss
zero
B + C ($1.5 billion)
FIGURE 10.13
Impact of an Acreage Limitation Program
The government could support a price of $10 per bushel by offering farmers cash for planting less acreage, reducing output to 5 billion bushels. With no acreage limitation program,
the sum of consumer and producer surplus is $54 billion, the maximum net benefit possible
in the market. The program decreases consumer surplus by $11 billion, increases producer
surplus by $14 billion, has a negative impact of $4.5 billion on the government budget, and
reduces the net benefit by $1.5 billion (the deadweight loss).
To maintain a price of $10 per bushel, the government could buy the extra 3 billion
bushels to eliminate the excess supply. When the government purchases are added to
the market demand (see the curve labeled D government purchases in Figure 10.14),
the equilibrium price will be $10 (at point W ). Under this government purchase program, consumer surplus measured by the area under the original market demand
curve D will decrease by $11 billion and producer surplus will increase by $14 billion,
Impact of Program
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Price (dollars per bushel)
$20
F
S
Support
price
$10
$8
$7
W
B G
C
A
D + government
J
purchases
E
H
I
$2
D
5
6
8
10
Quantity (billions of bushels per year)
With No Program
With Government
Purchase Program
Impact of Program
Consumer surplus
A+B+F
($36 billion)
F ($25 billion)
–A – B
( –$11 billion)
Producer surplus
C+E
($18 billion)
A+B+C+E+G
($32 billion)
A+B+G
($14 billion)
Impact on government budget
zero
–B – C – G – H –
I – J (– $30 billion)
–B – C – G – H –
I – J (–$30 billion)
Net benefits
(consumer surplus + producer
surplus – government expenditures)
A+B+C+E+F
($54 billion)
A+E+F–H–I–J
($27 billion)
–B – C – H – I – J
(–$27 billion)
Deadweight loss
zero
B+C+H+I+J
($27 billion)
FIGURE 10.14 Impact of a Government Purchase Program
The government could support a price of $10 per bushel with a government purchase program, buying up the excess supply of 3 billion bushels. With no program, the sum of consumer and producer surplus is $54 billion, the maximum net benefit possible in the market.
The program decreases consumer surplus by $11 billion, increases producer surplus by $14
billion, has a negative impact of $30 billion on the government budget, and reduces the net
benefit by $27 billion (the deadweight loss).
both the same as with the acreage limitation program discussed in the previous section.
Government expenditures, however, will be much greater than the $4.5 billion with
the acreage limitation program—$30 billion (3 billion bushels ⫻ $10 per bushel ⫽
areas B ⫹ C ⫹ G ⫹ H ⫹ I ⫹ J ). This means that the net economic benefit will be
much smaller ($27 billion, versus $52.5 billion with the acreage limitation program)
1 0 . 4 P R I C E S U P P O RT S I N T H E AG R I C U LT U R A L S E C TO R
421
and the deadweight loss much greater ($27 billion, versus $1.5 billion with the acreage
limitation program).
The government could try to reduce the cost of the program by selling some of
its 3 billion bushels elsewhere in the world (e.g., by selling at a low price to countries
in need). But if some of what it sells finds its way back into the U.S. market, the price
in the U.S. market could be driven down, thereby lowering farmers’ producer surplus
and working against the goal of the program.
Government purchase programs are more costly and less efficient than acreage
limitation programs.15 Often a government must spend much more than one dollar to
increase farmers’ producer surplus by a dollar. Nevertheless, many countries resort to
government purchase programs, and they are often more palatable politically than
direct cash payments to farmers.
A P P L I C A T I O N
10.6
A Bailout of the King of Cheeses
The Italian cheese Parmigiano Reggiano (Parmesan) is
often called the “King of Cheeses” for its high quality
and versatility for cooking.16 While there are cheeses
produced elsewhere (e.g., Wisconsin) that attempt to
mimic the flavor of Parmesan, many connoisseurs do
not feel that they are of the same quality. Real Italian
Parmesan is manufactured under strict regulations.
The cheese is made from the milk of a certain type
of cow that can only live on farms in a specific area
surrounding the northern Italian city of Parma. The
method of making the cheese is also strictly regulated.
In 2008, 430 small companies made official Parmesan
cheese. The cheese is produced in wheels that weigh
35 kilograms (about 75 pounds) each. According to industry estimates, the average cost of producing a wheel
of Parmesan was at least 8 euros ($12) per kilo. About
20 percent of the cheese was exported.
Unfortunately for Parmesan cheese manufacturers, while the cost of inputs (especially milk) had risen
during the first decade of the new millennium, the
market price fell for several years in a row. At the end
of 2008 Parmesan cheese sold for about 7.4 euros per
kilo, and many of the makers faced the threat of
15
bankruptcy. The Italian government responded in
December 2008 by announcing that it would purchase
100,000 wheels of Parmesan (as well as 100,000
wheels of a similar cheese called Gran Padano) in an
effort to raise the market price and help the industry.
The effects of this Parmesan bailout would be
very similar to the analysis illustrated in Figure 10.14.
The government purchases would move the industry
equilibrium from point G to point W. Producer surplus
would rise, consumer surplus would fall, and government expenditures for the bailout were reported to
be about 50 million euros. While this program would
benefit Parmesan producers as intended, it would
create a deadweight loss in the market.
As noted in the discussion of government purchase programs, the purchase program would not
succeed in supporting the price of Parmesan cheese if
the cheese purchased by the government were then
resold in the market. This would shift the demand
curve back toward its original location, with an equilibrium at point G in Figure 10.14. The Italian government therefore announced that it would donate the
cheese to charities that presumably would not have
purchased Parmesan cheese themselves and would
not resell the donated cheese.
If we think in terms of general equilibrium (see Chapter 16), the government purchase program in one
sector is likely to create even more deadweight loss in other sectors of the economy because larger taxes
will have to be collected elsewhere to finance the program.
16
Data in this application are largely drawn from the article, “Blessed Are (Some of ) the Cheesemakers,”
Robert Mackey, New York Times (The Lede), December 19, 2008.
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10.5
Consumers in a country will want to import a good when the world price of the good
IMPORT
Q U OTA S A N D
TA R I F F S
C O M P E T I T I V E M A R K E T S : A P P L I C AT I O N S
is below the equilibrium price in the domestic market with no imports. This leads
many governments to impose import quotas and tariffs in order to support the price
of a good in the domestic market, especially when the world price is quite low and unrestricted imports would hurt domestic producers. Quotas and tariffs lead to higher
domestic prices, enabling domestic producers to expand production and earn higher
profits. In this section, we will see that quotas and tariffs increase domestic producer
surplus and reduce domestic consumer surplus. We will also see that these forms of
government intervention lead to deadweight losses by reducing the amount of total
domestic surplus (producer surplus plus consumer surplus, or net economic benefit).
Q U OTA S
A quota is a restriction on the total amount of a good that can be imported into a
country—that is, a quota is a restriction on free trade, which would allow unlimited
imports of the good. In the extreme case, a quota can take the form of a complete prohibition on imports of the good (i.e., the allowed quota of imports is zero); more
often, a quota restricts imports to some positive amount of the good.
Figure 10.15 compares the domestic market for a good (the same market depicted
in Figure 10.14) in three cases: a trade prohibition (quota 0), free trade (no quota),
and a quota of 3 million units per year. We can use Figure 10.15 to compare the three
cases in terms of domestic consumer surplus, producer surplus (domestic and foreign),
domestic net economic benefits, and deadweight loss.
With a complete prohibition on trade, market equilibrium will be at the intersection of the domestic demand and supply curves, at a price of $8 per unit and with a
market-clearing quantity of 6 million units per year. Domestic consumer surplus will
be the area below the demand curve and above the equilibrium price of $8 (consumer
surplus area A), domestic producer surplus will be the area above the supply curve
and below the equilibrium price (producer surplus areas B F L), the domestic
net benefits will be the sum of domestic consumer surplus and domestic producer surplus (net benefits areas A B F L), and the deadweight loss will be the difference between net benefits with free trade (which, as we will see, is areas A B C
E F G H J K L) and net benefits with a complete prohibition on trade
(deadweight loss areas C E G H J K ).
Suppose now that foreign producers are willing to supply any quantity of the
good at a price of Pw $4 per unit. We will refer to $4 per unit as the world price.
You should think of the world price as being that price that is just sufficient to cover
foreign producers’ average cost of producing the good and delivering it to the domestic market. Perfect competition among foreign producers drives the price in the
global market to this level. Since the world price is below the equilibrium price in the
domestic market with no trade ($8), domestic consumers will want to import the
good and under a regime of free trade, they would be able to do so. At a price of $4,
domestic demand will be Q5 8 million units per year (at the intersection of Pw and
the demand curve), but domestic producers will be willing to supply only Q1 2 million units per year (at the intersection of Pw and the supply curve). Thus, to satisfy
the domestic demand, 6 million units per year would have to be imported (8 million
units demanded domestically 2 million units supplied domestically 6 million
units imported).
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1 0 . 5 I M P O RT Q U OTA S A N D TA R I F F S
Price (dollars per unit)
$20
Domestic supply
A
$8
B
C
$6
Pw = $4
F
E
H
G
J
K
L
$2
Domestic demand
Q1 = 2
Q2 = 4
Q3 = 6
Q5 = 8
Q4 = 7
10
Quantity (millions of units per year)
With Quota
Free Trade
(with no quota)
Impact of Quota
Trade Prohibition
Quota = 3 Million
Impact of Trade
Impact of Quota = 3
(quota = 0)
Units Per Year
Prohibition
Million Units Per Year
Consumer
surplus
(domestic)
A+B+C+
E+F+G+H+
J+K
A
A+B+C+E
–B – C – E – F –
G–H–J–K
–F – G – H – J – K
Producer
surplus
(domestic)
L
B+F+L
F+L
B+F
F
Net benefits
(domestic)
(consumer
surplus + domestic
producer surplus)
A+B+C+E+
F+G+H+J+
K+L
A+B+F+L
A+B+C+E+
F+L
–C – E – G –
H–J–K
–G – H – J – K
Deadweight
loss
zero
C+E+G+H+
J+K
G+H+J+K
C+E+G+H+
J+K
G+H+J+K
Producer surplus
(foreign)
zero
zero
H+J
zero
H+J
FIGURE 10.15
Impact of a Trade Prohibition versus Free Trade versus a Quota of 3 Million Units per Year
With a trade prohibition, the market would be in equilibrium at a price of $8 per unit and a quantity of Q3 6 million units per year. With free trade, the good would sell at the world price Pw $4 per unit, with 2 million units supplied domestically and 6 million units imported, for a total quantity of Q5 8 million units per year. With a quota of
3 million units per year, the government could support a price of $6 per unit, with 4 million units supplied domestically and 3 million units imported, for a total quantity of Q4 7 million units per year. Compared with free trade, a
trade prohibition decreases domestic consumer surplus, increases domestic producer surplus, decreases net benefit,
and increases deadweight loss; the quota does the same, but less dramatically, while also generating a producer
surplus for foreign suppliers.
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10.7
Sweet Deal: The U.S. Sugar Quota
Program17
One of Chicago’s many distinctions is that it is the center
of candy production in the United States. In the late
1990s, the candy industry employed 10,000 people in the
Chicago metropolitan area. For this reason, in early 2006
politicians and businesspeople eagerly awaited the
study completed by the U.S. Department of Commerce,
known as the Valentine’s Day Report (released on
February 14, 2006), which would document the impact
of the U.S. sugar quota program on U.S. consumers, U.S.
candy producers, and jobs in the U.S. candy industry.
The U.S. sugar quotas, which have been in effect
since 1981, restrict the amount of sugar that sugargrowing countries can sell in the United States. The
countries with the largest quotas are the Dominican
Republic, Brazil, and the Philippines. As a result of the
quotas, U.S. consumers pay a higher price for sugar than
if they had been able to purchase sugar at the prevailing
price in the world market. According to the Valentine’s
Day report, “Over the last 25 years, the U.S. price of
wholesale refined sugar has been on average two to
three times the world price, and in 2004, the world refined price was 10.9 cents per pound compared to the
U.S. price of 23.5 cents per pound” (p. 3). This, of course,
is good news for producers of sugar, who are shielded
from the effects of fluctuations in the world market
price. It is also good news for companies that produce
substitutes for sugar: Demand for their products goes up
because the price of sugar in the United States is higher
than it would have been otherwise. Archer Daniels
Midland, a leading food processing company, at one
time ran an advertisement on Sunday morning news
programs pointing out how much of a bargain sugar
was for U.S. consumers. They did so not because they produced sugar, but because they produced high-fructose
corn syrup, a substitute for sugar in, among other things,
the production of soft drinks. Convincing U.S. consumers
that sugar is a bargain is a good strategy for companies
that benefit when the price of sugar is high.
End consumers who purchase refined sugar for
the purpose of cooking or sweetening foods such as
17
cereal or fruit are clearly harmed by the elevated
prices due to the sugar quotas. But they are also indirectly harmed because manufacturers of products
such as breakfast cereal, candy, and ice cream also pay
a high price for the sugar they purchase, and this high
price is, at least in part, passed along to consumers of
these products. The Valentine’s Day report cites a
study by the Government Accountability Office and
the U.S. International Trade Commission that pegged
the economic loss to sugar cane refiners, food manufacturers, and end consumers at $1.9 billion in 1998.
In addition to harming end consumers, the U.S.
sugar quota program hurts employment in the industries that consume sugar. The Valentine’s Day report
suggests that employment in industries that consume
sugar fell by more than 10,000 jobs between 1997 and
2002. By contrast, employment in non–sugar-consuming
industries increased by more than 30,000 over the same
period. Sugar quotas have hit the Chicago area especially hard. The Valentine’s Day Report points out that
Chicago lost more than 4,000 jobs between 1991 and
2001 in the candy, gum, cereal, and bakery industries,
a decline of 27 percent. The number of manufacturing
jobs in Illinois decreased during this period, but only
by 7 percent. The shutdowns of Brach’s Candy’s Chicago
operation in 2003 and Fannie May’s Chicago operation in 2004 provided a vivid illustration to Chicagoarea politicians and Chicago voters of the cost of the
U.S. sugar quota program.
The Valentine’s Day report shone a light on a program that, to many people, had been obscure or unknown. With the U.S. Commerce Department having
now documented the significant negative economic effects of the quotas and the Central American Free Trade
Agreement (approved by the United States in 2005)
having resulted in reductions in quotas to allow additional sugar imports from Central America, it seems possible that the U.S. sugar quota may eventually be eliminated. However, sugar producers, as well as companies
that produce substitutes for sugar, remain powerful
advocates for keeping sugar quotas in place, and any
attempt to overturn them will have to face their strong
opposition to eliminating their “sweet deal.”
This example is based on U.S. Department of Commerce, U.S. International Trade Commission,
“Employment Changes in U.S. Food Manufacturing: The Impact of the Sugar Price” (February 1996);
“Sugar Daddy; Quotas and the U.S. Government,” Case 5-204-255 Kellogg School of Management
(2002); “U.S. Sugar Rules Costly,” Chicago Tribune (February 12, 2006), Section 3, p. 3.
1 0 . 5 I M P O RT Q U OTA S A N D TA R I F F S
What is the impact of free trade? Domestic consumer surplus will be the area
below the demand curve and above Pw (consumer surplus areas A B C E
F G H J K ), domestic producer surplus will be the area above the
supply curve and below that price (producer surplus area L), the domestic net benefits will be the sum of domestic consumer surplus and domestic producer surplus (net
benefits areas A B C E F G H J K L), and there will be no
deadweight loss. Thus, domestic consumer surplus is much greater than it is with a
trade prohibition, but domestic producer surplus is much smaller.
Since domestic producers stand to lose with free trade, they often attempt to restrict or even eliminate imports. We have seen how the complete elimination of imports through a trade prohibition benefits producers. Now let’s examine the impact of
a partial restriction on imports, through a quota that allows the import of some maximum number of units per year.
Suppose the government wants to support a domestic price of $6 per unit (as a
sort of compromise, say, between the interests of domestic consumers, who would
enjoy a low price of $4 with free trade, and the interests of domestic producers, who
would benefit from a high price of $8 with no trade). To accomplish this, the government can set a quota of 3 million units per year. To see why, note that the equilibrium
price in the domestic market will be the one that clears the market—that is, the price
that makes total supply (domestic and foreign) equal to domestic demand. At a price
of $6, consumers will demand Q4 7 million units per year (at the intersection of that
price with the demand curve), but domestic producers will be willing to supply only
4 million units per year (at the intersection of the price with the supply curve). Thus,
to satisfy domestic demand at that price, 3 million units per year would have to be imported (7 million units demanded domestically 4 million units supplied domestically 3 million units imported).
What is the impact of this quota? Domestic consumer surplus will be the area
below the demand curve and above the price of $6 (consumer surplus areas A B
C E), domestic producer surplus will be the area above the supply curve and
below that price (producer surplus areas F L), the domestic net benefits will be
the sum of domestic consumer surplus and domestic producer surplus (net benefits
areas A B C E F L), and the deadweight loss will be the difference between net benefits with free trade and net benefits with the quota (deadweight loss
areas G H J K ). In addition, foreign suppliers enjoy a producer surplus of their
own under the quota, because they can sell the good at a price of $6 when they would
have been willing to sell it at a price of $4.
In sum, with a quota, domestic consumer surplus is less than it is with free trade
but more than with a trade prohibition, while domestic producer surplus is more than
with free trade but less than with a trade prohibition, and foreign suppliers gain some
producer surplus.
TA R I F F S
A tariff is a tax on an imported good. Like a quota, a tariff restricts imports, and the
government can use a tariff to achieve the same objective achieved with a quota—to
support the domestic price of the good. For instance, in the market we have been
discussing, the government could eliminate imports (as it could do with a trade
prohibition—i.e., a quota of zero) by charging a tariff of $5 per unit. This would raise
the domestic price of the imported good to $9 per unit (Pw of $4 tariff of $5 $9).
In that case, no quantity of the good would be imported because no consumers would
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Price (dollars per unit)
$20
Domestic supply
A
$8
B
Pw + $2 tariff = $6
Pw = $4
C
F
G
H
E
J
K
L
$2
Domestic demand
Q1 = 2
Q2 = 4
Q3 = 6
Q5 = 8
Q4 = 7
10
Quantity (millions of units per year)
Free Trade
(with no tariff)
With Tariff
Impact of Tariff
Consumer surplus (domestic)
A+B+C+E+F+
G+H+J+K
A+B+C+E
–F – G – H – J – K
Producer surplus (domestic)
L
F+L
F
Impact on government budget
zero
H+J
H+J
Net benefits (domestic)
(consumer surplus + domestic
producer surplus + impact on
government budget)
A+B+C+E+F+
G+H+J+K+L
A+B+C+E+
F+L
–G – H – J – K
Deadweight loss
zero
G+K
G+K
Producer surplus (foreign)
zero
zero
zero
FIGURE 10.16 Impact of a Tariff of $2 per Unit versus Free Trade
With free trade, the good would sell at the world price Pw $4 per unit, with 2 million units
supplied domestically and 6 million units imported, for a total quantity of Q5 8 million
units per year. By imposing a tariff of $2 per unit, the government could support a price of
$6 per unit, with 4 million units supplied domestically and 3 million units imported, for a
total quantity of Q4 7 million units per year. Compared with free trade, a tariff has much
the same impact as a quota (see Figure 10.15), but rather than generating a producer surplus
for foreign suppliers, it generates revenues for the government, which the government can
use to benefit the domestic economy.
1 0 . 5 I M P O RT Q U OTA S A N D TA R I F F S
427
buy the good at that price (domestic producers would satisfy consumer demand at a
price of $8). Thus, if a tariff is larger than the difference between the domestic price
with no trade and the world price (i.e., if the tariff in our example were larger than
$4), nothing will be imported.
Suppose the government wants to achieve the same objective discussed in the preceding section—to support a domestic price of $6 per unit. Figure 10.16 shows that
the government could do this by setting a tariff of $2 per unit. The explanation of why
this works is exactly parallel with the explanation of why a quota of 3 million units per
year works. At a price of $6, consumers will demand Q4 7 million units per year, but
domestic producers will be willing to supply only 4 million units per year. To satisfy
domestic demand at that price, 3 million units per year would have to imported. Thus,
a tariff of $2 per unit creates the same equilibrium as an import quota of 3 million
units per year.
The overall impact of this tariff is very similar, but not identical, to the impact of
the quota. As shown by the tables in Figures 10.15 and 10.16, domestic consumer surplus and domestic producer surplus are the same in the two cases. However, what
would have been a gain in producer surplus to foreign suppliers under a quota is
instead a positive impact on the domestic government budget with a tariff. This is because the government collects the revenues from the tariff. The size of those revenues
is equal to the tariff ($2) times the number of units imported (3 million), or $6 million (areas H J in the two figures).
Thus, with a tariff, as with a quota, domestic consumer surplus is less than it is
with free trade but more than with a trade prohibition, while domestic producer surplus is more than with free trade but less than with a trade prohibition. In addition,
and in contrast to the situation with a quota, the government can benefit the economy
by redistributing the revenues from the tariff, so the deadweight loss is lower with the
tariff than under a quota.
A P P L I C A T I O N
10.8
Dumping
In the past decade some countries have complained
that other countries have subsidized their own industries to help them gain a larger share of the world
market. For example, it has often been alleged that
Japanese producers of steel are selling in foreign
markets at a price below their cost (a practice known
as dumping), in part because of subsidies from the
Japanese government. In this application, we study
the effects of dumping.
Suppose that the world price of steel delivered
to the United States is Pw, set in a competitive world
market in which price, average cost, and marginal
cost are equal. If a foreign government provides a
subsidy of $S per unit to its producers, domestic consumers will be able to import steel at a price Pw S,
as Figure 10.17 illustrates. Under free trade (with no
dumping) imports would be Q3 Q2. However, with
the new, lower domestic price with dumping, imports
will expand to Q4 Q1.
How will dumping affect the domestic market?
Domestic consumers will benefit: their surplus will increase by A B C H I. However, domestic producers will be quite unhappy: Their surplus will fall by A
I. Among other things, domestic producers will note
that dumping keeps workers on the job in the country
engaging in dumping, while unemployment is likely to
rise among steel workers at home. That is why it is often
said that dumping leads to an export of jobs from the
domestic country to the country subsidizing its industry.
In practice, it is not easy to establish that dumping is occurring because one needs proof that firms
are selling at a price below cost. It may be especially
difficult to gather data on the costs of production for
foreign firms.
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Price (dollars per ton)
Domestic supply
F
E
Pw
I
A
Pw – S
J
H
B
C
G
Domestic demand
Q1
Q2
Q3
Q4
Quantity (millions of tons per year)
Free Trade
(with no dumping)
With Dumping
Impact of Dumping
Consumer surplus
E+F
A+B+C+E+F+H+I
A+B+C+H+I
Domestic producer surplus
A+G+I
G
–A – I
Net benefits (domestic)
(consumer surplus +
producer surplus)
A+E+F+G+I
A+B+C+E+F+
G+H+I
B+C+H
zero
–B – C – H – I – J
–B – C – H – I – J
Impact on foreign
government budget
FIGURE 10.17
Impact of Dumping
With free trade (no dumping), Q3 million tons of steel would be consumed in the domestic
market, selling at the world price Pw, with Q2 million tons supplied domestically and Q3 Q2
million tons imported. With dumping, domestic consumption would rise to Q4 million tons
and the price would fall to Pw S, with only Q1 million tons supplied domestically and with
imports increasing to Q4 Q1 million tons. Domestically, dumping would increase consumer
surplus, decrease producer surplus, and increase net benefits. The increase in net benefit
partly reflects the subsidy that the foreign government is paying to its producers.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 0 . 6
D
Effects of an Import Tariff
The domestic demand for DVD players is
given by Q d 100 P, and the domestic supply is
given by Q s P where Q s and Q d measure quantities in
thousands of DVD players. DVD players can currently
be freely imported at the world price of $20. The government is planning to impose a tariff of $10 per unit on
imported DVD players.
429
1 0 . 5 I M P O RT Q U OTA S A N D TA R I F F S
Problem
Solution The accompanying graph shows the do-
DVDs at a price of $30. They would demand 70,000
units, and domestic suppliers would produce 30,000
units. Therefore 40,000 units would be imported.
With the tariff, domestic producer surplus would
increase by area G ($250,000). The government would
collect revenues represented by area F from the imports
($400,000).
mestic supply and demand curves. With a tariff of $10 per
unit, domestic consumers would be able to buy imported
Similar Problems:
With the tariff, how many units would be
imported? How much would domestic producer surplus
change if the government introduces a $10 import duty
per DVD player? How much revenue would the domestic government collect from the imports of DVD players?
Price (dollars per unit)
$100
10.27, 10.28
S
$50
$30
G
F
$20
D
20
A P P L I C A T I O N
30
70
80
Thousands of DVD players
10.9
Tariffs, Tires, and Trade Wars
Many U.S. manufacturing industries have struggled
for years to compete against foreign competition. Over
the last several decades, the share of the U.S. economy
devoted to manufacturing has declined dramatically.
One of the most important sources of foreign competition for manufacturing has been China, which can
often produce goods at lower cost, particularly because
of the low wages of Chinese workers.
In 2008 the U.S. economy plunged into recession.
It was also an election year, and the candidates for
president all tried to win votes from various constituents, including labor unions. During his campaign,
Barack Obama used tough rhetoric about restricting
free international trade, and in the election he won
the majority of votes of union members.
18
Q
In September 2009, President Obama announced
imposition of a 35 percent tariff on automobile and
light-truck tires imported from China.18 The tariff would
decrease to 30 percent the next year, and 25 percent
the year after. The decision was seen as a victory for
the United Steelworkers Union, since that union represents most workers in the tire industry. From 2004
to 2008, imports of tires from China to the U.S. increased threefold, and Chinese companies increased
their market share from about 5 percent to 17 percent. At the same time, four American tire factories
closed, with additional closures expected.
President Obama’s decision was the first time
that the United States had applied a special provision
of the 2001 agreement in which the country agreed
to support China’s entry into the World Trade
Organization. Under that provision, workers or firms
Edmund Andrews, “U.S. Adds Tariff on Chinese Tires.” New York Times (September 11, 2009).
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that feel they have suffered a “market disruption”
from Chinese imports can ask the government for protection. In this case the union requested protection for
the industry, but the Tire Industry Association did not.
The International Trade Commission, an independent
government agency, analyzed the request and voted
4–2 to recommend imposition of the tariffs.
When one country imposes limits or tariffs on imports of a product from another country, the other
country often responds by imposing its own restrictions on products from the original nation in retaliation. In this case, China retaliated immediately. The
U.S. announcement was made on a Friday. Over the
weekend, the Chinese commerce ministry made statements about the harm that a trade war would cause
for both nations. On the following Monday, the ministry announced that it was considering tariffs on
imports of automotive and chicken meat products
shipped from the United States to China. It planned
to investigate whether these products were being
subsidized and “dumped” onto the Chinese market.
Assuming that it reached this conclusion, the Chinese
government was expected to impose tariffs on these
products from the United States. At the time of this
writing, it was too early to tell whether this would
escalate into a broader trade war.
CHAPTER SUMMARY
• In a competitive market each producer acts in its own
self-interest, deciding whether to be in the market and,
if so, how much to produce to maximize its own profit.
Similarly, each consumer also acts in his or her own selfinterest, maximizing utility to determine how many
units of the good to buy. Even though there is no social
planner telling producers and consumers how to behave,
the output in a competitive market maximizes net economic benefits (as measured by the sum of the surpluses).
It is as though there is an “invisible hand” guiding a competitive market to the efficient level of production and
consumption.
• Government intervention can take many forms, including excise taxes and subsidies, minimum and maximum price regulation, production quotas, price support
programs, and quotas and tariffs on imports. For some
kinds of government intervention (such as taxes and subsidies), the market will clear. For other types of intervention (such as price ceilings, price floors, and production
quotas), the market will not clear. When the market does
not clear, we must understand who is participating in the
market when we measure consumer and producer surplus.
• When an excise tax is imposed in a market, the price
consumers pay usually rises by less than the amount of
the tax, and the price producers receive usually falls by
less than the amount of the tax. The incidence of a tax
measures the impact of the tax on the price consumers
pay versus the price that sellers receive. When demand
is rather inelastic and supply is relatively elastic, the incidence of an excise tax will be larger for consumers than
for producers. When the relative magnitudes of the
elasticities are reversed, the incidence of the tax will be
larger for producers than for consumers.
• Government intervention in competitive markets
usually leads to a deadweight loss. Deadweight loss is an
economic inefficiency that arises when consumers and
producers do not capture potential net benefits.
• Government intervention in competitive markets
often redistributes income from one part of the economy to another. If the government collects revenues
through taxes or tariffs, the receipts are part of the net
benefit to the economy because the revenues can be redistributed. Similarly, net flows away from the government are a part of the cost of a program.
• An excise tax leads to a deadweight loss because the
market produces less than the efficient level. A tax also
reduces both consumer and producer surplus. (LBD
Exercise 10.1)
• When the government pays a subsidy for each unit
produced, the market produces more than the efficient
level, leading to a deadweight loss. A subsidy increases
both consumer and producer surplus, but these gains are
less than the government’s cost to pay for the subsidy.
(LBD Exercise 10.2)
• With a binding price ceiling (i.e., a ceiling below the
free-market price), the amount exchanged in the market
will be less than the efficient level because producers
restrict supply. There will be excess demand in the market,
and consumers who value the good the most may not be
able to purchase the good. (LBD Exercise 10.3)
PROBLEMS
• With a binding price floor (i.e., a floor above the
free-market price), the amount exchanged in the market
will be less than the efficient level because consumers
buy less. There will be excess supply in the market, and
the lowest-cost producers may not be those who supply
the good. (LBD Exercise 10.4)
• A production quota raises the price consumers pay
by limiting the output in the market. Although one
would normally expect producer surplus to rise with
such a quota, this need not always occur. Because the
market does not clear with a production quota, there is
431
no guarantee that the suppliers serving the market are
the ones with the lowest cost. (LBD Exercise 10.5)
• Acreage limitation and government purchase programs
have often been used to support prices in the agricultural
sector. These programs can be quite costly to the government and also may introduce large deadweight losses.
• Governments may resort to import quotas and tariffs
to enhance producer surplus for domestic suppliers.
These forms of intervention reduce consumer surplus
and create deadweight loss for the domestic economy.
(LBD Exercise 10.6)
REVIEW QUESTIONS
1. What is the significance of the “invisible hand’’ in a
competitive market?
2. What is the size of the deadweight loss in a competitive market with no government intervention?
3. What is meant by the incidence of a tax? How is the
incidence of an excise tax related to the elasticities of
supply and demand in a market?
4. In the competitive market for hard liquor, the demand
is relatively inelastic and the supply is relatively elastic.
Will the incidence of an excise tax of $T be greater for
consumers or producers?
5. Gizmos are produced and sold in a competitive market. When there is no tax, the equilibrium price is $100
per gizmo. The own-price elasticity of demand for gizmos is known to be about –0.9, and the own-price elasticity of supply is about 1.2. In commenting on a proposed
excise tax of $10 per gizmo, a newspaper article states
that “the tax will probably drive the price of gizmos up by
about $10.” Is this a reasonable conclusion?
6. The cheese-making industry in Castoria is competitive, with an upward-sloping supply curve and a downward-sloping demand curve. The government gives
cheese producers a subsidy of $T for each kilogram of
cheese they make. Will consumer surplus increase? Will
producer surplus increase? Will there be a deadweight
loss?
7. Will a price ceiling always increase consumer surplus?
Will a price floor always increase producer surplus?
8. Will a production quota in a competitive market
always increase producer surplus?
9. Why are agricultural price support programs, such
as acreage limitation and government purchase programs, often very costly to implement?
10. If an import tariff and an import quota lead to the
same price in a competitive market, which one will lead
to a larger domestic deadweight loss?
11. Why does a market clear when the government imposes an excise tax of $T per unit?
12. Why does a market clear when the government
gives producers a subsidy of $S per unit?
13. Why does the market not clear with a production
quota?
14. With a price floor, will the most efficient producers
necessarily be the ones supplying the market?
PROBLEMS
10.1. In a competitive market with no government
intervention, the equilibrium price is $10 and the equilibrium quantity is 10,000 units. Explain whether the
market will clear under each of the following forms of
government intervention:
a) The government imposes an excise tax of $1 per unit.
b) The government pays a subsidy of $5 per unit
produced.
c) The government sets a price floor of $12.
d) The government sets a price ceiling of $8.
e) The government sets a production quota, allowing
only 5,000 units to be produced.
10.2. In Learning-By-Doing Exercise 10.1 we examined the effects of an excise tax of $6 per unit. Repeat that
exercise for an excise tax of $3.
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10.3. Gadgets are produced and sold in a competitive
market. When there is no tax, the equilibrium price is
$20 per gadget. The own-price elasticity of demand for
gadgets is 0.5. If an excise tax of $4 leads to an increase
in the price of gadgets to $24, what must be true about
the own-price elasticity of supply for gadgets?
10.4. When gasoline prices reached a price of $2.00 per
gallon, public policy makers considered cutting excise
taxes by $0.10 per gallon to lower prices for the consumer.
In discussing the effects of the proposed tax reduction, a
news commentator stated that the effect of tax reduction
should lead to a price of about $1.90 per gallon, and, that
if the price did not drop by as much, it would be evidence
that oil companies are somehow conspiring to keep gasoline prices high. Evaluate this claim.
10.5. Consider the market for crude oil. Suppose the demand curve is described by Q d 100 P, where Q d is the
quantity buyers will purchase when the price they pay is P
(measured in dollars per barrel). The equation representing the supply curve is Q S P/3, where Q S is the quantity
that producers will supply when the price they receive is P.
The market for crude oil is initially in equilibrium, with no
tax and no subsidy. Because it regards the price of oil as too
high, the government wishes to help buyers by announcing that it will give producers a subsidy of $4 per barrel. A
local television station reporter announces that the subsidy
should lower the price consumers pay by $4 per barrel.
Analyze the reporter’s claim by determining the price buyers pay before and after the subsidy, and provide intuition
to explain why the reporter is correct or incorrect.
10.6. The table in Application 10.1 indicates that revenues from gasoline taxes will increase by about $14 billion
(from $56 billion to about $69 billion per year) if the gasoline tax is raised from $0.40 to $0.50 per gallon. Using the
supply and demand curves in Application 10.1, show that
the equilibrium quantity, price consumers pay, price producers receive, and tax receipts are as indicated in the table
when the tax is $0.50 per gallon. Draw a graph illustrating
the equilibrium when the tax is $0.50 per gallon.
10.7. In a competitive market, there is currently no tax,
and the equilibrium price is $40. The market has an
upward-sloping supply curve. The government is about
to impose an excise tax of $5 per unit. In the new equilibrium with the tax, what price will producers receive and
consumers pay if the demand curve is
a) Perfectly elastic?
b) Perfectly inelastic?
Illustrate your answers graphically.
10.8. In a competitive market, there is currently no tax,
and the equilibrium price is $60. The market has a
downward-sloping demand curve. The government is
about to impose an excise tax of $4 per unit. In the new
equilibrium with the tax, what price will producers receive
and consumers pay if the supply curve is
a) Perfectly elastic?
b) Perfectly inelastic?
Illustrate your answers graphically.
10.9. The current equilibrium price in a competitive
market is $100. The price elasticity of demand is 4 and
the price elasticity of supply is 2. If an excise tax of $3
per unit is imposed, how much would you expect the
equilibrium price paid by consumers to change? How
much would you expect the equilibrium price received by
producers to change?
10.10. Suppose that the market for cigarettes in a particular town has the following supply and demand curves:
Q S P; Q D 50 P, where the quantities are measured
in thousands of units. Suppose that the town council
needs to raise $300,000 in revenue and decides to do this
by taxing the cigarette market. What should the excise
tax be in order to raise the required amount of money?
10.11. Assume that a competitive market has an
upward-sloping supply curve and a downward-sloping
demand curve, both of which are linear. A tax of size $T
is currently imposed in the market. Suppose the tax is
doubled. By what multiple will the deadweight loss increase? (You may assume that at the new tax, the equilibrium quantity is positive.)
10.12. Refer to the accompanying diagram depicting a
competitive market. If the government imposes a price
ceiling of P1, using the areas in the graph below, identify
a) The most that consumers can gain from such a move.
b) The most that consumers can lose from such a move.
In other words, provide a maximum and a minimum limit
to the possible change in consumer surplus from the imposition of this price ceiling.
P
Supply
A
C
B
D
E
F
P1
Demand
100
200
300
Q
PROBLEMS
10.13. In a perfectly competitive market, the market demand curve is given by Q d 200 5P d, and the market
supply curve is given by Q d 35P s.
a) Find the equilibrium market price and quantity demanded and supplied in the absence of price controls.
b) Suppose a price ceiling of $2 per unit is imposed.
What is the quantity supplied with a price ceiling of this
magnitude? What is the size of the shortage created by
the price ceiling?
c) Find the consumer surplus and producer surplus in
the absence of a price ceiling. What is the net economic
benefit in the absence of the price ceiling?
d) Find the consumer surplus and producer surplus
under the price ceiling. Assume that rationing of the
scarce good is as efficient as possible. What is the net
economic benefit in this case? Does the price ceiling result in a deadweight loss? If so, how much is it?
e) Find the consumer surplus and producer surplus
under the price ceiling, assuming that the rationing of
the scarce good is as inefficient as possible. What is the net
economic benefit in this case? Does the price ceiling result in a deadweight loss? If so, how much is it?
For the next three questions, use the following information. The market for gizmos is competitive, with an
upward-sloping supply curve and a downward-sloping
demand curve. With no government intervention, the
equilibrium price would be $25, and the equilibrium
quantity would be 10,000 gizmos. Consider the following
programs of government intervention:
Program I: The government imposes an excise tax of $2
per gizmo.
Program II: The government provides a subsidy of $2
per gizmo for gizmo producers.
433
demanded (in millions of bushels) when the price consumers pay is P d. The supply curve is
Qs e
4 P s, when P s 4
0, when P s 6 4
where Q s is the quantity supplied (in millions of bushels)
when the price producers receive is P s.
a) What are the equilibrium price and quantity?
b) At the equilibrium in part (a), what is consumer surplus? producer surplus? deadweight loss? Show all of
these graphically.
c) Suppose the government imposes an excise tax of $2
per unit to raise government revenues. What will the
new equilibrium quantity be? What price will buyers
pay? What price will sellers receive?
d) At the equilibrium in part (c), what is consumer surplus? producer surplus? the impact on the government
budget (here a positive number, the government tax receipts)? deadweight loss? Show all of these graphically.
e) Suppose the government has a change of heart about the
importance of corn revenues to the happiness of the
Pulmonian farmers. The tax is removed, and a subsidy of $1
per unit is granted to corn producers. What will the equilibrium quantity be? What price will the buyer pay? What
amount (including the subsidy) will corn farmers receive?
f ) At the equilibrium in part (e), what is consumer surplus?
producer surplus? What will be the total cost to the government? deadweight loss? Show all of these graphically.
g) Verify that for your answers to parts (b), (d), and
(f ) the following sum is always the same: consumer
surplus producer surplus budgetary impact deadweight loss. Why is the sum equal in all three cases?
10.16. Which of these programs would surely lead to
an increase in consumer surplus? Briefly explain.
10.18. In a perfectly competitive market, the market demand and market supply curves are given by Q d 1,000
10P d and Q s 30P s. Suppose the government provides
a subsidy of $20 per unit to all sellers in the market.
a) Find the equilibrium quantity demanded and supplied;
find the equilibrium market price paid by buyers; find the
equilibrium after-subsidy price received by firms.
b) Find the consumer surplus and producer surplus in
the absence of the subsidy. What is the net economic
benefit in the absence of a subsidy?
c) Find the consumer surplus and producer surplus in
the presence of the subsidy. What is the impact of the
subsidy on the government budget? What is the net economic benefit under the subsidy program?
d) Does the subsidy result in a deadweight loss? If so,
how much is it?
10.17. Suppose the market for corn in Pulmonia is
competitive. No imports and exports are possible. The
demand curve is Q d 10 P d, where, Q d is the quantity
10.19. In a perfectly competitive market, the market
demand curve is Q d 10 P d, and the market supply
curve is Q s 1.5P s.
Program III: The government imposes a price floor of $30.
Program IV: The government imposes a price ceiling of
$20.
Program V: The government allows no more than 8,000
gizmos to be produced.
10.14. Which of these programs would lead to less than
10,000 units exchanged in the market? Briefly explain.
10.15. Under which of these programs will the market
clear? Briefly explain.
CHAPTER 10
C O M P E T I T I V E M A R K E T S : A P P L I C AT I O N S
a) Verify that the market equilibrium price and quantity
in the absence of government intervention are P d P s
4 and Q d Q s 6.
b) Consider two possible government interventions: (1) A
price ceiling of $1 per unit; (2) a subsidy of $5 per unit paid
to producers. Verify that the equilibrium market price paid
by consumers under the subsidy equals $1, the same as the
price ceiling. Are the quantities supplied and demanded
the same under each government intervention?
c) How will consumer surplus differ in these different
government interventions?
d) For which form of intervention will we expect the
product to be purchased by consumers with the highest
willingness to pay?
e) Which government intervention results in the lower
deadweight loss and why?
10.20. Consider a perfectly competitive market in
which the market demand curve is given by Q d 20
2P d and the market supply curve is given by Q s 2P s.
a) Find the equilibrium price and quantity in the absence
of government intervention.
b) Suppose the government imposes a price ceiling of $3
per unit. How much is supplied?
c) Suppose, as an alternative, the government imposes a
production quota limiting the quantity supplied to six
units. What is the market price under this type of intervention? Is the quantity supplied under the price ceiling
greater than, less than, or the same as the quantity under
the production quota?
d) Assuming that under price controls rationing is as
efficient as possible and under the quota, the allocation is as
efficient as possible, under which program is the deadweight
loss larger: the price ceiling or the production quota?
e) Assuming that under price controls rationing is as inefficient as possible, while under the quota the allocation
is as efficient as possible, under which program is the
b) Using areas in the graph, answer the following.
deadweight loss larger: the price ceiling or the production quota?
f ) Assuming that under price controls rationing is as inefficient as possible, while under the quota the allocation
is as inefficient as possible, under which program is the
deadweight loss larger: the price ceiling or the production quota?
10.21. Figure 10.18 shows the supply and demand
curves for cigarettes. The equilibrium price in the market
is $2 per pack if the government does not intervene, and
the quantity exchanged in the market is 1,000 million
packs. Suppose the government has decided to discourage
smoking and is considering two possible policies that
would reduce the quantity sold to 600 million packs. The
two policies are (i) a tax on cigarettes and (ii) a law setting
a minimum price for cigarettes. Analyze each of the policies, using the graph and filling in the figure’s table.
S
Price (dollars per pack)
434
F
$3.00
E
G
C
L
$2.00
$1.00
$0.60
B
R
A
H
T
K
N
M
S
J
I
U
D
600
1000 1200
Quantity (millions of packs)
FIGURE 10.18
Tax on Cigarettes versus Minimum
Price
a) What is the size of the tax per unit that would achieve
the government’s target of 600 million packs sold in the
market? What minimum price would achieve the target?
Explain.
Tax
What price per pack would consumers pay?
What price per pack would producers receive?
What area represents consumer surplus?
What area represents the largest producer surplus possible under the policy?
What area represents the smallest producer surplus possible under the policy?
What area represents government receipts?
What area represents smallest deadweight loss possible under the policy?
Minimum Price
435
PROBLEMS
10.22. Consider a market with an upward-sloping supply curve and a downward-sloping demand curve. Under
a government purchase program, which of the following
statements are true, and which are false?
(a) The increase in producer surplus will exceed the size
of the government expenditure.
(b) Consumer surplus will increase.
(c) The size of the government expenditure will exceed
the size of the deadweight loss.
10.23. The market demand for sorghum is given by
Q d 500 10P d, while the market supply curve is given
by Q s 40P s. The demand and supply curves are shown
at right. The government would like to increase the income of farmers and is considering two alternative government interventions: an acreage limitation program
and a government purchase program.
a) What is the equilibrium market price in the absence
of government intervention?
b) The government’s goal is to increase the price of
sorghum to $15 per unit. This is the support price. How
much would be demanded at a price of $15 unit? How
With No Program
Consumer surplus
Producer surplus
Impact on the
government budget
Net benefits (consumer
surplus producer
surplus government
expenditure)
Deadweight loss
much would farmers want to supply at a price of $15 per
unit? How much would the government need to pay
farmers in order for them to voluntarily restrict their output of sorghum to the level demanded at $15 per unit?
50
S
D
0
500
c) Fill in the following table for the acreage limitation
program:
With Acreage
Limitation Program
Impact of Program
436
CHAPTER 10
C O M P E T I T I V E M A R K E T S : A P P L I C AT I O N S
d) As an alternative way to support a price of $15, suppose the government purchases the difference between
the quantity demanded at a price of $15 and the quantity
supplied. How much does the government spend on this
price support program?
e) Fill in the following table for the government purchases program:
With Government
Purchase Program
With No Program
Impact of Program
Consumer surplus
Producer surplus
Impact on the
government budget
Net benefits (consumer
surplus producer
surplus government
expenditure)
Deadweight loss
10.24. Suppose that in the domestic market for computer chips the demand is P d 110 Q d, where Q d is the
number of units of chips demanded domestically when
the price is P d. The domestic supply is P s 10 Q s,
where Q s is the number of units of chips supplied domestically when domestic suppliers receive a price P s.
Foreign suppliers would be willing to supply any number
of chips at a price of $30. The government is contemplating three possible policies:
Policy I: The government decides to ban imports of chips.
Policy II: Foreign suppliers are allowed to import chips
(with no tariff ).
Policy III: The government allows imports, but imposes
a tariff of $10 per unit.
10.25. The domestic demand curve for portable radios is given by Q d 5000 100P, where Q d is the
number of radios that would be purchased when the
price is P. The domestic supply curve for radios is given
by Q s 150P, where Q s is the quantity of radios that
would be produced domestically if the price were P.
Suppose radios can be obtained in the world market at
a price of $10 per radio. Domestic radio producers have
successfully lobbied Congress to impose a tariff of $5
per radio.
a) Draw a graph illustrating the free trade equilibrium (with no tariff ). Clearly illustrate the equilibrium
price.
b) By how much would the tariff increase producer surplus for domestic radio suppliers?
Fill in the table in Figure 10.19, giving numerical answers.
Policy I
Ban Imports
Policy
How many units of chips would be consumed domestically?
How many units of chips would be produced domestically?
What is the size of domestic producer surplus?
What is the size of consumer surplus?
What is the size of government receipts?
FIGURE 10.19
Government Policies for Computer Chip Imports
Policy II
No Tariff
Policy III
Import Tariff
PROBLEMS
c) How much would the government collect in tariff
revenues?
d) What is the deadweight loss from the tariff?
10.26. Suppose that the supply curve in a market is upward sloping and that the demand curve is totally inelastic.
In a free market the price is $30 per ton. If an excise tax
of $2 per ton is imposed in the market, what will be the
resulting deadweight loss?
10.27. Suppose that the domestic demand for television
sets is described by Q d 40,000 180P and that the
supply is given by Q s 20P. If televisions can be freely
imported at a price of $160, how many televisions would
be produced in the domestic market? By how much
would domestic producer surplus and deadweight loss
change if the government introduces a $20 tariff per
television set? What if the tariff was $70?
10.28. Suppose that the domestic demand for television sets is described by Q d 40,000 180P and that the
supply is given by Q s 20P. Televisions can currently be
437
freely imported at the world price of $160. Suppose the
government bans the import of television sets. How
much would domestic producer surplus and deadweight
loss change?
10.29. Suppose that demand and supply curves in the
market for corn are Q d 20,000 50P and Q s 30P.
Suppose that the government would like to see the price
at $300 per unit and is prepared to artificially increase
demand by initiating a government purchase program.
How much would the government need to spend to
achieve this? What is the total deadweight loss if the
government is successful in its objective?
10.30. Suppose that demand and supply curves in the
market for corn are Q d 20,000 50P and Q s 30P.
Suppose that the government would like to see the price
at $300 per unit and would like to do so with an acreage
limitation program. How much would the government
need to spend to achieve this? What is the total deadweight loss at the point where the government is successful in its objective?
11
MONOPOLY AND
MONOPSONY
11.1
P R O F I T M A X I M I Z AT I O N
BY A MONOPOLIST
Is the DeBeers Diamond
APPLICATION 11.1
Monopoly Forever?
11.2
T H E I M P O RTA N C E O F P R I C E
ELASTICITY OF DEMAND
APPLICATION 11.2
Chewing Gum, Baby Food,
and the IEPR
APPLICATION 11.3
Market Power in the Breakfast
Cereal Industry
11.3
C O M PA R AT I V E S TAT I C S
FOR MONOPOLISTS
APPLICATION 11.4
Parking Meter Pricing
in Chicago
No Smoking Gun for
Cigarette Producers
APPLICATION 11.5
11.4
M O N O P O LY W I T H M U LT I P L E
PLANTS AND MARKETS
APPLICATION 11.6
Is a Cartel as Efficient
as a Monopoly?
11.5
T H E W E L FA R E E C O N O M I C S
O F M O N O P O LY
The Deadweight Loss in the
Norwegian Cement Industry
APPLICATION 11.7
11.6
W H Y D O M O N O P O LY
MARKETS EXIST?
APPLICATION 11.8
United States of America
versus Microsoft
11.7
MONOPSONY
438
APPLICATION 11.9
Is Wal-Mart a Monopsony?
Why Do Firms Play Monopoly?
“Book your place in space now and join around 250 Virgin Galactic astronauts who will venture into space.
Tickets cost $200,000 and deposits start at $20,000.”1 With these words on its website, Virgin Galactic
describes its plan to make suborbital space travel available to almost anyone who wants to travel into
space. Passengers will first undergo three days of preflight preparation and training at the Virgin Galactic
New Mexico Spaceport. Passengers then will fly on Virgin Galactic’s SpaceShipTwo, a 60-foot-long vehicle
designed to carry six passengers and two crew members on a two-hour journey that will reach an altitude
of 110 kilometers, 10 kilometers beyond the boundary that marks the beginning of space.
If all tests go well with Sir Richard Branson’s new flight system, Virgin Galactic will be operating the
first manned commercial spaceship in the near future. As the only provider of this service, Virgin Galactic
will have a monopoly in its market. A monopoly market consists of a single seller facing many buyers.
Initially, when Virgin Galactic is the only supplier of commercial space travel, it will be a pure monopoly,
serving 100 percent of the market because it faces no rivals. Its production decisions are made in a setting
vastly different from the environment in a perfectly competitive market, where each firm is a “price taker”
because its actions have an imperceptible impact on the market price. In contrast, Virgin Galactic knows
that the number of customers willing to book flights will surely depend on the price it charges, and it can
therefore set the price of a flight. Virgin Galactic will thus be a “price maker” in the market.
While pure monopolies are not widespread,
many markets operate under near-monopoly conditions, in which a single firm accounts for an overwhelming share of sales. For example, the German
firm Hauni Maschinenbau has a global market share
of over 90 percent for cigarette-making machines.
Another German company, Konig and Bauer,
produces 95 percent of the worldwide supply of
money-printing presses. Within the United States,
Microsoft Windows accounts for over 90 percent
of the market for operating systems for personal
computers. Even Virgin Galactic, which may serve the
entire market when it launches its space transport
service, may become a near-monopoly within a few
years when other firms enter the market.
Whether a firm produces as a near-monopoly or
a pure monopoly, it must recognize that its output
decision critically affects the market price for its
product. For example, if the firm reduces its rate of
1
http://www.virgingalactic.com/booking/, accessed February 16,
2010.
439
440
CHAPTER 11
M O N O P O LY A N D M O N O P S O N Y
production, the price of the product will probably rise. Of course, a firm, by itself (even Virgin Galactic), can
only raise the price so much. At some price no one will buy the product at all. Thus, the monopolist must
recognize that the properties of the demand curve—in particular, the price elasticity of market demand—will
affect the price it can set in the market.
When an individual agent can affect the price that prevails in the market, we say the agent has market
power. A monopsony market consists of a single buyer purchasing a product from many suppliers.
Monopsonies most frequently arise in markets for inputs, such as raw materials or industrial components.
They also arise in industries such as aerospace, where the buyer is often a government agency, such as the
U.S. Department of Defense or NASA.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Explain how a monopolist chooses the level of its output (and thus, its price) to maximize profit.
• Calculate a monopolist’s profit-maximizing price and quantity given information about demand and cost.
• Compare the market equilibrium in a competitive market with the profit-maximizing choices of a
monopolist.
• Determine how a monopolist with more than one plant allocates its production among those plants.
• Explain how a monopsonist chooses its inputs to maximize profit.
• Calculate a monopsonist’s profit-maximizing price and quantity given information about demand
and cost.
• Compare the market equilibrium in a competitive market with the profit-maximizing choices of a
monopsonist.
• Explain how the choices of a monopolist or a monopsonist lead to economic inefficiency in a market.
11.1
PROFIT
MAXIMIZATION
BY A
MONOPOLIST
A
firm in a perfectly competitive market has an inconsequential impact on the market
price and thus takes it as given. By contrast, a monopolist sets the market price for its
product. So what would stop the monopolist from setting an infinitely high price? The
answer is that the monopolist must take account of the market demand curve: The
higher the price it sets, the fewer units of its product it will sell; the lower the price it
sets, the more units it will sell. Thus, the monopolist’s market demand curve is downward sloping, as shown in Figure 11.1. The profit-maximizing monopolist’s problem is
finding the optimal trade-off between volume (the number of units it sells) and margin
(the differential between price and marginal cost on the units it sells). The logic we develop to analyze this volume–margin trade-off will apply in the nonmonopoly market
settings (oligopoly and monopolistic competition) that we study in later chapters.
T H E P R O F I T- M A X I M I Z AT I O N C O N D I T I O N
Suppose a monopolist faces the market demand curve D in Figure 11.1. The equation
of this demand curve is P(Q) 12 Q. (Q is expressed in millions of ounces per year,
and P is expressed in dollars per ounce.) To sell 2 million ounces, the monopolist
1 1 . 1 P R O F I T M A X I M I Z AT I O N B Y A M O N O P O L I S T
441
Price (dollars per ounce)
$12
$10
$7
D
0
5
2
12
Quantity (millions of ounces per year)
FIGURE 11.1 The Monopolist’s Demand
Curve Is the Market Demand Curve
The market demand curve is D. To sell more, the
monopolist must charge less. But at what quantity
will the monopolist maximize profit?
would charge a price of $10 per ounce. But to sell a higher quantity such as 5 million
ounces, the monopolist would have to lower its price to $7 per ounce.
As we move along the monopolist’s demand curve, different quantities and their
associated prices generate different amounts of total revenue for the monopolist. Total
revenue is price times quantity, so in this case the monopolist’s total revenue is TR(Q)
P(Q) ⫻ Q ⫽ 12Q ⫺ Q2.
Let’s further suppose that the monopolist’s total cost of production is given by the
equation TC(Q) ⫽ (1/2)Q2. Table 11.1 shows quantity, price, total revenue, total cost,
and profit for this monopolist. Figure 11.2(a) illustrates total revenue, total cost, and
profit graphically, revealing that TC increases as Q increases. By contrast, TR and
profit first rise as Q increases but then fall. The monopolist’s profit is maximized at
the peak of the profit hill, which occurs at Q ⫽ 4 million ounces.
For quantities less than Q ⫽ 4 million, increasing the output increases total revenues
more than it increases total cost, which moves the firm up its profit hill. As Figure 11.2(b)
shows, over this range of output, the monopolist’s marginal revenue exceeds its marginal
TABLE 11.1
Total Revenue, Cost, and Profit for a Monopolist
Q (million ounces)
P ($/oz.)
TR ($ million)
TC ($ million)
0
1
2
3
4
5
6
7
8
9
10
12
11
10
9
8
7
6
5
4
3
2
0
11.00
20.00
27.00
32.00
35.00
36.00
35.00
32.00
27.00
20.00
0
0.50
2.00
4.50
8.00
12.50
18.00
24.50
32.00
40.50
50.00
Profit ($ million)
0
10.50
18.00
22.50
24.00
22.50
18.00
10.50
0
13.50
30.00
CHAPTER 11
M O N O P O LY A N D M O N O P S O N Y
Total revenue, total cost, and
profit (millions of dollars per year)
442
TC
$24
TR
Profit
0
4
8
Quantity (millions of ounces per year)
12
(a)
MC
FIGURE 11.2 Profit Maximization
by a Monopolist
Panel (a): Total cost TC increases as Q increases.
Total revenue TR first increases and then decreases, and so does profit. The monopolist’s
profit is maximized at Q ⫽ 4 million ounces.
Panel (b): The monopolist’s profit-maximization
condition is MR ⫽ MC, where the marginal
revenue and marginal cost curves intersect.
Price (dollars per ounce)
$12
MR
0
4
Quantity (millions of ounces per year)
D
12
(b)
cost: MR MC. For quantities greater than Q 4 million, producing less output increases profit. Over this range, decreasing quantity decreases total cost faster than it
decreases total revenue, which also moves the firm up its profit hill. Over this range
of output, the monopolist’s marginal revenue is less than its marginal cost: MR ⬍ MC.
Let’s summarize what this discussion implies:
profit-maximization
condition for a
monopolist The condition that says that a
monopolist maximizes profit
by producing a quantity at
which marginal revenue
equals marginal cost.
• If the firm produces a quantity at which MR MC, the firm cannot be maximizing its profit because it could increase its output and its profit would go up.
• If the firm produces a quantity at which MR ⬍ MC, the firm cannot be maximizing its profit because it could decrease its output and its profit would go up.
• Thus, the only situation at which the monopolist cannot improve its profit by
increasing or decreasing output is where marginal revenue equals marginal cost.
That is, if Q* denotes the profit-maximizing output, then
MR(Q*) ⫽ MC(Q*)
(11.1)
Equation (11.1) is the profit-maximization condition for a monopolist. Figure 11.2(b)
shows this condition graphically: the quantity at which marginal revenue equals marginal
cost occurs where MR and MC cross.
1 1 . 1 P R O F I T M A X I M I Z AT I O N B Y A M O N O P O L I S T
A P P L I C A T I O N
11.1
Is the DeBeers Diamond Monopoly
Forever?2
DeBeers is a South Africa-based company that, until
the late 1990s, had a near monopoly on the sale of
diamonds worldwide. DeBeers had exclusive rights
to mining in Africa, producing about 80 percent of
the quantity and over 95 percent of the dollar value
of diamonds worldwide. Most diamonds were sold
through its London office. By effectively managing a
cartel of the major producers in Africa, DeBeers maximized profits by reducing the quantity of diamonds
sold, thereby raising prices. As one might expect, as a
near monopolist in the market for newly mined diamonds, DeBeers made enormous profits for many
years.
New developments since that time have threatened DeBeers’s monopoly. DeBeers also had the rights
to sell diamonds mined in the Soviet Union. However,
when the Soviet Union collapsed, DeBeers was unable
to enforce those agreements. The flow of Russian
diamonds increased dramatically, outside of DeBeers’s
control. Several jewelry companies, including Tiffany,
integrated backward into mining to avoid acquiring
diamonds from DeBeers. In 2004 Namibia passed a law
requiring miners to sell a percentage of their diamonds
to local polishers, also outside of DeBeers’s influence.
Other African nations were increasingly challenging
the dominance of DeBeers over the distribution and
sale of such a valuable commodity mined in their countries. DeBeers’s market share has gradually decreased
over this time.
A new development may be of even greater
concern for DeBeers: synthetic diamonds. Natural dia-
monds are formed when carbon is under intense
pressure under the Earth’s surface for hundreds of
millions of years. Recently, scientists have discovered
how to create diamonds in less than a week by putting carbon under extremely high pressure in a laboratory. The first synthetic diamonds were deemed
poor substitutes for natural diamonds in jewelry, but
they did prove to be excellent substitutes in industrial
applications (where diamonds are used for cutting
because of their extremely hard surfaces). By 2007,
synthetic diamonds had captured 90 percent of the industrial diamond market from DeBeers. Worse still for
DeBeers, makers of synthetic diamonds have improved
their products to such an extent that they are now
often indistinguishable from natural diamonds, even
to professional jewelers.
It will be interesting to see what effects synthetic
diamonds will have on the market for diamonds in
jewelry. Currently, most jewelers and customers have
a strong preference for natural diamonds, even
though synthetic ones are chemically identical and
are indistinguishable. Apparently, the “authenticity”
of natural diamonds still has sentimental value. The
market price of synthetic diamonds for jewelry is
about 30 percent of the price for natural diamonds.
However, preferences may change over time as consumers become more accustomed to synthetic diamonds and see that they are functionally equivalent
and much cheaper. If that happens, DeBeers will lose
a large part of its market power. DeBeers still controls
a large fraction of the supply of natural diamonds,
but it may be forced to dramatically cut prices (and
increase output it is willing to sell) in order to meet
the new competition.
The profit-maximization condition in equation (11.1) is a general one, applying
to both monopolists and perfectly competitive firms. As we showed in Chapter 9, in a
perfectly competitive market, a price-taking firm maximizes profit by producing a
quantity at which marginal cost equals marginal revenue (MC MR), and as we have
just shown, the profit-maximizing monopolist must do the same.
2
443
David McAdams & Cate Reavis, “DeBeers’s Diamond Dilemma,” Case 07-045, MIT Sloan School of
Management, 2008.
CHAPTER 11
M O N O P O LY A N D M O N O P S O N Y
A C L O S E R L O O K AT M A R G I N A L R E V E N U E :
MARGINAL UNITS AND INFRAMARGINAL UNITS
As we also showed in Chapter 9, for a price-taking firm, marginal revenue equals the
market price. For a monopolist, however, marginal revenue is not equal to market price.
To see why, let’s take another look at the demand curve D for our monopolist, in
Figure 11.3. Suppose the monopolist initially produces 2 million ounces, charging a
price of $10 per ounce. The total revenue it gets at this price is 2 million ⫻ $10, which
corresponds to area I ⫹ area II. Now suppose the monopolist contemplates producing a larger output, 5 million ounces. To sell this quantity, it must lower its price to
$7 per ounce, as dictated by the market demand curve. The monopolist’s total revenue
is now equal to area II ⫹ area III. Thus, the change in the monopolist’s revenue when
it increases output from 2 million ounces to 4 million ounces is area III minus area I.
Let’s interpret what each of these areas means:
• Area III represents the additional revenue the monopolist gets from the additional
3 million ounces of output it sells when it lowers its price to $7: $7 ⫻ (5 ⫺ 2) million ⫽ $21 million. The extra 3 million ounces are called the marginal units.
• Area I represents the revenue the monopolist sacrifices on the 2 million ounces it
could have sold at the higher price of $10: ($10 ⫺ $7) ⫻ 2 million ⫽ $6 million.
These 2 million ounces are called the inframarginal units.
When the monopolist lowers its price and raises its output, the change in total
revenue, ⌬TR, is the sum of the revenue gained on the marginal units minus the
$12
Price (dollars per ounce)
444
$10
I
$7
II
III
D
0
2
5
12
Quantity (millions of ounces per year)
FIGURE 11.3 The Change in Total Revenue When the Monopolist Increases Output
To increase output from 2 million to 5 million ounces per year, the monopolist must decrease price from $10 to $7 per ounce. The gain in revenue due to the increased output of
3 million units (the marginal units) is equal to area III, while the revenue sacrificed on the
2 million units (the inframarginal units) it could have sold at the higher price is equal to
area I. Thus, the change in total revenue equals area III ⫺ area I.
1 1 . 1 P R O F I T M A X I M I Z AT I O N B Y A M O N O P O L I S T
revenue sacrificed on the inframarginal units: ⌬TR ⫽ area III ⫺ area I $21 million
$6 million $15 million. Or, put another way, the monopolist’s total revenues go up
at a rate of $15 million/3 million ounces $5 per ounce.
To derive a general expression for marginal revenue, note that in Figure 11.3:3
Area III price ⫻ change in quantity ⫽ P⌬Q
Area I ⫽ ⫺quantity ⫻ change in price ⫽ ⫺Q⌬P
Thus, the change in the monopolist’s total revenue is: ⌬TR ⫽ area III ⫺ area I ⫽
P⌬Q ⫹ Q⌬P.
If we divide this change in total revenue by the change in quantity, we get the rate
of change in total revenue with respect to quantity, or marginal revenue:
MR ⫽
P¢Q ⫹ Q¢P
¢TR
¢P
⫽
⫽P⫹Q
¢Q
¢Q
¢Q
(11.2)
Equation (11.2) indicates that marginal revenue consists of two parts. The first
part, P, corresponds to the increase in revenue due to higher volume—the marginal
units. The second part, Q(⌬P/⌬Q) (which is negative, since ⌬P is negative), corresponds to the decrease in revenue due to the reduced price of the inframarginal
units. Since Q(⌬P/⌬Q) ⬍ 0, then MR ⬍ P. That is, the marginal revenue is less than
the price the monopolist can charge to sell that quantity, for any quantity greater
than 0.
When Q ⫽ 0, equation (11.2) implies that marginal revenue and price are equal.
This makes sense in light of Figure 11.3. Suppose the monopolist charges a price of
$12 per ounce and thus sells zero output. To increase its output, the monopolist has
to lower its price, but starting at Q ⫽ 0, it has no inframarginal units. That is, per
equation (11.2), marginal revenue equals price plus Q(⌬P/⌬Q), but when Q ⫽ 0,
Q(⌬P/⌬Q) ⫽ 0, and marginal revenue equals price.
Note that marginal revenue can either be positive or negative. It is negative if the
increased revenue the firm gets from selling additional volume is more than offset by
the decrease in revenue caused by the reduction in price on units that it could have
sold at a higher price. In fact, the greater the quantity, the more likely it is that marginal revenue will be negative because the reduced price (needed to sell more output)
affects more inframarginal units.
AV E R AG E R E V E N U E A N D M A R G I N A L R E V E N U E
In previous chapters, we usually contrasted the average of something with the marginal of the same thing (e.g., average product versus marginal product, average cost
versus marginal cost). For a monopolist, it is important to contrast average revenue
with marginal revenue because this will help explain why the monopolist’s marginal
revenue curve MR is not the same as its demand curve D, as shown in Figure 11.4 (b)
[and first illustrated in Figure 11.2(b)].
3
We put a minus sign in front of this expression for area I because if price goes down, as in Figure 11.2,
the change in price will be negative. The minus sign ensures that the calculated area is a positive number.
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CHAPTER 11
TR (millions of dollars per year)
446
M O N O P O LY A N D M O N O P S O N Y
$35
When P =
$7/ounce,
Q = 5, so
TR = $35
TR
0
5
6
Quantity (millions of ounces per year)
Price, AR, MR (dollars per ounce)
(a)
$12
$10
When Q = 5, P = $7/ounce
and TR = $35. Thus, AR = $35/5 = $7
∆P
$7
∆P/∆Q = –$1/ounce, so when Q = 5 and
P = $7/ounce, MR = $7 + (–$1)5 = $2/ounce
$2
0
∆Q
2
5
MR
D = AR
6
12
Quantity (millions of ounces per year)
(b)
FIGURE 11.4 Total, Average, and Marginal Revenue
The demand curve D and the average revenue curve AR coincide. The marginal revenue
curve MR lies below the demand curve. The slope of the demand curve is ⌬P/⌬Q ⫽ ⫺1;
for example, if price decreases by $3 per ounce (from $10 to $7), quantity increases by
3 million ounces per year (from 2 million to 5 million). When price P ⫽ $7 per ounce
and quantity Q ⫽ 5 million ounces per year:
• Panel (a)—Total revenue TR ⫽ P ⫻ Q ⫽ 7 ⫻ 5 ⫽ $35 million per year.
• Panel (b)—Average revenue AR ⫽ TR/Q ⫽ 35/5 ⫽ $7 per ounce.
Marginal revenue MR ⫽ P ⫹ Q(⌬P/⌬Q) ⫽ 7 ⫹ 5(⫺1) ⫽ $2 per ounce.
The total revenue curve in panel (a) reaches its maximum when Q ⫽ 6, the same quantity
at which MR ⫽ 0 in panel (b).
average revenue Total
revenue per unit of output
(i.e., the ratio of total
revenue to quantity).
The monopolist’s average revenue is the ratio of total revenue to quantity:
AR TR/Q. Since total revenue is price times quantity, AR (P ⫻ Q)/Q ⫽ P. Thus,
average revenue is equal to price. And, since the price P(Q) the monopolist can
charge to sell any quantity of output Q is determined by the market demand curve,
the monopolist’s average revenue curve coincides with the market demand curve:
AR(Q) ⫽ P(Q).
1 1 . 1 P R O F I T M A X I M I Z AT I O N B Y A M O N O P O L I S T
447
Combining these insights with the discussion in the preceding section, we can see
that, if output is positive (Q 0):
• Marginal revenue is less than price (MR ⬍ P ).
• Because average revenue is equal to price, marginal revenue is less than average
revenue (MR ⬍ AR).
• Since the average revenue curve coincides with the demand curve, the marginal
revenue curve must lie below the demand curve.
Figure 11.4 shows the relationships among price, quantity, total revenue, average revenue, and marginal revenue.
The relationship between average revenue and marginal revenue is consistent
with other average–marginal relationships we have seen elsewhere in the book. When
the average of something is falling, the marginal of that thing must be below the
average. Because market demand slopes downward (i.e., is falling) and the average revenue curve corresponds to the demand curve, the marginal revenue curve must be
below the average revenue curve.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 1 . 1
D
Marginal and Average Revenue for a Linear Demand Curve
Suppose that the equation of the market
demand curve is P a bQ.
Problem What are the expressions for the average
and marginal revenues curves?
Solution Average revenue coincides with the demand curve. Thus, AR ⫽ a ⫺ bQ.
Per equation (11.2), marginal revenue is
MR(Q) P ⫹ Q
¢P
¢Q
Now note that ⌬P/⌬Q b (since P a bQ is in the
general form of a linear equation). Substituting into the
equation above:
Thus, the marginal revenue curve for a linear demand
curve is also linear. In fact, it has the same P-intercept
as the demand curve (i.e., at a), with twice the slope.
This implies that the marginal revenue curve intersects
the Q-axis halfway between the origin and the horizontal intercept of the demand curve, which occurs at
Q a/(2b). For quantities greater than this halfway
point, marginal revenue not only lies below the demand
curve, it is also negative. Notice that the shape of the
marginal curve in Figure 11.4(b) is consistent with these
properties.
Similar Problems:
11.1, 11.2
MR(Q) a bQ ⫹ Q( ⫺ b)
a 2bQ
T H E P R O F I T- M A X I M I Z AT I O N C O N D I T I O N
S H O W N G R A P H I C A L LY
Figure 11.5 illustrates the profit-maximization condition MR MC for our monopolist.
The marginal revenue curve MR is decreasing and lies below the demand curve D (which
is also the average revenue curve) for all positive output levels. The marginal cost curve
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CHAPTER 11
M O N O P O LY A N D M O N O P S O N Y
FIGURE 11.5
The Monopolist’s ProfitMaximization Condition
The profit-maximizing output is 4 million ounces
per year, where MC MR. To sell that output, the
monopolist will set a price of $8 per ounce (as indicated by the demand curve D). Total revenue is
areas B ⫹ E ⫹ F. Total cost is area F. Profit (total
revenue minus total cost) is areas B ⫹ E.
Consumer surplus is area A.
Price (dollars per ounce)
$12
MC
A
$8
B
AC
$4
E
$2
0
F
MR
D
4
Quantity (millions of ounces per year)
MC is a straight line from the origin, as is the average cost curve AC. For all positive
output levels, the marginal cost curve lies above the average cost curve.
The profit-maximizing quantity is the quantity at the point where MR and MC
intersect: 4 million ounces per year. The profit-maximizing price is the price at
which that quantity meets the demand curve: $8 per ounce (at that price, the quantity demanded is 4 million ounces per year). At this profit-maximization condition,
profit equals total revenue minus total cost. Total revenue is price (or average revenue) times quantity (areas B ⫹ E ⫹ F ), and total cost is average cost times quantity
(area F ). Thus, profit equals areas B ⫹ E, or $24 million, which corresponds with
Table 11.1.
Figure 11.5 illustrates three important points about the equilibrium in a monopoly market:
• First, the monopolist’s profit-maximizing price ($8) exceeds the marginal cost
of the last unit supplied ($4). This differs from the outcome in a perfectly
competitive market, in which price equals the marginal cost of the last unit
supplied.
• Second, the monopolist’s economic profits can be positive. This is in contrast to
a perfectly competitive firm in a long-run equilibrium, because the monopolist
does not face the threat of free entry that drives economic profits to zero in
competitive markets.
• Third, even though the monopolist raises price above marginal cost and earns
positive economic profits, consumers still enjoy some benefits at the monopoly
equilibrium. The consumer surplus at the equilibrium in Figure 11.5 is the area
between price and the demand curve, or area A, which equals $8 million. The
total economic benefit at the monopoly equilibrium is the sum of consumer surplus and the monopolist’s producer surplus, which is equal to areas A ⫹ B ⫹ E,
or $32 million per year.
1 1 . 1 P R O F I T M A X I M I Z AT I O N B Y A M O N O P O L I S T
S
E
449
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 1 . 2
D
Applying the Monopolist’s Profit-Maximization Condition
The equation of the monopolist’s demand
curve in Figure 11.5 is P 12 Q, and the equation of
marginal cost is MC Q, where Q is expressed in millions of ounces.
Problem What are the profit-maximizing quantity
and price for the monopolist?
Solution To solve this problem, (1) find the marginal
revenue curve, (2) equate marginal revenue to marginal
cost to find the profit-maximizing quantity, and (3) substitute this quantity back into the demand curve to find
the profit-maximizing price.
The monopolist’s demand curve has the same form
as the demand curve in Learning-By-Doing Exercise 11.1
(P a bQ). Therefore, as in that exercise, our monopolist’s marginal revenue curve has the same vertical intercept as the demand curve (i.e., 12) and twice the slope:
MR 12 2Q. The profit-maximization condition is
MR MC, or 12 2Q Q. Thus, the profit-maximizing
quantity is Q 4 (i.e., 4 million ounces). Substituting
this result back into the equation for the demand curve,
we find that the profit-maximizing price P 12 4
8 (i.e., $8 per ounce). These results, of course, correspond
with the graphical solution of the monopolist’s profitmaximization problem shown in Figure 11.5.
Similar Problems:
11.5, 11.6, 11.7, 11.8, 11.9,
11.10, 11.12, 11.13, 11.14
A M O N O P O L I S T D O E S N OT H AV E A S U P P LY C U RV E
A perfectly competitive firm takes the market price as given and chooses a profitmaximizing quantity. The fact that the perfect competitor views price as exogenous
allows us to construct the firm’s supply schedule, by taking each possible market price
and associating it with the corresponding profit-maximizing quantity.
For the monopolist, however, price is endogenous, not exogenous. That is, the monopolist determines both quantity and price. Depending on the shape of the demand
curve, the monopolist might supply the same quantity at two different prices or different quantities at the same price. The unique association between price and quantity that exists for a perfectly competitive firm does not exist for a monopolist. Thus,
a monopolist does not have a supply curve.
Figure 11.6 illustrates this point. For demand curve D1, the profit-maximizing
quantity is 5 million units per year, and the profit-maximizing price is $15 per unit. If
Price (dollars per unit)
MC
$20
$15
$10
D1
MR2
0
MR1
5
Quantity (millions of units per year)
D2
FIGURE 11.6 The
Monopolist Does Not Have a
Supply Curve
When the demand curve is
D1, the monopolist’s profitmaximizing quantity is 5 and the
profit-maximizing price is $15.
When the demand curve is D2,
the profit-maximizing quantity is
also 5, but the profit-maximizing
price is $20. Thus, the monopolist
might sell the same quantity at
different prices, depending on
demand.
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the monopolist’s demand curve shifts to D2, the profit-maximizing quantity continues
to be 5 million units per year, but the profit-maximizing price is now $20 per unit. It
is thus possible, depending on market demand, for a monopolist to sell a given profitmaximizing quantity (5 million units in Figure 11.6) at different prices ($15 and $20).
Therefore, no unique supply curve exists for a monopolist.
11.2
We have just seen that the monopolist uses the market demand curve to set price.
THE
I M P O R TA N C E
OF PRICE
ELASTICITY
OF DEMAND
We have also seen that the monopolist’s profit-maximizing price exceeds the marginal
cost of the last unit supplied. In this section, we explore in more detail how the nature
of the demand curve affects the gap between the monopolist’s profit-maximizing price
and the marginal cost. In particular, we will see that this gap is influenced in a very
important way by the price elasticity of demand.
PRICE ELASTICITY OF DEMAND
A N D T H E P R O F I T- M A X I M I Z I N G P R I C E
Price (dollars per unit)
Figure 11.7 shows why the price elasticity of demand plays such an important role in
the monopolist’s profit-maximization condition. Figure 11.7(a) shows the profitmaximizing price PA and quantity QA in a particular monopoly market, A. Figure 11.7(b)
shows another monopoly market, B, in which demand is less sensitive to price. In particular, we constructed the demand curve in market B by pivoting the demand curve
in market A around the profit-maximizing price and quantity in market A. That is,
demand curve DB is less price elastic than demand curve DA at the profit-maximizing
price PA for market A. Comparing the two markets, we see that the gap between the
PB
PA
0
DA
MRA
QA
Quantity (units per year)
(a) Market A
DB
DA
MRB
MC
MC
0
QB
QA
Quantity (units per year)
(b) Market B
FIGURE 11.7 How Price Elasticity of Demand Affects Monopoly Pricing
In market A, the profit-maximizing price is PA. In market B, where demand is less price elastic at
the price PA, the profit-maximizing monopoly price is PB. The difference between the profitmaximizing price and the marginal cost MC is smaller when demand is more price elastic.
1 1 . 2 T H E I M P O RTA N C E O F P R I C E E L A S T I C I T Y O F D E M A N D
profit-maximizing price and marginal cost is much less in monopoly market A, in
which demand is relatively more price elastic, than it is in market B, where demand is
relatively less price elastic. This shows us that the price elasticity of demand plays an
important role in determining the extent to which the monopolist can raise price
above marginal cost.
This insight suggests an important point about the role of indirect competition
from outside an industry. Any real-world monopolist will typically face some sort of
competition from outside its industry. If there are especially close substitutes for the
monopolist’s product, consumers are likely to be relatively price sensitive, and the monopolist will be unable to set its price very much above its marginal cost. The firm will
be a monopoly, but the threat of substitute products will not allow it to translate that
monopoly into a large markup of price over marginal cost. This would explain why a
monopolist, despite having the market to itself, might not set outrageously high
prices. This reflects a recognition of price elasticity of demand: By setting too high a
price, a monopolist will lose customers to other products.
MARGINAL REVENUE AND PRICE ELASTICITY
OF DEMAND
Let’s now formalize the relationship between the price elasticity of demand and the
monopolist’s markup of price over marginal cost by deriving an equation that shows
how they are related. As a first step, we need to restate equation (11.2) for marginal
revenue.
MR ⫽ P ⫹ Q
¢P
¢Q
By rearranging terms in this formula, we can write marginal revenue in terms of the
price elasticity of demand, ⑀Q,P :4
MR ⫽ P a1 ⫹
1
b
⑀Q,P
(11.3)
This formula shows how marginal revenue depends on the price elasticity of demand.
Since ⑀Q,P ⬍ 0, the formula also confirms our earlier conclusion that MR ⬍ P, and it
reveals another important set of relationships between price elasticity of demand and
4
To derive this expression, factor P out of equation (11.2), giving
MR ⫽ P a1 ⫹
Q ¢P
b
P ¢Q
Now recall that the price elasticity of demand is given by the formula ⑀Q, P ⫽ (¢Q/¢P )(P/Q). Thus, the
term (Q/P)(⌬P/⌬Q) is equal to 1/⑀Q,P, that is, the reciprocal of the price elasticity of demand. Making this
substitution gives us
MR ⫽ P a1 ⫹
1
b
⑀Q,P
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CHAPTER 11
M O N O P O LY A N D M O N O P S O N Y
marginal revenue (and therefore between total revenue and price), as shown in the following table:
Relationship between
Region of Demand Curve
Marginal Revenue and ⑀Q,P
Total Revenue and Price
Elastic (⫺q ⬍ ⑀Q,P ⬍ ⫺1)
MR ⬎ 0
[because 1 ⫹ (1/⑀Q,P ) ⬎ 0]
The monopolist can increase total revenue by
decreasing price (and
thereby increasing quantity) by a small amount.
Unitary elastic (⑀Q,P ⫽ ⫺1)
MR ⫽ 0
[because 1 ⫹ (1/⑀Q,P ) ⫽ 0]
The monopolist’s total
revenue will not change
when price (or quantity)
is changed by a small
amount.
Inelastic (⫺1 ⬍ ⑀Q,P ⬍ 0)
MR ⬍ 0
[because 1 ⫹ (1/⑀Q,P ) ⬍ 0]
The monopolist can increase total revenue by
increasing price (and
thereby decreasing quantity) by a small amount.
This table reflects our discussion in Chapter 2 of how a firm’s total revenue responds to a price change. The relationship between marginal revenue and price elasticity of demand shown in the table is illustrated in Figure 11.8.
P = a – bQ
FIGURE 11.8
Marginal Revenue
and Price Elasticity of Demand for a
Linear Demand Curve
Where demand is elastic, marginal
revenue is positive. Where demand is
unitary elastic, marginal revenue is
zero (i.e., MR crosses the horizontal
axis). Where demand is inelastic,
marginal revenue is negative.
P (dollars per unit)
a
Ela
sti
cr
eg
ion
Demand is elastic when Q < a/(2b):
–∞ < εQ,P < –1, MR > 0
Demand is unitary elastic when Q = a/(2b):
εQ,P = –1, MR = 0
Demand is inelastic when a/(2b) < Q < a/b:
–1 < εQ,P < 0, MR < 0
Ine
las
MR
0
D
a
2b
Quantity Q (units per year)
tic
reg
ion
a
b
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1 1 . 2 T H E I M P O RTA N C E O F P R I C E E L A S T I C I T Y O F D E M A N D
MARGINAL COST AND PRICE ELASTICITY OF
DEMAND: THE INVERSE ELASTICITY PRICING RULE
The relationship between marginal revenue and the price elasticity of demand gives
us another way to express the monopolist’s profit-maximization condition, in terms of
marginal cost. Per equation (11.1), at the profit-maximizing price P * and quantity Q*,
MR(Q*) MC(Q*). Therefore, per equation (11.3),
MC(Q *) P * a1 ⫹
1
b
⑀Q,P
If we let MC * stand for MC(Q*) and rearrange this expression algebraically, we get
P * MC *
1
⑀Q, P
P*
(11.4)
The left-hand side of equation (11.4) is the monopolist’s optimal markup of price over
marginal cost, expressed as a percentage of the price. For this reason, equation (11.4) is
called the inverse elasticity pricing rule (IEPR). The IEPR states that the monopolist’s
optimal markup of price above marginal cost, expressed as a percentage of price, is equal
to minus the inverse of the price elasticity of demand. The IEPR tells us that the price
elasticity of demand plays a vital role in determining what price a monopolist should
charge to maximize profits. Specifically, the IEPR summarizes the relationship between
price elasticity of demand and the monopoly price that we saw in Figure 11.7: The more
price elastic the monopolist’s demand, the smaller will be the optimal markup.
Learning-By-Doing Exercises 11.3 and 11.4 show that, if we know the price elasticity of demand, we can apply the IEPR to compute the profit-maximizing monopoly price.
S
E
inverse elasticity
pricing rule (IEPR) The
rule stating that the difference between the profitmaximizing price, and
marginal cost, expressed
as a percentage of price, is
equal to minus the inverse
of the price elasticity of
demand.
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 1 . 3
D
Computing the Optimal Monopoly Price for a Constant Elasticity
Demand Curve
The general form of a constant elasticity demand curve
is Q aPb. At every point on such a curve, the price
elasticity of demand equals b.5 Suppose a monopolist
has a constant marginal cost MC $50.
(a) The price elasticity of demand ⑀Q,P 2. Thus,
P 50
1
P
2
P $100
Problem
(b) Price elasticity of demand ⑀Q,P 5. Thus,
(a) What is the monopolist’s optimal price if its constant
elasticity demand curve is Q 100P2?
P 50
1
P
5
(b) What is the monopolist’s optimal price if its constant
elasticity demand curve is Q 100P5?
P $62.50
Solution
Notice that when demand is more elastic, the monopolist’s profit-maximizing price goes down (holding
marginal cost constant).
For both parts of this problem, we use the IEPR [equation (11.4)] to compute the answer.
Similar Problems:
5
See Chapter 2 and its appendix for discussion of constant elasticity demand curves.
11.17, 11.18, 11.19
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E
M O N O P O LY A N D M O N O P S O N Y
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 1 . 4
D
Computing the Optimal Monopoly Price for a Linear Demand Curve
Along a linear demand curve, the price
elasticity of demand is not constant. Nevertheless, we
can still use the IEPR to compute the profit-maximizing
price (and then use that result to compute the profitmaximizing quantity). Also, we can get the same results
by applying the profit-maximizing condition expressed
in equation (11.1)—MC MR.
Suppose a monopolist has a constant marginal cost
MC $50 and faces the demand curve P 100 Q/2
(which can be rewritten as Q 200 2P).
Problem
(a) Find the profit-maximizing price and quantity for
the monopolist using the IEPR.
(b) Find the profit-maximizing price and quantity for
the monopolist by equating MR to MC.
Solution
(a) For a linear demand curve, the price elasticity of
demand is given by a formula derived from the general
expression for elasticity, ⑀Q,P ⫽ (¢Q/¢P )( P/Q ).6 In this
particular example, ⌬Q/⌬P ⫽ ⫺2, so
⑀Q,P ⫽ ⫺2
P
Q
Thus, the IEPR for this example is
200 ⫺ 2P
1
P ⫺ 50
⫽
⫽⫺
2P
P
2P
⫺(200 ⫺ 2P)
If we multiply each side of this expression by 2P, we
get a simple linear equation: 2P ⫺ 100 ⫽ 200 ⫺ 2P, or
P ⫽ 75. Thus, the profit-maximizing monopoly price is
$75. We find the profit-maximizing monopoly quantity
by substituting this price into the demand curve:
Q ⫽ 200 ⫺ 2(75) ⫽ 50.
(b) To solve the problem by equating MR and MC, recall Learning-By-Doing Exercise 11.1. In that exercise,
we showed that, for a linear demand curve of the form
P ⫽ a ⫺ bQ, marginal revenue MR ⫽ a ⫺ 2bQ. In this
example, then, MR ⫽ 100 ⫺ Q. Since MR ⫽ MC and
MC ⫽ 50, 50 ⫽ 100 ⫺ Q, or Q ⫽ 50. Substituting this
quantity back into the demand curve, we find that P ⫽
100 ⫺ 50/2 ⫽ 75.
Thus, the IEPR and the MR ⫽ MC condition give
the same results for the profit-maximizing price and
quantity (this is as it should be, of course, since the IEPR
was derived from the MR ⫽ MC condition). Also, note
that for a linear demand curve, where price elasticity of
demand is not constant, we have to begin with the general formula for ⑀Q,P when applying the IEPR.
Similar Problem:
11.11
Since Q ⫽ 200 ⫺ 2P,
2P
200 ⫺ 2P
⑀Q,P ⫽ ⫺
T H E M O N O P O L I S T A LWAYS P R O D U C E S O N T H E
E L A S T I C R E G I O N O F T H E M A R K E T D E M A N D C U RV E
Although a monopolist could, in theory, set its price anywhere along the market
demand curve, a profit-maximizing monopolist will only want to operate on the elastic
region of the market demand curve (i.e., the region in which the price elasticity of
demand ⑀Q,P is between ⫺1 and ⫺q). Figure 11.9 illustrates why. If you were a monopolist and you contemplated operating at a point such as A at which demand was inelastic, you could always increase profit by raising your price, reducing your quantity,
and moving to point B. When you move from point A to point B, your total revenue
goes up by the difference between area I and area II, and your total costs go down because you are producing less. If your total revenue goes up and your total costs go
6
For discussion of how the price elasticity of demand varies along a linear demand curve, see Chapter 2.
Price (dollars per unit)
1 1 . 2 T H E I M P O RTA N C E O F P R I C E E L A S T I C I T Y O F D E M A N D
B
PB
I
A
PA
II
QB
0
D
QA
Quantity (units per year)
MR
A P P L I C A T I O N
FIGURE 11.9 Why a Profit-Maximizing
Monopolist Will Not Operate on the Inelastic Region of the Market Demand Curve
At point A, on the inelastic region of the
demand curve D, the monopolist is charging price PA, and selling quantity QA. If the
monopolist raises price to PB and decreases
quantity to QB, thereby moving to point B
on the elastic region of the demand curve,
total revenue increases by area I ⫺ area II,
and total costs go down because the
monopolist is producing less. Thus, the
monopolist’s profits must go up.
11.2
Chewing Gum, Baby Food,
and the IEPR
Supermarkets are not monopolists, but many consumers
often shop at the same supermarket week after
week.7 This suggests that supermarkets have the ability to mark up prices above marginal costs, an ability
that they evidently take advantage of. For most grocery products, the difference between the retail price
that the shopper pays to the supermarket and the
wholesale price that the supermarket pays to its suppliers (manufacturers or distributors) ranges between
10 and 40 percent. Interestingly, though, these markups
differ systematically across product categories in
7
455
almost every grocery store, and markups within a
particular product category remain fairly stable over
time. For example, the retail markup on candy and
chewing gum in most grocery stores is usually between 30 and 40 percent, while the markup on
baby food and disposable diapers is usually less than
10 percent.
The IEPR can help us understand why the
markups for chewing gum and candy are so different
from the markups for baby food and disposable diapers. Retailers believe that chewing gum and candy
are impulse purchase items. That is, consumers often
decide to purchase these products on the basis of
whims or momentary urges once they are inside the
store, usually without thinking much about their prices.
Margaret Slade reports that grocery-store marketing managers believe that fewer than 10 percent of
households engage in comparison shopping among local grocery stores to find the lowest-priced items.
For the 90 percent of consumers who frequent the same store each week, their choice of store is thought
to be determined by location (proximity to home or work) and by the quality of the store (e.g., product
variety, freshness of produce). See M. Slade, “Product Rivalry with Multiple Strategic Weapons: An Analysis
of Price and Advertising Competition,” Journal of Economics and Management Strategy (Fall 1995): 445–476.
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By contrast, retailers believe that baby food and disposable diapers are not purchased impulsively. They
believe that most consumers of these products put
considerable thought into their purchase decisions
and pay close attention to price when deciding how
much to buy. This suggests that the demand for
chewing gum and candy is less price elastic than the
demand for baby food and disposable diapers. If so,
the IEPR implies that we should see precisely what
we do see: higher markups for chewing gum and
candy than for baby food and disposable diapers.
For these products, at least, grocery stores seem to
set retail prices in a manner that is broadly consistent
with the IEPR.
down, your profit goes up. Thus, at any point on the inelastic region of the market
demand curve, the monopolist can always find a point on the elastic region that gives
it a higher profit.
We can use the IEPR to reach the same conclusion. To see why, we start with the
(perhaps obvious) observation that marginal cost is positive. This implies that the
term 1 ⫹ (1/⑀Q,P) in equation (11.3) must also be positive. But the only way this term
can be positive is if ⑀Q,P is between ⫺1 and ⫺q, that is, if demand is price elastic.
Thus, the IEPR implies that the monopolist’s profit-maximizing price and quantity
occur along the elastic region of the market demand curve.
T H E I E P R A P P L I E S N OT O N LY TO M O N O P O L I S T S
product differentiation
A situation in which two or
more products possess
attributes that, in the minds
of consumers, set the products apart from one another
and make them less than
perfect substitutes.
The IEPR applies to any firm that faces a downward-sloping demand for its product,
not just to monopolists. Consider, for example, the pricing problem Coca-Cola faces.
Coca-Cola does not have a monopoly in the U.S. cola market: Pepsi is an important
competitor. Still, Coca-Cola and Pepsi are not perfectly competitive firms. In other
words, if Coca-Cola raised its price, it would not lose all its sales to Pepsi, and if it
lowered its price, it would not steal all of Pepsi’s business. The reason for this is that
the two colas exhibit product differentiation, a condition in which two or more
products possess attributes that, in the minds of consumers, set the products apart
from one another and make them less than perfect substitutes. Some people prefer the
sweeter taste of Pepsi to the less sweet taste of Coke and would continue to buy Pepsi
even if it cost more than Coke. You might prefer the taste of Coke. Or you might be
indifferent about the taste but prefer Coca-Cola’s packaging or advertisements.
Differentiated products will have downward-sloping demand curves, even though
the sellers of the products are not monopolists. The optimal pricing decision for a
seller of a differentiated product can thus be characterized by a rule very much like
the IEPR. For example, the optimal price markups for Coca-Cola and Pepsi (denoted
by A and I, respectively) would be described by
P A MC A
1
A
⑀
P
QA, PA
P I ⫺ MC I
1
I
⑀QI , PI
P
In these formulas, ⑀QA, PA and ⑀QI,PI are not market-level price elasticities of demand.
Rather, they are the brand-level price elasticities of demand for Coca-Cola and Pepsi.
1 1 . 2 T H E I M P O RTA N C E O F P R I C E E L A S T I C I T Y O F D E M A N D
457
Thus, ⑀QA,PA tells us the sensitivity of Coca-Cola’s demand to Coca-Cola’s price, holding all other factors affecting Coke’s demand (including Pepsi’s price) fixed.8
Q UA N T I F Y I N G M A R K E T P O W E R : T H E L E R N E R I N D E X
When a firm faces a downward-sloping demand curve, either because it is a monopolist or (like Coca-Cola) it produces a differentiated product, the firm will have some
control over the market price it sets. For a monopoly, the ability to set the market
price is constrained by competition from substitute products. In the case of differentiated products, a firm’s direct competitors constrain its pricing freedom (e.g., Pepsi’s
price limits the price Coca-Cola can charge).
When a firm can exercise some degree of control over its price in the market, we
say that it has market power.9 Note that perfectly competitive firms do not have
market power. Because perfectly competitive firms produce at the point where price
equals marginal cost, while monopolists or producers of differentiated products will,
in general, charge prices that exceed marginal cost, a natural measure of market power
is the percentage markup of price over marginal cost, (P MC )/P (the left-hand side
of the IEPR). This measure was suggested by the economist Abba Lerner and is called
the Lerner Index of market power.
The Lerner Index ranges from 0 to 1 (or from 0 to 100 percent). It is zero for a
perfectly competitive industry. It is positive for any industry that departs from perfect
competition. The IEPR tells us that in the equilibrium in a monopoly market, the
Lerner Index will be inversely related to the market price elasticity of demand. As
we’ve discussed, an important driver of the price elasticity of demand is the threat of
substitute products outside the industry. If a monopoly market faces strong competition from substitute products, the Lerner Index can still be low. In other words, a firm
might have a monopoly, but its market power might still be weak.
A P P L I C A T I O N
power of an individual economic agent to affect the
price that prevails in the
market.
Lerner Index of market
power A measure of monopoly power; the percentage markup of price over
marginal cost (P MC )/P.
11.3
Market Power in the Breakfast
Cereal Industry
The breakfast cereal is dominated by four large sellers —
General Mills, Kellogg, Post, and Quaker Oats. Do
these firms have some market power in the breakfast cereal market? If so, how is that reflected in
prices? The availability of supermarket scanner data
now allows economists to study questions such as
these with very good data. Such data were used by
Benaissa Chidmi and Rigoberto Lopez to calculate
8
market power The
Lerner Indices and determinants of prices for 37
brands of breakfast cereals sold in supermarkets in
Boston.10 They estimated an average price markup of
28 percent over marginal cost. Table 11.2 provides
some examples of estimated Lerner Indices. Corn
Flakes had the highest percentage markups, while
Cookie Crisps had the lowest. Markups and elasticities of demand varied substantially across cereal
brands, and also across supermarket chains. For example, markups were higher at chains with higher
market share, suggesting that more efficient stores
with lower marginal costs gain market share. The
See Chapter 2 for a detailed discussion of the difference between the brand-level and market-level price
elasticity of demand.
9
Monopolists and sellers of differentiated products are not the only kinds of firms with market power, as
you will learn in Chapter 13.
10
Benaissa Chidmi and Rigoberto Lopez, “Brand-Supermarket Demand for Breakfast Cereals and Retail
Competition,” American Journal of Agricultural Economics 27 (May 2007): 324 –337.
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TABLE 11.2
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Sample Lerner Ratios for Breakfast Cereals across Department Store Chains
Supermarket Chain
Cereal Brand
Stop &
Shop
Shaw’s
Demoulas
Star
Market
Average
Kellogg Corn Flakes
General Mills Cheerios
Post Grape Nuts
Quaker Cap N’ Crunch
Nabisco Spoon Size Shredded Wheat
Ralston Cookie Crisp
Average
47.97
32.01
43.4
29.64
34.02
18.24
30.07
42.9
32.13
40.25
29.27
33.67
18.98
28.79
40.43
26.61
38.11
27.91
31.1
19.99
27.51
38.78
26.08
38.72
23.61
30.15
15.52
25.52
42.52
29.21
40.12
27.61
32.24
18.18
27.97
Source: Chidmi & Lopez, 2007.
researchers found that sales were highly sensitive to
price, with own-price elasticities of demand ranging
from about ⫺2.4 for Corn Flakes to ⫺7.1 for Cookie
Crisps. However, they also estimated very low crossprice elasticities when comparing sales across brands
of cereal or across supermarket chains. In other words,
consumers have relatively strong brand and supermarket chain loyalty.
In an earlier study of breakfast cereals, economist
Aviv Nevo used data on cereal prices, product characteristics, consumer demographics (like household income),
and estimated elasticities of demand to compute the
Lerner Indices under two scenarios: one in which cereal
11.3
COMPARATIVE
STATICS FOR
MONOPOLISTS
producers act collectively as a profit-maximizing monopolist, and the other in which producers compete as
independent firms in a market with differentiated
products.11 Nevo concluded that, in a collusive industry,
one would expect to observe Lerner Indices for an individual brand in the range of 65–70. In an industry in
which firms acted more competitively, he determined
that the Lerner Indices would be around 40–44. It turns
out that the actual Lerner Index for the industry in the
mid-1990s was about 45. He thus concluded that market
power in the industry seems to arise because brands are
differentiated products, not because of collusion
among manufacturers.
N
ow that we have explored how the monopolist determines its profit-maximizing
quantity and price and the role that the price elasticity of demand plays in that determination, we are ready to examine how shifts in demand or cost affect the monopolist’s decisions.
SHIFTS IN MARKET DEMAND
Comparative Statics
Figure 11.10 illustrates how a rightward shift in market demand affects the monopolist’s choice of price and quantity. In both panels, we assume that quantity demanded
increases at all market prices (i.e., the original demand curve D0 and the new demand
curve D1 do not intersect) and that the rightward shift in the demand curve results in
a rightward shift in the marginal revenue curve (from MR0 to MR1).
11
A. Nevo, “Measuring Market Power in the Ready-to-Eat Breakfast Cereal Industry,” Econometrica 69
(March 2001): 307–342. Computation of cereal markups in this scenario requires using oligopoly theory,
which you will study in Chapter 13.
459
1 1 . 3 C O M PA R AT I V E S TAT I C S F O R M O N O P O L I S T S
$12
$10
D1
MR0 MR1
0
2
Price (dollars per unit)
Price (dollars per unit)
MC
$10
$9
D0
MR0
3
0
2
MC
6
D1
D0
MR1
Quantity (millions of units per year)
Quantity (millions of units per year)
(a)
$13
(b)
FIGURE 11.10
How a Shift in Demand Affects the Monopolist’s Profit-Maximizing
Quantity and Price
In both panels, a rightward shift in demand (from D0 to D1) causes the profit-maximizing quantity
to increase. In panel (a), where marginal cost MC increases as quantity increases, the profitmaximizing price also goes up. But in panel (b), where marginal cost decreases as quantity
increases, the profit-maximizing price goes down.
In Figure 11.10(a), marginal cost MC increases as quantity increases. In this case,
the increase in demand causes an increase in both the optimal quantity (from 2 million to 3 million units per year) and the optimal price (from $10 to $12 per unit).
In Figure 11.10(b), in contrast, marginal cost decreases as quantity increases. This
still causes the optimal quantity to increase (from 2 million to 6 million units per
year), but it causes the optimal price to decrease (from $10 to $9 per unit), even
though the monopolist can charge a higher price for any given quantity than before
demand increased—for example, before the increase in demand, the monopolist could
sell 2 million units at a price of $10, and after the increase the monopolist could sell
the same 2 million units at a price of $13. However, the monopolist would choose not
to do this because it can maximize profit by selling 6 million units at a price of $9. The
figure shows that, when marginal cost decreases as quantity increases, a rightward
shift in demand may lead the monopolist to lower the price.
In general, as long as the rightward shift in the demand curve results in a rightward shift in the marginal revenue curve, the increase in demand will increase the
monopolist’s optimal quantity. The rightward shift in marginal revenue guarantees
that the intersection of marginal revenue and marginal cost will occur at a quantity
that is higher than the initial one. Similarly, a decrease in demand accompanied by a
corresponding leftward shift in the marginal revenue curve will always decrease the
monopolist’s optimal quantity. However, the impact of a shift in demand on the
optimal market price will (in general) depend on whether marginal cost increases or
decreases as quantity increases.
The Monopoly Midpoint Rule
For a monopolist facing a constant marginal cost and a linear demand curve, there is
a convenient formula for determining the profit-maximizing price: the monopoly
midpoint rule. As shown in Figure 11.11, the monopoly midpoint rule tells us that
monopoly midpoint
rule A rule that states
that the optimal price is
halfway between the vertical intercept of the demand
curve (i.e., the choke price)
and the vertical intercept of
the marginal cost curve.
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the optimal price P * is halfway between the vertical intercept of the demand curve,
a (i.e., the choke price), and the vertical intercept of the marginal cost curve, c. This
implies that an increase in the choke price of ⌬a would cause a corresponding increase
of half that amount (⌬a/2) in the market price. (That is, if the choke price increases
by $10, the monopolist will increase the market price by $5.) Thus, as we see
in Learning-By-Doing Exercise 11.5, the monopoly midpoint rule can be stated as
P * ⫽ (a ⫹ c)/2.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 1 . 5
D
Computing the Optimal Price Using the Monopoly Midpoint Rule
MR ⫽ MC
Suppose a monopolist faces a linear market
demand curve P ⫽ a ⫺ bQ and has a constant marginal
cost MC ⫽ c (as illustrated in Figure 11.11).
a ⫺ 2bQ* ⫽ c
Q* ⫽
Problem What is the monopolist’s profit-maximizing
quantity and price?
a⫺c
2b
We can find the monopolist’s optimal price P * by substituting this optimal quantity back into the demand curve:
P* ⫽ a ⫺ b a
Solution For this demand curve, the monopolist’s
marginal revenue curve is MR ⫽ a ⫺ 2bQ. We equate
this expression to marginal cost and solve for the monopolist’s optimal quantity Q*:
a⫺c
1
1
a⫹c
b⫽a⫺ a⫹ c⫽
2b
2
2
2
Similar Problem:
11.25
Demand curve D is P = a – bQ
P (dollars per unit)
a
MC = c
MR = a – 2bQ
Profit-maximizing price P * = (a + c)/2, halfway between the
choke price a and the marginal cost c
P*=
a+c
2
MC
c
D
0
Q (units per year)
MR
FIGURE 11.11
The Monopoly Midpoint Rule
When the monopolist has a linear demand curve and constant marginal cost, the profit-maximizing
price P* is halfway between the vertical intercept of the marginal cost curve c and the choke
price a.
1 1 . 3 C O M PA R AT I V E S TAT I C S F O R M O N O P O L I S T S
A P P L I C A T I O N
11.4
Parking Meter Pricing in Chicago
In 2009 the city of Chicago outsourced its parking
meters, selling the rights to install, operate, and collect
the profits from the meters to the private firm
Chicago Parking Meters (CPM). Meter rates were substantially increased throughout the city, to great
protest from citizens. As of January 2010, the meter
rate was $4.50 per hour in the Loop business district.
In other busy downtown neighborhoods the rate was
$2.50 per hour, while in less busy areas it was $1.25.
The monopoly midpoint rule shows why it might
make sense for CPM to increase the price in busy
areas. Since drivers can park in garages or take cabs
or public transportation, the company is not a mo-
nopolist. However, the convenience of driving one’s
car and parking right on the street means that CPM
faces a downward-sloping demand curve. It is reasonable to assume that the marginal cost of operating
an additional parking meter is approximately the
same in each neighborhood. However, the demand
curve for parking in the Loop probably lies above and
to the right of the demand curve for parking in other
parts of Chicago. This is because of congestion and
because more drivers have urgent business and so are
willing to pay more for the convenience of street
parking. Given all of this, the monopoly midpoint
rule implies that CPM can increase its profits by
charging higher prices in the Loop, and lower prices
in less busy neighborhoods.
SHIFTS IN MARGINAL COST
Comparative Statics
Price (dollars per unit)
The IEPR suggests that an increase in marginal cost will increase the profit-maximizing
price and, because the demand curve has a negative slope, decrease the profit-maximizing
quantity. Figure 11.12 confirms this intuition. An upward shift in the monopolist’s marginal cost curve increases price and decreases output because the point of intersection
between the marginal revenue curve and the marginal cost curve moves upward and leftward. (Similarly, a downward shift in marginal cost would induce an increase in the monopolist’s profit-maximizing quantity and a decrease in the profit-maximizing price.)
MC1
$9
$8
MC0
FIGURE 11.12
MR
0
4
461
D
6
Quantity (millions of units per year)
How an Increase in
Marginal Cost Changes the Monopoly
Equilibrium
When the monopolist’s marginal cost
curve shifts from MC0 to MC1, the profitmaximizing quantity falls from 6 million to
4 million units per year and price goes up
from $8 to $9 per unit.
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M O N O P O LY A N D M O N O P S O N Y
Total Revenue
(millions of dollars per year)
462
$32
$27
Marginal
revenue
is positive
Marginal
revenue
is negative
TR
0
3
4
Quantity (millions of ounces per year)
(a)
FIGURE 11.13
An Increase in
Marginal Cost Must Decrease the
Monopolist’s Total Revenue
Panel (b) shows that an upward shift in
the marginal cost decreases the monopolist’s optimal quantity from 4 million to
3 million ounces per year. Because the
monopolist always operates on the elastic
region of market demand, the monopolist
operates in the region in which total revenue goes down as output goes down.
The decrease in the profit-maximizing
output thus decreases total revenue from
$32 million to $27 million.
Price (dollars per ounce)
MC1
MC0
$9
$8
MR
0
3
D
4
Quantity (millions of ounces per year)
(b)
How the Revenue Impact of a Shift in Marginal Cost Can Tell Us Whether
Firms Are Behaving as a Profit-Maximizing Monopolist
As Application 11.3 illustrates, firms in an industry with just a few producers are occasionally accused of acting collusively (i.e., collectively acting as a profit-maximizing monopolist). Apart from documentary evidence that firms acted in concert to fix prices, is
there any way to tell whether such an accusation is true? The answer is yes. By looking
at the impact of a shift in marginal cost on the industry’s total revenue, we might be able
to refute the claim that firms in the industry are colluding. Figure 11.13 shows why.
Figure 11.13 illustrates what happens when our monopolist faces an increase in its
marginal cost from MC0 to MC1. When marginal cost shifts upward, the monopolist reduces its output. Since the monopolist operates on the elastic range of the demand curve,
where marginal revenue is positive, the monopolist must be on the upward-sloping part of
its total revenue hill, as shown in Figure 11.13(a). As the monopolist reduces its output in
response to the upward shift in marginal cost, it moves down the total revenue hill, and its
total revenues thus decrease. This illustrates the following comparative statics results:12
12
See J. Panzar and J. Rosse, “Testing for Monopoly Equilibrium,” Journal of Industrial Economics (1987)
for further exploration of the implications of these comparative statics results.
463
1 1 . 4 M O N O P O LY W I T H M U LT I P L E P L A N T S A N D M A R K E T S
A P P L I C A T I O N
11.5
No Smoking Gun for Cigarette
Producers
The cigarette industry is one of the most highly concentrated in the U.S. economy. In the 1990s, the four
largest firms accounted for more than 92 percent of
industry sales. Throughout most of the twentieth century, firms in the cigarette industry displayed remarkable pricing discipline. Twice a year (generally in June
and December), one of the dominant firms announced
its intention to raise the list prices of its cigarettes,
and within days the other cigarette manufacturers
followed with increases of their own. Since the 1970s,
either Philip Morris or RJR has generally been the
price leader. Such discipline has made cigarettes one
of the most profitable businesses in the American
economy. The success of such pricing coordination naturally raises the question of whether the big tobacco
companies have collectively acted as a profit-maximizing
monopolist.
Daniel Sullivan explored this question using the
comparative statics analysis that we just described.13
Using statistical methods, Sullivan studied how prices,
quantities, and revenues over the period 1955–1982
changed in response to changes in state excise taxes.
His research led him to conclude that observed industry outcomes during this period were inconsistent
with the hypothesis that cigarette firms were jointly
acting as a profit-maximizing monopolist.
If cigarette producers do not act as a profitmaximizing cartel, why do they appear to be so profitable? As we will see in Chapter 13, one answer is
that firms in an industry with only a few producers
can still be highly profitable, even if they do not replicate the outcome that a profit-maximizing monopolist
would attain. This is another reminder that market
power and monopoly are not synonymous.
• An upward shift in marginal cost reduces the profit-maximizing monopolist’s
total revenue.
• A downward shift in marginal cost increases the profit-maximizing monopolist’s
total revenue.
We could use these comparative statics results to refute the hypothesis that firms
in a nonmonopoly industry are collectively acting as a profit-maximizing monopolist.
Suppose, for example, that we discovered that an increase in the federal excise tax on
beer resulted in an increase in overall total revenue in the brewing industry. Because
our comparative statics analysis tells us that industry revenue could not have increased if
beer firms were collectively acting as a monopolist, the fact that industry revenue did
increase suggests that beer firms were not acting collusively.
Many firms operate more than one production facility or serve more than one market. 11.4
For example, an electric utility, such as Chicago’s Commonwealth Edison, often uses
several power plants for generating electricity. The theory of monopoly can be easily extended to cover the case of a multiplant firm. We first consider the choice of output by a
monopolist with two plants. We then consider how the analysis applies to a cartel. Finally,
we examine how a monopolist would choose output if it serves more than one market.
OUTPUT CHOICE WITH TWO PLANTS
Consider a monopolist with two plants, with marginal cost functions MC1 and MC2.
The monopolist’s output choice problem consists of two parts: How much should it
produce overall, and how should it divide its production between its two plants?
13
D. Sullivan, “Testing Hypotheses about Firm Behavior in the Cigarette Industry,” Journal of Political
Economy ( June 1985): 586–597.
M O N O P O LY
WITH
M U LT I P L E
PLANTS AND
MARKETS
FIGURE 11.14
CHAPTER 11
Profit
Maximization by a Multiplant
Monopolist
The monopolist’s multiplant
marginal cost curve MCT is the
horizontal sum of the individual
plant’s marginal cost curves MC1
and MC2. The monopolist’s optimal total output of 3.75 million
units per year occurs at MR
MCT, where the optimal price is
$6.25 per unit. Plant 1 produces
1.25 million units of the total
output, and plant 2 produces
2.5 million units.
multiplant marginal
cost curve The horizontal sum of the marginal cost
curves of individual plants.
Price (dollars per unit)
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M O N O P O LY A N D M O N O P S O N Y
MC1 (plant 1)
Profitmaximizing
price
G
$6.25
$6.00
$3.00
C
MC2 (plant 2)
MCT
E
B
A
H
I
F
MR
0
1.25
Output
produced
in plant 1
2.5 3 3.75
D
6
9
Output
Profit-maximizing
produced total output
in plant 2
Quantity (millions of units per year)
Suppose the firm plans to produce 6 million units, with the output equally divided
between plants 1 and 2. Figure 11.14 shows that at an output of 3 million units, plant
1 has a higher marginal cost than plant 2: $6 per unit versus $3 per unit (point B versus point A). Under these circumstances, there is a simple way for the firm to reduce
its total costs (while holding revenues fixed): Increase output at plant 2, and decrease
output at plant 1 by the same amount. Increasing output at plant 2 increases costs at
a rate of $3 per unit, but decreasing output at plant 2 saves costs at a rate of $6 per
unit. Reallocating production away from plant 1 toward plant 2 reduces the firm’s
total production costs. Since reallocation is always profitable whenever the firm operates at a point at which the marginal costs of the plants differ, we conclude that a
profit-maximizing firm will always allocate output among the plants so as to keep their
marginal costs equal.
This insight allows us to construct a marginal cost schedule for a multiplant firm.
Consider, again, Figure 11.14, and pick any possible level of marginal cost, such as $6.
To attain this level of marginal cost at both plants, the firm would produce 3 million
units in plant 1 (point B) and 6 million units in plant 2 (point C ). Thus, it can attain
a marginal cost of $6 when it produces a total output of 9 million units (point E ). The
curve MCT —the multiplant marginal cost curve—traces out the set of points generated by horizontally summing the marginal cost curves of the individual plants.
Having derived the multiplant marginal cost curve, the answer to the first question—
how much should the monopolist produce in total—is relatively easy to find. The
monopolist equates marginal revenue to its multiplant marginal cost curve, MR MCT.
In Figure 11.14, this occurs at a total output of 3.75 million units (point F ). The optimal
price corresponding to this output is $6.25 (point G).
Thus, we have determined the monopolist’s profit-maximizing total quantity and
price. But determining the division of production between the two plants is somewhat
more complex. Graphically, each plant produces at a level defined by the intersection
of its marginal cost curve with a line drawn horizontally from the point of intersection of MR and MCT (i.e., from point F ). Thus, plant 1 produces 1.25 million units
per year (point H ) and plant 2 produces 2.5 million units per year (point I ). LearningBy-Doing Exercise 11.6 shows how to derive all these results algebraically.
1 1 . 4 M O N O P O LY W I T H M U LT I P L E P L A N T S A N D M A R K E T S
S
E
465
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 1 . 6
D
Determining the Optimal Output, Price, and Division of Production
for a Multiplant Monopolist
Suppose a monopolist faces a demand curve given by
P 120 3Q. The monopolist has two plants. The first
has a marginal cost curve given by MC1 10 ⫹ 20Q1,
and the second plant’s marginal cost curve is given by
MC2 60 ⫹ 5Q2.
Problem
(a) Find the monopolist’s optimal total quantity and price.
(b) Find the optimal division of the monopolist’s quantity between its two plants.
Solution
(a) First, let’s construct the monopolist’s multiplant
marginal cost curve MCT, the horizontal sum of MC1
and MC2. To find the equation of MCT, you cannot just
add MC1 and MC2 as follows: 10 ⫹ 20Q ⫹ 60 ⫹ 5Q
70 ⫹ 25Q. This is incorrect because it gives us the vertical
sum of the two curves. To get the horizontal sum, we first
must invert each marginal cost curve by expressing Q as
a function of MC:
1
1
1
Q1 ⫹ Q2 ⫹
MCT ⫹ ⫺12 ⫹ MCT
2
20
5
12.5 ⫹ 0.25MCT
If we let Q Q1 ⫹ Q2 denote the monopolist’s total
output, we can now solve this equation for MCT :
Q 12.5 ⫹ 0.25MCT, or MCT 50 ⫹ 4Q.
Now we can equate marginal revenue to marginal
cost in order to find the monopolist’s profit-maximizing
quantity and price: MR MCT, or 120 6Q 50 ⫹ 4Q,
or Q 7. We can find the optimal price by substituting
this quantity back into the demand curve: P 120
3(7) 99.
(b) To find the division of output across the monopolist’s
plants, we first determine the monopolist’s marginal cost
at the optimal quantity of Q 7: MCT 50 ⫹ 4(7) 78.
Now we can use the inverted marginal cost curves
we derived above to find out how much we must produce
at each plant to attain a marginal cost of 78 at each plant:
1
1
Q1 ⫹
(78) 3.4
2
20
1
1
Q1 ⫹
MC1
2
20
Q2 12 ⫹
1
MC2
5
Now we can add these two equations to get the horizontal sum of MC1 and MC2:
Q2 12 ⫹
1
(78) 3.6
5
Thus, of the total quantity of 7, plant 1 produces 3.4
units, while plant 2 produces 3.6 units.
Similar Problems:
11.21, 11.22
OUTPUT CHOICE WITH TWO MARKETS
Now consider a monopolist that serves two markets. In this section we will assume
that the monopolist must charge the same price in both markets. (In Chapter 12 we
will consider how the firm might behave if it can “price discriminate” by charging different prices in different markets.) The demand in market 1 is Q1(P), where Q1 is the
quantity demanded in market 1 when the price is P. Similarly, the quantity demanded
in market 2 when the price is P is Q2(P). The firm’s total cost of production depends
on the total amount produced, Q, where Q Q1(P) ⫹ Q2(P). The firm’s total cost of
production is C(Q), and its marginal cost is MC(Q). What price should the firm set if
it wishes to maximize profit in both markets together?
The firm’s profits in both markets will be the difference between the total revenues
in the two markets and the costs C(Q). To find the firm’s total revenues in both markets,
the firm will need to determine its aggregate demand Q Q1(P) ⫹ Q2(P). Graphically,
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CHAPTER 11
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this total demand is simply the horizontal sum of the demands in the two markets. Once
the aggregate demand is known, the firm will use the optimal quantity choice rule by
setting the marginal revenue for the aggregate demand equal to the marginal cost
MC(Q). The optimal price is then determined from the aggregate demand curve.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 1 . 7
D
Determining the Optimal Output and Price for a Monopolist Serving
Two Markets
Sky Tour is the only firm allowed to provide parasailing
service on an island in the Caribbean. The firm knows
that there are two types of customers: those visiting the
island on business and those on vacation. The firm can
charge whatever price it wishes for a parasailing trip, but
it is required to charge the same price P to all customers.
The demand for a parasailing trip by business customers
is Q1(P) 180 P. The demand by customers on vacation
is Q2(P) 120 P. The firm’s marginal cost of providing
a parasailing trip is MC(Q) 30.
Problem How many trips will the firm provide, and
what price will the firm charge if it wishes to maximize
profits?
Solution First, let’s analyze the aggregate demand
that the firm faces. The choke prices for business and
vacation customers are, respectively, 180 and 120. Thus,
when the price is between 120 and 180, only business customers will purchase a parasailing trip, and the aggregate
demand will be Q 180 P. When the price is less than
120, both types of customers will demand service, and the
aggregate demand will be Q 300 2P. To summarize:
When P ⭐ 120, aggregate demand is Q ⫽ 300 ⫺ 2P,
or in inverse form, P ⫽ 150 ⫺ 0.5Q. The marginal
revenue will then be MR ⫽ 150 ⫺ Q.
First let’s consider the possibility that the optimal
price is greater than 120. (As we shall see, this will turn out
not to be the case.) Assume P is greater than 120. Let’s see
what happens when we set MR ⫽ MC; 180 ⫺ 2Q ⫽ 30, so
that Q ⫽ 75. The optimal price would be P ⫽ 180 ⫺ 75 ⫽
105. But this price is not greater than 120 (as we had assumed), so the assumption that P is greater than 120 is
not correct.
Let’s consider the second possibility that the optimal price is less than 120. So we now assume P is less
than 120. Let’s see what happens when we set MR ⫽
MC; 150 ⫺ Q ⫽ 30, so that Q ⫽ 120. The optimal price
would be P ⫽ 150 ⫺ (0.5)(120) ⫽ 90. So the assumption
that P is less than 90 is correct. The firm should charge
a price of 90, and it will provide 120 trips. Business customers will demand 90 trips, and vacation customers will
purchase 30 trips.
Similar Problems: 11.24, 11.25
When 120 ⭐ P ⭐ 180, aggregate demand is Q ⫽
180 ⫺ P, or in inverse form, P ⫽ 180 ⫺ Q. The
marginal revenue will then be MR ⫽ 180 ⫺ 2Q.
P R O F I T M A X I M I Z AT I O N B Y A C A R T E L
cartel A group of
producers that collusively
determines the price and
output in a market.
A cartel is a group of producers that collusively determine the price and output in a
market. One of history’s most famous (or notorious) cartels is the Organization of
Petroleum Exporting Countries, or OPEC, whose members include some of the
world’s largest oil producers, such as Saudi Arabia, Kuwait, Iran, and Venezuela.
Sometimes cartels are even sanctioned by government. For example, in the early
1980s, the 17 firms in Japan’s electric cable industry received permission from Japan’s
Ministry of International Trade and Industry to act as a cartel. The cartel’s stated goal
was to reduce industry output in order to raise price and increase industry profits.
When a cartel works as its members intend, it acts as a single monopoly firm that
maximizes total industry profit. The problem a cartel faces in allocating output levels
across individual producers is identical to the problem faced by a multiplant monopolist
1 1 . 4 M O N O P O LY W I T H M U LT I P L E P L A N T S A N D M A R K E T S
467
in allocating output across its individual plants. Thus, the conditions for profit maximization by a cartel are identical to those for a multiplant monopolist. To illustrate,
suppose a cartel consists of two firms, with marginal cost functions MC1(Q1) and
MC2(Q2). At the profit-maximizing solution, the cartel allocates production between
the two firms so that marginal costs are equal and the common marginal cost equals
the industrywide marginal revenue. Mathematically, letting Q* be the optimal total
output for the cartel as a whole, and letting Q1* and Q2* be the optimal outputs of the
individual cartel members, we can express the profit-maximization condition of the
cartel as follows:14
MR(Q*) MC1(Q*1)
MR(Q*) MC2(Q*2)
Price (dollars per unit)
Figure 11.15 (with curves identical to those in Figure 11.14) illustrates the solution
to the cartel’s profit-maximization problem. In this example, the profit-maximizing
cartel output occurs at 3.75 million units per year, and the profit-maximizing price is
$6.25 per unit (again as in Figure 11.14, illustrating that the cartel’s profit-maximization
problem is identical to that of a multiplant monopolist). The cartel then allocates
production across its members to equalize marginal costs across firms. Notice that
the firm with the higher marginal cost schedule (firm 1) is allocated the smaller share
of total cartel output (1.25 million units versus 2.5 million units for firm 2). Thus,
the cartel does not necessarily divide up the market equally among its members: The
low-marginal-cost firms supply a bigger share of total cartel output than do the highmarginal-cost firms.
MC1 (firm 1)
Profitmaximizing
price
MC2 (firm 2)
MCT (cartel)
$6.25
Common
marginal
cost for
each
firm in
cartel
MR
0
1.25
2.5
Output
produced
by firm 1
D
3.75
Output
Profit-maximizing
produced total output
by firm 2
Quantity (millions of units per year)
14
We can also express the cartel’s profit-maximization condition as an IEPR, where P* is the cartel’s
optimal price:
P* MC1(Q*1)
P*
P* MC2(Q*2)
P*
1
⑀Q,P
FIGURE 11.15
Profit
Maximization by a Cartel
The cartel’s marginal cost
curve MCT is the horizontal
sum of MC1 and MC2, the
marginal cost curves of the
individual firms in the cartel.
The cartel’s optimal total
output of 3.75 million units
per year occurs at MR
MCT, where the optimal
price is $6.25 per unit. Firm
1 produces 1.25 million units
of the total output, and firm
2 produces 2.5 million units.
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CHAPTER 11
A P P L I C A T I O N
M O N O P O LY A N D M O N O P S O N Y
11.6
Is a Cartel as Efficient
as a Monopoly?
The goal of a cartel is to coordinate production and
pricing among firms that would otherwise compete,
to exercise joint monopoly power. If the cartel operates perfectly, the result should be monopoly prices,
output, and profits. In practice, rarely if ever are
cartels able to achieve this goal. For example, several studies have analyzed the behavior of OPEC,
the oil cartel of the Organization of Petroleum
Exporting Countries, and concluded that it produces
at higher output levels and lower prices than if it
were a pure monopoly.15 One reason for this is that
OPEC requires coordination across many nations,
each of which has incentives to grab as large a share
of OPEC output and profits as possible. Negotiating
and enforcing agreements across many nations is
very difficult.
A much simpler cartel was the Norwegian cement
industry. In 1923, the three Norwegian cement firms
were given the legal right to act as a cartel. They set
up a coordinating office to allocate production across
the three firms. Such a cartel should be much easier to
enforce than OPEC. An agreement among three firms
is much easier to strike, monitor, and enforce.
Moreover, there are no international issues at stake
since the three firms are in the same country, whereas
OPEC production decisions must balance political and
diplomatic factors. The Norwegian example is also
ideal for study, since in 1968 the three firms were allowed
to merge and form a monopoly.
Economists Lars-Hendrik Röller and Frode Steen
analyzed output and profitability of the cartel and
subsequent monopoly.16 They found that cartel profits were substantially below those of the monopoly
and that profits had been declining for years. An
important reason why the cartel failed, despite so
many institutional advantages, was the output sharing rule the cartel chose. The cartel first decided on
the total level of output to be sold. Any output not
15
sold domestically was exported at world cement
prices. The cartel allocated total output to the three
firms in proportion to their share of total production
capacity.
First note that this rule is not the efficient one, as
it does not allocate output based on marginal costs of
each firm. Worse for the cartel, though, was the incentives that this created. Since excess output was
exported, each firm had an incentive to expand its
output in order to gain a larger share of cartel profits
from Norwegian sales. In fact, the firms had an incentive to expand even if marginal costs exceeded the
world price for cement. Why? Although a firm would
lose money on some export sales, it would capture additional cartel profits on domestic sales. Indeed, Röller
and Steen found that the three firms gradually increased output over time, and cartel prices ended up
close to competitive world prices. The firms also
earned losses on their exports by 1968.
After the firms merged into a monopoly, expansion of total capacity stopped rising. Domestic cement
prices rose, as did total profits for the combined firms.
Interestingly, because the firms had expanded beyond
efficient levels (marginal cost above world prices), the
economists estimate that overall efficiency was higher
with monopoly than with the cartel! In other words,
the losses in consumer surplus from the monopoly
were smaller than the gains from eliminating losses
on exports. However, total efficiency would have
been higher still if the industry had been competitive.
See Application 11.7 for some estimates of these welfare effects.
What might the cartel have done differently?
First, it could have limited total output. By allowing
the firms to export excess output, the firms expanded
not only beyond the monopoly level of output, but
beyond the competitive level. Second, it could have
allocated output based on relative marginal costs, as
described in this section. Of course, given the apparently sympathetic legal regime, it would have profited even more by merging to form a monopoly back
in 1923.
See, for example, S. Martin, Industrial Economics: Economic Analysis and Public Policy (New York:
Macmillan, 1988), pp. 137–138.
16
Lars-Hendrik Röller and Frode Steen, “On the Workings of a Cartel: Evidence from the Norwegian
Cement Industry,” American Economic Review 96 (March 2006): 321–338.
469
1 1 . 5 T H E W E L FA R E E C O N O M I C S O F M O N O P O LY
I
n Chapter 10 we showed that a perfectly competitive equilibrium maximizes social
welfare (net economic benefit). We also showed that departures from perfectly competitive equilibrium create deadweight losses. As we will see, the monopoly equilibrium does not, in general, correspond to the perfectly competitive equilibrium. For
that reason, the monopoly equilibrium entails a deadweight loss as well.
11.5
THE WELFARE
ECONOMICS
OF MONOPOLY
T H E M O N O P O LY E Q U I L I B R I U M D I F F E R S F R O M
T H E P E R F E C T LY C O M P E T I T I V E E Q U I L I B R I U M
Figure 11.16 shows the equilibrium in a perfectly competitive market. The competitive equilibrium price is $5.00 per unit, where the industry supply curve S intersects
the demand curve D. The equilibrium quantity is 1,000 units.
Suppose this industry was monopolized (we might imagine a single firm acquiring all of the perfect competitive firms, keeping some in operation and shutting
down the rest). Now recall from Chapters 9 and 10 that the industry supply curve
in a competitive market tells us the marginal cost of supplying units to the market.
Price (dollars per unit)
$15
Monopoly profit-maximizing price point J
Perfectly competitive equilibrium: point K
A
Monopoly
price
$9
Perfectly
competitive
price
$5
J
B
K
G
E
$3
S, MC
F
H
MR
0
600
D
1,000
FIGURE 11.16
Perfectly
competitive
output
Quantity (units per year)
Monopoly
output
Perfect Competition
Monopoly
Impact of
Monopoly
Consumer surplus
A+B+F
A
–B–F
Producer surplus
E+G+H
B+E+H
B–G
Net economic benefit
A+B+E+F+G+H
A+B+E+H
–F–G
Monopoly
Equilibrium versus Perfectly
Competitive Equilibrium
The profit-maximizing monopoly
quantity is 600 units per year, and
the profit-maximizing monopoly
price is $9 per unit. In a perfectly
competitive market, the equilibrium quantity is 1,000 units and
the equilibrium price is $5. At the
monopoly equilibrium, consumer
surplus is A and producer surplus
is B ⫹ E ⫹ H. Consumer surplus
in the competitive market is
A ⫹ B ⫹ F, while producer surplus
is E ⫹ G ⫹ H. The deadweight loss
due to monopoly is thus F ⫹ G.
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For example, as shown in Figure 11.16, if a perfectly competitive industry supplied
600 units, the supply curve would tell us the marginal cost of the 600th unit: $3.
When the industry is monopolized, the supply curve S now becomes the monopolist’s marginal cost curve, MC. Given this, the profit-maximizing monopoly equilibrium occurs where MR MC, at a quantity of 600 units and price of $9 per unit.
We can see from Figure 11.16 how the monopoly equilibrium (point J ) and the
competitive equilibrium (point K ) differ: The monopoly price is higher than the
perfectly competitive price, and the monopolist supplies less output than the perfectly competitive industry does.
A P P L I C A T I O N
11.7
The Deadweight Loss in the
Norwegian Cement Industry
In Application 11.6 we presented the example of the
Norwegian cement industry cartel, which later became a monopoly. The study by economists Röller and
Steen also estimated the relative efficiency of the
cartel, a monopoly, and a third case of “Cournot” competition. We will examine Cournot rivalry in Chapter 13.
This is a form of imperfect competition between firms
in a concentrated industry. While Cournot competition does not result in perfect efficiency, it does tend
to result in output and prices that are closer to those
of perfect competition than one would observe in a
monopoly or a well-functioning cartel. Therefore, estimates of the welfare effects of the three cases (one
hypothetical but the other two actually observed in
the Norwegian cement industry) provide some idea of
the welfare costs of both monopoly and cartel. Note
that these estimates will understate the deadweight
loss because Cournot competition itself involves some
inefficiency as well.
Table 11.3 shows estimates of the changes in producer and consumer surplus, and the net effect of
both, in a move in the industry from cartel to monopoly.17 Consumers include foreign buyers of cement
exported from Norway. The table also shows estimates of these for a movement from the cartel to hypothetical Cournot competition. Numbers are for the
changes in one year. The deadweight loss of monopoly or cartel would be even larger over the course of
many years. And as mentioned earlier, these estimates
understate the deadweight loss of monopoly or cartel
because Cournot competition is less efficient than
perfect competition. Moving from a cartel or monopoly to Cournot rivalry would lead to welfare gains of
about $187 to $203 million per year.
TABLE 11.3 Deadweight Loss of Monopoly and Cartel, Norwegian Cement
Industry ($US millions)
Change in
From Cartel to Cournot Competition
From Cartel to Monopoly
Producer
Surplus
Consumer
Surplus
Net
$153.1
$ 68.6
$340.2
$ 52.7
$187.1
$ 16.0
Source: Röller & Steen, 2006.
17
Based on Table 3 in Röller & Steen, “On the Workings of a Cartel,” p. 336. Numbers were adjusted for
inflation, using the 1968–2009 consumer price index from the government agency Statistics Norway.
They were then converted to U.S. dollars using the exchange rate at the end of 2009.
471
1 1 . 6 W H Y D O M O N O P O LY M A R K E T S E X I S T ?
M O N O P O LY D E A D W E I G H T L O S S
How does the difference between the monopoly and competitive equilibria affect
economic benefits in this market? In Figure 11.16, the consumer surplus with a profitmaximizing monopolist is area A. The monopolist’s producer surplus is the accumulation of the difference between the monopolist’s price and the marginal cost of each
unit it produces. This corresponds to areas B ⫹ E ⫹ H. Thus, the net economic benefit at the monopoly equilibrium is A ⫹ B ⫹ E ⫹ H. In the perfectly competitive market, consumer surplus is areas A ⫹ B ⫹ F and producer surplus is areas E ⫹ G ⫹ H.
Net economic benefit under perfect competition is thus A ⫹ B ⫹ E ⫹ F ⫹ G ⫹ H.
The table in Figure 11.16 compares the net benefits under monopoly and perfect
competition. It shows that the net economic benefit under perfect competition exceeds the net economic benefit under monopoly by an amount equal to areas F ⫹ G.
This difference is the deadweight loss due to monopoly. This deadweight loss is
analogous to the deadweight losses you saw in Chapter 10. It represents the difference
between the net economic benefit that would arise if the market were perfectly competitive and the net benefit attained with the monopoly. In Figure 11.16, the monopoly
deadweight loss arises because the monopolist does not produce units of output between 600 and 1,000 for which consumers’ marginal willingness to pay (represented
by the demand curve) exceeds marginal cost. Production of these units enhances total
economic benefit, but production also reduces the monopolist’s profit. Therefore, the
monopolist does not produce them.
deadweight loss due
to monopoly The difference between the net
economic benefit that
would arise if the market
were perfectly competitive
and the net economic benefit attained at the monopoly
equilibrium.
R E N T- S E E K I N G AC T I V I T I E S
The table in Figure 11.16 might understate the monopoly deadweight loss. Because a
monopolist often earns positive economic profits, you might expect that firms would
have an incentive to acquire monopoly power. For example, during the 1990s, cable
television companies spent millions lobbying Congress to preserve regulations that
limit the ability of satellite broadcasters to compete with traditional cable service.
Activities aimed at creating or preserving monopoly power are called rent-seeking
activities. Expenditures on rent-seeking activities can represent an important social
cost of monopoly that the table does not reflect.
The incentive to engage in rent-seeking activities gets stronger the greater the
potential monopoly profit (areas B ⫹ E ⫹ H in Figure 11.16). Indeed, the monopoly
profit represents the maximum a firm would be willing to spend on rent-seeking activities to protect its monopoly. If a firm spent this maximum amount, the deadweight loss
from monopoly would be the sum of monopoly profit B ⫹ E ⫹ H and the traditional
deadweight loss F ⫹ G. If the monopolist engages in rent-seeking activities to acquire
or preserve its monopoly position, F ⫹ G represents a lower bound on the deadweight
loss from monopoly, while B ⫹ E ⫹ F ⫹ G ⫹ H represents an upper bound.
W
e have studied how a profit-maximizing monopolist determines its quantity and
price. And because its quantity and price differ from the perfectly competitive equilibrium, we have seen that the monopoly equilibrium creates a deadweight loss. But how
do monopolies arise in the first place? Why, for example, does BSkyB have a monopoly on satellite broadcasting in the United Kingdom? Why does Microsoft Windows
have nearly 100 percent of the market for personal computer operating systems? In this
section we explore why monopoly markets might arise. To do so, we first study the concept of a natural monopoly. Then, we explore the notion of barriers to entry.
rent-seeking activities
Activities aimed at creating
or preserving monopoly
power.
11.6
WHY DO
M O N O P O LY
MARKETS
EXIST?
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N AT U R A L M O N O P O LY
market in which, for any
relevant level of industry
output, the total cost incurred by a single firm producing that output is less
than the combined total
cost that two or more
firms would incur if they
divided that output among
themselves.
FIGURE 11.17
A market is a natural monopoly if, for any relevant level of industry output, the total
cost a single firm producing that output would incur is less than the combined total
cost that two or more firms would incur if they divided that output among them. A
good example of a natural monopoly is satellite television broadcasting. If, for example, two firms split a market consisting of 50 million subscribers, each must incur the
cost of buying, launching, and maintaining a satellite to provide service to its 25 million subscribers. But if a single firm serves the entire market, the satellite that served
25 million subscribers can just as well serve 50 million subscribers. That is, the cost of
the satellite is fixed: It does not go up as the number of subscribers goes up. A single
firm needs just one satellite to serve the market, while two independent firms would
need two satellites to serve the same number of subscribers overall.
Figure 11.17 shows a natural monopoly market. The market demand curve is D,
and each firm has access to a technology that generates a long-run average cost curve
AC. For any output less than 10,000 units per year, a single firm can produce output
more cheaply than two or more firms could. To illustrate why, consider an output level
Q 9,000 units per year. A single firm’s total cost of producing 9,000 units per year
is TC(9,000) 9,000 ⫻ AC(9,000) ⫽ $9,000, since AC(9,000) ⫽ $1. Suppose we divided this output equally between two firms. The total cost of production would be
9,000 ⫻ AC(4,500) ⫽ $11,800, since AC(4,500) ⫽ $1.20. Thus, it is more expensive
to split production of 9,000 units of output among two firms than it is to produce all
9,000 units in a single firm.
Note that, in Figure 11.17, some levels of output along the demand curve can be
produced more cheaply by two firms than one (e.g., Q ⫽ 12,000). However, such output levels would be demanded only at prices less than the minimum level of average
cost. Thus, they would not be profitable. At all relevant levels of market demand—
that is, all levels of market demand that could be profitably produced—the total cost
of production is minimized when one firm serves the entire market.
Natural Monopoly
Market
Any output level less than 10,000 units
per year can be produced most cheaply
by a single firm. For example, a single
firm can produce an output of 9,000
units for an average cost of $1 per unit.
Two firms, each producing 4,500 units,
would incur an average cost of $1.20 per
unit. Two firms could produce 12,000 units
at a lower total cost than one firm could.
However, this level of output would not
be profitable because the price P12 at
which 12,000 units would be demanded
is less than the minimum level of average cost.
Cost (dollars per unit)
natural monopoly A
AC
$1.20
$1.00
D
P12
0
4,500 6,000
9,000 10,000
Quantity (units per year)
12,000
473
1 1 . 6 W H Y D O M O N O P O LY M A R K E T S E X I S T ?
If one firm can serve a market at lower total cost than two or more firms, we
would expect that the market would eventually become monopolized. This is what
happened in the satellite broadcasting market in the United Kingdom. Two firms entered that market in the early 1990s: British Satellite Broadcasting and Sky Television.
But with both firms in the market, neither could make a profit. In fact, at one point
both companies were losing more than $1 million a day. Eventually, the two firms
merged, forming the satellite television monopolist BSkyB, which, since the merger,
has become profitable.
The analysis in Figure 11.17 implies two important points about natural monopoly markets. First, a necessary condition for natural monopoly is that the average cost
curve must decrease with output over some range. That is, natural monopoly markets
must involve economies of scale. In the example of satellite broadcasting, the fixed
cost of the satellite and its associated infrastructure gives rise to significant economies
of scale. Second, whether a market is a natural monopoly depends not only on technological conditions (the shape of the AC curve) but also on demand conditions. A
market might be a natural monopoly when demand is low but not when demand is
high. This would explain why the satellite broadcasting market in the United
Kingdom contains just one firm (BSkyB), while the much larger U.S. market can accommodate several competitors.
BA R R I E R S TO E N T RY
A natural monopoly is an example of a more general phenomenon known as barriers
to entry. Barriers to entry are factors that allow an incumbent firm to earn positive
economic profits, while at the same time making it unprofitable for newcomers to
enter the industry. Perfectly competitive markets have no barriers to entry: When incumbent firms earn positive profits, new firms enter the industry, driving profits to
zero. But barriers to entry are essential for a firm to remain a monopolist. Without
the protection of barriers to entry, a monopoly or cartel that earned positive economic
profits would attract new market entry, and competition would then dissipate industry profit.
Barriers to entry can be structural, legal, or strategic. Structural barriers to
entry exist when incumbent firms have cost or marketing advantages that would make
it unattractive for a new firm to enter the industry and compete against them. The interaction of economies of scale and market demand that gives rise to a natural monopoly market is an example of a structural barrier to entry. The Internet auction market
provides an example of another type of structural entry barrier, this one based on positive network externalities. As noted in Chapter 5, positive network externalities arise
when a firm’s product is more attractive to a given consumer the more the product is
used by other consumers. The auction site of market leader eBay is attractive to auction buyers because there are so many items offered for sale and there are often several
sellers of the same item. Auction sellers like eBay because there are so many buyers.
The sheer volume of transactions on eBay, in and of itself, is an important part of
eBay’s appeal. This network externality creates a significant barrier to entry. A newcomer seeking to establish its own Internet auction site (to make money, as eBay does,
through commissions on transactions) would face an enormous challenge: Lacking
the critical mass that eBay possesses, it would simply not be as attractive a site. This
barrier to entry explains why some very savvy Internet companies, including
Amazon.com and Yahoo, found it difficult to establish their own auction sites to compete against eBay.
barriers to entry
Factors that allow an incumbent firm to earn positive economic profits while
making it unprofitable
for newcomers to enter
the industry.
structural barriers to
entry Barriers to entry
that exist when incumbent
firms have cost or demand
advantages that would
make it unattractive
for a new firm to enter
the industry.
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CHAPTER 11
legal barriers to entry
Legal barriers to entry exist when an incumbent firm is legally protected against
competition. Patents are an important legal barrier to entry. Government regulations
can also create legal barriers to entry. For example, between 1994 and 1999, the company Network Solutions had a government-sanctioned monopoly in the business of
registering domain names on the Internet.
Strategic barriers to entry result when an incumbent firm takes explicit steps to
deter entry. An example of a strategic barrier to entry would be the development of a
reputation over time as a firm that will aggressively defend its market against encroachment by new entrants (e.g., by starting a price war if a new firm chooses to
come into the market). Polaroid’s aggressive response to Kodak’s entry into the instant
photography market in the 1970s is an illustration of this strategy.
Barriers to entry that exist
when an incumbent firm is
legally protected against
competition.
strategic barriers to
entry Barriers to entry
that result when an incumbent firm takes explicit
steps to deter entry.
A P P L I C A T I O N
M O N O P O LY A N D M O N O P S O N Y
11.8
United States of America
versus Microsoft
Between October 1998 and June 1999, one of
America’s best-known and most successful companies,
Microsoft, went on trial for violating the U.S. antitrust
statutes. The U.S. government accused Microsoft of
employing tactics aimed at monopolizing the market
for operating systems for personal computers (PCs). In
the opinion of the U.S. district court, “Microsoft . . . engaged in a concerted series of actions designed to protect the applications barrier to entry, and hence its
monopoly power, from a variety of . . . threats, including Netscape’s Web browser and Sun’s implementation
of Java. Many of these actions have harmed consumers
in ways that are immediate and easily discernible.”18
What does the court mean by the term applications barrier to entry? This phrase appears repeatedly
in the court’s opinion in this case. The court uses the
term applications barrier to entry to describe a barrier
to entry in the market for PC operating systems based
on positive network externalities. This barrier, in the
court’s opinion, allowed Microsoft Windows to monopolize the market for operating systems for Intelcompatible PCs. The court described the applications
barrier to entry this way:
The fact that there is a multitude of people using
Windows makes the product more attractive to
consumers. The large installed base attracts corporate customers who want to use an operating system that new employees are already likely to know
how to use, and it attracts academic consumers
18
who want to use software that will allow them to
share files easily with colleagues at other institutions. The main reason that demand for Windows
experiences positive network effects, however, is
that the size of Windows’ installed base impels ISVs
[companies that write software applications] to write
applications first and foremost to Windows. . . . The
large body of applications thus reinforces the
demand for Windows, augmenting Microsoft’s dominant position and thereby perpetuating ISV incentives to write applications principally for Windows.
This self-reinforcing cycle is often referred to as a
“positive feedback loop.”
What for Microsoft is a positive feedback loop
is for would-be competitors a vicious cycle. For just
as Microsoft’s large market share creates incentives
for ISVs to develop applications first and foremost
for Windows, the small or non-existent market share
of an aspiring competitor makes it prohibitively expensive for the aspirant to develop its PC operating
system into an acceptable substitute for Windows
(pp. 18–19).
In the court’s opinion—an opinion that Microsoft
strongly disputed before settling the case in 2001—
many of Microsoft’s actions toward competitors, such
as Netscape and Sun, were attempts to preserve this
applications barrier to entry. For example, in the
summer of 1995, Microsoft attempted to convince
Netscape to drop efforts to develop a Web browser
that could have served as a platform for Internetbased software applications. The court believed that
Microsoft did this in order to remove a threat to the
applications barrier to entry that sustained Windows’
dominance.
This quote comes from p. 204 of the United States of America v. Microsoft, United States District Court
for the District of Columbia, Findings of Fact.
475
11.7 MONOPSONY
A
monopsony market is a market consisting of a single buyer that can purchase
from many sellers. We call this single buyer a monopsonist. For example, until 1976,
major league baseball players were not allowed to bargain with more than one team
simultaneously. Thus, each baseball team was a monopsonist in the baseball players
market. As in this case, a monopsonist could be a firm that constitutes the only potential buyer of an input. Or a monopsonist can be an individual or organization that is
the only buyer of a finished product. For example, the U.S. government is the monopsonist in the market for U.S. military uniforms. In this section, we study a firm that is
a monopsonist in the market for one of its inputs.
11.7
MONOPSONY
monopsony market
A market consisting of a single buyer and many sellers.
T H E M O N O P S O N I S T ’ S P R O F I T- M A X I M I Z AT I O N
CONDITION
Wage rate, w (dollars per hour of labor)
Let’s imagine a firm whose production function depends on a single input L. The
firm’s total output is Q f (L). You might, for example, imagine that L is the quantity
of labor a coal mine employs. If the mine size is fixed, the amount Q of coal produced
per month depends only on the amount L of labor hired. Imagine that this firm is a
perfect competitor in the market for coal (e.g., it sells its coal in a national or global
market) and thus takes the market price P as given. The coal company’s total revenue
is thus Pf (L). The marginal revenue product of labor—denoted by MRPL—is the
additional revenue that the firm gets when it employs an additional unit of labor. Since
the firm is a price taker in its output market, marginal revenue product is the market
price times the marginal product of labor: MRPL P ⫻ MPL ⫽ P (⌬Q/⌬L).
Now suppose that our coal mine is the only employer of labor in its region.
Hence, it acts as a monopsonist in the labor market. The supply of labor in the coal
company’s region of operation is described by the labor supply curve w(L) shown in
Figure 11.18, telling us the quantity of labor that will be supplied at any wage. This
curve can also be interpreted in inverse form: It tells us the wage that is necessary to
induce a given amount of labor to be offered in the market.
Since the labor supply curve is upward sloping, the monopsonist knows that it
must pay a higher wage rate when it wants to hire more labor. For example if the
marginal revenue
product of labor The
additional revenue that a
firm gets when it employs
an additional unit of labor.
Marginal expenditure
on labor MEL
w(L)
T
$12
$10
$8
FIGURE 11.18
II
I
Marginal revenue product
of labor MRPL
0
3,000 4,000 5,000
Quantity of labor, L (hours per week)
Profit
Maximization by a Monopsonist
The monopsonist maximizes profit
when its marginal revenue product of
labor equals its marginal expenditure
on labor, at the intersection of MRPL
and MEL—that is, by employing a
quantity of labor L ⫽ 3,000 hours per
week. To elicit this supply of labor,
the firm must pay a wage rate
w ⫽ $8 per hour.
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CHAPTER 11
marginal expenditure
on labor The rate at
which a firm’s total cost
goes up, per unit of labor,
as it hires more labor.
M O N O P O LY A N D M O N O P S O N Y
monopsonist desires to employ 1,000 more hours of labor per week above an initial
level of 4,000 hours per week, it will have to increase the wage from $10 per hour to
$12 per hour to do so, as shown in Figure 11.18. The firm’s total cost is the firm’s total
expenditure on labor: TC wL. The firm’s marginal expenditure on labor—
denoted by MEL—is the rate at which the firm’s total cost goes up, per unit of labor,
as it employs more labor. Figure 11.18 reveals that this additional cost has two components: areas I and II. Area I (w⌬L) represents the extra cost that comes from employing more workers. Area II (L⌬w) is the extra cost that comes from having to raise
the wage for all units of labor that would have been supplied at the initial wage rate
of $10. The marginal expenditure on labor is thus:
MEL ⫽
⫽
¢TC
area I ⫹ area II
⫽
¢L
¢L
w¢L ⫹ L¢w
¢L
⫽w⫹L
¢w
¢L
Since the supply curve for labor is upward sloping, ⌬w/⌬L ⬎ 0. The marginal expenditure curve therefore lies above the labor supply curve, as Figure 11.18 shows.
The coal mine’s profit-maximization problem is to choose a quantity of labor L to
maximize total profit , which is the difference between total revenue and total cost:
⫽ Pf (L) ⫺ wL. The firm will maximize profit at the point at which marginal revenue product of labor equals marginal expenditure on labor: MRPL ⫽ MEL. The
profit maximum occurs in Figure 11.18 at a quantity of labor equal to 3,000 hours per
week. The wage rate needed to induce this supply of labor is $8 per hour, which is less
than the marginal expenditure on labor at L ⫽ 3,000, at point T in the figure.
Why does the monopsonist fail to maximize profit if it hires more than 3,000 hours
per week? Consider what happens if it hires a 4,000th unit of labor. As Figure 11.18
shows, when L ⫽ 4,000, MEL ⬎ MRPL. The additional expenditure on that unit of
labor exceeds the additional revenues from the additional output that labor produces.
The firm would be better off not hiring that unit of labor (or any amount of labor
higher than 3,000 hours).
Similarly, the firm would not want to hire less than 3,000 hours of labor. If the
firm hired only 2,000 units, an additional unit of labor would bring in additional revenues that exceed the additional expenditures (MRPL ⬎ MEL).
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 1 . 8
D
Applying the Monopsonist’s Profit-Maximization Condition
Suppose that a monopsonist’s only input is
labor and its production function is Q ⫽ 5L, where L is
the quantity of labor (expressed in thousands of hours
per week). Suppose, too, that the monopsonist can sell
all the output it wants at a market price of $10 per unit
and that the supply curve it faces for labor is w ⫽ 2 ⫹ 2L.
Problem How much labor would the monopsonist
hire, and what wage rate would it pay, to maximize profit?
Solution The monopsonist maximizes profit by
employing a quantity of labor corresponding to the
point where the marginal revenue product of labor
equals the marginal expenditure on labor.
The marginal expenditure on labor is MEL ⫽ w ⫹
L(⌬w/⌬L), where ⌬w/⌬L is the slope of the labor supply
curve. In this case, ⌬w/⌬L ⫽ 2. Now we can substitute
this value for ⌬w/⌬L and the value for w given by the
11.7 MONOPSONY
labor supply curve into the equation for MEL: MEL
(2 ⫹ 2L) ⫹ 2L 2 ⫹ 4L.
The marginal revenue product of labor MRPL is
price ($10) times the marginal product of labor MPL
⌬Q/⌬L ⫽ 5. Thus, MRPL ⫽ 10 ⫻ 5 ⫽ 50.
Now we can equate MEL and MRPL: 2 ⫹ 4L ⫽ 50,
or L ⫽ 12. And we can substitute this result back into the
477
labor supply curve: w ⫽ 2 ⫹ 2(12) ⫽ 26. Thus, the
monopsonist’s profit-maximizing condition is to employ
12,000 hours of labor per week at a wage rate of $26
per hour.
Similar Problems: 11.28, 11.29, 11.31
AN INVERSE ELASTICITY PRICING RULE
FOR MONOPSONY
The monopoly equilibrium condition, MR MC, gave rise to an inverse elasticity
pricing rule (IEPR), as we saw above. The monopsony equilibrium condition, MRPL
MEL, also gives rise to an inverse elasticity pricing rule. The key elasticity in this rule
is the elasticity of labor supply, ⑀L,w, the percentage change in labor supplied per a 1 percent change in the wage rate.19
The IEPR in a monopsony market is
MRPL w
1
w
⑀L,w
In words, this condition says that the percentage deviation between the marginal revenue product of labor and the wage rate is equal to the inverse of the elasticity of labor
supply.
A P P L I C A T I O N
11.9
Is Wal-Mart a Monopsony?
Wal-Mart is the world’s largest private company, with
the highest revenue and over 2 million employees
worldwide. The firm was founded in 1969. For most of
its history it has employed a strategy of opening stores
primarily in smaller metropolitan areas and rural communities. In such areas Wal-Mart is often the largest
retail store by a considerable margin. The company is
often criticized because small local stores find it difficult
to compete with Wal-Mart’s broad product range and
low prices, especially in rural communities. Smaller stores
often close their doors after Wal-Mart enters a market.
Because Wal-Mart stores often dominate local
retail shopping and are major employers in local labor
markets, Wal-Mart may have monopsony power in
some areas. Economists Alessandro Bonanno and
Rigoberto Lopez studied this question using data on
wages and employment in 2006 from almost every
19
county in the contiguous 48 United States.20 They
estimated the IEPR for a monopsony firm described in
this section of the text, the percentage markdown of
wages compared to marginal product of labor.
For the United States as a whole, Bonnano and
Lopez estimate a markdown of about 2 percent, suggesting that Wal-Mart has little monopsony power on
average. However, their estimates by county tell a
different story. In metropolitan markets and in more
heavily populated areas (such as the northeastern
United States), Wal-Mart’s wage IEPR is not significantly different from zero. In smaller and more rural
communities, the firm pays a markdown of as much as
5 percent. The markdown tends to be greater in rural
towns in south-central states. Why might these findings make sense? Larger cities and more densely populated areas are likely to provide more job opportunities for employees, making it more difficult for any
firm to act as a monopsonist in the market for labor.
This is analogous to the price elasticity of supply that we discussed in Chapters 2 and 9.
Alessandro Bonanno and Rigoberto Lopez, “Wal-Mart’s Monopsony Power in Local Labor Markets,”
presented at the 2008 annual meeting of the American Agricultural Economics Association, http://
ageconsearch.umn.edu/handle/6219 (accessed February 16, 2010).
20
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CHAPTER 11
M O N O P O LY A N D M O N O P S O N Y
Why is this IEPR significant? One important reason is that this condition distinguishes monopsony labor markets from perfectly competitive labor markets. In a perfectly competitive labor market, in which many firms purchase labor services, each
firm would take the price of labor w as given. Each firm would thus maximize its profits by choosing a quantity of labor that equates the marginal revenue product of labor
with the wage rate: MRPL w. In a monopsony labor market, by contrast, the
monopsony firm pays a wage that is less than the marginal revenue product. The IEPR
tells us that the amount by which the wage falls short of the marginal revenue product is determined by the inverse elasticity of labor supply.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 1 . 9
D
Applying the Inverse Elasticity Rule for a Monopsonist
A firm produces output, measured by Q,
which is sold in a market in which the price P 12, regardless of the size of Q. The output is produced using
only one input, labor (measured by L); the production
function is Q(L) L. Labor is supplied by competitive
suppliers, and everywhere along the supply curve the
elasticity of supply is 2. The firm is a monopsonist in the
labor market.
Problem How much lower is the wage rate paid by
the monopsonist than the wage rate the firm would
charge if it behaved as a perfect competitor?
Solution Each unit of labor produces 1 unit of output
Thus, MRPL P(MPL) 12. The perfectly competitive
firm would pay a wage rate equal to the marginal product
of labor, which is 12.
Now let’s consider how the firm sets the wage rate
if it behaves as a monopsonist. Since the elasticity of
supply of labor is constant, we can use the inverse elasticity rule for a monopsonist: [MRPL w]/w 1/eL,w.
The inverse elasticity rule for the monopsonist then
becomes [12 w]/w 1/2. Thus, the monopsonist
would pay a wage rate of 8, which is a third less than the
wage rate in a perfectly competitive market.
Similar Problems: 11.30, 11.32
(MPL 1), each of which can be sold at a price of 12.
MONOPSONY DEADWEIGHT LOSS
Just as monopoly results in a deadweight loss, so does monopsony. To see why, consider the monopsony equilibrium in Figure 11.19, where our monopsonistic coal mining firm pays a wage rate of $8 per hour and employs a total quantity of labor of 3,000
hours per week (the same condition illustrated in Figure 11.18). In this monopsonistic market, the coal mining firm is a “consumer” of labor services, while the workers
are the “producers” of labor services. The coal firm’s profit equals total revenue less
total expenditures on labor. Total revenue from selling output is the area under the
marginal revenue product of labor curve MRPL up to the optimal labor supply of
3,000, or areas A ⫹ B ⫹ C ⫹ D ⫹ E. The firm’s total cost of labor is areas D ⫹ E, so
the coal firm’s profit, or equivalently, its consumer surplus, is areas A ⫹ B ⫹ C.
The labor suppliers’ producer surplus is the difference between total wages received
and the total opportunity cost of the labor supplied. Total wage payments equal areas
D ⫹ E. The opportunity cost of labor supply is reflected in the labor supply curve.
The area underneath the labor supply curve w(L) up to the quantity of 3,000—area
E—represents the total compensation needed to elicit that supply of labor, which corresponds to the economic value workers receive in their best outside opportunity.
11.7 MONOPSONY
Wage rate, w (dollars per hour of labor)
MEL
479
Labor
supply curve
w(L)
A
Perfectly
competitive
wage
$12
Monopsony
wage
$8
B
F
C
G
D
E
MRPL
3,000
5,000
FIGURE 11.19
Perfectly
competitive
labor supply
Quantity of labor, L (hours per week)
Monopsony
labor supply
Perfect Competition
Monopsony
Impact of
Monopsony
Consumer surplus
A+B+F
A+B+C
C–F
Producer surplus
C+D+G
D
–C–G
Net economic benefit
A+B+C+D+F+G
A+B+C+D
–F–G
Monopsony
Equilibrium versus Perfectly
Competitive Equilibrium
The profit-maximizing monopsony
quantity of labor is 3,000 hours per
week, and the profit-maximizing
wage rate is $8 per hour. In a perfectly competitive market, the
equilibrium quantity of labor is
5,000 hours per week, and the
equilibrium wage rate is $12 per
hour. At the monopsony equilibrium, net economic benefit is
A ⫹ B ⫹ C ⫹ D. At the perfectly
competitive equilibrium, net economic benefit is A ⫹ B ⫹ C ⫹ D ⫹
F ⫹ G. The deadweight loss due
to monopsony is thus F ⫹ G.
That outside opportunity might be the value of the leisure a worker enjoys by not
working, or it might be the wage he or she would get if he or she migrated from the
region to another labor market. Thus, producer surplus is areas D ⫹ E ⫺ E area D.
The sum of producer and consumer surplus (net economic benefit) thus equals areas
A ⫹ B ⫹ C ⫹ D.
If the market for labor were perfectly competitive, the market clearing price of
labor would equal $12 per hour, and the corresponding quantity of labor would be
5,000 hours per week. Thus, a monopsony market results in an underemployment of
the input—in this case, labor—relative to the competitive market outcome. In a competitive market, consumer surplus equals areas A ⫹ B ⫹ F, while producer surplus
equals areas C ⫹ D ⫹ G. As the table in Figure 11.19 reveals, monopsony transfers
surplus from the owners of the input to the buyers of the input—in this case, from
workers to the coal mining firm. Since the monopsonist uses fewer units of the
input than a competitive market would use, there is a deadweight loss. The table in
Figure 11.19 shows that this deadweight loss is areas F ⫹ G.
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M O N O P O LY A N D M O N O P S O N Y
CHAPTER SUMMARY
• A monopoly market consists of a single seller facing
many buyers.
• In setting its price, the monopolist must take account
of the market demand curve, which is downward sloping.
The higher the price it sets, the fewer units of product it
will sell. The lower the price it sets, the more units it
will sell.
• A monopolist maximizes profit by producing a quantity of output at which marginal cost equals marginal
revenue. (LBD Exercise 11.2)
• When increasing output, the monopolist’s marginal
revenue consists of two parts: an increase in revenue
(equal to the market price) corresponding to the sale of
the marginal units and a decrease in revenue corresponding to the sale of the inframarginal units.
• When output is positive, the monopolist’s marginal
revenue is less than its average revenue, and the marginal revenue curve lies below the market demand curve.
(LBD Exercise 11.1)
• A monopolist does not have a supply curve.
• The inverse elasticity pricing rule (IEPR) states that
the difference between the profit-maximizing price and
marginal cost, as a percentage of price, is equal to minus
the inverse of the price elasticity of market demand.
(LBD Exercises 11.3, 11.4)
• The IEPR implies that a profit-maximizing monopolist facing positive marginal cost produces only on the
elastic portion of the market demand curve.
• When a firm can control its price in the market, we
say that it has market power. The IEPR applies not only
to a monopolist but to any firm that has market power,
such as a firm that competes in an industry with differentiated products.
• If an increase (i.e., rightward shift) in demand results
in a rightward shift in the marginal revenue curve, the
increase in demand will also increase the monopolist’s
equilibrium quantity. The monopolist’s price might go
up or down. (LBD Exercise 11.5)
• An increase (upward shift) in marginal cost will increase the monopolist’s profit-maximizing price and
decrease its profit-maximizing quantity.
• A profit-maximizing firm with multiple plants will
always allocate output among the plants so as to keep
their marginal costs equal. The multiplant monopolist
equates marginal revenue with an overall marginal cost
curve, which is found by horizontally summing the marginal cost curves of the monopolist’s individual plants.
(LBD Exercise 11.6)
• A cartel maximizes profit in the same way as a multiplant monopolist. Thus, to maximize overall profit, not
all cartel members will necessarily produce the same
output.
• A profit-maximizing firm that must charge the same
price in two different markets will first find the aggregate demand curve by horizontally summing the demand curves in the two markets. It will then choose its
output so that the marginal cost equals the marginal
revenue for the aggregate demand. The optimal price is
then determined from the aggregate demand curve.
(LBD Exercise 11.7)
• The monopolist produces less output than a perfectly competitive industry would produce in equilibrium. This implies that monopoly pricing entails a deadweight loss. Rent-seeking activities (activities aimed at
creating or preserving monopoly power) can increase
the deadweight loss from monopoly.
• Monopoly markets exist either because the market is
a natural monopoly (where one seller will have lower
total costs than multiple sellers) or because of barriers to
entry, which make it unprofitable for newcomers to
enter the market.
• A monopsony market consists of a single buyer facing many sellers.
• A profit-maximizing monopsonist will buy a quantity
of the input (e.g., labor) at which the marginal revenue
product of the input equals the marginal expenditure on
the input. The price that the monopsonist then pays for
the input is determined from the supply curve of the
input. (LBD Exercise 11.8)
• The IEPR in a monopsony market states that the
percentage difference between the marginal revenue
product of the input and the price of the input, as a percentage of the input price, is equal to the inverse of the
elasticity of the input supply. (LBD Exercise 11.9)
• Like monopoly, the monopsony equilibrium entails a
deadweight loss compared to the perfectly competitive
market outcome.
PROBLEMS
481
REVIEW QUESTIONS
1. Why is the demand curve facing a monopolist the
market demand curve?
6. What is the IEPR? How does it relate to the monopolist’s profit-maximizing condition, MR MC?
2. The marginal revenue for a perfectly competitive
firm is equal to the market price. Why is the marginal
revenue for a monopolist less than the market price for
positive quantities of output?
7. Evaluate the following statement: Toyota faces
competition from many other firms in the world market
for automobiles; therefore, Toyota cannot have market
power.
3. Why can a monopolist’s marginal revenue be negative for some levels of output? Why is marginal revenue
negative when market demand is price inelastic?
8. What rule does a multiplant monopolist use to allocate output among its plants? Would a multiplant perfect
competitor use the same rule?
4. Assume that the monopolist’s marginal cost is positive at all levels of output.
9. Why does the monopoly equilibrium give rise to a
deadweight loss?
a) True or false: When the monopolist operates on the
inelastic region of the market demand curve, it can always increase profit by producing less output.
10. How does a monopsonist differ from a monopolist?
Could a firm be both a monopsonist and a monopolist?
b) True or false: When the monopolist operates on the
elastic region of the market demand curve, it can always
increase profit by producing more output.
11. What is a monopsonist’s marginal expenditure
function? Why does a monopsonist’s marginal expenditure exceed the input price at positive quantities of the
input?
5. At the quantity of output at which the monopolist
maximizes total profit, is the monopolist’s total revenue
maximized? Explain.
12. Why does the monopsony equilibrium give rise to
a deadweight loss?
PROBLEMS
11.1. Suppose that the market demand curve is given
by Q 100 5P.
a) What is the inverse market demand curve?
b) What is the average revenue function for a monopolist in this market?
c) What is the marginal revenue function that corresponds to this demand curve?
11.2. The market demand curve for a monopolist is
given by P 40 2Q.
a) What is the marginal revenue function for the firm?
b) What is the maximum possible revenue that the firm
can earn?
11.3. Show that the price elasticity of demand is 1 if
and only if the marginal revenue is zero.
11.4. Suppose that Intel has a monopoly in the market
for microprocessors in Brazil. During the year 2005, it
faces a market demand curve given by P 9 Q, where
Q is millions of microprocessors sold per year. Suppose
you know nothing about Intel’s costs of production.
Assuming that Intel acts as a profit-maximizing monopolist,
would it ever sell 7 million microprocessors in Brazil
in 2005?
11.5. A monopolist operates in an industry where the
demand curve is given by Q 1000 20P. The monopolist’s constant marginal cost is $8. What is the monopolist’s profit-maximizing price?
11.6. Suppose that United Airlines has a monopoly on
the route between Chicago and Omaha, Nebraska.
During the winter (December–March), the monthly demand on this route is given by P a1 bQ. During the
summer ( June–August), the monthly demand is given by
P a2 bQ, where a2 ⬎ a1. Assuming that United’s marginal cost function is the same in both the summer and
the winter, and assuming that the marginal cost function
is independent of the quantity Q of passengers served,
will United charge a higher price in the summer or in the
winter?
11.7. A monopolist operates with the following data on
cost and demand. It has a total fixed cost of $1,400 and a
total variable cost of Q2, where Q is the number of units
of output it produces. The firm’s demand curve is
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CHAPTER 11
M O N O P O LY A N D M O N O P S O N Y
P $120 2Q. The size of its sunk cost is $600. The firm
expects the conditions of demand and cost to continue in
the foreseeable future.
a) What is the firm’s profit if it operates and it maximizes
profit?
b) Should the firm continue to operate in the short run,
or should it shut down? Explain.
11.8. A monopolist operates with a fixed cost and a
variable cost. Part of the fixed cost is sunk, and part
nonsunk. How will the sunk and nonsunk fixed costs affect the firm’s decisions as it tries to maximize profit in
the short run?
11.9. Under what conditions will a profit-maximizing
monopolist and a revenue-maximizing monopolist set
the same price?
11.10. Assume that a monopolist sells a product with
the cost function C F ⫹ 20Q, where C is total cost, F
is a fixed cost, and Q is the level of output. The inverse
demand function is P 60 Q, where P is the price in
the market. The firm will earn zero economic profit
when it charges a price of 30 (this is not the price that
maximizes profit). How much profit does the firm earn
when it charges the price that maximizes profit?
11.11. Assume that a monopolist sells a product with a
total cost function TC 1,200 ⫹ 0.5Q2 and a corresponding marginal cost function MC Q. The market demand
curve is given by the equation P 300 Q.
a) Find the profit-maximizing output and price for this
monopolist. Is the monopolist profitable?
b) Calculate the price elasticity of demand at the monopolist’s profit-maximizing price. Also calculate the marginal cost at the monopolist’s profit-maximizing output.
Verify that the IEPR holds.
11.12. A monopolist faces a demand curve P 210 4Q
and initially faces a constant marginal cost MC 10.
a) Calculate the profit-maximizing monopoly quantity
and compute the monopolist’s total revenue at the optimal price.
b) Suppose that the monopolist’s marginal cost increases
to MC 20. Verify that the monopolist’s total revenue
goes down.
c) Suppose that all firms in a perfectly competitive
equilibrium had a constant marginal cost MC 10.
Find the long-run perfectly competitive industry price
and quantity.
d) Suppose that all firms’ marginal costs increased to
MC 20. Verify that the increase in marginal cost causes
total industry revenue to go up.
11.13. A monopolist serves a market in which the
demand is P 120 2Q. It has a fixed cost of 300. Its
marginal cost is 10 for the first 15 units (MC 10 when
0 ⭐ Q ⭐ 15). If it wants to produce more than 15 units,
it must pay overtime wages to its workers, and its marginal cost is then 20. What is the maximum amount of
profit the firm can earn?
11.14. A monopolist faces the demand function P ⫽
100 ⫺ Q ⫹ I, where I is average consumer income in the
monopolist’s market. Suppose we know that the monopolist’s marginal cost function is not downward sloping. If
consumer income goes up, will the monopolist charge a
higher price, a lower price, or the same price?
11.15. Two monopolists in different markets have
identical, constant marginal cost functions.
a) Suppose each faces a linear demand curve and the two
curves are parallel. Which monopolist will have the
higher markup (ratio of P to MC): the one whose demand curve is closer to the origin or the one whose demand curve is farther from the origin?
b) Suppose their linear demand curves have identical
vertical intercepts but different slopes. Which monopolist will have a higher markup: the one with the flatter
demand curve or the one with the steeper demand curve?
c) Suppose their linear demand curves have identical
horizontal intercepts but different slopes. Which
monopolist will have a higher markup: the one with the
flatter demand curve or the one with the steeper demand curve?
11.16. Suppose a monopolist faces the market demand
function P ⫽ a ⫺ bQ. Its marginal cost is given by MC ⫽
c ⫹ eQ. Assume that a ⬎ c and 2b ⫹ e ⬎ 0.
a) Derive an expression for the monopolist’s optimal
quantity and price in terms of a, b, c, and e.
b) Show that an increase in c (which corresponds to an
upward parallel shift in marginal cost) or a decrease in
a (which corresponds to a leftward parallel shift in demand) must decrease the equilibrium quantity of output.
c) Show that when e ⭓ 0, an increase in a must increase
the equilibrium price.
11.17. Suppose a monopolist has the demand function
Q ⫽ 1,000P⫺3. What is the monopolist’s optimal
markup of price above marginal cost?
11.18. Suppose a monopolist has an inverse demand
function given by P ⫽ 100Q⫺1/2. What is the monopolist’s optimal markup of price above marginal cost?
11.19. The marginal cost of preparing a large latte in a
specialty coffee house is $1. The firm’s market research
reveals that the elasticity of demand for its large lattes is
constant, with a value of about ⫺1.3. If the firm wants to
maximize profit from the sale of large lattes, about what
price should the firm charge?
PROBLEMS
11.20. The following diagram shows the average cost
curve and the marginal revenue curve for a monopolist in
a particular industry. What range of quantities could it be
possible to observe this firm producing, assuming that the
firm maximizes profit? You can read your answers off the
graph, and therefore approximate values are permissible.
400
MR
MR, AC
300
a) Find the monopolist’s profit-maximizing price and
output at each plant.
b) How would your answer to part (a) change if MC2
(Q2) 4?
11.23. A monopolist producing only one product has
two plants with the following marginal cost functions:
MC1 20 ⫹ 2Q1 and MC2 10 ⫹ 5Q2, where MC1 and
MC2 are the marginal costs in plants 1 and 2, and Q1 and
Q2 are the levels of output in each plant, respectively. If
the firm is maximizing profits and is producing Q2 4,
what is Q1?
11.24. Suppose that you are hired as a consultant to a
firm producing a therapeutic drug protected by a patent
that gives a firm a monopoly in two markets. The drug
can be transported between the two markets at no cost, so
the firm must charge the same price in both markets. The
demand schedule in the first market is P1 200 2Q1,
where P1 is the price of the product and Q1 is the amount
sold in the market. In the second market, the demand is
P2 140 Q2, where P2 is the price and Q2 the quantity.
The firm’s overall marginal cost is MC 20 ⫹ Q1 ⫹ Q2.
What price should the firm charge?
200
AC
100
0
5
483
10
15
20
25
30
Q
11.21. Imagine that Gillette has a monopoly in the
market for razor blades in Mexico. The market demand
curve for blades in Mexico is P 968 20Q, where P is
the price of blades in cents and Q is annual demand for
blades expressed in millions. Gillette has two plants in
which it can produce blades for the Mexican market: one
in Los Angeles and one in Mexico City. In its L.A. plant,
Gillette can produce any quantity of blades it wants at a
marginal cost of 8 cents per blade. Letting Q1 and MC1
denote the output and marginal cost at the L.A. plant, we
have MC1(Q1) 8. The Mexican plant has a marginal
cost function given by MC2(Q2) 1 ⫹ 0.5Q2.
a) Find Gillette’s profit-maximizing price and quantity of
output for the Mexican market overall. How will Gillette
allocate production between its Mexican plant and its
U.S. plant?
b) Suppose Gillette’s L.A. plant had a marginal cost of
10 cents rather than 8 cents per blade. How would your
answer to part (a) change?
11.22. Market demand is P 64 (Q/7). A multiplant
monopolist operates three plants, with marginal cost
functions:
MC1(Q1) 4Q1
MC2(Q2) 2 ⫹ 2Q2
MC3(Q3) 6 ⫹ Q3
11.25. A firm has a monopoly in the production of a
software application in Europe. The demand schedule in
Europe is Q1 120 P, where Q1 is the amount sold in
Europe when the price is P. The firm’s marginal cost is 20.
a) What price would the firm choose if it wishes to
maximize profits?
b) Now suppose the firm also receives a patent for the
application in the United States. The demand for the
application in the United States is Q2 240 2P, where
Q2 is the quantity sold when the price is P. Because it
costs essentially nothing to transport software over the
Internet, the firm must charge the same price in Europe
and the United States. What price would maximize the
firm’s profit?
c) Use the monopoly midpoint rule (Learning-By-Doing
Exercise 11.5) to explain the relationship between your
answers to parts (a) and (b).
11.26. Suppose that a monopolist’s market demand is
given by P 100 2Q and that marginal cost is given
by MC Q/2.
a) Calculate the profit-maximizing monopoly price and
quantity.
b) Calculate the price and quantity that arise under perfect competition with a supply curve P Q/2.
c) Compare consumer and producer surplus under
monopoly versus marginal cost pricing. What is the
deadweight loss due to monopoly?
d) Suppose market demand is given by P 180 4Q.
What is the deadweight loss due to monopoly now?
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Explain why this deadweight loss differs from that in
part (c).
11.27. The demand curve for a certain good is
P 100 Q. The marginal cost for a monopolist is
MC(Q) Q, for Q ⭐ 30. The maximum that can be supplied in this market is Q ⫽ 30, that is, the marginal cost
is infinite for Q ⬎ 30.
a) What price will the profit-maximizing monopolist set?
b) What is the deadweight loss due to monopoly in this
market?
11.28. A coal mine operates with a production function
Q ⫽ L/2, where L is the quantity of labor it employs and
Q is total output. The firm is a price taker in the output
market, where the price is currently 32. The firm is a
monopsonist in the labor market, where the supply curve
for labor is w ⫽ 4L.
a) What is the monopsonist’s marginal expenditure
function, MEL?
b) Calculate the monopsonist’s optimal quantity of labor.
What wage rate must the monopsonist pay to attract this
quantity of labor?
c) What is the deadweight loss due to monopsony in this
market?
11.29. A firm produces output, measured by Q, which
is sold in a market in which the price P ⫽ 20, regardless
of the size of Q. The output is produced using only one
input, labor (measured by L); the production function is
Q(L) ⫽ L. There are many suppliers of labor, and the
supply schedule is w ⫽ 2L, where w is the wage rate. The
firm is a monopsonist in the labor market.
a) What wage rate will the monopsonist pay?
b) How much extra profit does the firm earn when it
pays labor as a monopsonist instead of paying the wage
rate that would be observed in a perfectly competitive
market?
11.30. A firm produces output, measured by Q, which
is sold in a market in which the price is 4, regardless of the
size of Q. The output is produced using only one input,
labor (measured by L); the production function is Q(L) ⫽
10L. Labor is supplied by competitive suppliers, and
everywhere along the supply curve the elasticity of supply
is 3. The firm is a monopsonist in the labor market.
What wage rate will it pay its workers?
11.31. National Hospital is the only employer of nurses
in the country of Castoria, and it acts as a profitmaximizing monopsonist in the market for nursing labor.
The marginal revenue product for nurses is w ⫽ 50 ⫺
2N, where w is the wage rate and N is the number of
nurses employed (measured in hundreds of nurses).
Nursing services are provided according to the supply
schedule w ⫽ 14 ⫹ 2N.
a) How many nurses does National Hospital employ, and
what wage will National pay its nurses?
b) What is the deadweight loss arising from monopsony?
11.32. A hospital is a monopsonist in the market for
nursing services in a city. At its profit-maximizing input
combination, the elasticity of supply for nursing services
is ⫹1. What does this tell you about the magnitude of the
marginal revenue product of labor relative to the wage
that the firm is currently paying its workers?
12
CAPTURING
SURPLUS
12.1
CAPTURING SURPLUS
APPLICATION 12.1
Dizzying Disneyland Pricing
APPLICATION 12.2
Education in the First Degree
APPLICATION 12.3
Block Pricing in Electricity
12.2
F I R S T- D E G R E E P R I C E
D I S C R I M I N AT I O N : M A K I N G T H E
M O S T F R O M E AC H C O N S U M E R
12.3
SECOND-DEGREE PRICE
D I S C R I M I N AT I O N : Q UA N T I T Y
DISCOUNTS
12.4
THIRD-DEGREE PRICE
D I S C R I M I N AT I O N : D I F F E R E N T P R I C E S
FOR DIFFERENT MARKET SEGMENTS
Forward Integrate
to Price Discriminate
APPLICATION 12.5 Fencing in the Price of Flight
APPLICATION 12.6 Can You “Damage”
the Metropolitan Museum of Art?
APPLICATION 12.4
12.5
TYING (TIE-IN SALES)
APPLICATION 12.7
Bunding Cable
APPLICATION 12.8
Advertising on Google
12.6
A DV E RT I S I N G
Why Did Your Ticket Cost So Much Less Than Mine?
American Airlines typically operates one nonstop flight per day in each direction between Chicago, Illinois
and Brussels, Belgium. If you had visited the company’s website (aa.com) on March 15, 2010 to consider
booking a roundtrip for a week in an upcoming month, you would have been presented with a vast array
485
TABLE 12.1
Prices for American Airlines Roundtrip, Chicago to Brussels and Returna
Chicago to Brussels
Flight AA 88
Return to Chicago
Flight AA 89
Economy
Super Saver
Economy
Saver
Economy
Flexible
Saturday, April 3
Saturday, April 10
Tuesday, April 13
Saturday, July 10
Tuesday, July 13
Saturday, April 10
Saturday, April 17
Tuesday, April 20
Saturday, July 17
Tuesday, July 20
$1,128.90
$1,193.90
$ 854.90
$1,270.90
$1,361.90
$3,491.90
$3,491.90
$3,491.90
$3,491.90
$3,491.90
$3,408.90
$3,924.90
$3,924.90
$3,924.90
$3,924.90
a
Prices quoted for booking on Monday, March 15, 2010, for travel on dates indicated in the table. Data were accessed from
the American Airlines website (aa.com). Prices do not include additional fees, such as charges for checked baggage.
of choices of dates and fares. Table 12.1 illustrates just a few of the options for flights, all within coach
class. Of course, American also offered business class service at generally higher prices (not shown in the
table) for those who wished to travel in more luxurious style.
Why might a passenger pay a higher price for a ticket in the coach cabin when a lower priced ticket is
available? And why might an airline offer different types of fares for tickets in the same class of service?
Airlines recognize that any given flight has many different types of travelers. Some passengers, especially
those traveling on business, need to travel to specific destinations at a particular time, even if the fares are
expensive. Other travelers, such as families going on vacation, will be much more sensitive to ticket prices.
To avoid high fares, they may be willing to alter the timing of their vacation (for example, leaving on April
13 rather than on April 10) or even travel to another destination. To take advantage of attractive fares,
they may be willing to purchase their tickets weeks or even months in advance.
Many passengers may also be willing to live with the reduced flexibility and less preferential treatment
that often comes with lower-priced tickets. For example, the Super Saver tickets for travel between
Chicago and Brussels were not refundable, in contrast to the tickets purchased at the higher fares. In
general, passengers traveling on more expensive fares may have many other advantages, including more
flexibility in changing flights, special boarding privileges, reduced or no charges for checked baggage,
expedited baggage delivery at the destination, and higher priority for accommodation on alternative
flights when a flight is canceled.
An airline faces a balancing act. The airline wants
to fill the plane because empty seats yield no revenue.
It could sell many seats well in advance of the flight at
low discount fares. However, the flight might then have
no seats available to accommodate last-minute travelers
who would pay a premium for a seat. When an airline
knows that it can influence the number of travelers on
a given flight by changing its fares, it has market power.
It employs a system of yield management to fill the
plane with travelers in the most profitable way. Yield
management helps the airline determine how many
seats it should allocate to each type of fare.
486
487
12.1 CAPTURING SURPLUS
In Chapter 11 we saw that managing a firm with market power is more complex than managing a
perfectly competitive firm. In a perfectly competitive market, managers cannot control the prices of
inputs or outputs. They can only determine the amounts of inputs they will purchase and outputs they
will produce. However, the managers of a firm with market power must know something about the
relationship between the quantity demanded, the quality of the output it produces, and the price it
sets. A firm with market power may be able to increase its profits by charging more than one price for
its product through price discrimination.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Explain how a firm with market power can capture more surplus by engaging in price discrimination—
that is, by charging consumers different prices for a good.
• Demonstrate why a firm must have information about reservation prices or elasticities of demand and
be able to prevent resale to succeed with price discrimination.
• Analyze three types (degrees) of price discrimination, and show how price discrimination affects prices,
consumer surplus, and producer surplus.
• Explain why firms often create different versions of a product: a low-quality, low-price version that
appeals to price-sensitive consumers, and a high-quality, high-price version to appeal to less pricesensitive consumers.
• Show how a firm can capture more surplus if it bundles two related products together and sells them
as a package.
• Explain how a firm can use advertising, a form of nonprice competition, to create and capture surplus.
Although advertising can increase the demand for a product, it is costly. You will be able to show
how decisions about the level of advertising and pricing should be made if the firm is to capture
more surplus.
In Chapter 11, the monopolists in our examples charge all consumers the same price 12.1
per unit of output. To maximize profit, a monopolist facing a downward-sloping demand curve D produces a quantity of output Qm corresponding to the point at which
marginal revenue MR equals marginal cost MC; the monopolist charges the price
Pm that induces consumers to buy the quantity Qm. In this situation, as shown in
Figure 12.1, the maximum amount of producer surplus that the monopolist can capture is represented by areas G ⫹ H ⫹ K ⫹ L. The monopolist does not capture the
consumer surplus represented by areas E ⫹ F (consumers capture those benefits). In
addition, the deadweight loss represented by areas J ⫹ N is potential economic benefit that neither the monopolist nor consumers capture. This deadweight loss arises
because there are consumers between points A and B on the demand curve who will
not buy the good at price Pm, although they would buy additional units up to quantity Q1 at lower prices greater than or equal to the marginal cost (i.e., at prices between Pm and P1).
CAPTURING
SURPLUS
CHAPTER 12
CAPTURING SURPLUS
FIGURE 12.1 Monopoly with Uniform Pricing
A profit-maximizing monopolist charging a uniform
price would choose the price Pm and sell Qm. Its
producer surplus would be the area G ⫹ H ⫹ K ⫹ L.
However, some consumer surplus (area E ⫹ F ) escapes
the producer. In addition, the deadweight loss (area
J ⫹ N ) represents potential surplus that neither the
producer nor consumers capture.
price discrimination
The practice of charging
consumers different prices
for the same good or service.
Price ($ per unit)
488
E
F
A
Pm
G
P1
MC
H
K
J
L
B
N
MR (uniform price)
0
Qm
D
Q1
Quantity (units per year)
Price discrimination (charging different prices for different consumers) offers the
monopolist, or any firm with market power, an opportunity to capture more surplus.
There are three basic types of price discrimination:
• First-degree price discrimination. The firm tries to price each unit at the consumer’s reservation price (i.e., the maximum price that the consumer is willing
to pay for that unit). For example, when a firm sells a product at an auction, it
hopes that consumers will bid up the price until the consumer with the highest
reservation price pays that price for the product. The seller hopes that the price
will be close to the maximum amount the winner is willing to pay for the good.
• Second-degree price discrimination. The firm offers consumers quantity
discounts—the price per unit goes down if the consumer buys more units. For
example, a software firm might set a price of $50 per unit for consumers buying
between 1 and 9 copies of a computer game, a price of $40 per unit for 10 to
99 copies, and a price of $30 per unit for 100⫹ copies.
• Third-degree price discrimination. The firm identifies different consumer
groups, or segments, in the market, each with a different demand curve. Then,
to maximize profit, the firm sets a price for each segment by equating marginal
revenue and marginal cost or, equivalently, by using the inverse elasticity pricing
rule (IEPR, as discussed in Chapter 11).1 For example, if an airline identifies business and vacation travelers as segments having different demand curves for flights
on the same route, it can charge a different price for each segment—say, $500 per
ticket for business travelers and only $200 per ticket for vacation travelers.
first-degree price
discrimination The
practice of attempting to
price each unit at the consumer’s reservation price
(i.e., the consumer’s
maximum willingness to
pay for that unit).
second-degree price
discrimination The
practice of offering consumers a quantity discount.
third-degree price
discrimination The
practice of charging different
uniform prices to different
consumer groups or
segments in a market.
Certain market features must be present for a firm to capture more surplus with
price discrimination:
• A firm must have some market power to price discriminate. In other words, the demand
curve the firm faces must be downward sloping. If the firm has no market power,
1
The inverse elasticity pricing rule is (Pi ⫺ MCi )/Pi ⫽ ⫺1/⑀Qi ,Pi , where Pi is the price of product i, MCi
is the marginal cost, and ⑀Qi,Pi is the firm’s own price elasticity of demand for the product.
12.1 CAPTURING SURPLUS
489
it is a price taker, and thus has no ability to set different prices for different units
of output. As we suggested in Chapter 11, market power is present in many
markets. In many industries there are only a few producers, and each producer
may have some control over the price of its output. For example, in the airline
industry, each company knows that it can attract more customers if it lowers its
price. Even though an airline is not a monopolist, it may still have market power.
• The firm must have some information about the different amounts people will pay for
its product. The firm must know how reservation prices or elasticities of demand
differ across consumers.
• A firm must be able to prevent resale, or arbitrage. If the firm cannot prevent resale,
then a customer who buys at a low price can act as a middleman, buying at a low
price and reselling the good to other customers who are willing to pay more for
it. In that case, the middleman, not the firm that sells the good initially, captures
the surplus.
A P P L I C A T I O N
12.1
Dizzying Disneyland Pricing
Disneyland, located in Anaheim, California, attracts
approximately 15 million visitors per year, making it
second only to Disney World in Florida among the
world’s most popular theme parks. Disney employs
many of the techniques that we will analyze in this
chapter to capture more surplus from its customers, including price discrimination, bundling, and advertising.
The simplest entry pass to Disneyland is the 1-day
ticket, priced at $72 as of March 2010.2 However,
Disney offers a variety of prices tailored to the different types of visitors. Customers between 3 and 9
years of age pay only $62 for the 1-day ticket. Residents
of Southern California are eligible for discounts, as
are members of the American Automobile Association.
As this was written, Disney was offering $40 discounts
to those who purchased their tickets online at
Disney’s website. Group rates are available with lower
prices per person. These are examples of third-degree
price discrimination, with different types of customers
being charged different prices for entry into the park.
Disney also offers quantity discounts, a form of
second-degree price discrimination. It is possible to
purchase tickets that allow entry into the park for differing lengths of time, ranging from 2 days to 6 days.
The more days allowed on the pass, the lower is the
2
per-day cost of the ticket. Disney also sells an annual
pass with a “Deluxe” option that can be used 315
days of the year, but precludes entry on selected very
popular days. Consumers can purchase a “Premium”
option that can be used whenever the park is open.
You may also purchase tickets that bundle entry
to Disneyland with other goods. One example is the
“1 Day Park Hopper” ticket, which allows you to visit
both Disneyland and the adjacent Disney’s California
Adventures Park. A standard ticket to either would
cost $72, but the Hopper ticket costs only $97 ($87 for
those age 3–9), far less than the price of two standard
tickets. In addition, discounted entry to either park is
available if you stay at the Disneyland Resort Hotel,
or if you purchase selected vacation packages offered
by travel agents.
In the past Disney employed other pricing strategies. From its opening in 1955 until 1982, Disney
charged customers a relatively low flat fee to enter
the park and then required visitors to buy individual
tickets for each ride. The price it charged for a ride
depended on the popularity and excitement of the
ride. Its tickets ranged from the least expensive A
rides to the most expensive E rides. In fact, the colloquial term an E ticket used to describe the best of
something stems from this system of pricing.
http://disneyland.disney.go.com/disneyland/en_US/reserve/ticketListing?name=TicketListingPage&bhcp=1,
accessed March 17, 2010.
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CAPTURING SURPLUS
T
o understand first-degree price discrimination, think of the demand schedule for a
product as a willingness-to-pay schedule because the demand curve represents the
amounts consumers are willing to pay for the units they purchase. Since the demand
curve slopes downward, the consumer buying the first unit is willing to pay a higher
price than the consumer buying the second unit. The maximum willingness to pay declines with each successive unit purchased.
First-degree price discrimination is ideal from the seller’s viewpoint. If the seller
can perfectly implement first-degree price discrimination, it will price each unit at the
maximum amount the consumer of that unit is willing to pay.3
Suppose that you own a particular line of designer jeans and that all of the customers in the market walk in to your store. When each customer enters, suppose further that you can see indelibly and truthfully stamped on her forehead the maximum
amount she is willing to pay for a pair of your jeans. Once all of the customers are in
your store, you will know the demand curve for your jeans, as shown in Figure 12.2
(the curves in this figure are identical to those in Figure 12.1).
Price ($ per unit)
F I R S T- D E G R E E
PRICE
DISCRIMINATION: MAKING
THE M O S T
F R O M E AC H
CONSUMER
CHAPTER 12
E
G
P1
FIGURE 12.2
Uniform
Pricing versus First-Degree
Price Discrimination
With uniform pricing, the producer sells Qm units at price
Pm. In this situation, the producer does not capture all of
the consumer surplus and there
is a deadweight loss. With firstdegree price discrimination,
the producer sells Q1 units
(i.e., all the units for which the
price is equal to or greater
than P1, where price equals
marginal cost). The producer
sells each unit to the consumer
with the highest reservation
price for that unit, at that
price. The producer captures
all the surplus and there is no
deadweight loss.
3
F
A
Pm
MC
H
J
L
K
N
Qm
0
B
MR (uniform price)
Q1
D
Quantity (units per year)
First-Degree Price
Discrimination
Uniform Pricing
Consumer surplus
E+F
zero
Producer surplus
G+H+K+L
E+F+G+H+J+K+L+N
Total surplus
E+F+G+H+K+L
E+F+G+H+J+K+L+N
Deadweight loss
J+N
zero
For this reason, some texts call first-degree price discrimination perfect price discrimination.
1 2 . 2 F I R S T- D E G R E E P R I C E D I S C R I M I N AT I O N
How would you price your jeans to maximize your profits? You would charge the
customer with the highest reservation price (the one at the top of the demand curve) a
price just equal to her reservation price. For example, suppose she is willing to pay up
to $100 for a pair of your jeans. You would then charge her $100 and capture all of the
surplus for yourself.4 Similarly, if the person with the second highest reservation price is
willing to pay $99, you would charge that person $99 and capture all of the surplus for
that pair of jeans as well. If you can perfectly price discriminate, you would be able to
sell every pair of jeans at the reservation price for the consumer buying that pair.
How many pairs of jeans would you sell? If your marginal cost and demand schedules are as in Figure 12.2, you will sell Q1 units because the price you receive exceeds
the marginal cost of production for each unit sold up to Q1. You will not sell any more
units because the marginal cost would exceed the price for additional units. Your producer surplus will then be represented by the area between the demand curve and the
marginal cost curve (areas E ⫹ F ⫹ G ⫹ H ⫹ J ⫹ K ⫹ L ⫹ N).5 Consumers will receive no surplus because you, the producer, have captured all of it.
We can use this example to illustrate the three preconditions for price discrimination described above. First, the seller must have market power—that is, the demand
curve for its designer jeans must be downward sloping. The seller need not be a
monopolist in the designer jeans market because other stores may sell other brands
of designer jeans.
Second, the seller must know something about how willingness to pay varies
across consumers. In this example, we assume that we can observe willingness to pay
just by looking at the amount displayed on the customer’s forehead. In the real world,
it is harder to learn about willingness to pay. If you ask a customer about her willingness to pay, she will not want to tell you the truth if she thinks you will charge her a
price equal to her willingness to pay. A consumer would like to tell you that she has a
low willingness to pay, so that she can capture some consumer surplus herself. Often
sellers can learn something about willingness to pay based on knowledge of where a
person lives and works, how she dresses or speaks, the kind of car she drives, or how
much money she makes. The information may not perfectly reveal a consumer’s willingness to pay, but it can help the seller to capture more surplus than it could without
such information.
Third, the seller must prevent resale. In this example, suppose the only people
who walk in to your store have reservation prices of $50 or less. Those with a higher
willingness to pay wait outside the store. If you sell jeans for $50 or less, the customers
who buy the jeans can become middlemen. They can walk out the door and resell
jeans to those with a higher willingness to pay. Because of resale, you will fail to capture some of the surplus. Instead, middlemen will capture some of the surplus.
As we see in Figure 12.2, there is a deadweight loss when a monopolist charges a
uniform price. What can we say about the deadweight loss with first-degree price discrimination? In Figure 12.2, note that every customer who is receiving the good
(those to the left of Q1) has a willingness to pay exceeding or equal to the marginal
cost of production. And every customer who does not purchase the good (those to the
4
As a finer point, you might note that a customer with a reservation price of $100 is just indifferent
between buying the jeans and not buying if you charge her $100. To make sure that she buys the jeans,
you might therefore charge her $99.99. She will have a consumer surplus of $0.01, and you, the producer,
will capture all the rest of the surplus. As a practical matter, we will assume that she buys the jeans if you
charge her $100.
5
As we saw in Chapter 9, producer surplus is the difference between revenue and nonsunk cost. Here we
are assuming that any fixed costs are sunk.
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right of Q1) has a willingness to pay below marginal cost. Perfect first-degree price
discrimination therefore leads to an economically efficient level of output—in other
words, there is no deadweight loss.6
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D
Capturing Surplus: Uniform Pricing versus First-Degree Price Discrimination
In this exercise we will see how a monopolist
can capture more surplus with first-degree price discrimination than with a uniform price. Suppose a monopolist has
a constant marginal cost MC ⫽ 2 and faces the demand
curve P ⫽ 20 ⫺ Q, as shown in Figure 12.3. There are no
fixed costs.
Problem
(a) Suppose price discrimination is not allowed (or is not
possible). How large will the producer surplus be?
Price ($ per unit)
$20
$11
$2
(b) Suppose the firm can engage in perfect first-degree
price discrimination. How large will the producer surplus be?
Solution
(a) The marginal revenue curve is MR ⫽ P ⫹ (⌬P/
⌬Q)Q ⫽ (20 ⫺ Q) ⫹ (⫺1)Q ⫽ 20 ⫺ 2Q. To find the optimal quantity, we set marginal revenue equal to marginal
cost. Thus, 20 ⫺ 2Q ⫽ 2, or Q ⫽ 9. Substituting this into
the demand curve, we find that P ⫽ 20 ⫺ 9 ⫽ 11.
W
R
T
Z
M
O
N
X
Y
9
MR (with
MC
D
18
20
uniform pricing)
Quantity (units per year)
FIGURE 12.3 Capturing Surplus: Uniform Pricing versus First-Degree Price Discrimination
With uniform pricing, the firm produces 9 units (corresponding to the intersection of the marginal cost curve MC and the marginal revenue curve MR). It sells these units at a price of $11
per unit, capturing a producer surplus of $81 (area RTMZ ). With perfect first-degree price discrimination, the firm produces 18 units (corresponding to the intersection of MC and the
demand curve D), capturing a producer surplus of $162 (area WXZ ).
6
Although perfect first-degree price discrimination leads to an efficient market (with zero deadweight
loss), not everyone would be happy with this outcome. In particular, consumers would not be happy
because all of the surplus goes to producers. What is efficient may not always be viewed as “fair” or
“equitable” by all the participants in a market. For more on the potential conflicts between the two, see
Edward E. Zajac, Political Economy of Fairness (Cambridge, MA: MIT Press, 1995).
1 2 . 2 F I R S T- D E G R E E P R I C E D I S C R I M I N AT I O N
Since there are no fixed costs, producer surplus (PS )
is revenue less total variable cost, which is equal to marginal cost times quantity, or 2Q. Since revenue is price
times quantity, PS ⫽ PQ ⫺ 2Q ⫽ (11)(9) ⫺ 2(9) ⫽ 81. In
Figure 12.3, producer surplus is the revenue (area
ORTN ) less the variable cost (the area under the marginal cost curve, OZMN ). Producer surplus is thus area
RTMZ.
(b) With first-degree price discrimination, the firm will
supply all the units it can sell at a price equal to or
greater than the marginal cost. That is, it will produce a
quantity corresponding to the point where the demand
curve and the marginal cost curve intersect. To find that
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493
quantity, we equate the demand curve and the marginal
cost curve: 20 ⫺ Q ⫽ 2, or Q ⫽ 18. Total revenue is the
area below the demand curve for all units produced (area
OWXY ), which equals 198 (area of triangle WXZ plus
area of rectangle OZXY ). Total variable cost is marginal
cost times quantity: 2(18) ⫽ 36.
Producer surplus is total revenue less total variable
cost: 198 ⫺ 36 ⫽ 162. In Figure 12.3, this corresponds
to area OWXY (total revenue) less area OZXY (total variable cost) ⫽ area WXZ (producer surplus).
Thus, perfect first-degree price discrimination increases producer surplus by 81 over uniform pricing.
Similar Problems: 12.2, 12.3, 12.4, 12.5
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 2 . 2
D
Where Is the Marginal Revenue Curve with First-Degree Price Discrimination?
In Chapter 11 we saw that, with uniform
pricing, the marginal revenue curve is MR ⫽ P ⫹
(⌬P/⌬Q)Q.
Problem Where is the marginal revenue curve when
the firm engages in perfect first-degree price discrimination? Does marginal revenue equal marginal cost at the
output the firm chooses?
Solution The expression for the marginal revenue
with uniform pricing MR ⫽ P ⫹ (⌬P/⌬Q)Q tells us that
marginal revenue is the sum of two effects. When the
firm sells one more unit, (1) revenues go up because the
firm receives the price P for that unit, and (2) revenues
are reduced because the price falls by ⌬P/⌬Q for all of
the Q units the firm is already selling.
With perfect first-degree price discrimination, only
the first effect is present. When the firm sells one more
unit, it receives the price P for that unit. However, it
does not have to reduce its price on all the other units it
is already selling. So the marginal revenue curve with
first-degree price discrimination is just MR ⫽ P. The
marginal revenue curve is the same as the demand curve.
With first-degree price discrimination, the seller in
Figure 12.3 is choosing output so that marginal revenue
equals marginal cost. But now the seller chooses the
level of output at which the marginal cost and demand
curves intersect (Q ⫽ 18). At this level of output, the
marginal revenue from the last unit sold is the price of
the unit ($2). The producer is maximizing profit because
the marginal revenue just covers the marginal cost of
that unit. The producer would not want to sell any fewer
units than 18 because marginal revenue would be greater
than marginal cost. Similarly, the seller would not want
to sell any more units than 18 because marginal revenue
would be less than marginal cost.
Similar Problems: 12.6, 12.7
Examples of first-degree price discrimination are plentiful. Consider what happens
when you walk through a flea market, or try to buy a car or a house. Sellers often try
to assess your willingness to pay based on what they observe about you. A seller may
ask more than you are willing to pay initially, but adjust the price as he bargains with
you and learns more about you. (Of course, you are simultaneously trying to increase
your consumer surplus by trying to find out how low the seller will go!) Auctions are
also designed to push sales prices closer to a buyer’s willingness to pay. While the
highest bidder for an object of art or a tract of land may not have to pay as much as
the bidder is willing to pay, the seller hopes to capture as much of the surplus as possible
by making potential buyers compete for the good being sold.
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12.2
Education in the First Degree
A college education in the United States can be expensive. Tuition costs more than $150,000 for four years at
many private colleges and universities, and often more
than $60,000 at state-supported colleges. Colleges
are naturally concerned about whether families of
prospective students can afford such large expenses.
Some types of financial aid are based on merit,
recognizing a student’s academic performance. More
often, at the undergraduate level, financial aid is
based on a family’s financial need. The amount a student’s family will be required to contribute toward
college expenses will be based on how much money
the family has saved and expects to earn, how much
the college estimates that the student can afford in
student loans, as well as the cost of the education at
a particular institution.
How do colleges determine how much you should
be willing to pay for a college education? Before being
considered for many types of aid, students must supply
information about their family finances on forms such
as the Free Application for Federal Student Aid
(FAFSA). Colleges then use a government-sponsored
formula to calculate the amount the family is expected
to contribute toward college expenses. This is called
the Expected Family Contribution (EFC). If the EFC is
equal to or more than the cost at a particular college,
then the student will probably be ineligible for much
financial aid. However, if the projected cost of an education at a college exceeds the EFC, then the student
will probably qualify for assistance, maybe even enough
to meet the full costs.
In 1991 the U.S. Department of Justice filed a
lawsuit against universities in the Ivy League and the
Massachusetts Institute of Technology, alleging a conspiracy to fix “prices”— student aid — in violation of the
Sherman Act.7 In the late 1980s over 20 colleges held annual meetings to discuss aid offers that they would be
making to their current and newly admitted students.
The Justice Department argued that this cooperation
served to reduce competition for students. The Ivy
League schools signed a consent decree to stop the
7
meetings, but MIT refused to sign and took the case to
trial. MIT argued that financial aid was a “gift” to students and that, as a nonprofit it was not subject to
the Sherman Act. In 1992 MIT lost the case. However,
Congress soon passed the Higher Education Act of 1992,
legalizing much of the conduct in question. In 1993, MIT
won a reversal of the court decision on appeal, at which
point the Justice Department settled. Colleges are now
allowed to engage in most of the conduct that had
been in contention during the trial.
Princeton University made news in 2001 when it
announced a new “no loan” financial aid policy. All financial aid decisions at Princeton since that year have
been made with the assumption that no Princeton
student will be expected to take out any student loans
to pay for college. Instead of student loans, Princeton
now gives grants of equivalent value to all students
whose financial situation requires them. The average
student at a four-year college borrows about $15,000
over four-years, so Princeton’s policy is quite generous
compared to that of its competitors. Princeton stated
that its goal was to increase enrollment of low- and
middle-income students, and that the program has
been successful in doing so. A few other colleges (like
Williams and Dartmouth) had adopted a “no loan”
policy in recent years; however, after university endowments plummeted during the Great Recession of
2008–2009, they reinstituted loans for some students.
When colleges base the amount of financial aid
they give you on your ability to pay, they are engaging in first-degree price discrimination. Although no
college is a monopolist, each knows that the demand
for the education it offers is downward sloping. The
number of students who would like to attend a college rises as the price the college charges (for room,
board, and tuition, less any financial aid) fall. To price
discriminate, colleges must have information on willingness to pay. Although colleges may not be able to
get an exact measure of the amount a family will be
willing to expend, that amount is probably highly related to the calculated EFC. Finally, colleges don’t
have to worry about “resale” because you cannot sell
your college education to someone else.
Gustavo Bamberger and Dennis Carlton, “Antitrust & Higher Education: MIT Financial Aid (1993),”
in John Kwoka and Lawrence White, The Antitrust Revolution: Economics, Competition, and Policy, New York:
Oxford University Press, 2003.
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I
n many markets each consumer buys more than one unit of the good or service in a
given time period. For example, each month consumers buy many units of electricity
and water. People who commute to work on mass transit systems make many trips a
month. And many airline travelers are frequent flyers.
Sellers know that each customer’s demand curve for a good is typically downward
sloping. In other words, the customer’s willingness to pay decreases as successive units
are purchased. A seller may use this information to capture extra surplus by offering
quantity discounts to consumers.
However, not every form of quantity discounting represents price discrimination.
Often sellers offer quantity discounts because it costs them less to sell a larger quantity. For example, a pizza that serves four people usually sells for less than twice the
price of a pizza for two people. Labor, cooking, and packaging costs are not very sensitive to the size of the pizza. The pricing reflects the fact that the cost per ounce is
lower for a large pizza.
What, then, characterizes quantity discounting with second-degree price discrimination? One distinguishing feature of second-degree price discrimination is that the
amount consumers pay for the good or service actually depends on two or more
prices. For example, many consumers buy their telephone service under a multipart
tariff (a tariff, or price, that consists of two or more separate prices). Thus, you might
pay a price of $20 a month (a subscription charge) just to be hooked up to the telephone
system, even if you never make a call. In addition, you might pay another price of
5 cents per call for local calls (a usage charge).
In this section we will consider two different ways in which sellers can use quantity discounting to capture surplus. First, we will look at block pricing (like the software firm’s pricing system for computer games, discussed in Section 12.1). We will
then take a more detailed look at pricing with subscription and usage charges.
12.3
SECONDDEGREE PRICE
DISCRIMIN AT I O N :
Q UA N T I T Y
DISCOUNTS
BLOCK PRICING
Suppose there is only one consumer in the market for electricity. The consumer’s
demand curve and the marginal cost curve are the same as in Figure 12.3: Demand is
P ⫽ 20 ⫺ Q and marginal cost is MC ⫽ 2, as shown in Figure 12.4. As we saw in
Learning-By-Doing Exercise 12.1, with uniform pricing, the price that maximizes
profit is P ⫽ $11 per unit of electricity. At this price, the consumer buys 9 units, and
the firm captures a producer surplus of $81.
Now suppose that the firm offers a quantity discount—for example, charging $11
per unit for the first 9 units the consumer buys and $8 per unit for any additional
units. As we can see in Figure 12.4, in this situation the consumer will buy 3 additional
units, for a total of 12 units, and the firm will capture additional producer surplus of
$18 (area JKLM ), for a total producer surplus of $99.
This pricing schedule is an example of a block tariff. (It is a kind of multipart tariff because it consists of two prices, one price for the first 9 units and another price for
additional units.) We can see that this type of quantity discounting represents seconddegree price discrimination because the firm’s marginal cost is constant at 2—that is,
it doesn’t cost the firm less to sell a larger quantity (unlike in the pizza example discussed above).
Now we can ask: What is this firm’s optimal block tariff (the block tariff that maximizes producer surplus)? For simplicity’s sake, we’ll assume that the firm’s tariff will
consist of only two blocks.
block tariff A form of
second-degree price discrimination in which the
consumer pays one price
for units consumed in the
first block of output (up to
a given quantity) and a
different (usually lower)
price for any additional
units consumed in the
second block.
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FIGURE 12.4
Uniform Pricing versus
Second-Degree Price Discrimination
With uniform pricing, the firm captures a
producer surplus of $81 (equal to area
RTMZ ). With a block tariff, the firm charges
a price of $11 for the first 9 units a consumer
purchases and a price of $8 for the three additional units. This example of second-degree
price discrimination lets the firm capture a
producer surplus of $99 (areas RTMZ ⫹ JKLM).
Price ($ per units)
$20
11
W
T
R
J
K
Z
M
L
O
N
8
2
X
MC
D An individual consumer's
Y
9
12
demand for electricity
18
20
Quantity (units per year)
In Figure 12.5 (with the same demand and marginal cost curves as Figure 12.4),
P1 and Q1 represent the optimal price and quantity for the first block, while P2 and
(Q2 ⫺ Q1) represent the optimal price and quantity for the second block. Calculating
the optimal block tariff will involve three steps:
1. Expressing Q2 in terms of Q1.
2. Expressing producer surplus (PS ) in terms of Q1.
3. Finding the value of Q1 that maximizes PS, using that value to calculate P1 and
Q2, and using the value of Q2 to calculate P2.
Step 1. The segment BE is what’s left of the consumer’s demand curve after
purchasing the first block Q1. The marginal revenue curve associated
An individual consumer's
demand for electricity
Price ($ per unit)
$20
FIGURE 12.5
Optimizing Producer Surplus
with Second-Degree Price Discrimination
With the optimal block tariff (assuming only two
blocks), the firm sells 6 units at a price of $14 per
unit and 6 additional units at a price of $8 per
unit. This maximizes producer surplus at $108
(the shaded area ABFKLZ ).
P1 = 14
A
B
F
P2 = 8
2
K
L
Z
MC
N
0
6
12
Q1
Q2
E
18
Quantity (units per year)
20
1 2 . 3 S E C O N D - D E G R E E P R I C E D I S C R I M I N AT I O N : Q UA N T I T Y D I S C O U N T S
497
with this part of the demand curve is the segment BN. Since the second
block will be sold at a single, uniform price, the optimal quantity for
this second block will correspond to the intersection of the marginal
revenue curve and the marginal cost curve MC, at Q2. Since the demand curve is linear, the marginal revenue curve has twice the slope of
the demand curve, and Q2 must lie halfway between Q1 and 18 (as we
showed when deriving the monopoly midpoint rule in Chapter 11—see
Learning-By-Doing Exercise 11.5). That is, Q2 ⫽ (Q1 ⫹ 18)/2.
Step 2. Producer surplus is total revenue minus total variable cost. The revenue
from the first block is P1Q1, the revenue from the second block is
P2(Q2 ⫺ Q1), and total variable cost is 2Q2. Thus, producer surplus
PS ⫽ P1Q1 ⫹ P2(Q2 ⫺ Q1) ⫺ 2Q2. The demand equation tells us
that P1 ⫽ 20 ⫺ Q1 and that P2 ⫽ 20 ⫺ Q2, which means that
PS ⫽ (20 ⫺ Q1)Q1 ⫹ (20 ⫺ Q2)(Q2 ⫺ Q1) ⫺ 2Q2, which reduces
to PS ⫽ ⫺(3/4)(Q1 ⫺6)2 ⫹ 108.
Step 3. Since the expression (3/4)(Q1 ⫺ 6)2 is negative for any value of Q1 other
than 6, PS is maximized (at 108) when this expression equals zero, or
when Q1 ⫽ 6. Thus, the optimal quantity for the first block Q1 ⫽ 6
units of electricity, with an optimal price P1 ⫽ 20 ⫺ 6 ⫽ $14 per unit;
the optimal quantity for the second block is then Q2 ⫽ (6 ⫹ 18)/2 ⫽ 12
units, with an optimal price P2 ⫽ 20 ⫺ 12 ⫽ $8 per unit; and the
maximum producer surplus is $108.8
In this example, second-degree price discrimination with the optimal block tariff
(assuming just two blocks) increased producer surplus by $27 over producer surplus
with uniform pricing ($108 versus $81).
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D
Increasing Profits with a Block Tariff
Softco is a software company that sells a
patented computer program to businesses. Each business it serves has the demand for Softco’s product: P ⫽
70 ⫺ 0.5Q. The marginal cost for each program is $10.
Assume there are no fixed costs.
Problem
(a) If Softco sells its program at a uniform price, what
price would maximize profit? How many units would it
sell to each business customer? How much profit would
it earn from each business customer?
(b) Softco would like to know if it is possible to improve
its profit by implementing block pricing. Suppose that
8
Softco were to sell the first block at the price you determined in (a), and that the quantity for that block is the
quantity you determined in (a). Find the profit-maximizing
quantity and price per unit for the second block. How
much extra profit would Softco earn from each of its
business customers?
(c) Do you think Softco could earn even more profits
with a set of prices and quantities for the two blocks different from those in part (b)? Explain.
Solution
(a) The marginal revenue for each customer is MR ⫽
70 ⫺ Q. We can find the optimal quantity by setting
One can also find the optimal block tariffs using calculus. As above, PS ⫽ (20 ⫺ Q1)Q1 ⫹ (20 ⫺ Q2) ⭈
(Q2 ⫺ Q1) ⫺ 2Q2. If we set the partial derivative of PS with respect to Q1 equal to zero, we find that Q2 ⫽
2Q1. If we set the partial derivative of PS with respect to Q2 equal to zero, we find that 18 ⫺ 2Q2 ⫹ Q1 ⫽ 0.
Then we solve these two equations in two unknowns to find that Q1 ⫽ 6 and Q2 ⫽ 12, from which we
can calculate the block prices and the producer surplus. For more on the use of derivatives to find a maximum, see the Mathematical Appendix at the end of the book.
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MR ⫽ MC: 70 ⫺ Q ⫽ 10, or Q ⫽ 60. The uniform price
that maximizes profit is P ⫽ 70 ⫺ 0.5(60) ⫽ $40. The
revenue will be PQ ⫽ $40(60) ⫽ $2,400. Since the marginal cost is $10 for each unit, and there are no fixed
costs, the total cost is $600. The profit from each customer is $1,800.
(b) In the first block, P1 ⫽ $40 and Q1 ⫽ 60 units. In
other words, Softco sells each of the first 60 units at a
price of $40.
How do we find the optimal price in the second
block, given the price and quantity in the first block? We
can represent the marginal willingness to pay for each
unit beyond Q1 ⫽ 60 as P ⫽ 70 ⫺ 0.5(60 ⫹ Q2) ⫽ 40 ⫺
0.5Q2. The associated marginal revenue is then MR ⫽
40 ⫺ Q2. The price that maximizes profit in the second
block is MR ⫽ MC: 40 ⫺ Q2 ⫽ 10, so Q2 ⫽ 30 and P2 ⫽
40 ⫺ 0.5(30) ⫽ $25.
In summary, Softco sells the first 60 units at a price
of $40 apiece, and it sells any quantity above 60 at $25
apiece. Softco still earns $1,800 from each customer
from the first block, as shown in (a). The additional revenues from the second block are P2Q2 ⫽ (25)(30) ⫽
$750. Its additional costs from sales in the second block
are $300. Therefore, the second block has increased
profit by $450 per customer.
(c) The exercise in (b) calculates the optimal price in the
second block, given that the price in the first block is
$40. However, as the discussion in the text suggests,
Softco could do even better if it chooses a price different
from $40 in the first block. We will leave the calculation
of the optimal price in the first block as an exercise at the
end of the chapter.
Similar Problems:
12.8, 12.9, 12.11
Now let’s take a look at how quantity discounts affect the consumer’s average
expenditure per unit (sometimes called the average outlay), which is equal to the total
outlay E divided by the total quantity purchased Q.
As long as the consumer purchases 6 or fewer units, the price of each unit is $14.
In that case, the consumer’s total outlay will be $14Q. For purchases of more than 6 units,
the total outlay will be $14(6) ⫹ $8(Q ⫺ 6):
E⫽ e
$14Q, if Q ⱕ 6
$84 ⫹ $8(Q ⫺ 6),
if Q 7 6
Thus, the consumer’s average outlay schedule is
$14, if Q ⱕ 6
E
⫽ • $84 ⫹ $8(Q ⫺ 6) ,
Q
Q
nonlinear outlay
schedule An expenditure schedule in which the
average outlay (expenditure)
changes with the number
of units purchased.
if Q 7 6
An outlay schedule like this is said to be nonlinear. A nonlinear outlay schedule
is one in which the average outlay changes as the number of units purchased changes.
Second-degree price discrimination results in nonlinear outlay schedules because the
consumer is charged different prices for different quantities purchased. Figure 12.6
illustrates the nonlinear outlay schedule in our example. As long as the consumer purchases 6 units or fewer, the average outlay curve AO is a horizontal line at $14 per
unit. For additional quantities, the average outlay curve slopes downward (i.e., the average outlay decreases). Thus, if the consumer buys 8 units, the average outlay is
$12.50 (point B); if the consumer buys 10 units, the average outlay is $11.60 (point C ).
S U B S C R I P T I O N A N D U S AG E C H A R G E S
At the beginning of Section 12.3, we considered an example in which a consumer pays
a subscription charge of $20 per month for telephone service (just to be hooked up to
the telephone system) and a usage charge of $0.05 per call for local calls. You can see
1 2 . 3 S E C O N D - D E G R E E P R I C E D I S C R I M I N AT I O N : Q UA N T I T Y D I S C O U N T S
Average expenditure
(average outlay)
$16
Average outlay (dollars per unit)
14
B
12.50
11.60
C
AO
10
8
6
FIGURE 12.6 Nonlinear Outlay
Schedule
With the block tariff illustrated in Figure 12.5,
the average expenditure per unit is constant
($14 per unit) up to a quantity of 6 units. If
the consumer buys more than 6 units, the
average expenditure declines. Since the average outlay curve AO is not a straight line,
it is called nonlinear.
4
2
0
2
4
6
8
Quantity purchased (units)
A P P L I C A T I O N
10
12
12.3
Block Pricing in Electricity
When a power company sells electricity with a block
tariff, it does not know each individual’s demand
schedule. However, it does know that some customers
have larger demands for electricity than others. It also
knows that each consumer’s demand curve is down-
Price ($ per call)
499
ward sloping, so that a lower price will stimulate that
consumer to purchase more electricity.
Suppose the market has two customers, Mr. Large
and Mr. Small, with the demand curves shown in
Figure 12.7. If the company charges a uniform price P1
for all units of electricity sold, Mr. Small will buy Q1S
units of electricity per month, and Mr. Large will
Additional consumer
surplus
P1
I
P2
Additional
producer
surplus
II
MC
DSmall
Q1S
DLarge
Q1L Q2L
Quantity (calls per month)
FIGURE 12.7 Benefits of
Block Pricing for Electricity
With uniform price P1 per unit of
electricity, Mr. Small buys Q1S units
and Mr. Large buys Q1L units. With
block pricing (P1 per unit for the
first Q1L units, P2 per unit for additional units), Mr. Small’s situation
doesn’t change: he still buys Q1S
units at P1 per unit, with the same
consumer surplus. But Mr. Large
now buys a total of Q2L units. His
consumer surplus goes up by area
I, and the company’s producer
surplus goes up by area II.
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purchase Q1L units. But suppose the company introduces a block tariff, charging P1 per unit for the first
Q1L units purchased and a lower price P2 per unit for
any additional units. How will the block pricing affect
Mr. Small, Mr. Large, and the electric power company?
Mr. Small’s purchases are unchanged because he
does not purchase enough electricity to take advantage of the lower block price P2. He still buys Q1S
units at a price P1, and his consumer surplus is therefore the same as it was under the uniform pricing
system. Mr. Large, however, will expand his consump-
that this is a system of quantity discounting by considering the consumer’s average
cost per call. If the consumer makes two calls per month, the bill will be $20 ⫹ $0.10 ⫽
$20.10, and the average outlay per call will be $10.05. In contrast, if the consumer
makes 200 calls per month, the bill will be $20 ⫹ $10 ⫽ $30, but the average outlay
per call will be only $0.15.
How might a firm use subscription and usage charges to capture more surplus?
Let’s consider a simple example in which all consumers are alike, each having a demand for telephone service like the one shown in Figure 12.8. Assume the telephone
company incurs a marginal cost of $0.05 for each call. The company could make sure
FIGURE 12.8 Subscriber and Usage Charges
Each consumer has the demand curve D for telephone service, and the telephone company incurs
a marginal cost of $0.05 for each call. If the company sets a usage charge of $0.05 for each call,
the consumer would make Q1 calls each month
and realize a consumer surplus of S1. The telephone
company could capture virtually all the consumer
surplus by implementing a monthly subscription
charge of slightly less than S1 dollars.
9
Price ($ per call)
Pareto superior An
allocation of resources that
makes at least one participant in the market better
off and no one worse off.
tion of electricity from Q1L to Q2L units, increasing his
consumer surplus by area I. And the company will be
better off because its producer surplus will increase
by area II.
This example illustrates an important potential
benefit of block tariffs. If we start with a uniform
price that is different from marginal cost, then introducing a block tariff leads to a Pareto superior allocation
of resources. A Pareto superior allocation of resources
makes at least one participant in the market better
off and no one else worse off.9
S1
MC
$0.05
D
Q1
Quantity (calls per month)
For more on this topic, see R. D. Willig, “Pareto Superior Nonlinear Outlay Schedules,” Bell Journal
of Economics 9 (1978): 56–69. With respect to the market for electricity, the argument for the Pareto
superiority of nonlinear outlay schedules is clearest when the consumers are end users of electricity
(e.g., households). The argument is a bit more complex when the purchasers of electricity are firms that
compete with one another in some market. One of the complications arises because quantity discounts
from block pricing could conceivably allow a larger, less efficient firm to produce with lower costs than a
smaller, more efficient firm, because the larger firm can purchase electricity at a lower average price.
Pareto superiority is named for the Italian economist Vilfredo Pareto (1848–1923).
501
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that there is no deadweight loss if it sets a usage charge of $0.05 for each call the consumer makes. The consumer will make Q1 calls each month, and his consumer surplus will be area S1. The telephone company could then capture consumer surplus by
implementing a monthly subscription charge. As long as the subscription charge is
less than S1 dollars, the consumer will continue to buy telephone service.
In this example, the consumer would be indifferent between subscribing and not
subscribing if the firm sets a subscription charge equal to S1. To ensure that each consumer subscribes, the firm could set the subscription charge to be slightly less than S1,
thus capturing virtually all the surplus.
In the real world, however, the firm cannot so easily capture all the surplus, for
two reasons. First, demand differs from one consumer to the next. If the firm increases
its subscription and usage charges to capture more surplus from consumers with large
demands, some consumers with small demands will not buy the service at all. The firm
therefore needs to know how many consumers have large demands and how many
have small demands.
In addition, although the firm may know that there are different types of consumers, it may not know which consumers are large and which are small users of telephone service. Firms therefore often offer customers a menu of subscription and usage
charges, and then allow each consumer to select the best combination. For example, a
cellular telephone company may offer one package with a monthly subscription charge
of $20 and a usage charge of $0.25 per call. It may also offer another package with a
subscription charge of $30 and a usage charge of $0.20 per call. A consumer who expects to make fewer than 200 calls per month will prefer the first package, while a consumer who expects to make more than 200 calls per month will prefer the second.10
Where else have you encountered subscription and user charges? Consider club
memberships. The subscription charge is the fee charged for membership in the club.
The usage charges are the fees you pay when you use the club. For example, when you
join a music club, you often pay a membership fee and then pay a certain amount for
every CD or MP3 you buy. Members of a country club pay a membership fee and then
pay usage fees to use the golf course or the tennis courts. Some computer networks
charge you a subscription fee to have access to a service and then a usage charge for
every minute you actually use the network.
I
f a firm can identify different consumer groups, or segments, in a market and can estimate each segment’s demand curve, the firm can practice third-degree price discrimination by setting a profit-maximizing price for each segment.
T W O D I F F E R E N T S E G M E N T S, T W O D I F F E R E N T P R I C E S
For an example of third-degree price discrimination, consider the difference in the
prices U.S. railroads charge for transporting coal versus grain. In the United States,
railroad transportation rates were largely deregulated in the 1980s,11 and since that
10
For more on second-degree price discrimination, see Robert B. Wilson, Nonlinear Pricing (New York:
Oxford University Press, 1992), and S. J. Brown and D. S. Sibley, The Theory of Public Utility Pricing
(New York: Cambridge University Press, 1986).
11
For a good discussion of regulatory reform in the railroad industry, see Ted Keeler, Railroads, Freight,
and Public Policy (Washington, DC: The Brookings Institution, 1983), and Tony Gomez-Ibanez and Cliff
Winston, eds., Transportation Economics and Policy: A Handbook in Honor of John Meyer (Washington, DC:
The Brookings Institution, 1999).
12.4
THIRD-DEGREE
PRICE
DISCRIMINATION:
DIFFERENT
PRICES FOR
DIFFERENT
MARKET
SEGMENTS
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$38
(dollars per ton)
Pc, price of shipping grain
Pc, price of shipping coal (dollars per ton)
time railroads have charged different prices for transporting different kinds of goods.
Coal and grain, however, are both bulk commodities; they are loaded into cars with
no special handling or packaging. Also, a car loaded with grain weighs about the same
as a car loaded with coal (typically, around 100 tons), so the marginal cost of moving
a ton of either commodity over a given distance is about the same.12 Yet railroads
charge two or three times as much to move coal as they do to move grain. Why is this
the case?
The answer lies in the differences in the demands for moving coal and grain.
Railroads face more competition from barges and trucks when they carry grain. For
example, grain shipped from Iowa to port facilities in New Orleans can be moved by
barges along the Mississippi River or along highways by trucks. Therefore, the demand for rail transport services by shippers of grain is sensitive to the price a railroad
charges. Figure 12.9(b) illustrates this price sensitivity in the demand curve Dg faced
by a railroad firm for transporting grain. If the railroad charges too high a price, many
grain shippers will not use rail service.
Coal, in contrast, is often shipped over much longer distances (e.g., from coalproducing regions in Wyoming to electric power companies in Arkansas and
Louisiana), and railroads have a cost advantage over trucks for such long shipments.
Furthermore, there are few options for moving the coal by water because most coal
mines are not located near canals or navigable rivers, so there is little competition
Pc = 24
10
MRc
0
14 19
(tons per year)
MC
Dg, demand for grain
Dc, demand for coal
shipments by rail
MRg
8
38
Qc, quantity of coal
(a)
$14
Pg = 12
shipments by rail
28
56
Qg, quantity of grain
(b)
(tons per year)
FIGURE 12.9 Pricing Coal and Grain Transport by Rail: Third-Degree Price Discrimination
The demand for rail transport of coal is much less price sensitive than the demand for rail
transport of grain. Railroads can exploit this fact, using third-degree price discrimination to
set a much higher profit-maximizing price for coal than for grain, even though the marginal
costs of transporting the two goods are the same.
12
One can measure the output of a freight transportation company in more than one way. One measure
commonly used in the United States is the ton-mile, which refers to the movement of one ton of the
commodity over one mile. In other parts of the world, output is often measured by ton-kilometers.
1 2 . 4 T H I R D - D E G R E E P R I C E D I S C R I M I N AT I O N
503
from barge transport. Figure 12.9(a) illustrates the demand curve Dc for rail transport
services by shippers of coal. Since coal shippers are more dependent on rail transport
than are grain shippers, they are willing to pay more for rail service.
Figure 12.9 reflects the assumption that the marginal cost is the same ($10) for
moving either coal or grain. But because of the difference in price sensitivity, the
profit-maximizing price (found by equating MR and MC) is much higher for coal ($24
per ton-mile) than for grain ($12 per ton-mile). As this example shows, railroads have
little trouble implementing price discrimination in the movement of coal and grain.
Once they have an idea about the nature of the demands for the rail services, they can
price discriminate without having to worry about resale. They know who buys coal
transport services (e.g., electric utilities) and who buys grain transport. An electric
utility wanting to buy coal is not likely to find ways of transporting coal at a price
lower than the railroad charges.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 2 . 4
D
Third-Degree Price Discrimination in Railroad Transport
Suppose a railroad faces the demand
curves for transporting coal and grain shown in Figure
12.9. For coal, Pc ⫽ 38 ⫺ Qc , where Qc is the amount of
coal moved when the transport price for coal is Pc. For
grain, Pg ⫽ 14 ⫺ 0.25Qg, where Qg is the amount of
grain shipped when the transport price for grain is Pg.
The marginal cost for moving either commodity is $10.
Problem Equate marginal revenue and marginal cost
to find the profit-maximizing rates for coal and grain
transport.
Solution For coal, the marginal revenue curve is
MRc ⫽ 38 ⫺ 2Qc. Now we equate marginal revenue to
A P P L I C A T I O N
Similar Problems:
12.14, 12.15, 12.16, 12.17,
12.20, 12.21, 12.22
12.4
Forward Integrate to Price
Discriminate
At the beginning of this chapter, we pointed out that
a firm needs to be able to prevent resale if it is to
price discriminate successfully. One interesting strategy for doing this is forward integration, whereby a
firm moves into the same business that its customers
are in. For example, in the mid-1990s Intel, a manufacturer of microprocessors, considered following a
13
marginal cost: 38 ⫺ 2Qc ⫽ 10, or Qc ⫽ 14. Substituting
this into the equation for the demand curve, we find:
Pc ⫽ 38 ⫺ 14 ⫽ 24. The profit-maximizing rate for
transporting coal is $24 per ton-mile.
For grain, the marginal revenue curve is MRg ⫽
14 ⫺ 0.5Qg. Now we equate marginal revenue to marginal cost: 14 ⫺ 0.5Qg ⫽ 10, or Qg ⫽ 8. Substituting this
into the equation for the demand curve, we find: Pg ⫽
14 ⫺ 0.25(8) ⫽ 12. The profit-maximizing rate for
transporting grain is $12 per ton-mile.
forward-integration strategy when it contemplated
making personal computers (manufacturers of which
purchase microprocessors from Intel).
Alcoa, a monopolistic producer of primary aluminum ingot until the 1930s, used forward integration in order to engage in price discrimination and
prevent resale.13 Alcoa knew that aluminum was particularly valuable in some uses because of its metallurgical properties. For example, it is a light metal, making it desirable in the manufacturing of airplane
wings. It also has special “tensile” properties (relating
See Martin Perry, “Forward Integration by Alcoa: 1888–1930,” Journal of Industrial Economics 29
(1980): 37–53.
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to how it stretches when it bears a load), making it especially useful in cables for bridges. Since other materials could not be substituted for aluminum in these
uses, Alcoa knew that the demand for primary aluminum was relatively inelastic for its sales to manufacturers of airplane wings and bridge cable.
In the manufacturing of other products, the advantages of aluminum over other materials are less
important. For example, aluminum can be used to
make pots and pans. But so can copper, steel, or cast
iron. Given these substitutes for aluminum, Alcoa’s
demand for primary aluminum in making cookware
was relatively elastic.
Alcoa wanted to use third-degree price discrimination by selling aluminum at a high price to cable
and aircraft manufacturers and at a low price to makers of cookware. However, Alcoa knew it would have
to worry about resale if it sold aluminum externally
at two prices. If it announced that cookware buyers
could purchase primary aluminum at a low price,
every buyer (including makers of airplane wings and
cable) would claim to be a cookware manufacturer.
Even if Alcoa knew that a buyer made cookware,
what would prevent that buyer from reselling the
aluminum at a higher price to a maker of airplane
wings?
To prevent resale, Alcoa decided to make aluminum pots and pans itself (that is, it integrated forward into the cookware business). It could then provide aluminum to its own cookware manufacturing
division at a low price. It did not sell primary aluminum
to any external buyers at the low price. Its only external sales were at high prices. By vertically integrating,
Alcoa could price discriminate and prevent resale.
SCREENING
screening A process for
sorting consumers based on
a consumer characteristic
that (1) the firm can see
(such as age or status) and
(2) is strongly related to a
consumer characteristic
that the firm cannot see
but would like to observe
(such as willingness to pay
or elasticity of demand).
Have you ever wondered why businesses, such as movie theaters, airlines, urban mass
transit authorities, and restaurants, often give discounts to senior citizens and students? One possible answer to this question is that this form of price discrimination
helps businesses capture more surplus.14 Most students and many older people, particularly those who are retired, live on limited incomes. Both students and senior citizens typically have more free time to shop around than many people who work full
time. Consequently, senior citizens and students often have relatively elastic demands
for goods and services. The inverse elasticity pricing rule therefore suggests that businesses ought to set prices lower for these consumers.
Businesses often use observable characteristics, such as age and student status, as
screening mechanisms. Screening sorts consumers based on consumer characteristics
that (1) the firm can observe (such as age or student status) and (2) that are strongly
related to other consumer characteristics that the firm cannot observe but would like
to observe (such as willingness to pay or elasticity of demand). For example, the movie
theater manager would like to see the consumer’s elasticity of demand or willingness
to pay when he walks up to the ticket counter, but she cannot observe that information directly. If she were to ask the consumer how much he would be willing to pay,
he might lie, knowing that the manager might charge a higher price if he reveals that
he has a high willingness to pay.
However, the manager can observe characteristics such as the consumer’s age or
student status. Most students and senior citizens have more elastic demands, so the
manager can set lower prices for these consumer segments. To prevent arbitrage, the
manager can require the consumer to present an identity card to verify age or student
status when the consumer enters the theater.
14
There are surely other reasons to offer discounts to senior citizens and students. For example, regulators
of urban mass transit systems may view a lower price for these consumers as a socially noble objective,
perhaps as a means of creating more purchasing power for deserving sets of consumers.
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S
E
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D
Third-Degree Price Discrimination for Airline Tickets
According to Table 2.2, the estimated
price elasticity of demand for coach class airline tickets
for business travelers is ⑀QB, PB ⫽ ⫺1.15, while for vacation (leisure) travelers it is ⑀QV, PV ⫽ ⫺1.52.15 Suppose
an airline facing these demand elasticities wants to use
third-degree price discrimination to maximize profit, by
setting the price of a business travel ticket to PB and the
price of a vacation travel ticket to PV. Also suppose that
the airline faces the same marginal cost MC for both
types of travelers.
Problem Use the inverse elasticity pricing rule [IEPR;
see equation (11.4)] to determine the ratio PB /PV.
The IEPR also tells us that (PV ⫺ MC )/PV ⫽
⫺(1/⑀QV,PV ). Substituting the value for ⑀QV,PV given
above and again solving for MC, we find: MC ⫽
0.342PV.
Now we can equate these two expressions for MC:
0.13PB ⫽ 0.342PV. Rearranging terms, we find that
PB/PV ⫽ 0.342/0.130 ⫽ 2.63.
Thus, the airline will maximize profit by charging
2.63 times as much for a business travel ticket as it
charges for a vacation travel ticket (the exact prices of
the tickets will depend on the marginal cost).
Similar Problems:
12.13, 12.18, 12.19
Solution The IEPR tells us that (PB ⫺ MC )/
PB ⫽ ⫺(1/⑀QB,PB ). Now we substitute the value for
⑀QB,PB given above and solve for MC: MC ⫽ 0.13PB.
We see many other examples of screening in everyday life, including the two types
discussed below: intertemporal price discrimination and coupons and rebates.
Intertemporal Price Discrimination
Many services are sold at different prices depending on the season, the time of day, or
the elapsed time since the product was introduced. For example, telephone companies
often set higher prices during the day, when they know consumers and enterprises
must conduct business. Similarly, electricity prices often vary by the time of day, generally being set higher when demand is at its peak.
In other cases, consumers may want to be “the first one on the block” to own a new
computer product, to purchase a new home sound system, or to see a new movie. Sellers
know that such people will pay more to get the product early, and sellers therefore often
use time (early sales) as a screening mechanism, pricing goods higher when they are first
introduced. For example, buyers often paid several hundred dollars for a four-function
calculator (a hand calculator that could add, subtract, multiply, and divide) when they
were first introduced in the 1960s. A few years later, such simple calculators were often
available for a few dollars.16 We can observe similar trends today with computers. Often
the price of a new model may fall by 50 percent within a year of its introduction.
Of course, price discrimination is not the only reason for setting a higher price
early in the life of a product. The price of a product may fall over time because manufacturing costs fall. As the price of a type of computer chip falls over time, the price
of a computer model using that chip can also be expected to fall. Also, as newer, faster
computers become available, the demand for an older model will fall, leading to a
lower price for the older model.
15
Although on many international flights there exists a separate business class section of the airplane,
many domestic flights lack this distinction, so that most business travelers fly coach.
16
See N. Stokey, “Intertemporal Price Discrimination,” Quarterly Journal of Economics 94 (1979): 355–371.
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Coupons and Rebates
Almost any Sunday newspaper carries coupons that you can redeem at a store for discounts on items. Brand managers often offer coupons on new products, food products,
pet food, toilet paper, and toothpaste. If you have a coupon, you pay a lower net price
(the retail price less the value of the coupon) than you would without a coupon. A
rebate is similar to a coupon, but is typically offered on the package containing the
product you purchase. For example, you may buy a package of batteries for $5.00. On
the package is a printed form that you can fill out and send to the manufacturer to
receive a $1.50 rebate in the mail.
Researchers have suggested that coupons and rebates are often used to price discriminate in consumer product markets. The basic idea is this: Brand managers know that
people who are willing to take the time to collect and redeem coupons or redeem rebate
certificates are likely to be more sensitive to price than consumers who do not.17 In other
words, coupons and rebates are screening mechanisms. They offer a lower net price to
those consumers who are likely to have more price elastic demands for the product.
Once again, price discrimination is not the only possible reason for offering
coupons or rebates. For example, firms may offer them to induce consumers to try a
product, hoping that an initial purchase will lead to more sales later.
T H I R D - D E G R E E P R I C E D I S C R I M I N AT I O N
W I T H C A PAC I T Y C O N S T R A I N T S
In many settings in which firms engage in third-degree price discrimination, firms
face constraints on how many customers can be served in a given period. Examples
would include airlines, rental car companies, cruise lines, and hotels. The presence of
a capacity constraint does not change the fundamental insight that firms with market
power can benefit from engaging in price discrimination. However, capacity constraints complicate the determination of the profit-maximizing prices and quantities.
To illustrate profit-maximizing price discrimination with capacity constraints,
consider a firm that faces two market segments. For simplicity, assume that the firm
has a constant marginal cost MC in each segment. Suppose that the firm has tentatively decided to charge prices P1 and P2 in the two segments, which would result in
sales of Q1 and Q2 units in each segment. Suppose, further, that Q1 ⫹ Q2 equals the
firm’s available capacity; in other words, the capacity constraint is binding. Finally, let
MR1 and MR2 denote the marginal revenues in each segment, given the currently
planned prices and quantities.
Now, suppose it was the case that MR1 ⫺ MC MR2 ⫺ MC, or equivalently,
MR1 MR2. Recalling that marginal revenue is the change in the firm’s total revenue
from selling one more unit (and also the change in total revenue from selling one less
unit), the fact that MR1 MR2 tells us that if the firm sold one more unit in market
segment 1 and one fewer unit in market segment 2 (thus, keeping its total output equal
to its available capacity), total revenue would go up in market segment 1 by more than
total revenue would go down in market segment 2. Since marginal cost is the same in
each segment, by selling one more unit to segment 1 and one fewer unit to segment 2,
the firm would leave its costs unchanged, and the shift of one unit from segment 2 to
17
Marketing studies show that consumers who use coupons to buy products typically have a more elastic
demand than consumers who do not use coupons. See, for example, C. Narasimhan, “A Price Discrimination
Theory of Coupons,’’ Marketing Science (Spring 1984): 128–147.
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507
segment 1 would thus increase the firm’s total profit. The way the firm would engineer this increase in profit would be to decrease price in segment 1 by just enough to
increase the quantity demanded by one unit and increase the price in segment 2 by
just enough to decrease the quantity demanded by one unit.
Analogous reasoning would imply that when MR2 ⬎ MR1, the firm can increase
profits by selling one more unit in market segment 2 (reducing the price by just enough
to do so) and selling one less unit in market segment 1 (increasing the price by just
enough to do so). We have just seen that whenever MR2 ⬎ MR1 or MR1 ⬎ MR2, the
current set of quantities and prices are not profit-maximizing. It therefore follows that
when the firm faces a capacity constraint, the only situation consistent with profitmaximizing behavior is when the quantities and prices are such that MR1 ⫽ MR2. In
other words, profit-maximizing price discrimination subject to capacity constraints requires that the marginal revenues be equated across the market segments the firm serves.
The condition that marginal revenues must be equated across markets may strike
you as a bit abstract. After all, how would actual firms ever be able to determine
whether this condition is satisfied? But real firms in businesses such as airlines and
hotels attempt to equate marginal revenues every day. As discussed in the chapter
opener, airlines and hotels (as well as other companies such as rental car companies
and cruiselines) use a sophisticated set of optimization processes collectively known as
yield management to determine the profit-maximizing way to allocate scarce capacity
aboard an airplane or in a hotel. In these industries even small changes in the way
scarce capacity is allocated can translate into large increases in profits. Thus, skill at
yield management—bringing those marginal revenues into alignment—is an important determinant of success in industries that operate in the face of capacity constraints.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 2 . 6
D
Price Discrimination Subject to Capacity Constraints
This exercise shows you how to determine
the profit-maximizing prices and quantities for a firm
that wants to engage in third-degree price discrimination but operates with a capacity constraint.
Suppose that the demand curve in market segment 1
is Q1 ⫽ 200 ⫺ 2P1 and the demand curve in market segment 2 is Q2 ⫽ 250 ⫺ P2. The marginal cost of selling
in each market segment is $10 per unit. The firm’s overall capacity is 150 units.
Problem What are the profit-maximizing quantities
and prices in each market segment?
Solution Let’s begin by determining the marginal
revenue functions in each market segment. In market
segment 1, we have Q1 ⫽ 200 ⫺ 2P1, which implies an
inverse demand curve P1 ⫽ 100 ⫺ 1/2Q1, which in turn
gives us a marginal revenue function MR1 ⫽ 100 ⫺ Q1.
In market segment 2 we have an inverse demand curve
P2 ⫽ 250 ⫺ Q2, which gives us a marginal revenue function
MR2 ⫽ 250 ⫺ 2Q2. Equating the marginal revenue functions gives us one equation in two unknowns, Q1 and Q2:
100 ⫺ Q1 ⫽ 250 ⫺ 2Q2
The second equation that must hold is the firm’s
total production must add up to its total capacity:
Q1 ⫹ Q2 ⫽ 150
Therefore, we have a system of two linear equations
in two unknowns. Using straightforward algebra, we find
that the solution to this system is: Q1 ⫽ 50 and Q2 ⫽ 100.
Substituting these quantities back into the respective
inverse demand curves gives us P1 ⫽ 75 and P2 ⫽ 150.
Note that the marginal revenue from each segment
is 50, well in excess of the marginal cost of 10. Thus, the
firm will want to operate at capacity.
Similar Problems: 12.23, 12.24, 12.25, 12.26
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IMPLEMENTING THE SCHEME OF PRICE
D I S C R I M I N AT I O N : B U I L D I N G “ F E N C E S ”
Even if a firm has figured out a way to screen consumers, it still faces the issue of
implementing the desired scheme of price discrimination. That is, how can the firm
ensure that the consumers who are targeted to pay the high price actually pay the high
price and that consumers who are targeted to pay the low price actually pay the low
price? The upper panel of Figure 12.10 illustrates the issue. The figure depicts the situation of a firm that faces two market segments. The vertical axis measures the price
P charged in each segment. In the market segment consisting of price-sensitive consumers (let’s call this segment, group beta), it charges a price of $50. In the market
segment consisting of less price-sensitive consumers (let’s call this segment, group
alpha), it charges a price of $125. Suppose initially that the product offered for sale to
each consumer group has the same quality. Quality is measured along the horizontal
Panel A
P, Price
A
$125/unit
$50/unit
B
q, quality level
Panel B
P, Price
directions of
decreasing preference
uA
A
$125/unit
directions of
increasing preference
uB
$50/unit
B
C
FIGURE 12.10
q, quality level
Building a “Fence” to Implement a Scheme of Price Discrimination
Panel A shows the case of a firm that offers a product of the same quality at different prices. Panel
B shows how the firm can build a “fence” by offering a high-quality version of the product at a
high price (point A) and a low-quality version of the good at a low price (point C). Group alpha
consumers (low-price sensitivity and high-quality sensitivity) prefer version A to version C, while
group beta consumers (high-price sensitivity and low quality sensitivity) prefer version C to version A).
1 2 . 4 T H I R D - D E G R E E P R I C E D I S C R I M I N AT I O N
axis in Figure 12.10 and is denoted by q. We interpret product quality broadly. It could
refer to tangible characteristics of product performance (e.g., the speed of a laser
printer), but it also could refer to factors relating to the amount of hassle that the customer must go through in purchasing the product or getting it serviced (e.g., the more
hassle, the lower is q).
In the initial situation, where the quality of the product sold to each group is the
same, one of several things could happen. If the low-price version of the product is
readily available to all, then consumers in group alpha will buy at the lower price.
(Consumers in this group may not be as price sensitive as consumers in the other
group, but if a completely equivalent product is readily available at a lower price, then
why pay full price!) This is why some Broadway insiders have been concerned about
the recent trend toward variable pricing of tickets to Broadway shows (variable pricing is the theater business’s term for price discrimination).18 Some believe that if discount tickets become too easily available, then there will be no full-price buyers, and
all that variable pricing will have done is to lower prices across the board.
If the low-price good is not easily obtained in a direct manner by the less pricesensitive consumers, what might happen is that the availability of two quality-equivalent
versions of the good at different prices might attract bootleggers: individuals who buy
units of the good at the low price and then resell them (either directly or through intermediaries) to the less price-sensitive consumers at a price that is high enough so
that the bootlegger makes a profit, but not as high as the high price being asked by
the seller. This is what happens with textbooks. Publishers understand that the
Chinese market is generally more price sensitive than the U.S. market for (Englishlanguage) textbooks, and so they charge lower prices for international editions sold in
China. But because there is often virtually no difference between the international
edition (except for possibly a sticker that says something to the effect that the book
cannot be sold in the United States), it pays for bootleggers to purchase international
editions at a low price and ship them back to the United States with the sticker removed. That is how books that were intended to be international editions sold in
China end up on the shelves of university bookstores in the United States.
If all consumers end up purchasing the good at the low price, the firm cannot implement its scheme of price discrimination and cannot capture the extra profit that is
generated through that scheme. So what can a firm do? Somehow it needs to build
what Robert Dolan and Hermann Simon call a “fence,” which keeps the less pricesensitive consumers from being able and/or willing to purchase the low-price version
of the good.19
One way that the firm may be able to build a fence is to exploit a common correlation: The least price-sensitive consumers also tend to be the most quality sensitive.
That is, the least price-sensitive consumers will typically be willing to pay a higher
price premium for a given increment to quality than the more-price-sensitive consumers. The bottom panel of Figure 12.10 shows how to build the fence. The line
labeled uA is an indifference curve for a group alpha consumer. It shows all of the combinations of price and quality pairs (or what we will call “offers”) that a consumer in
this group views as equivalent to the quality-price offer at point A—the firm’s actual
offer to group alpha consumers. Quality-price offers located to the northwest of point
A are less preferred by consumers in this group to the offer at point A (these offers
18
See “How Much Did Your Seat Cost?” New York Times ( July 20, 2003).
Robert J. Dolan and Hermann Simon, Power Pricing: How Managing Price Transforms the Bottom Line
(New York: The Free Press, 1996), p. 122.
19
509
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12.5
Fencing in the Price of Flight
Airlines typically sell tickets at a variety of fares, as we
saw at the beginning of the chapter when we looked
at American Airlines’ flights between Chicago and
Brussels. Third-degree price discrimination in one of
the strategies airlines use to fill the plane with travelers in the most profitable way. Airlines often charge
different prices for seats in the same class of service,
such as coach class, even though the marginal cost of
serving a passenger is about the same for all passengers. Different customers are willing to pay different
amounts for tickets. For example, people traveling on
vacation often can book their tickets weeks or even
months in advance of the flight, and they are willing
to shop around for the best price. They may even decide to choose their destinations based on the availability of relatively inexpensive tickets. Thus, vacation
travelers are usually quite sensitive to price, especially
if the vacation involves the whole family. In contrast,
passengers traveling on business are often less sensitive to the price of the ticket. When business requires
that a passenger be in London for an important meeting on Monday at 8:00 AM, the traveler will make the
trip even if the fare is expensive.
An airline knows that it serves different types of
customers, including business customers with a typically relatively inelastic demand, and vacation travelers with relatively elastic demand. Since the marginal
costs of service are similar, the inverse elasticity rule
suggests that an airline would like to charge a higher
price for business travelers.
How does the airline implement price discrimination? Although it knows that there are different
types of travelers, it does not know the specific type
of any customer. It could ask the customer to reveal
his or her type with a direct question, “Are you traveling on business or pleasure?” But if travelers knew
they would be quoted a lower price by identifying
themselves as vacation travelers, the response would
often not be truthful. Economists say that information is asymmetric: The customer knows his or her
type, but the airline does not.
How does the airline design a mechanism to implement price discrimination in the face of the informational asymmetry? It builds a set of fences. Restrictions
on fares are ways of “degrading” or “damaging” product quality. A nonrefundable fare is of lower quality
than a refundable fare. A fare that requires that you to
pay for checked baggage is also of lower quality. So are
fares that require you to stay over a Saturday night or
to purchase the ticket 14 days in advance. In building
fences, the airline is creating different versions of its
product, with low-quality, low-price tickets that appeal
to price-sensitive customers, and high-quality, highprice tickets that appeal to less price-sensitive customers. Customers are then induced to self-select into
the product type designed for them.
involve a higher price and/or a lower quality), while quality-price pairs located to the
southeast of point A are preferred to point A by group alpha consumers.
The line labeled uB is an indifference curve for group beta consumers (high price
sensitivity and low quality sensitivity) and can be interpreted the same way as uA.
Notice that at the point at which uA and uB cross, uA is steeper than uB. This illustrates
that starting from a given quality-price offer, group alpha consumers are willing to pay
more for a given increment to quality than are the consumers in group beta.
The shaded area that lies to the east of uB and the west of uA is critical for building the desired fence. Consumers in group beta prefer any quality-price offer in this
range to the offer at point B. Thus, consumers in group beta are made better off if the
firm makes the low-price offer point C rather than point B. Moreover, group beta consumers prefer offer C to offer A.
By contrast, consumers in group alpha prefer quality-price offer A to qualityprice offer C. Thus, they will purchase the high-price–high-quality version of the
1 2 . 4 T H I R D - D E G R E E P R I C E D I S C R I M I N AT I O N
product. Notice what the firm has done: By reducing the quality of the low-price
offer, the firm has made it unattractive for group alpha consumers to choose that
offer. But because group beta consumers are more tolerant of quality degradations,
they are willing to choose the low-quality version (and indeed, prefer this version to
the one that they would have chosen if the firm had not differentiated the quality of
the two offers).
A strategy of selling two (or more) versions of the product with different quality
levels at different prices is known as versioning. A particularly interesting type of
versioning is what Raymond Deneckere and Preston McAfee refer to as a damaged
goods strategy.20 Under a damaged goods strategy, a firm creates a low-end version
of its full-priced good by deliberately damaging the product: deliberately removing
features or reducing performance characteristics so that the product works less well
than its full-price counterpart. Ironically, if damaging the product requires an additional step in the production process, the marginal cost of producing the damaged
good can actually be higher than the marginal cost of the high-end version of the
product. This cost differential will be worth incurring if it is less than the gain in profits the firm achieves as a result of successfully building a fence that allows its scheme
of price discrimination to be implemented.
Deneckere and McAfee provide a number of examples of damaged goods. Two of
the most interesting are:
• IBM’s Laser Printer E. IBM’s primary laser printer in the early 1990s was called
the LaserPrinter. In May 1990 it introduced the LaserPrinter E. The two products were practically the same except the LaserPrinter E printed text at half the
speed of the LaserPrinter. This was done by adding chips to LaserPrinter E that
had the sole effect of causing the printer to pause, thereby slowing it down!
• Intel’s 486SX. The 486 was the new generation microprocessor introduced by
Intel in the early 1990s. Once its competitor, AMD, introduced a fast version of
the 386 microprocessor, Intel introduced a low-end version of the 486, known
as the 486SX, while at the same time renaming the original (high-end) version
the 486DX. Deneckere and McAfee note that the 486SX was the exact same
product as the 486DX except that the math co-processor was disabled, making
the low-end SX actually more expensive to produce than the high-end DX!
In some cases, implementation of the price discrimination scheme by the building of
fences is closely bound up in the screening of consumer types. Coupons are an excellent example of this. Willingness to take the time to find, cut out, and accumulate
coupons correlates with a consumer’s price sensitivity (more price-sensitive consumers are willing to do these activities; less price-sensitive consumers are not). In
this sense, coupons serve as a screening mechanism. At the same time, they act as a
fence that keeps those consumers whom the firm wants to charge full price from purchasing the good at a low price. This is because coupons create a hassle-factor in the
purchase of the good that is far more salient to consumers with low price sensitivity
than to consumers who are more sensitive to price and willing to go to great lengths
to get a discount.
20
Raymond J. Deneckere, and Preston McAfee, “Damaged Goods,” Journal of Economics and Management
Strategy, 5, no. 2 (Summer 1996), pp. 149–174.
511
versioning A strategy
of selling two or more
versions of a product with
different quality levels at
different prices.
damaged goods
strategy A versioning
strategy in which the firm
creates a low-end version
of its full-price good by
deliberately damaging the
product.
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12.6
Can You “Damage” the
Metropolitan Museum of Art? 21
Located in the heart of New York City’s Central Park,
the Metropolitan Museum of Art (or the Met) is one
of the most heavily visited museums in the world, and
it is almost certainly one of the top two or three art
museums in the world.
As you probably know, many art museums do not
require visitors to pay an admissions fee; instead, they
suggest a voluntary contribution. The Met has chosen
an interesting twist on this approach. Above the
ticket kiosks in the entrance to the museum is a sign
that reads:
Adults $15
Seniors and students $10
If look above the prices, you will see, in very
(very) small letters, the word, “Recommended.”
The Met is actually employing a type of damaged
goods strategy. To see this, think about what the sign
could have said:
Adults who are willing to pay full price, $15; if
not, you are free to pay less. Seniors and students
who are willing to pay full price, $10; if not, you
are free to pay less.
This sign accurately reflects the Met’s policy. But if the
Met were to use that sign, it is fair to say that a great
many people would pay less than full price (though
probably not all—there are some who undoubtedly
feel strongly about supporting a great institution like
the Met). What the Met has done is to make it a hassle
12.5
TYING (TIE-IN
SALES)
tying (tie-in sales) A
sales practice that allows a
customer to buy one product
(the tying product) only if
that customer agrees to
buy another product (the
tied product).
to pay less. It does so in three ways. First, as described above, the small print on the Met’s price sign
makes it difficult to see that the admissions fee is not
required. Second, the Met uses the word “recommended” rather than the more common words “suggested” or “voluntary” used in museums that do not
require an admissions fee. The difference is perhaps
subtle, but the idea is that the term recommended
makes the admissions fee seem “more mandatory.”
Finally, while the admissions fee truly is voluntary,
those who don’t pay risk being glared at by the ticket
agents manning the kiosks in the Great Hall of the
museum. While those who are highly motivated to
pay a low price might be willing to put up with this
hassle, those who are more inclined to pay full price
might well conclude that it is worth doing so in order
to avoid a reproachful look from the ticket agent.
And so, the Met, in effect, offers two versions of
its core product, access to the museum. The full-price
version requires no squinting at the sign, entails no
worry about whether or not the admissions fee is
required, and results in no embarrassment when obtaining a ticket to enter the museum. And then there
is the damaged version, which requires effort to read
the sign and parse the words, and the risk of a condescending look from a ticket agent. Undoubtedly
there are some consumers who, by virtue of their income, or simply the desire to get the best deal possible, do not pay full price. But there certainly must be
many others who would pay less than full price if
there was no onus in doing so, but who are motivated to pay full price because the damaged version
of access to the Met is sufficiently unattractive.
A
nother technique that firms use to capture surplus is tying. Tying (also called tie-in
sales) refers to a sales practice that allows a customer to buy one product (the “tying”
product) only if she agrees to buy another product (the “tied” product) as well.
Often, tying is used when customers differ by the frequency with which they wish
to use a product. For example, suppose a firm has a patent on a copy machine with
some unique features. Such a patent may give the firm some market power because
the patent prevents other firms from selling the same kind of machine. The firm would
like to price discriminate, setting a higher price for customers who make 15,000 copies
21
This application is based on “Seeing Art: What’s It Worth to You,” New York Times ( July 21, 2006), p. 25.
12.5 TYING (TIE-IN SALES)
513
per month than for customers making only 4,000 copies. However, it may be impossible for the firm to know how many copies a customer will make.
How, then, can the firm use its market power in copying machines to capture surplus? The firm might tie the sale of the machine to the purchase of materials used to
make copies, such as copying paper. For example, the firm could sell its copier under
a “requirements contract,” that is, a contract that requires a purchaser of a copy machine
to buy all copying paper from the firm. By setting a price for the paper that exceeds
the cost of making it, the firm can generate higher profits.
Tying often enables a firm to extend its market power from the tying product to
the tied product, as in the copier example. Without the tie-in sale, the firm could
probably not make any extranormal return in the market for copying paper. The market
for copying paper would be competitive because no special technology is involved in
making paper. If the firm wants to sell copying paper at a price higher than the competitive price, it must make sure that its customers do not buy the paper from other
companies. For example, it might try to enforce tying by informing users of the copy
machine that the warranty on the machine remains valid only if customers use the
firm’s copying paper.22
Tying arrangements often lead to disputes. The manufacturer of a computer
printer may want to require users to buy its own ink cartridges. The printer manufacturer may argue that the tie-in is necessary to guarantee that the ink will not damage
or jam the printer and that such quality control is necessary to protect the reputation
of the manufacturer. Other manufacturers who want to sell ink cartridges may feel
that the tie-in violates antitrust laws by illegally foreclosing them from the market.
With large profits at stake, the battle over tying arrangements often ends up in court.
In the United States the primary law addressing tying arrangements is the
Clayton Act, Section 3. The law has been interpreted in a series of cases over the
years. In practice, the courts often try to determine what the relevant market is for
the tied product, and to measure the seller’s share of that market. Some requirements
contracts have been found to be legal, usually when the seller of a tied product has
only a small share of the market. As F. M. Scherer notes, “Requirements contracts
negotiated by sellers possessing a very small share of the relevant market do stand a
good chance of escaping challenge, and not all challenged contracts have been found
illegal.”23 However, in other cases, tying is illegal. For example, when McDonald’s
sells a franchise, it cannot require its franchisee to buy supplies such as napkins and
cups from McDonald’s. The franchisees can buy cups from any supplier whose products
meet standards set by McDonald’s.
BUNDLING
Bundling refers to tie-in sales in which customers are required to purchase goods in
a package, goods that they cannot buy separately. For example, when you subscribe to
cable television, you typically have to buy a “package” of channels together, rather
than subscribing to each channel individually. When you go to Disney World, the
22
The practice of charging more to customers who use a product more is often called metering. A copy
machine, for example, typically has a device (a meter) that counts the number of copies made. When the
seller of the machine performs maintenance, it can determine how many copies have been made.
23
See F. M. Scherer, Industrial Market Structure and Economic Performance (Chicago: Rand McNally, 1980),
pp. 585–586.
bundling A type of tie-in
sale in which a firm requires
customers who buy one of
its products also to simultaneously buy another of its
products.
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CAPTURING SURPLUS
TABLE 12.2 Bundling Can Increase Profit When Customer Preferences
Are Negatively Correlated
Reservation Price
(maximum willingness to pay)
Customer 1
Customer 2
Marginal cost
Computer
Monitor
$1,200
$1,500
$1,000
$600
$400
$300
ticket you buy at the park entrance gives you admission and entitles you to go on all
the rides inside the park.24 A computer manufacturer may offer you a bundle that includes both a computer (a central processing unit) and a monitor.
Why do firms sometimes sell two or more items as a package instead of separately? Bundling can increase profits when customers have different tastes (different
willingnesses to pay) for the two products and when the firm cannot price discriminate. To see how this practice can be used to increase producer surplus, let’s consider
a company that sells two different products: a computer and a computer monitor. The
marginal cost of the computer is $1,000, and the marginal cost of the monitor is $300.
For simplicity, suppose only two customers are in the market, but the firm cannot
price discriminate. Table 12.2 shows how much each customer is willing to pay for a
computer and for a monitor. Both customers might like to buy a new computer and
a new monitor. However, either customer might like to buy a new computer alone
(perhaps already having an old monitor) or a new monitor alone (perhaps for use
with an old computer). Customer 1 would pay up to $1,200 for a computer and
$600 for a monitor. Customer 2 would pay up to $1,500 for a computer and $400 for
a monitor.
First, let’s see how much profit the firm can earn if it does not bundle the computer and the monitor. What price should it set for the computer (Pc)? If the firm sets
Pc ⫽ 1,500, it will sell only one computer (to customer 2) and earn a profit of $500
(equal to the price, $1,500, less the marginal cost of the computer, $1,000).25 If it sets
Pc ⫽ $1,200, it will sell two computers (one to each customer) and earn a profit of
$400 ($200 from each computer). So it should set the price of the computer at $1,500.
What price should it set for the monitor (Pm)? If the firm sets Pm ⫽ $600, it will
sell only one monitor (to customer 1) and earn a profit of $300 (equal to the price,
$600, less the marginal cost of the monitor, $300). If it sets Pm ⫽ $400, it will sell two
monitors (one to each customer) and earn a profit of $200 ($100 from each monitor).
24
Bundling is a kind of tying, but not all tying involves bundling. For example, as described above, a tying
arrangement might require a customer who buys a copy machine from a manufacturer also to buy all
copying paper from the manufacturer. The machine and the paper are not bundled because a customer
could buy paper without buying a machine. In contrast, in the Disney World bundling example, the
customer cannot buy admission to the park without also buying entitlement to the rides. Nor can the
customer buy entitlement to the rides without buying admission.
25
The reservation price for customer 1 is $1,500. Strictly speaking, if the manufacturer sets a computer
price Pc ⫽ $1,500, customer 1 will be indifferent between buying and not buying. Here we will suppose
that a customer buys when the price equals the maximum willingness to pay. (The firm could always cut
the price by one cent to ensure that it makes the sale.)
12.5 TYING (TIE-IN SALES)
TABLE 12.3 Bundling Does Not Increase Profit When Customer
Preferences Are Positively Correlated
Reservation Price
(maximum willingness to pay)
Customer 1
Customer 2
Marginal cost
Computer
Monitor
$1,200
$1,500
$1,000
$400
$600
$300
The best the firm can do without bundling is to set Pc ⫽ $1,500 and Pm ⫽ $600.
It will then earn a total profit of $800, $500 from the computer sales and $300 from
the monitor sales.
Now consider the option to bundle the computer and the monitor, selling the two
components in a single package. What is the maximum profit it can earn? Customer
1 would be willing to pay up to $1,800 for the package, and customer 2 would pay up
to $1,900. If the bundle is sold at Pb ⫽ $1,900, only customer 2 will buy the bundle.
The revenue would be $1,900, and the cost would be $1,300 ($1,000 for the computer
and $300 for the monitor). Thus, the profit would be $600.
However, the firm can do better by setting the price of the bundle at Pb ⫽ $1,800.
For each package sold, the profit will be $500, equal to the revenue of $1,800, less the
cost of $1,300. Both customers will buy the bundle, and the total profit will be $1,000.
Thus, the manufacturer will maximize profit by selling a bundle at Pb ⫽ $1,800.
Bundling has increased profit from $800 (without bundling) to $1,000 (with bundling).
Why does bundling work to increase profit? The key is that the customers’ demands are negatively correlated. The negative correlation means that customer 2 is willing to pay more than customer 1 for the computer, while customer 1 is willing to pay
more than customer 2 for the monitor. By bundling the goods, the manufacturer is inducing the consumers to take both products when they might not otherwise do so.
To see why the negative correlation of customer demands is important, let’s see
what happens if the customer demands are positively correlated. Suppose the customer
demands are as shown in Table 12.3. Here the customer preferences are positively
correlated because customer 2 is willing to pay more for a monitor, and more for a
computer, than customer 1.
If the manufacturer does not bundle, it maximizes profit by selling computers at
$1,500, earning a profit of $500 from each computer sold. Only customer 2 buys a
computer at this price. The most the firm can earn in the monitor market is a profit of
$300, and it earns this by selling monitors at $600. Only customer 2 buys a monitor.
Total profit will be $800. (You should verify that it would be less profitable for the firm
to sell either a computer or a monitor at a price low enough to attract customer 1.)
If the manufacturer offers the computer and monitor as a bundle, the best the
firm can do is to set the price at $2,100, earning a profit of $800. Therefore, bundling
does not increase the firm’s profits.
MIXED BUNDLING
In practice, firms often allow customers to purchase components individually, as well
as offering a bundle. For example, you can purchase a computer from Dell with or
without a monitor. This is called mixed bundling. To see why mixed bundling might
515
516
CHAPTER 12
TABLE 12.4
CAPTURING SURPLUS
Mixed Bundling Can Increase Profit
Reservation Price
(maximum willingness to pay)
Customer 1
Customer 2
Customer 3
Customer 4
Marginal cost
Computer
Monitor
$ 900
$ 1,100
$ 1,300
$1,500
$1,000
$800
$600
$400
$200
$300
be the most profitable strategy for a firm, consider the example illustrated in Table 12.4.
In this example, each of the four customers is willing to pay $1,700 for a bundle. Their
demands are negatively correlated because a customer who is willing to pay more for
a computer is willing to pay less for a monitor. However, as we shall see, the manufacturer will not maximize profits by offering only a bundle at a price of $1,700.
To see what the optimal strategy will be, let’s consider three options.
• Option 1: No bundling. If the manufacturer does not bundle, it maximizes profit by
selling computers at $1,300 and monitors at $600. When the price of a computer
is $1,300, customers 3 and 4 will buy computers. The firm’s profit from computers
will be $600 because two computers are sold, the price of each is $1,300, and the
cost of each is $1,000. When the price of a monitor is $600, customers 1 and 2
will buy monitors. The firm’s profit from monitors will also be $600 because two
monitors are sold, the price of each is $600, and the marginal cost of each is $300.
The total profit will be $1,200.
• Option 2: Pure bundling (selling only a bundle). If the manufacturer offers the
computer and monitor as a bundle, priced at $1,700, all four customers buy the
bundle. On each bundle the profit will be $400 (the revenue of $1,700 less the
marginal cost of $1,300). The total profit will therefore be $1,600.
• Option 3: Mixed bundling. Here the manufacturer offers customers three options.
It sells a computer separately at one price (Pc), sells a monitor separately at
another price (Pm), and offers a package with a computer and a monitor at a
bundled price (Pb).
Why is the firm’s optimal strategy to offer mixed bundling in this example? This
pricing strategy discourages any customer from buying a component when the customer’s willingness to pay is less than the marginal cost of that component.
Note that customer 1 is only willing to pay $900 for a computer, which is less than
the marginal cost of the computer. It will therefore not be profitable for the firm to
sell a computer to customer 1. If customer 1 buys a bundle at $1,700, the firm makes
a profit of $400 (i.e., $1,700 revenue less $1,300 cost) from the sale of that bundle. If
the customer buys the bundle, he earns a surplus of zero dollars.
However, the firm can make more profit from customer 1 by selling the monitor
separately. The firm could induce customer 1 to buy the monitor separately by pricing
it to give him more consumer surplus than the customer would get from the bundle.
If the manufacturer prices the monitor separately at $799, customer 1 will buy it, and
the sale of that monitor generates a profit of $499 for the firm. The firm is better off
12.5 TYING (TIE-IN SALES)
517
(by $99) when the customer buys only the monitor instead of the bundle. And the customer is better off buying only the monitor, earning a consumer surplus of $1 (equal
to her willingness to pay for a monitor, $800, less the price of the monitor, $799). So
the firm should set Pm ⫽ $799.
Similarly, customer 4 is only willing to pay $200 for a monitor, which is less than
the marginal cost of the monitor. It will therefore not be profitable for the firm to sell
a monitor to customer 4. Customer 4 will be happier purchasing only the computer
at $1,499 (earning $1 of consumer surplus) instead of the bundle at $1,700 (earning a
surplus of zero). The sale of the computer separately to customer 4 generates a profit
of $499 for the firm, in contrast to the $400 profit it would have earned if customer
4 had bought the bundle. The firm should set Pc ⫽ $1,499.
Finally, customers 2 and 3 have negatively correlated demands. Further, the
amounts that they are willing to pay for each component exceed the marginal cost.
The firm would therefore like to sell them a bundle. It should offer a package with a
computer and a monitor at Pb ⫽ $1,700.
In sum, with mixed bundling, customer 4 buys the computer separately, customer
1 takes the monitor alone, and customers 2 and 3 buy the bundle. Total profit is
$1,798. The profit is higher with mixed bundling than it would be with no bundling
($1,200) or selling only a bundle ($1,600).
A P P L I C A T I O N
12.7
Bundling Cable
Cable television companies such as Comcast offer a
variety of bundled packages of their products. For example, in Chicago a customer can subscribe to the
basic Digital Starter package for about $25 per
month. This package provides only local television stations, children’s programs and weather stations. For
$45, the Digital Preferred option adds over 100 television channels and 45 music-only channels. The Digital
Preferred Plus package adds about 50 more channels,
including the premium channels HBO and Starz, for
$99. Finally, Digital Premier adds all premium channels, a sports entertainment package, and 50 more
channels, for $115. Any of these packages can also be
combined with Internet access, telephone service, or
both (all using the same cable into the home). In practice, Comcast actually offers choices with mixed bundles, allowing customers to choose among not only
collections of cable channels, but also packages that
include digital voice service and high-speed Internet
service.
A common complaint about the cable packages is
that most customers regularly view only a small fraction of the channels provided. Why would Comcast
offer 100 channels for a fixed price rather than allowing the customer to pick and choose her favorite
channels, paying lower a la carte prices for each? The
answer lies in the economics of bundling.
Consider a simple example where there are two
consumers, Kathryn and Mike, and two channels, the
Food Network and Travel Channel. Kathryn’s favorite
channel is the Food Network, while Mike’s is the
Travel Channel. Kathryn gets $30 worth of utility per
month from the Food Network, but only $5 from the
Travel Channel. Mike gets $30 utility from the Travel
Channel but only $5 from the Food Network. The
maximum revenue that Comcast could get for each
channel (without bundling) would be to charge $30
for each channel and provide a single channel to
each customer. However, if they bundle the channels
together, they can charge $35 to both customers for
a package of both channels. As long as the marginal
cost of providing a second channel to a customer is
lower than $5 (and Comcast’s marginal cost of adding
one channel for a customer is probably very low for
many channels), then bundling will be more profitable for Comcast. For example, if the marginal cost
is zero in the example, then Comcast’s profit will increase by $10 by bundling the stations as a package.
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12.6
A DV E R T I S I N G
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S
o far in this chapter, we have examined how a firm can capture surplus with pricing
strategies. We now show how a firm with market power can also create and capture surplus with nonprice strategies, such as by choosing the amount of advertising for its product.
By advertising, a seller hopes to increase the demand for its product, shifting the
demand curve rightward and creating more surplus in the market. However, the firm
must also recognize that advertising is costly. Only by correctly choosing the level of
advertising can the firm capture as much surplus as possible.
Figure 12.11 illustrates the effects of advertising, assuming that the firm cannot
price discriminate and that advertising expenditures affect the firm’s fixed costs but
not its marginal cost of production (e.g., it is reasonable to assume that the marginal
cost curve is not affected by advertising).
If the firm does not advertise at all, the demand and marginal revenue curves for
its product are D0 and MR0. The average and marginal cost curves are AC0 and MC.
The firm produces Q0 and sells at a price P0. The maximum profit the firm can earn
with no advertising is areas I ⫹ II.
If the firm spends A1 dollars on advertising, the demand curve for its product
shifts to the right, to D1, and the marginal revenue curve becomes MR1. Since advertising adds to the firm’s total costs, the average cost curve rises to AC1. To maximize
profits, the firm produces Q1 and sells at a price P1. For the demand and cost curves
depicted in the figure, it is clearly profitable for the firm to advertise. When it spends
A1 on advertising, the maximum profit the firm can earn increases to areas II ⫹ III.
For a firm to maximize profit by advertising (expenditure on advertising A ⬎ 0)
and producing a positive quantity (Q ⬎ 0), two conditions must hold:
1. When output Q is chosen optimally, the change in total revenue from the last
unit produced ⌬TR /⌬Q (i.e., the marginal revenue MRQ) must equal the marginal cost of that last unit ⌬TC/⌬Q (denoted by MCQ). The requirement that
MRQ ⫽ MCQ is the usual optimal quantity choice rule for a monopolist, as we
saw in Chapter 11. We can write the optimal quantity choice equivalently as the
inverse elasticity pricing rule:
P ⫺ MCQ
P
1
⫽⫺
(12.1)
⑀Q,P
FIGURE 12.11 Effects
of Advertising
When the firm does not
advertise (D0, MR0, AC0, Q0,
P0), its maximum profit is
areas I ⫹ II. When the firms
spends A1 dollars on advertising (D1, MR1, AC1, Q1, P1),
its maximum profit is
areas II ⫹ III.
Price ($ per unit)
Profit with A1
dollars of advertising
P1
MC
III
P0
AC1
II
I
AC0
Profit
without
advertising
D0 MR1
MR0
Q0
Q1
Quantity (units per year)
D1
1 2 . 6 A DV E RT I S I N G
where P is the price of the product and ⑀Q,P is the price elasticity of demand for
the firm’s product.
2. When the level of expenditure on advertising A is chosen optimally, the marginal
revenue from the last dollar spent on advertising ⌬TR/⌬A (denoted by MRA)
must equal the marginal cost that the firm incurs when it spends an additional
dollar on advertising ⌬TC/⌬ A (denoted by MCA).
Why must MRA ⫽ MCA at a profit maximum? If at the current level of advertising
MRA ⬎ MCA, an additional unit of advertising would increase revenues by more than it
would increase cost. Therefore, the firm could increase profit by advertising more. By
similar reasoning, if MRA ⬍ MCA, the firm could increase profit by advertising less.
Assuming that price is held constant, we can represent the condition that
MRA ⫽ MCA in another way. First we ask, how does a change in the level of advertising affect the total revenue for the firm? If the demand for the product is Q(P, A) (i.e.,
the quantity demanded depends on both price and advertising), the firm’s total revenue
is TR ⫽ PQ(P, A). When advertising expenditures go up by a small amount (⌬ A), the
change in total revenue (⌬TR) will be equal to the price P times the change in quantity
demanded as advertising increases (⌬Q). Thus, ⌬TR ⫽ P⌬Q. If we divide both sides
by ⌬ A, we get ⌬TR/⌬ A ⫽ P(⌬Q/⌬ A). Since ⌬TR/⌬ A ⫽ MRA, the marginal revenue
from advertising is MRA ⫽ P(⌬Q/⌬ A).
Then we ask, how does a change in the level of advertising expenditure affect the
total cost for the firm? The total cost is TC ⫽ C(Q(P, A)) ⫹ A. The marginal cost from
another dollar of advertising is ⌬TC/⌬ A ⫽ MCA. When the firm increases advertising by a small amount (⌬ A), two things happen to costs: advertising expenditures go
up by ⌬ A, and the quantity demanded goes up by ⌬Q. When the firm produces this
extra quantity, production costs will increase by (MCQ)(⌬Q). Thus the impact of the
extra advertising on total cost is ⌬TC ⫽ MCQ(⌬Q) ⫹ ⌬ A. If we divide both sides by
⌬ A, we get ⌬TC/⌬ A ⫽ MCQ(⌬Q/⌬ A) ⫹ 1. Since ⌬TC/⌬ A ⫽ MCA, the marginal cost
of advertising is MCA ⫽ MCQ(⌬Q/⌬ A) ⫹ 1.
Since MRA ⫽ MCA, we can equate these two expressions: P(⌬Q/⌬ A) ⫽
MCQ(⌬Q/⌬ A) ⫹ 1.
Now consider a measure called the advertising elasticity of demand (denoted by ⑀Q, A),
which tells us the percentage increase in quantity demanded that would result from a 1 percent increase in advertising: ⑀Q, A ⫽ (¢Q/¢ A)(A/Q), which we can rewrite as ¢Q/¢ A ⫽
Q⑀Q, A /A. Substituting this expression for ¢Q/¢ A into the equation above, we find
Pa
Q⑀Q, A
Q⑀
b ⫽ MCQ a Q, A b ⫹ 1
A
A
Multiplying both sides by A:
PQ⑀Q, A ⫽ MCQQ⑀Q, A ⫹ A
Dividing by ⑀Q, A:
PQ ⫽ MCQQ ⫹
A
⑀Q, A
Rearranging terms and factoring out Q:
Q(P ⫺ MCQ) ⫽
Dividing by Q:
P ⫺ MCQ ⫽
A
⑀Q, A
1 A
⑀Q, A Q
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And then dividing by P:
P MCQ
P
1 A
⑀Q, A PQ
(12.2)
Because the left-hand sides of equations (12.1) and (12.2) are the same (the Lerner
Index) it must be true that:
1
⑀Q,P
1
⑀Q, A
A
PQ
Multiplying both sides by ⑀Q, A gives
⑀Q, A
A
⑀Q,P
PQ
(12.3)
The left-hand side of equation (12.3) is the ratio of advertising expenditures A to
sales revenues PQ. The right-hand side is the negative ratio of the advertising elasticity of demand to the own price elasticity of demand. If you think about it, this relationship simply makes good business sense. Suppose you examined two markets with
approximately the same own price elasticity of demand, but greatly different advertising elasticities of demand. In the market in which demand is highly sensitive to the
amount of advertising, you would expect the advertising-to-sales ratio to be higher
compared to the market with a low elasticity of demand for advertising.26
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 2 . 7
D
Markup and Advertising-to-Sales Ratio
Suppose you own a restaurant specializing
in fine steak dinners, and you want to maximize your
profits. Your marketing studies have revealed that your
own price elasticity of demand is 1.5 and that your advertising elasticity of demand is 0.1. Assume that these
elasticities are constant, even if you change your price
and your level of advertising.
Problem
(a) Interpret the advertising elasticity of demand.
(b) How much should you mark up your price over
marginal cost of your dinners? What should your
advertising-to-sales ratio be?
tures will increase quantity demanded by about onetenth of 1 percent.
(b) The inverse elasticity pricing rule, equation (12.1),
states (P MCQ)/P 1/⑀Q,P (1/1.5) 2/3. Thus,
P MCQ (2/3)P, or P 3MCQ. The dinners should
be priced at three times marginal cost. According to equation (12.3), the optimal advertising-to-sales ratio should be
A/ (PQ) ⑀Q, A /⑀Q,P (0.1)/(1.5) 0.067. Thus,
your advertising expenses should be 6.7 percent of your
sales revenues.
Similar Problems: 12.29, 12.30
Solution
(a) The advertising elasticity of demand, ⑀Q, A 0.1,
implies that a 1 percent increase in advertising expendi-
26
For important early work on advertising, including some of the main insights discussed in this section,
see R. Dorfman and P. Steiner, “Optimal Advertising and Optimal Quality,” American Economic Review 44
(December 1954): 826–836.
C H A P T E R S U M M A RY
A P P L I C A T I O N
521
12.8
Advertising on Google
Google has been the most popular online search engine for several years. In 2009, it accounted for approximately 60 percent of online searches, roughly 4
times larger than its nearest competitor, Yahoo. Its
earnings were well over $22 billion. Yet when you use
Google, you do not pay them a dime. So how did the
company earn $22 billion? The answer is that nearly
all of Google’s revenues come from charging for the
ads (such as the Sponsored Links) that it places on its
Web page. Google accounts for roughly 60 percent of
all Internet advertising revenues.
Internet advertising is still a small fraction of the
total advertising industry (about 5 to 6 percent), but it
is growing rapidly. There are two reasons for this. One
is that the cost of placing ads on websites is very low,
since they can be replicated at nearly zero marginal
cost and delivered digitally. A more subtle but very
important benefit of Internet advertising is that advertisers can often target more directly the type of customers that they are trying to reach with their ads.
Internet access providers and search engines like
Google have specific information about the interests
of specific customers by tracking their Internet usage
or search requests over time. For example, if you repeatedly use Google to find economics articles,
Google learns of your interest in economics. It can use
that information to help advertisers locate consumers
who are most likely to be interested in their ads. In
addition, when you enter a search request, Google
can sell Sponsored Links that match the search request
on the page that displays the search results.
In terms of our discussion of the benefits of
advertising, tracking customer Internet and search
activity means that firms can stimulate the demand
for their products at much lower cost because they
do not waste resources on customers who are less
likely to have interest in their ads. In Figure 12.11, D1
and MR1 will both shift further to the right compared to D0 and MR0, increasing the area of A. Thus,
better targeting of ads to customers raises the advertising elasticity of demand, possibly very significantly. For this reason, many firms are increasing the
proportion of their advertising budget expenditures
on Internet ads.
CHAPTER SUMMARY
• A firm with market power can influence the price in the
market and capture surplus (i.e., increase profit). A firm
need not be a monopolist to have market power, but the demand curve the firm faces must be downward sloping.
• One way a firm may capture surplus is through price
discrimination—that is, by charging more than one price
for its product. There are three basic types of price discrimination: first-degree price discrimination, seconddegree price discrimination, and third-degree price discrimination. But for a firm to price discriminate at all,
three conditions are necessary: The firm must have market
power, the firm must have some information about how
reservation prices or elasticities of demand differ across
consumers, and the firm must be able to prevent resale.
• With first-degree price discrimination, the firm attempts to price each unit at the consumer’s reservation
price for that unit. The marginal revenue curve is therefore the same as the demand curve. First-degree price
discrimination allows the producer to capture all of the
surplus. (LBD Exercises 12.1, 12.2)
• Under second-degree price discrimination, the firm
offers consumers a quantity discount. With a block tariff (with two blocks), the consumer pays one price for
units consumed in the first block of output (up to a given
quantity) and a different (usually lower) price for any additional units. With a combination of subscription and
usage charges, the consumer pays an entry fee (the subscription charge) and then pays a specified price per unit
(the usage charge). (LBD Exercise 12.3)
• With third-degree price discrimination, the firm
identifies different consumer groups, or segments, in a
market, and then charges a price for each segment by
setting marginal revenue equal to marginal cost or,
equivalently, by using the inverse elasticity pricing rule.
Price is uniform within a segment but differs across
segments. (LBD Exercises 12.4, 12.5)
• To implement third-degree price discrimination, firms
sometimes use screening to infer how reservation prices or
elasticities of demand differ across consumers. Screening
sorts consumers based on a consumer characteristic that
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CAPTURING SURPLUS
the firm can see (e.g., age or status) and that is strongly related to a consumer characteristic that the firm cannot see
but would like to observe (e.g., willingness to pay or elasticity of demand).
• A firm that engages in third-degree price discrimation with capacity constraints will maximize its profit by
allocating its fixed capacity in such a way as to equate the
marginal revenues across its market segments. (LBD
Exercise 12.6)
• One way for a firm to implement a scheme of price
discrimination is to create different versions of the good:
a low-quality, low-price version that appeals to pricesensitive consumers and a high-quality, high-price version that appeals to less price-sensitive consumers.
• Tying allows a customer to buy one product (the
tying product) only if the customer agrees to buy another product (the tied product). The consumer might
buy the tied product without the tying product, but
not the reverse. Tying often enables a firm to extend
its market power from the tying product to the tied
product.
• Bundling refers to tying that requires customers to
purchase goods in a package. The customer cannot buy
the goods separately. Bundling may increase profits
when customers have negatively correlated demands. It
may be profitable to offer consumers the option of
“mixed bundling,” where they have a choice of buying
goods in a bundle or separately.
• Advertising can help a firm capture more surplus
when advertising increases the demand for a product.
However, advertising is costly. When a firm simultaneously chooses its level of output and the level of advertising, it must attempt to (1) equate the marginal revenue
from production to the marginal cost of production and
(2) equate the marginal revenue from advertising to the
marginal cost of advertising. When a firm maximizes
profit, the advertising-to-sales ratio equals the negative
of the ratio of the advertising and own price elasticities
of demand. (LBD Exercise 12.7)
REVIEW QUESTIONS
1. Why must a firm have at least some market power to
price discriminate?
2. Does a firm need to be a monopolist to price discriminate?
3. Why must a firm prevent resale if it is to price discriminate successfully?
4. What are the differences among first-degree, seconddegree, and third-degree price discrimination?
5. With first-degree price discrimination, why is the
marginal revenue curve the same as the demand curve?
6. How large will the deadweight loss be if a profitmaximizing firm engages in perfect first-degree price
discrimination?
7. What is the difference between a uniform price and a
nonuniform (nonlinear) price? Give an example of a nonlinear price.
8. Suppose a company is currently charging a uniform
price for its two products, creamy and crunchy peanut
butter. Will third-degree price discrimination necessarily
improve its profit? Would the firm ever be worse off with
price discrimination?
9. How might screening help a firm price discriminate?
Give an example of screening and explain how it works.
10. Why might a firm try to implement a tying arrangement? What is the difference between tying and
bundling?
11. How might bundling increase a firm’s profits? When
is bundling not likely to increase profits?
12. Even if a monopolist knows that advertising shifts
the demand curve for its product to the right, why might
it decide not to advertise at all? If it does advertise, what
factors determine how much advertising it will do?
PROBLEMS
12.1. Which of the following are examples of
first-degree, second-degree, or third-degree price discrimination?
a) The publishers of the Journal of Price Discrimination
charge a subscription price of $75 per year to individuals
and $300 per year to libraries.
b) The U.S. government auctions off leases on tracts of
land in the Gulf of Mexico. Oil companies bid for the
right to explore each tract of land and to extract oil.
c) Ye Olde Country Club charges golfers $12 to play the
first 9 holes of golf on a given day, $9 to play an additional 9 holes, and $6 to play 9 more holes.
PROBLEMS
d) The telephone company charges you $0.10 per minute
to make a long-distance call from Monday through
Saturday and $0.05 per minute on Sunday.
e) You can buy one computer disk for $10, a pack of 3 for
$27, or a pack of 10 for $75.
f ) When you fly from New York to Chicago, the airline
charges you $250 if you buy your ticket 14 days in
advance, but $350 if you buy the ticket on the day of
travel.
12.2. Suppose a profit-maximizing monopolist producing Q units of output faces the demand curve P ⫽ 20 ⫺ Q.
Its total cost when producing Q units of output is TC ⫽
24 ⫹ Q2. The fixed cost is sunk, and the marginal cost
curve is MC ⫽ 2Q.
a) If price discrimination is impossible, how large will the
profit be? How large will the producer surplus be?
b) Suppose the firm can engage in perfect first-degree
price discrimination. How large will the profit be? How
large is the producer surplus?
c) How much extra surplus does the producer capture
when it can engage in first-degree price discrimination
instead of charging a uniform price?
12.3. Suppose a monopolist producing Q units of
output faces the demand curve P ⫽ 20 ⫺ Q. Its total cost
when producing Q units of output is TC ⫽ F ⫹ Q2, where
F is a fixed cost. The marginal cost is MC ⫽ 2Q.
a) For what values of F can a profit-maximizing firm charging a uniform price earn at least zero economic profit?
b) For what values of F can a profit-maximizing firm engaging in perfect first-degree price discrimination earn at
least zero economic profit?
12.4. A firm serving a market operates with total variable cost TVC ⫽ Q2. The corresponding marginal cost is
MC ⫽ 2Q. The firm faces a market demand represented
by P ⫽ 40 ⫺ 3Q.
a) Suppose the firm sets the uniform price that maximizes
profit. What would that price be?
b) Suppose the firm were able to act as a perfect firstdegree price-discriminating monopolist. How much would
the firm’s profit increase compared with the uniform
profit-maximizing price you found in (a)?
12.5. A natural monopoly exists in an industry with a
demand schedule P ⫽ 100 ⫺ Q. The marginal revenue
schedule is then MR ⫽ 100 ⫺ 2Q. The monopolist operates with a fixed cost F, and a total variable cost TVC ⫽
20Q. The corresponding marginal cost is thus constant
and equal to 20.
a) Suppose the firm sets a uniform price to maximize
profit. What is the largest value of F for which the firm
could earn zero profit?
523
b) Suppose the firm is able to engage in perfect firstdegree price discrimination. What is the largest value of
F for which the firm could earn zero profit?
12.6. Suppose a monopolist is able to engage in perfect
first-degree price discrimination in a market. It can sell
the first unit at a price of 10 euros, the second at a price
of 9 euros, the third at a price of 8 euros, the fourth at a
price of 7 euros, the fifth at a price of 6 euros, and the
sixth at a price of 5 euros. It must sell whole units, not
fractions of units.
a) What is the firm’s total revenue when it produces
two units?
b) What is the total revenue when it produces three units?
c) What is the relationship between the price of the third
unit and the marginal revenue of the third unit?
d) What is the relationship between the price and the
marginal revenue of the fourth unit?
12.7. Suppose the monopolist in Problem 12.6 incurs a
marginal cost of 5.50 euros for every unit it produces.
The firm has no fixed costs.
a) How many units will it produce if it wants to maximize
its profit? (Remember, it must produce whole units.)
b) What will its profit be when it maximizes profit?
c) What will the deadweight loss be when it maximizes
profit? Explain.
12.8. Fore! is a seller of golf balls that wants to increase
its revenues by offering a quantity discount. For simplicity, assume that the firm sells to only one customer and
that the demand for Fore!’s golf balls is P ⫽ 100 ⫺ Q. Its
marginal cost is MC ⫽ 10. Suppose that Fore! sells the
first block of Q1 golf balls at a price of P1 per unit.
a) Find the profit-maximizing quantity and price per unit
for the second block if Q1 ⫽ 20 and P1 ⫽ 80.
b) Find the profit-maximizing quantity and price per unit
for the second block if Q1 ⫽ 30 and P1 ⫽ 70.
c) Find the profit-maximizing quantity and price per unit
for the second block if Q1 ⫽ 40 and P1 ⫽ 60.
d) Of the three options in parts (a) through (c), which
block tariff maximizes Fore!’s total profits?
12.9. Consider the manufacturer of golf balls in Problem 12.8. The firm faces the demand curve P ⫽ 100 ⫺ Q,
and operates with a marginal cost of 10 for all units produced. Among all the possible block tariffs (with two
blocks), what block tariff structure will maximize profit?
In other words, what choices of P1, Q1 for the first block
and P2, Q2 for the second block will maximize profit?
12.10. Suppose that you are a monopolist who produces gizmos, Z, with the total cost function C(Z ) ⫽ F ⫹
50Z, where F represents the firm’s fixed cost. Your marginal cost is MC ⫽ 50. Suppose also that there is only one
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consumer in the market for gizmos, and she has the
demand function P ⫽ 60 ⫺ Z.
a) If you use a constant per-unit price for gizmos, what
price maximizes your profits? What is the smallest value
of F such that you could earn positive profits at this price?
b) Suppose instead that you charge a per-unit price equal
to marginal cost, that is, P ⫽ MC ⫽ 50. How many units
would the customer purchase at this price? Illustrate your
answer in a graph (featuring the individual demand curve
and marginal cost).
c) Now consider charging the customer a “subscription
fee” of S in addition to a usage fee. If you set the usage
fee as in part (b), what is the largest fixed fee you could
charge the consumer, while ensuring that she is willing to
participate in this market?
d) For what values of F will you be able to earn positive
profits if you follow the pricing strategy you outlined in
part (c)? How does this relate to your answer in part (a)?
e) Suppose now that there are N consumers in the market for gizmos, each with the individual demand function
P ⫽ 60 ⫺ Z. Expressing your answer in terms of N, how
large can the fixed costs F be for you to still earn positive
profits if you use the above nonlinear pricing strategy.
12.11. In part (c) of Learning-By-Doing Exercise 12.3,
we suggested that the profit-maximizing structure for the
first and second blocks for Softco is something other
than the pricing structure we determined in part (b), selling the first 60 units at a price of $40 apiece, and selling
any quantity above 60 at $25 apiece. Find the structure
that maximizes profit.
12.12. Consider a market with 100 identical individuals, each with the demand schedule for electricity of P ⫽
10 ⫺ Q. They are served by an electric utility that operates with a fixed cost 1,200 and a constant marginal cost
of 2. A regulator would like to introduce a two-part tariff, where S is a fixed subscription charge and m is a usage
charge per unit of electricity consumed. How should the
regulator set S and m to maximize the sum of consumer
and producer surplus while allowing the firm to earn exactly zero economic profit?
12.13. A monopolist faces two market segments. In
each market segment, the demand curve is of the constant elasticity form. In market segment 1, the price elasticity of demand is ⫺3, while in market segment 2, the
price elasticity of demand is ⫺1.5. The monopolist has a
constant marginal cost of $5 per unit, which is the same
in each market segment. What is the monopolist’s profitmaximizing price in each segment?
12.14. Suppose that Acme Pharmaceutical Company
discovers a drug that cures the common cold. Acme has
plants in both the United States and Europe and can
manufacture the drug on either continent at a marginal
cost of 10. Assume there are no fixed costs. In Europe,
the demand for the drug is QE ⫽ 70 ⫺ PE, where QE is
the quantity demanded when the price in Europe is PE.
In the United States, the demand for the drug is QU ⫽
110 ⫺ PU, where QU is the quantity demanded when the
price in the United States is PU.
a) If the firm can engage in third-degree price discrimination, what price should it set on each continent to maximize its profit?
b) Assume now that it is illegal for the firm to price discriminate, so that it can charge only a single price P on
both continents. What price will it charge, and what
profits will it earn?
c) Will the total consumer and producer surplus in the
world be higher with price discrimination or without
price discrimination? Will the firm sell the drug on both
continents?
12.15. Consider Problem 12.14 with the following
change. Suppose the demand for the drug in Europe declines to QE ⫽ 30 ⫺ PE. If the firm cannot price discriminate, will it be in the firm’s interest to sell on both continents?
12.16. Consider Problem 12.14 with the following
change. Suppose the demand for the drug in Europe becomes QE ⫽ 55 ⫺ 0.5PE. Will third-degree price discrimination increase the firm’s profits?
12.17. Think about the problem that Acme faces in
Problem 12.14. Consider any demand curves for the drug
in Europe and in the United States. Will its profits ever
be lower with third-degree price discrimination than they
would be if price discrimination were impossible?
12.18. There is another way to solve Learning-ByDoing Exercise 12.5. Recall that marginal revenue can be
written as MR ⫽ P ⫹ (⌬P/⌬Q)Q. By factoring out P, we can
write MR ⫽ P[1 ⫹ (¢P/¢Q)(Q/P)] ⫽ P[1 ⫹ (1/⑀Q, P)].
Since third-degree price discrimination means that
marginal cost equals marginal revenue in each market
segment, the profit-maximizing regular and vacation
fares will be determined by MRR ⫽ MRV ⫽ MC.
(Remember the marginal cost of both classes of service
is assumed to be the same in the exercise.) Thus
PR [1 ⫹ (1/⑀QR,PR)] ⫽ PE [1 ⫹ (1/⑀QE,PE)] ⫽ MC. Use this
relationship to verify the answer given in the exercise.
12.19. J. Cigliano (“Price and Income Elasticities for
Airline Travel: The North Atlantic Market,” Business
Economics, September 1980) estimated the price elasticity
of demand for regular (full-fare) travel in coach class in
the North Atlantic market to be ⑀B ⫽ ⫺1.3. He also
found the price elasticity of demand for excursion (vacation) travel to be about ⑀V ⫽ ⫺1.8. Suppose Transatlantic
Airlines faces these price elasticities of demand, and that
525
PROBLEMS
the elasticities are constant; that is, they do not vary with
price. Since both are coach fares, you may also assume
that the marginal cost of service is about the same for
business and vacation travelers. Suppose an airline facing
these demand elasticities wants to set PR (the price of a
round-trip ticket to regular business travelers) and PV
(the price of a round-trip ticket to vacation travelers) to
maximize profit. What prices should the firm charge if
the marginal cost of a round trip is 200?
12.20. La Durazno is the only resort hotel on a small
desert island off the coast of South America. It faces two
market segments: bargain travelers and high-end travelers. The demand curve for bargain travelers is given by
Q1 400 2P1. The demand curve for high-end travelers is given by Q2 500 P2. In each equation, Q denotes the number of travelers of each type who stay at the
hotel each day, and P denotes the price of one room per
day. The marginal cost of serving an additional traveler
of either type is $20 per traveler per day.
10
PR
12.21. A pipeline transports gasoline from a refinery at
point A to destinations at R and T. The marginal cost of
transporting gasoline to each destination is MC 2. The
pipeline has a fixed cost of 160. The demand curve for the
transportation of gasoline from A to R is QR 100 10PR,
where QR is the number of units transported when PR is the
transport price per unit. The demand for pipeline movements from A to T will be 20 units as long as PT 12.
If PT 7 12, the customers at T will purchase gasoline
from another source, buying no gasoline shipped through
the pipeline. These demand curves are shown below.
a) If this firm were unable to engage in price discrimination (so that it can only choose a single P for the two markets), what would the profit-maximizing tariff be? What
level of profit would the firm realize?
b) If this firm were able to implement third-degree price
discrimination to maximize profits, what would the
profit-maximizing prices be? What level of profits would
the firm realize?
12
Demand for
transportation
from A to R
PT
100
a) Under the assumption that there is a positive demand
from each type of traveler, what is the equation of the
overall market demand curve facing the resort?
b) What is the profit-maximizing price under the
assumption that the resort must set a uniform price for all
travelers? For the purpose of this problem, you may
assume that at the profit-maximizing price, both types
of travelers are served. Under the uniform price, what
fraction of customers are bargain travelers, and what
fraction are high end?
c) Suppose that the resort can engage in third-degree
price discrimination based on whether a traveler is a
high-end traveler or a bargain traveler. What is the
profit-maximizing price in each segment? Under price
discrimination, what fraction of customers are bargain
travelers and what fraction are high end?
d) The management of La Durazno is probably unable to
determine, just from looking at a customer, whether he
or she is a high-end or bargain traveler. How might La
Durazno screen its customers so that it can charge the
profit-maximizing discriminatory prices you derived in
part (c)?
Demand for
transportation
from A to T
YR
20
YT
12.22. A seller produces output with a constant marginal
cost MC 2. Suppose there is one group of consumers
with the demand curve P1 16 Q1, and another with the
demand curve P2 10 (1/2)Q2.
a) If the seller can discriminate between the two markets,
what prices would she charge to each group of consumers? (You may want to exploit the monopoly midpoint rule from Learning-By-Doing Exercise 11.5.)
b) If the seller cannot discriminate, but instead must
charge the same price P1 P2 P to each consumer
group, what will be her profit-maximizing price?
c) Which, if any, consumer group benefits from price discrimination?
d) If instead P1 10 Q1, does either group benefit
from price discrimination?
12.23. A cruise line has space for 500 passengers on
each voyage. There are two market segments: elderly
passengers and younger passengers. The demand curve
for the elderly market segment is Q1 750 4P1. The
demand curve for the younger market segment is Q2
850 2P2. In each equation, Q denotes the number of
526
CHAPTER 12
CAPTURING SURPLUS
passengers on a cruise of a given length and P denotes
the price per day. The marginal cost of serving a passenger of either type is $40 per person per day.
Assuming the cruise line can price discriminate, what is
the profit-maximizing number of passengers of each
type? What is the profit-maximizing price for each type
of passenger?
12.24. An airline has 200 seats in the coach portion of
the cabin of an Airbus A340. It is attempting to determine
how many seats it should sell to business travelers and
how many to vacation travelers on a flight between
Chicago and Dubai that departs on Monday morning,
January 25. It has tentatively decided to sell 150 seats to
business travelers and 50 seats to vacation travelers at
$4,000 and $1,000, respectively. It also knows:
a) To sell an additional seat it sells to business travelers, it
would need to reduce price by $25. To reduce demand by
business travelers by one seat, it would need to increase
price by $25.
b) To reduce demand by one unit among vacation travelers, it would need to increase price by $5. To sell an additional seat to vacation travelers, it would need to reduce
price by $5.
Assuming that the marginal cost of carrying either type
of passenger is zero, is the current allocation of seats
profit maximizing? If not, would you sell more seats to
business travelers or vacation travelers?
12.25. A summer theater has a capacity of 200 seats
for its Saturday evening concerts. The marginal cost of
admitting a spectator is zero up to that capacity. The
theater wants to maximize profits and recognizes that
there are two kinds of customers. It offers discounts to
senior citizens and students, who generally are more
price sensitive than other customers. The demand curve
for tickets by seniors and students is described by P1
16 0.04Q1, where Q1 is the number of discount tickets
sold at a price of P1. The demand schedule for tickets by
customers who do not qualify for a discount is represented by P2 28 0.1Q2, where Q2 is the number of
nondiscount tickets sold at a price of P2. What are the
two prices that would maximize profit for the Saturday
evening concerts?
12.26. A small island near a major city has a beautiful
beach. The company that owns the island sells day passes
for the beach, including travel by ferry to and from the
beach. Because the beach is small, the company does not
want to sell more than 200 excursion tickets per day. The
company knows there are two kinds of visitors: those who
are willing to buy tickets a month in advance and those
who want to buy on the day of the trip. Those willing to
buy in advance are typically more price sensitive. The
demand curve for advance purchase excursion tickets is
described by P1 100 0.2Q1, where Q1 is the number
of advance purchase tickets sold at a price of P1. The demand schedule for tickets by day-of-travel excursions is
represented by P2 200 0.8Q2, where Q2 is the number of tickets sold at a price of P2.
a) Suppose the marginal cost of the ferry trip and use of
beach is 50 per customer. What prices should the firm
charge for its excursion tickets?
b) If the marginal cost were high enough, the firm would
want to sell fewer than 200 tickets. Suppose the marginal
cost of the ferry trip and use of beach is 80 per customer.
What prices should the firm charge for its beach excursion tickets?
12.27. You are the only European firm selling vacation
trips to the North Pole. You know only three customers
are in the market. You offer two services, round trip airfare and a stay at the Polar Bear Hotel. It costs you 300
euros to host a traveler at the Polar Bear and 300 euros
for the airfare. If you do not bundle the services, a customer might buy your airfare but not stay at the hotel. A
customer could also travel to the North Pole in some
other way (by private plane), but still stay at the Polar
Bear. The customers have the following reservation
prices for these services:
Reservation Prices (in euros)
Customer
Airfare
Hotel
1
2
3
100
500
800
800
500
100
a) If you do not bundle the hotel and airfare, what are the
optimal prices PA and PH, and what profits do you earn?
b) If you only sell the hotel and airfare in a bundle, what
is the optimal price of the bundle PB, and what profits do
you earn?
c) If you follow a strategy of mixed bundling, what are
the optimal prices of the separate hotel, the separate airfare, and the bundle (PA, PH, and PB, respectively) and
what profits do you earn?
12.28. You operate the only fast-food restaurant in
town, selling burgers and fries. There are only two customers, one of whom is on the Atkins diet and the other
on the Zone diet, whose willingness to pay for each item
is displayed in the following table. For simplicity, assume
you have zero fixed and marginal costs for each item.
Customers
Atkins dieters
Zone dieters
Burger
Fries
Burger and Fries
$8
$5
$x
$3
$(8 x)
$8
PROBLEMS
a) If x ⫽ 1 and you do not bundle the two products, what
are your profit-maximizing prices PB and PF? Calculate
total surplus under this outcome.
b) Now assume only that x ⬎ 0. Instead, suppose that
you hired an economist who tells you that the
profit-maximizing bundle price (for a burger and fries) is
$8, while if you sold the items individually (and did not
offer a bundle) your profit-maximizing price for fries
would be greater than $3. Using this information, what is
the range of possible values for x?
12.29. Suppose your company produces athletic
footwear. Marketing studies indicate that your own price
elasticity of demand is ⫺3 and that your advertising elasticity of demand is 0.5. You may assume these elasticities
527
to be approximately constant over a wide range of prices
and advertising expenses.
a) By how much should the company mark up price over
marginal cost for its footwear?
b) What should the company’s advertising-to-sales
ratio be?
12.30. The motor home industry consists of a small
number of large firms. In 2003, producers of motor
homes had an average advertising sales ratio of 1.8 percent. Assuming that the price elasticity of demand facing
a typical motor home producer is ⫺4, what is the advertising elasticity of demand facing a typical producer,
under the assumption that each producer has chosen its
price and advertising level to maximize profits?
13
MARKET STRUCTURE
AND COMPETITION
13.1
DESCRIBING AND MEASURING
M A R K E T S T RU C T U R E
Market Structure Metrics
for U.S. Manufacturing Industries
APPLICATION 13.1
13.2
O L I G O P O LY W I T H
HOMOGENEOUS PRODUCTS
Corn Syrup Capacity
Expansion Confirms Cournot
APPLICATION 13.2
13.3
DOMINANT FIRM MARKETS
APPLICATION 13.3
U.S. Steel: The Price of
Dominance
13.4
O L I G O P O LY W I T H H O R I Z O N TA L LY
D I F F E R E N T I AT E D P R O D U C T S
APPLICATION 13.4
APPLICATION 13.5
APPLICATION 13.6
Smartphone Wars
Chunnel versus Ferry
Wireless Number Portability
13.5
MONOPOLISTIC COMPETITION
APPLICATION 13.7
APPLICATION 13.8
Wine or Roses?
When a Good Doctor
Is Hard to Find
APPENDIX
T H E C O U R N OT E Q U I L I B R I U M
AND THE INVERSE ELASTICITY
P R I C I N G RU L E
Is Competition Always the Same? If Not, Why Not?
What brand of cola can you buy on your campus? If you are a student at Rutgers University or the University
of North Carolina, you can buy Pepsi but not Coca-Cola. If you attend the University of Arizona or Villanova
University, you can get Coca-Cola but not Pepsi. Your choice is limited because for nearly 20 years Coke and
528
Pepsi have been competing to sign exclusive distribution deals with colleges throughout the United States.
In 1992, for example, Pepsi paid Penn State $14 million to be the school’s official drink. Under this deal, no
other brand of soda could be sold anywhere on Penn State’s 21 campuses. Not to be outdone, in 1994
Coke paid $28 million to the University of Minnesota to be that university’s sole supplier of soft drinks and
to place signs at sports venues and in campus food service areas. The $28 million was used for scholarships,
student activities, and a women’s hockey team.
The “cola war” between Coke and Pepsi is an example of competition between a few firms whose
fortunes are closely intertwined. Moreover, Coca-Cola and Pepsi sell differentiated products. Although
most people view Coca-Cola and Pepsi Cola as similar products, few consider them identical products.
Indeed, many consumers have long-standing loyalties to either Coke or Pepsi. The desire to develop these
brand loyalties at an early age has led Coke and Pepsi to place such strategic importance on gaining exclusive
access to college campuses.
What forces drive the outcome of competitive battles in markets that have only a few sellers or in
which consumers see products as imperfect substitutes? Neither the theory of perfect competition that we
studied in Chapter 9 nor the theory of monopoly in Chapter 11 applies to the competitive battle between
the two soft drink giants.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Describe the conditions that characterize different types of market structures, including oligopoly
markets, dominant firm markets, and monopolistically competitive markets.
• Find the reaction function that shows how one firm sets its profit-maximizing quantity or price given
the quantity or price of the other firm.
• Sketch the reaction function for a
quantity-setting or price-setting
oligopoly firm.
• Compute the equilibrium in the Cournot
model of oligopoly and illustrate it
graphically.
• Explain how and why the Cournot
equilibrium differs from a Bertrand
equilibrium in a homogeneous products
oligopoly.
• Find the Stackelberg equilibrium and
explain how and why it differs from
the Cournot equilibrium.
• Compute the equilibrium in the
dominant firm model and illustrate it
graphically.
529
530
CHAPTER 13
M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
• Differentiate between horizontal product differentiation and vertical product differentiation.
• Explain how horizontal product differentiation affects the shape of a firm’s demand curve in a differentiated product oligopoly.
• Compute the Bertrand equilibrium in a differentiated product oligopoly and illustrate it graphically.
• Illustrate graphically the short-run and long-run equilibrium in a monopolistically competitive industry.
13.1
Market structures differ on two important dimensions: the number of firms and the
DESCRIBING
AND
MEASURING
MARKET
STRUCTURE
nature of product differentiation.1 Table 13.1 shows how different combinations of
these characteristics give rise to different market structures. Going across the table,
we move from competitive markets, in which there are many sellers, to oligopoly markets, in which there are just a few sellers, to monopoly markets, in which there is just
one seller. Reading down the table, we move from markets in which firms sell identical or nearly identical products to differentiated products markets in which firms sell
products that consumers view as distinctive. The table indicates the economic theory
that applies to each market structure and provides an example to which each of the
theories might apply. (Recall that we studied perfectly competitive markets in
Chapters 9 and 10 and monopoly markets in Chapter 11.)
In this chapter, we will study the four market structures that we have yet to
encounter:
• In homogeneous products oligopoly markets, a small number of firms sell
products that have virtually the same attributes, performance characteristics,
image, and (ultimately) price. For example, in the U.S. glass container industry,
the three largest firms—Owens-Illinois, Saint-Gobain, and Anchor—sell very
similar products and account for 82 percent of U.S. sales of bottles and jars.2 In
the global market for titanium dioxide (an inorganic pigment used to whiten
products such as paint and plastics), several large firms such as DuPont,
Millennium Inorganic, Huntsman, and Tronox sell products that are virtually
identical chemically.
homogeneous products
oligopoly markets
Markets in which a small
number of firms sell products that have virtually the
same attributes, performance
characteristics, image, and
(ultimately) price.
TABLE 13.1
Types of Market Structures
Number of Firms
Product Differentiation
Firms produce identical
products
Firms produce
differentiated products
a
Until 1999.
Many
Perfect competition
(Chapter 9)
Example: fresh-cut
rose market
Monopolistic competition
Example: local physicians
markets
1
2
Few
One Dominant
One
Dominant firm
Monopoly
Example: U.S. glass
container market
Example : U.S.
light bulb
market
(Chapter 11)
Example: Internet
domain name
registrationa
Differentiated products
oligopoly
No applicable
theory
Homogeneous
products oligopoly
Example: breakfast cereal
market
Recall that Chapter 11 introduced and briefly discussed the concept of product differentiation.
“Owens-Illinois,” Wikinvest, http://www.wikinvest.com/stock/Owens-Illinois_(OI) (accessed March 14, 2010).
1 3 . 1 D E S C R I B I N G A N D M E A S U R I N G M A R K E T S T RU C T U R E
531
• In dominant firm markets, one firm possesses a large share of the market but
competes against numerous small firms, each offering identical products. The
U.S. market for lightbulbs is a good example of a dominant firm market: many
small firms, including private-label manufacturers, compete in this market, but
General Electric holds a dominant market share, accounting for over 50 percent
of sales in the U.S. market.
• In differentiated products oligopoly markets, a small number of firms sell
products that are substitutes for each other but also differ from each other
in significant ways, including attributes, performance, packaging, and image.
Examples include the U.S. market for soft drinks where Coke and Pepsi are
archrivals, the U.S. market for breakfast cereals in which Kellogg, General Mills,
Post, and Quaker Oats sell more than 85 percent of all cereal purchased in the
United States, and the market for beer in Japan in which four firms, Asahi, Kirin,
Sapporo, and Suntory, account for nearly 100 percent of Japanese beer sales.
• Monopolistic competition refers to a market in which many firms produce
differentiated products that are sold to many buyers. Local markets for DVD
rentals, dry cleaning, and physician services are good examples of monopolistically competitive markets.
dominant firm markets
Economists use several different quantitative metrics to describe the structure of
a market. One common metric is the four-firm concentration ratio (or 4CR for short).
This metric calculates the share of industry sales revenue accounted for by the four
firms with the largest sales revenue in the industry.3 An industry whose sales are entirely due to just four firms would have a 4CR equal to 100. An industry in which the
four largest firms accounted for 3 percent, 2 percent, 2 percent, and 1 percent of sales,
respectively, would have a 4CR equal to 8 (3 ⫹ 2 ⫹ 2 ⫹ 1).
Another metric used to characterize market structure is the HerfindahlHirschman Index (or HHI for short). This index takes the market share of each firm
in the industry, squares it, and sums the squared market shares across all firms in the
industry. (A firm’s market share is its sales revenue divided by total industry sales; that
is, it is the share of industry sales accounted for by that firm.) In a monopoly, where a
single firm accounts for 100 percent of industry sales, the HHI 1002 10,000. This
is the maximum possible value of the HHI. In a fragmented industry in which, say,
1,000 identical firms each have 1/1,000 percent of industry sales, the HHI would
equal (1/1,000)2 added up 1,000 times or 1,000(1/1,000)2 0.001. As the number of
firms grows and their market shares shrink to 0, the HHI would approach 0. Thus,
the HHI takes on values between 0 and 10,000.4
We would expect that industries corresponding to the market structures described
in Table 13.1 would have broadly different 4CRs and HHIs. Perfectly competitive and
monopolistically competitive industries would be expected to have very low 4CRs and
HHIs. By contrast, monopoly and dominant firm markets would have quite large 4CRs
and HHIs (in fact, as just noted, a monopoly industry would have an HHI of 10,000,
and its 4CR would equal 100), while oligopoly industries (with either homogeneous or
differentiated products) would have intermediate 4CRs and HHIs.
3
The 4CR might also be based on other measures of firm size such as production output, capacity, or
employees.
4
In practice, the HHI is often computed for a subset of firms in the industry. For example, in Table 13.2,
the HHI is computed using the 50 largest firms. Including more firms with very small market shares
would not substantially change the value of the HHI.
Markets in which one firm
possesses a large share of
the market but competes
against numerous small
firms, each offering identical
products.
differentiated products
oligopoly markets
Markets in which a small
number of firms sell products that are substitutes for
each other but also differ
from each other in significant
ways, including attributes,
performance, packaging,
and image.
monopolistic competition Competition in a
market in which many firms
produce differentiated
products that are sold to
many buyers.
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CHAPTER 13
A P P L I C A T I O N
M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
13.1
Market Structure Metrics for U.S.
Manufacturing Industries
Table 13.2 shows the 4CR and HHI for a variety of U.S.
manufacturing industries for the year 2002. Some
industries, such as beer brewing, breakfast cereals,
guided missile and space vehicle manufacturing, and
glass containers are highly concentrated; that is, their
4CR is large. Such industries are dominated by a few large
firms. Beer brewing, breakfast cereals, and guided missile
and space vehicle manufacturing are good examples of
differentiated product oligopolies; they consist of few
firms that produce similar but not identical products. The
glass container industry is, as noted above, a good example of a homogeneous product oligopoly; firms in the industry produce largely similar products, and the three
largest firms account for over 80 percent of industry sales.
Other industries, such as curtain and drapery mills
and fabricated structural metal manufacturing, are
more fragmented. These industries contain thousands
of U.S. firms producing nearly identical products, and
each provides a good approximation to a perfectly
competitive industry.
Table 13.2 indicates that, by and large, the 4CR and
the HHI are highly correlated: When one metric is large,
the other is as well. However, occasionally one sees
some differences. For example, the electric lamp bulb
and parts manufacturing industry (whose principal
product is lightbulbs) and the glass container industry
have approximately the same 4CR, but the HHI for the
lightbulb industry is almost 200 points higher than the
HHI for the glass container industry. This is because, although the top four firms in the two industries account
for about the same share of industry sales, the lightbulb
industry consists of about 30 more smaller firms, and the
largest firm in that industry, GE Lighting, has a larger
market share than the largest firm in the glass container
industry, Owens-Illinois. Thus, the structure of the lightbulb industry is more asymmetric than that of the glass
container industry, a feature captured by the lightbulb
industry’s larger HHI. An advantage that the HHI has
over the 4CR as a measure of market structure is that it
is sensitive to such asymmetries among firms.
When evaluating market structure metrics, it is important to recognize the geographic scope of an industry.
An industry such as cement manufacturing is primarily
TABLE 13.2 Four-firm Concentration Ratios and Herfindahl-Hirschman Indices for Selected U.S.
Manufacturing Firms, 2002
Industry
Guided missiles and space vehicles
Cigarette manufacturing
Beer breweries
Electric lamp bulb and parts manufacturing
Glass container manufacturing
Primary aluminum manufacturing
Breakfast cereal manufacturing
Dog and cat food manufacturing
Ice manufacturing
Automatic vending machine manufacturing
Cement manufacturing
Curtain and drapery mills
Fabricated structural metal manufacturing
NAICS Codea
Total Number
of Companies
4CR
HHI
336414
312221
312120
335110
327213
331312
311230
311111
312113
333311
327310
314121
332312
13
15
349
57
22
26
45
176
425
106
131
1,778
3,569
96.0
95.3
90.8
88.5
88.3
85.3
78.4
64.2
42.9
42.3
38.7
16.1
8.9
nab
na
na
2,757.6
2,582.1
na
2,521.3
1,845.5
763.1
679.0
568.5
111.0
39.5
a
NAICS, the North American Industry Classification System, is the system the U.S. Census Bureau
uses to classify industries.
b
For industries with only a few firms, the Census Bureau does not publish the HHI because of
confidentiality concerns about disclosing data on the sales of individual companies.
Source: U.S. Census Bureau, Concentration Ratios: 2002, http://www.census.gov/epcd/www/
concentration.html (accessed March 10, 2010).
533
1 3 . 2 O L I G O P O LY W I T H H O M O G E N E O U S P R O D U C T S
regional. Although it is not highly concentrated nationally, in state or regional markets, there may be only two
or three large firms. By contrast, an industry such as
primary aluminum production is global. Although it
appears to be relatively concentrated in the United
States, U.S. firms compete with firms located all over the
world. On a global basis, the industry is more fragmented
and may even be approximately perfectly competitive.
I
n perfectly competitive and monopoly markets, firms do not have to worry about
their rivals. In a monopoly market, the monopolist has no rivals. In a perfectly competitive market, each seller is so small that it has an imperceptible competitive impact
on rival producers. A central feature of oligopoly markets, by contrast, is competitive
interdependence: The decisions of every firm significantly affect the profits of competitors. For example, in the world market for memory chips, Samsung recognizes
that the profit it gets from selling DRAM chips depends, in part, on the volume of
chips that key competitors such as NEC and Lucky Goldstar will produce. If Samsung’s
competitors increase output, the market price for DRAM chips is likely to fall; if they
decrease output, the market price will rise. In planning how many chips to produce
within its current facilities, or in deciding whether to expand or build new facilities,
Samsung’s management must forecast how much output NEC, Lucky Goldstar, and
other large semiconductor competitors are likely to produce. A central question of oligopoly theory, therefore, is how the close interdependence among firms in the market affects their behavior. Answering this question helps us understand the unique impact that an oligopoly market structure can have on prices, output levels, and profits.
13.2
O L I G O P O LY
WITH
HOMOGENEOUS
PRODUCTS
T H E C O U R N OT M O D E L O F O L I G O P O LY
Microeconomics offers several different models of oligopoly, based on different assumptions about how oligopolists might interact. Augustin Cournot developed the
first theory of oligopoly in 1838 in his book Researches into the Mathematical Principles
of the Theory of Wealth.5 Although Cournot’s model of oligopoly was part of a broader
mathematical treatment of microeconomics, including demand, monopoly, and taxes,
his theory of oligopoly was the most original part of his book and has had the greatest impact on the field of economics.
Profit Maximization by Cournot Firms
The Cournot model pertains to a homogeneous products oligopoly. Cournot initially
considered a duopoly market: a market in which there are just two firms. In Cournot’s
duopoly, the two firms produced mineral water. To give Cournot’s theory a more
modern feel, let’s imagine that the firms are Samsung and Lucky Goldstar (LG) and
that the product is DRAM chips.
Suppose that Samsung’s and LG’s DRAMs are identical and that their marginal
costs are also identical, so both firms will charge the same price. The only decision
each firm needs to make is how much to produce. The firms select their output
simultaneously, noncooperatively (without colluding with each other), and with no
5
A. Cournot, “On the Competition of Producers,” Chapter 7 in Researches into the Mathematical Principles
of the Theory of Wealth, translated by N. T. Bacon (New York: Macmillan, 1897).
duopoly market A
market in which there are
just two firms.
CHAPTER 13
residual demand curve
Price (dollars per unit)
In a Cournot model, the
curve that traces out the
relationship between the
market price and a firm’s
quantity when rival firms
hold their outputs fixed.
M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
knowledge of each other’s plans (without spying on each other). Once both firms
select their output, the market price instantly adjusts to clear the market. That is, given
the firms’ output choices, the market price becomes the price at which consumers are
willing to buy the firms’ combined output.
Each firm’s output choice depends on the market price, but the market price depends on the combined output of the two firms—that is, the market price isn’t known
until both firms have made their output choice. Therefore, each firm will make the
output choice that maximizes its profit based on its expectation of the other firm’s output choice. Thus, Samsung will choose the level of production that maximizes its
profits, given what it thinks LG’s output will be, and LG will choose the level of production that maximizes its profits, given what output it thinks Samsung will produce.
In the Cournot model, firms thus act as quantity takers.
Figure 13.1(a) shows Samsung’s output-choice problem. Suppose that Samsung
expects LG to produce 50 units of output. Then, the relationship between the market
price and Samsung’s output is given by the residual demand curve D50. A residual
demand curve traces out the relationship between the market price and a firm’s quantity when the other firm sells a fixed amount of output (50 units, in this case). The
residual demand curve D50 is the market demand curve (DM) shifted leftward by an
amount equal to LG’s output of 50. This ensures that when Samsung’s output is added
to LG’s output of 50, the price along the residual demand curve D50 equals the price
along the market demand curve DM when we combine the two firms’ outputs. For example, when LG produces 50 and Samsung produces 30, the price along the residual
demand curve is $20, which is also the price along the market demand curve DM when
total output equals 80. MR50 is the marginal revenue curve associated with D50. It
bears the same relationship to the residual demand curve that a monopolist’s marginal
revenue curve bears to a market demand curve.
MR50
50 units
$20
$10
0
D50
20 30
DM
80
Quantity (units per year)
(a) Samsung’s profit-maximization problem
when LG produces 50
MC
Price (dollars per unit)
534
20 units
$20
$10
0
MR20
D20
DM
35
Quantity (units per year)
(b) Samsung’s profit-maximization problem
when LG produces 20
FIGURE 13.1 Price Determination and Profit Maximization in the Cournot Model
Panel (a) shows that when Samsung produces 30 units and LG produces 50, the market price
will be $20. When LG produces 50 units, Samsung’s residual demand curve is D50, which is the
market demand curve shifted leftward by 50 units. The residual demand curve traces out the
quantity-price combinations that are available to Samsung when LG’s output is 50 units. Facing
this residual demand curve, Samsung maximizes its profits by producing 20 units, the point at
which its marginal revenue, MR50, equals its marginal cost, MC. This output is Samsung’s best
response when LG produces 50 units. Panel (b) shows that when LG produces 20 units, Samsung
faces residual demand and marginal revenue curves D20 and MR20, respectively, and maximizes
profit by producing 35 units, where MR20 MC.
MC
1 3 . 2 O L I G O P O LY W I T H H O M O G E N E O U S P R O D U C T S
535
Q2 (LG’ output, units per year)
90
A
50
45
E
30
B
20
RS
0
0
20
30 35
45
Q1 (Samsung’s output, units per year)
RLG
90
FIGURE 13.2 Cournot Reaction Functions
and Equilibrium
RS is Samsung’s reaction function. RLG is LG’s
reaction function. Point E, where the two
reaction functions intersect, is the Cournot
equilibrium. Points A and B on RS represent
the best responses for Samsung if LG produces
20 units and 50 units, respectively; these points
correspond to the profit-maximization solutions
shown in Figure 13.1.
Samsung acts as a monopolist relative to its residual demand curve when it chooses
its output. It thus equates MR50 to its marginal cost MC (which is assumed to be constant at $10 per unit). This occurs at an output of 20 units. An output of 20 units is
thus Samsung’s best response to an output of 50 units from LG. A Cournot firm’s
best response to a particular level of output by rival firms is the firm’s profit-maximizing
choice of output given the rival’s output. Figure 13.1(b) shows that when LG’s output
is 20 units, Samsung’s best response is to produce 35 units.
For every possible output that LG might choose, we could determine Samsung’s
profit-maximizing output as we did in Figure 13.1. The curve RS in Figure 13.2 summarizes Samsung’s profit-maximizing output choices. The curve RS is a reaction
function. It tells us a firm’s best response (i.e., profit-maximizing output choice) to
the output level of a rival firm. Figure 13.2 also graphs LG’s reaction function RLG.6
Note that both reaction functions are downward sloping. Thus, each firm’s profitmaximizing output choice becomes smaller as its rival produces more output.
best response A firm’s
profit-maximizing choice of
output given the level of
output by rival firms.
reaction function A
graph that shows a firm’s
best response (i.e., profitmaximizing choice of output or price) for each possible action of a rival firm.
Equilibrium in a Cournot Market
Under perfect competition, a key feature of the market equilibrium is that no firm has
an incentive to deviate from its profit-maximizing decision once the market equilibrium has been attained. The same is true of an equilibrium in a Cournot market: At a
Cournot equilibrium, each firm’s output is a best response to the other firm’s output
(i.e., in equilibrium, each firm is doing as well as it can given the other firm’s output).
Thus, neither firm has any after-the-fact reason to regret its output choice.7
6
If the firms are identical, why do their reaction functions appear different? The reason is that, in
Figure 13.2, the horizontal axis represents Samsung’s output and the vertical axis represents LG’s output.
Plotting both curves on the same graph makes one look like the inverse of the other. Algebraically, the
two reaction functions are identical (as is shown in Learning-By-Doing Exercise 13.1).
7
In Chapter 14, you will see that the Cournot equilibrium is a particular example of what is called a Nash
equilibrium. For this reason, some textbooks refer to the Cournot equilibrium as the Cournot-Nash
equilibrium or the Nash equilibrium in quantities.
Cournot equilibrium
An equilibrium in an oligopoly market in which
each firm chooses a profitmaximizing output given
the output chosen by
other firms.
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CHAPTER 13
M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
In Figure 13.2, the Cournot equilibrium occurs at point E, where the two reaction functions intersect—that is, when each firm produces 30 units. We know that this
is the equilibrium because we see from RS that when LG produces 30 units, Samsung’s
best response is to produce 30 units, and we see from RLG that when Samsung produces 30 units, LG’s best response is to produce 30 units, and as noted above, neither
firm has any regret about its output choice.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 3 . 1
D
Computing a Cournot Equilibrium
The market demand curve DM in Figure 13.1
is given by P 100 ⫺ Q1 ⫺ Q2, where Q1 is the amount
of output Samsung produces and Q2 is LG’s level of
output. The marginal cost of each firm is $10.
Problem
(a) Given this market demand curve, what is Samsung’s
profit-maximizing quantity when LG produces 50 units?
(b) What is Samsung’s profit-maximizing output when
LG produces an arbitrary output Q2 (i.e., what is the
equation of Samsung’s reaction function)?
(c) Compute the Cournot equilibrium quantities and
price in this market.
Solution
(a) We can compute Samsung’s best response using concepts from monopoly theory in Chapter 11. When LG
produces Q2 50, Samsung’s residual demand curve
is given by P 100 ⫺ Q1 ⫺ 50 50 ⫺ Q1. This is a linear demand curve, so the associated marginal revenue
curve (MR) is MR 50 ⫺ 2Q1. Equating this marginal
revenue to Samsung’s marginal cost yields 50 ⫺ 2Q1 10,
or Q1 20.
(b) Samsung’s residual demand curve is given by
P (100 ⫺ Q2) ⫺ Q1, where the parentheses highlight
the terms that Samsung views as fixed. This linear residual demand curve has a vertical intercept of (100 ⫺ Q2)
and a slope of ⫺1. As we learned in Chapter 11, the corresponding marginal revenue curve has the same vertical
intercept and twice the slope, or MR (100 ⫺ Q2) ⫺ 2Q1.
Equating marginal revenue to marginal cost yields
Samsung’s reaction function: (100 ⫺ Q2) ⫺ 2Q1 10, or
Q1 45 ⫺ Q2/2. (Using the same logic, we could compute LG’s reaction function as Q2 45 ⫺ Q1/2.)
(c) The Cournot equilibrium occurs where the two
reaction functions intersect. This corresponds to the pair
of outputs that simultaneously solve the two firm’s reaction functions (you should verify that the solution to this
system of equations is Q1 Q2 30). We find the equilibrium market price P* by substituting these quantities
into the market demand curve: P* 100 ⫺ 30 ⫺ 30 40.
Similar Problems:
13.4, 13.5, 13.6, 13.7, 13.8,
13.14, 13.15, 13.16, 13.17
How Do Firms Achieve the Cournot Equilibrium?
The Cournot theory is a static model of oligopoly: It does not explain how the firms
arrive at the output choices corresponding to the Cournot equilibrium.
Do the two firms have to be omniscient? Perhaps not. Consider Figure 13.3,
which illustrates how Samsung’s managers might reason:
Putting ourselves in LG’s shoes, we see that LG would never produce a quantity greater
than 45, because no matter what output we choose, a quantity greater than 45 never
maximizes LG’s profits. We can see this because LG’s reaction function RLG does not
“extend” above Q2 45.8
8
In the language of game theory that we will introduce in Chapter 14, we say that quantities greater than
Q2 45 are dominated strategies.
537
1 3 . 2 O L I G O P O LY W I T H H O M O G E N E O U S P R O D U C T S
FIGURE 13.3 How Firms Achieve a
Cournot Equilibrium
Samsung concludes that LG will produce
fewer than 45 units. This, in turn, induces Samsung to produce at least 22.5
units. Samsung reasons that LG will
figure this out and thus concludes that
LG will produce fewer than 33.75 units.
This, in turn, induces Samsung to produce at least 28.125 units. This thought
process ends with Samsung concluding
that LG will produce 30 units, leading
Samsung to produce 30 units. If LG goes
through a parallel thought process, both
firms will produce 30 units.
Q2 (LG's output, units per year)
90
RS (Samsung's reaction function)
45
Cournot equilibrium
33.75
30
E
RLG (LG's reaction function)
0
22.5 30
45
28.125
Q1 (Samsung's output, units per year)
If they are clever, Samsung’s managers would then conclude:
Given that LG will not produce more than 45, we should produce at least 22.5. Why?
Because we see from RS that any quantity less than 22.5 could never be profit maximizing for us given that LG will never produce more than 45.
But Samsung can go even deeper:
We should assume that LG has reasoned the same way we have—after all, they are just as
clever as we are. But if LG realizes that we will produce at least 22.5, LG would never
produce more than 33.75, as we see from RLG.
But, of course, Samsung’s managers can reason more deeply still:
Given that LG will produce no more than 33.75, we should produce at least 28.125.
Why? Because we see from RS that any quantity smaller than 28.125 could never be
profit maximizing for us given that LG will never produce more than 33.75.
Of course, you see where this is headed. As Samsung’s managers think through
LGs and their own profit-maximization problems, they will keep eliminating output
choices until they reach the Cournot equilibrium of 30 units for each firm.9 To be
sure, this is complicated reasoning, but it is no more complicated than what a smart
chess or bridge player uses against equally clever rivals. Seen this way, the Cournot
equilibrium is a natural outcome when both firms fully understand their interdependence and have confidence in each other’s rationality.
9
In Chapter 14, we will learn that in game theory, this approach to solving a game is called elimination of
dominated strategies.
90
538
CHAPTER 13
A P P L I C A T I O N
M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
13.2
Corn Syrup Capacity Expansion
Confirms Cournot
Michael Porter and Michael Spence’s study of the corn
wet-milling industry is an application of the Cournot
model to a real-world market.10 Firms in the corn wetmilling industry convert corn into corn starch and corn
syrup. The industry had been a stable oligopoly until
the early 1970s, but in 1972, a major development occurred: The production of high-fructose corn syrup
(HFCS) became commercially viable. HFCS can be used
instead of sugar to sweeten products, such as soft
drinks. With sugar prices expected to rise, a significant
market for HFCS beckoned. Firms in the corn wetmilling industry had to decide whether to add capacity to meet the expected demand.
Porter and Spence studied this capacity expansion
process by constructing a model of competitive behavior
based on an in-depth study of the 11 major competitors in the industry. They then used this model to cal-
TABLE 13.3
culate a Cournot equilibrium for the corn wet-milling
industry. In this equilibrium, each firm’s capacity
choice was an optimal response to its expectations
about rival firms’ capacity choices, and the total industry capacity expansion that resulted from these
optimal choices matched the expectations on which
firms based their decisions.
Based on their analysis, Porter and Spence concluded that at an industry equilibrium, a moderate
amount of additional capacity would be added to the
industry as a result of the commercialization of HFCS.
Table 13.3 shows the specific predictions of their
model compared with the pattern of capacity expansion that actually occurred.
Though not perfect, Porter and Spence’s calculated equilibrium was close to the actual capacity
expansion in the industry, particularly in 1973 and 1974.
Their research suggests that the Cournot model, when
adapted to specific industry conditions, can accurately
describe the dynamics of capacity expansion in a
homogeneous-product oligopoly.
Capacity Expansion in the Corn Wet-Milling Industry
Actual capacity expansiona
Predicted capacity expansion
1973
1974
1975
0.6
0.6
1.0
1.5
1.4
3.5
1976
6.2
3.5
Total
9.2
9.1
a
Billions of pounds.
The Cournot Equilibrium versus Monopoly Equilibrium
and Perfectly Competitive Equilibrium
In the Samsung–LG example above, the Cournot equilibrium price of $40 exceeds
each firm’s marginal cost of $10. Therefore, the Cournot equilibrium does not correspond to the perfectly competitive equilibrium. In general, then, Cournot firms exhibit market power.
But that does not imply that they can attain the monopoly or collusive equilibrium. Recall that industry output at the Cournot equilibrium in our example is
60 units, with each firm producing 30 units, as shown in Figure 13.4 (point E). This
output does not maximize industry profit. The monopoly outcome in this market occurs where marginal revenue equals marginal cost, which occurs at a market output of
45 units, and the corresponding monopoly price is $55.11 If Samsung and LG were to
10
M. Porter and A. M. Spence, “The Capacity Expansion Decision in a Growing Oligopoly: The Case of
Corn Wet Milling,” in J. J. McCall, ed., The Economics of Information and Uncertainty (Chicago: University
of Chicago Press, 1982), pp. 259–316.
11
You should verify this for yourself.
539
1 3 . 2 O L I G O P O LY W I T H H O M O G E N E O U S P R O D U C T S
FIGURE 13.4
Cournot Equilibrium
versus Monopoly Equilibrium
If Samsung and LG behave as a profitmaximizing cartel (monopoly) they will
produce a total of 45 units. Splitting
this equally gives each an output of
22.5. The cartel or monopoly equilibrium, point M, thus differs from the
Cournot equilibrium, point E.
Q2 (LG's output, units per year)
90
RS (Samsung's reaction function)
Profit-maximizing cartel (monopoly)
45
30
22.5
Cournot equilibrium
E
M
RLG (LG's reaction function)
0
22.5 30
45
90
Q1 (Samsung's output, units per year)
act as a profit-maximizing cartel, they would charge this price and split the market
evenly, each producing a quantity of 22.5 (point M ). By independently maximizing
their own profits, firms produce more total output than they would if they collusively
maximized industry profits. This is an important characteristic of oligopolistic industries: The pursuit of individual self-interest does not typically maximize the wellbeing of the industry as a whole.
The inability of the two firms to attain the collusive outcome occurs for the following reason. When one firm, say Samsung, expands its output, it reduces the market price and thus lowers LG’s sales revenue. Samsung does not care about lowering
its rival’s revenue because it is seeking to maximize its own profit, not total industry
profit. Thus, Samsung expands its production volume more aggressively than it would
if it were seeking to maximize industry profit. If both firms behave this way, the market price must be less than the monopoly price.
The smaller a firm’s share of industry sales is, the greater the divergence will be
between its private gain and the revenue destruction it causes by expanding its output.
This suggests that as the number of firms in the industry increases, the Cournot equilibrium diverges further from the monopoly outcome. Table 13.4 illustrates this point
TABLE 13.4
Cournot Equilibrium for Various Numbers of Firms
Number of Firms
Price
Market Quantity
Per-Firm Profit
Total Profit
1 (monopoly)
2
3
5
10
100
(perfect competition)
$55.0
$40.0
$32.5
$25.0
$18.2
$10.9
$10.0
45.0
60.0
67.5
75.0
81.8
89.1
90.0
$2,025
$ 900
$ 506
$ 225
$ 67
$
1
0
$2,025
$1,800
$ 1,519
$ 1,125
$ 669
$ 79
0
540
CHAPTER 13
M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
by showing equilibrium prices, outputs, and profits in a Cournot oligopoly with the
same demand and cost curves as in the Samsung–LG example.12 The equilibrium
price and profit per firm decline as the number of firms increases. In the extreme
case of a market with an infinite number of firms, per-firm and industry profits are
both zero.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 3 . 2
D
Computing the Cournot Equilibrium for Two or More Firms
with Linear Demand
Suppose that a market consists of N identical firms, that
the market demand curve is P a ⫺ bQ, and that each
firm’s marginal cost is c.
To find the Cournot equilibrium quantity per firm, we
solve this equation for Q1 (which we can rewrite as Q*,
representing the output of any arbitrary individual firm):
Problem
Q*
(a) What is the Cournot equilibrium quantity per firm?
(b) What are the equilibrium market quantity and price?
(b) Market quantity is N times an individual firm’s
quantity:
Solution
Q
(a) The residual demand curve for any one firm (call it
Firm 1) is P (a ⫺ bX) ⫺ bQ1, where X denotes the
combined output of the other N ⫺ 1 firms. Thus, Firm
1’s marginal revenue curve is MR (a ⫺ bX ) ⫺ 2bQ1. To
find Firm 1’s reaction function, we equate its marginal
revenue to marginal cost: (a ⫺ bX ) ⫺ 2bQ1 c, or
Q1
a⫺c
1
⫺ X
2b
2
Since the firms are identical, each will produce the same
amount. Thus, the value of X is N ⫺ 1 times Q1, so
Q1
a⫺c
1
⫺ [(N ⫺ 1)Q1 ]
2b
2
a⫺c
1
a
b
b
(N ⫹ 1)
a⫺c
N
a
b
b
(N ⫹ 1)
To find the equilibrium market price, we substitute this
value for Q into the equation for the demand curve:
Pa⫺b
a⫺c
a
N
N
a
b
⫹
c
b
N⫹1
N⫹1
(N ⫹ 1)
As N gets bigger, N/(N ⫹ 1) gets closer to 1, which
means that the Cournot equilibrium output approaches
the perfectly competitive output and the Cournot equilibrium price approaches the marginal cost c.
Similar Problems:
13.9, 13.10, 13.13
In Learning-By-Doing Exercise 13.1 and in other Learning-By-Doing exercises
in previous chapters, you saw how to compute the equilibrium quantity for an individual firm and the market equilibrium price and quantity in the case of a monopoly,
a Cournot duopoly, and perfect competition. If we perform those computations for
the scenario in Learning-By-Doing Exercise 13.2, we get the results summarized in
Table 13.5. As you can see from the table, these other three structures can be regarded
as special cases of the N-firm Cournot oligopoly, where N 1 (monopoly), N 2
(Cournot duopoly), and N q (perfect competition).
12
In Learning-By-Doing Exercise 13.2, you will learn how to calculate a Cournot equilibrium with more
than two firms.
1 3 . 2 O L I G O P O LY W I T H H O M O G E N E O U S P R O D U C T S
TABLE 13.5
Comparison of Equilibria
Market Structure
Price
Market Quantity
Per-Firm Quantity
Monopoly
1
1
a c
2
2
1
2
a c
b
1
2
a c
b
Cournot duopoly
1
2
a c
3
3
2
3
a c
b
1
3
a c
b
N-firm Cournot oligopoly
1
N
a
c
N 1
N 1
N
N 1
Perfect competition
c
a c
b
a c
b
1
N 1
a c
b
Virtually 0
Cournot Equilibrium and the IEPR
In Chapters 11 and 12, we saw how a monopolist’s profit-maximization condition
could be expressed as an inverse elasticity pricing rule (IEPR):
P * MC
1
⑀Q,P
P*
The left-hand side of this equation (the difference between the monopolist’s price and
marginal cost expressed as a percentage of price), which we referred to in Chapter 11
as the Lerner Index, is also termed the percentage contribution margin (PCM). Thus, the
equation says that the monopolist maximizes profit by setting its PCM equal to minus
one over the price elasticity of market demand. A modified version of this IEPR applies to the individual firms in an N-firm Cournot oligopoly where all the firms are
identical and their marginal cost is MC, in which case the PCM for each firm at the
Cournot equilibrium is
P * MC
1
1
⫻
⑀Q,P
P*
N
This modified IEPR provides a compelling link between market structure and
how firms perform in an oligopoly market. It implies that the more firms there are in
the industry, the smaller their percentage contribution margin will be. (This mirrors
the relationship shown in Table 13.4.) Recall from Chapter 11 that the Lerner Index
(or PCM) is commonly used to measure market power. The Cournot model thus implies that market power will go down as more firms compete in the market.
T H E B E R T R A N D M O D E L O F O L I G O P O LY
In the Cournot model, each firm selects a quantity to produce, and the resulting total
output determines the market price. Alternatively, one might imagine a market in
which each firm selects a price and stands ready to meet all the demand for its product at that price. This model of competition was first articulated by French mathematician Joseph Bertrand in 1883 in a review of Cournot’s book.13 Bertrand criticized
13
J. Bertrand, book reviews of Walras’s Theorie Mathematique de la Richese Sociale and Cournot’s Researches
sur les Principes Mathematiques de la Theorie des Richesses, reprinted as Chapter 2 in A. F. Daughety, ed.,
Cournot Oligopoly: Characterization and Applications (Cambridge, UK : Cambridge University Press, 1988).
541
Bertrand equilibrium
An equilibrium in which
each firm chooses a profitmaximizing price given the
price set by other firms.
CHAPTER 13
M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
Cournot’s assumption of quantity-taking behavior and argued that a more plausible
model of oligopoly was one in which each firm chose a price, taking as given the prices
of other firms. Once firms choose their prices, they will then adjust their production
to satisfy all of the demand that comes their way.14 If firms produce identical products, the firm that sets the lowest price captures the entire market demand, and the
other firms sell nothing.
To illustrate Bertrand price competition, let’s return to our Samsung–LG example. A Bertrand equilibrium occurs when each firm chooses a profit-maximizing
price, given the price set by the other firm. Recall from Figure 13.2 that at the
Cournot equilibrium each firm produced 30 units and sold them at a price of $40
(point E in Figure 13.5). Is this also the Bertrand equilibrium? The answer is no. To
see why, consider Samsung’s pricing problem in Figure 13.5. If Samsung takes LG’s
price as fixed at $40, Samsung’s demand curve DS is a broken line that coincides with
the market demand curve DM at prices below $40 and with the vertical axis at prices
above $40. If Samsung slightly undercut LG’s price by charging $39, it would steal all
of LG’s business and would also stimulate one unit of additional demand. Thus,
Samsung more than compensates for its lower price by more than doubling its volume. As a result, Samsung’s profit increases by area B (the gain from the additional
volume of output it sells) minus area A (the reduction in profit due to the fact that it
could have sold 30 units at the higher price of $40).
But note that prices of $39 for Samsung and $40 for LG cannot be an equilibrium
either because LG would gain by undercutting Samsung’s price. Indeed, as long as both
firms set prices that exceed their common marginal cost of $10, one firm can always
increase its profits by slightly undercutting its competitor. This implies that the only
possible equilibrium in the Bertrand model is achieved when each firm sets a price
equal to its marginal cost of $10. At this point, neither firm can do better by changing its
FIGURE 13.5
Bertrand Price Competition
If LG’s price is $40, Samsung’s demand curve
is the broken line DS. By setting a price of
$39, Samsung can increase its profit by area
B minus area A. This tells us that each firm
charging a price of $40, with each producing
30 units, is not the Bertrand equilibrium.
14
Samsung's price (dollars per unit)
542
$100
DM
Samsung's demand curve
DS
$40
$39
E
A
DS
B
MC
$10
0
30
60 61
90 100
Quantity (units per year)
Bertrand writes: “By treating (the quantities) as independent variables, (Cournot) assumes that the one
quantity happening to change by the will of the owner, the other would remain constant. The contrary is
obviously true.” Ibid., p. 77.
1 3 . 2 O L I G O P O LY W I T H H O M O G E N E O U S P R O D U C T S
price. If either firm lowers price further, it will lose money on each unit it sells. If
either firm raises price, it will sell nothing. Thus, in the Bertrand equilibrium, P
MC $10, and the resulting market demand is 90 units. Thus, unlike the Cournot
equilibrium with two firms, the Bertrand equilibrium with two firms results in the
same outcome as a perfectly competitive market with a large number of firms.
W H Y A R E T H E C O U R N OT A N D B E R T R A N D
EQUILIBRIA DIFFERENT?
The Cournot and Bertrand models make dramatically different predictions about
the quantities, prices, and profits that will arise under oligopolistic competition. In
the Cournot model, the equilibrium price is generally above marginal cost, and the
Cournot equilibrium approaches the perfectly competitive equilibrium only as the
number of competitors in the market becomes large. In the Bertrand model, by contrast,
competition between even two firms is enough to replicate the perfectly competitive
equilibrium. Why are these models so different, and how does each apply to the real
world?
One difference is that Cournot and Bertrand competition can be viewed as taking
place over different time frames. The Cournot model can be viewed as a long-run
capacity competition. From this perspective, firms first choose capacities and then
compete as price setters given these capacities. The result of this “two-stage” competition (first choose capacities and then choose prices) can be shown to be identical to
the Cournot equilibrium in quantities.15 In contrast, the Bertrand model can be
thought of as short-run price competition when both firms have more than enough
capacity to satisfy market demand at any price greater than or equal to marginal cost.
Another difference between the Cournot and Bertrand models is that they make
different assumptions about how a firm expects its rivals to react to its competitive
moves. The Cournot firm takes its competitors’ outputs as given and assumes that its
competitors will instantly match any price change the firm makes so that they can
keep their sales volumes constant. This expectation might make sense in industries
such as mining or chemical processing, in which firms typically can adjust their prices
more quickly than their rates of production. Because a firm cannot expect to “steal”
customers from its rivals by lowering its price, Cournot competitors behave less aggressively than Bertrand competitors. Thus, the Cournot equilibrium outcome, while
not the monopoly one, nevertheless results in positive profits and a price that exceeds
marginal cost.
By contrast, a Bertrand competitor believes that it can lure customers from its
rivals by small cuts in price, and it knows that it has sufficient production capacity to
be able to satisfy this additional demand. These beliefs might make sense in a market
such as the U.S. airline industry in the early 2000s, which had significant excess capacity. Many airlines at that time believed that they would fly their planes virtually
empty unless they cut their prices below their competitors. (Of course, if all firms in
the market think this way, each one will attempt to steal business from its competitors
through price cutting, with the result that prices drop to marginal cost.)
15
The idea that the Cournot equilibrium can (under some circumstances) emerge as the outcome of
a “two-stage game” in which firms first choose capacities and then choose prices is due to D. Kreps and
J. Scheinkman, “Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes,” Bell
Journal of Economics 14 (1983): 326–337.
543
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M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
T H E S TAC K E L B E R G M O D E L O F O L I G O P O LY
Stackelberg model of
oligopoly A situation in
which one firm acts as a
quantity leader, choosing its
quantity first, with all other
firms acting as followers.
In the Cournot model of quantity setting, both firms are assumed to choose their
quantities simultaneously. However, in some situations, it might be more natural to
assume that one firm chooses its quantity before the other firms make their choices.
This assumption may be especially natural if we think of the quantities as levels of production capacity. In many oligopolistic industries, capacity expansion decisions tend
to occur sequentially rather than simultaneously. For example, in the U.S. turbine
generator industry of the 1950s and 1960s, Westinghouse and Allis-Chalmers generally undertook major capacity expansions only after industry leader, General Electric,
had expanded its capacity.16
The Stackelberg model of oligopoly pertains to a situation in which one firm
acts as a quantity leader, choosing its quantity first, with all other firms acting as followers, making their quantity decisions after the leader has moved. To illustrate the
Stackelberg model, we will continue to use the example of the DRAM market, but
now we will assume that Samsung (Firm 1) acts as the Stackelberg leader and chooses
its output first, and LG (Firm 2) acts as the Stackelberg follower and chooses its output after the leader has made its choice.
We analyze the Stackelberg model by considering the follower’s profit-maximization
problem first. The follower, LG, observes the quantity Q1 chosen by the leader and
chooses a profit-maximizing response to this quantity. LG’s profit-maximizing response
to any Q1 selected by Samsung is given by LG’s reaction function from the Cournot
model. We derived this reaction function in Learning-By-Doing Exercise 13.1: Q2
45 ⫺ Q1/2, and we show its graph as RLG in Figure 13.6.
FIGURE 13.6
The Stackelberg Model and
the Follower’s Profit Maximization
The line RLG is LG’s reaction function. The table
in the upper right-hand corner shows the
market price and Samsung’s profits at various
points along this reaction function. In the
Stackelberg model, the leader (Samsung)
chooses the point on the reaction function of
the follower (LG) that makes the leader’s profits
as high as possible. This occurs at point S.
16
Q2 (LG's output, units per year)
90
45
37.5
A
Point on
LG's
reaction
function
Market
Price
A
C
S
F
G
$47.5/unit $562.50
$40/unit
$900.00
$32.5/unit $1,012.50
$25/unit
$900.00
$17.5/unit $562.50
C
30
S
22.5
F
15
G
7.5
0
0
Samsung's
Profit
RLG
15
30
45
60
75
90
Q1 (Samsung's output, units per year)
See Chapter 11 of Ralph Sultan, Pricing in the Electrical Oligopoly, Volume II (Cambridge, MA: Harvard
University Press, 1975).
1 3 . 2 O L I G O P O LY W I T H H O M O G E N E O U S P R O D U C T S
Now let’s consider what Samsung will do. If it understands that LG acts as a
profit-maximizer, it will recognize that LG will choose its output according to its
reaction function RLG. This means that by its choice of output, Q1, Samsung can, in
effect, place the industry somewhere along its rival’s reaction function. For example,
we can see from Figure 13.6 that if Samsung chose Q1 15, then LG would choose
an output of 37.5 units and the industry would end up at point A. If, by contrast,
Samsung chose Q1 60, then LG would choose an output of 15 and the industry
would end up at point F.
Which output should Samsung choose? It should choose the output that maximizes its profits. To illustrate where this profit-maximizing quantity is located, the
table in the upper right-hand corner of Figure 13.6 shows the market price and
Samsung’s profit at a variety of points along LG’s reaction function. For example, at
point A (where Samsung produces 15 units and LG’s best response is to produce 37.5
units) the market price is 100 ⫺ 15 ⫺ 37.5 $47.5 per unit, and Samsung’s profit
equals ($47.5 ⫺ $10) ⫻ 15 $562.50. Of the points shown, the quantity of output
that gives Samsung the highest profit is at point S, at which Samsung produces 45
units of output, which in turn induces LG to produce 22.5 units of output.
We can verify this with some calculations. Recall that the market demand curve
is given by the equation P 100 ⫺ Q1 ⫺ Q2. But since Q2 is chosen so that Q2
$45 ⫺ Q1/2, it follows that the market price will ultimately depend on Samsung’s
quantity choice: P 100 ⫺ Q1 ⫺ (45 ⫺ Q1/2), or P 55 ⫺ Q1/2. This expression can
be thought of as the residual demand curve faced by the Stackelberg leader in that it
tells the leader how the market price will vary as a function of its quantity choice,
taking into account the follower’s reaction to that quantity choice.
Finding Samsung’s optimal quantity choice is now straightforward. We identify
the marginal revenue curve corresponding to the leader’s residual demand curve, and
find the quantity that equates this marginal revenue to the leader’s marginal cost. The
associated marginal revenue curve is MR 55 ⫺ Q1, and equating this marginal revenue to Samsung’s marginal cost yields
55 ⫺ Q1 10, or Q1 45
In response to this choice of output by the leader, the follower chooses output level
Q2 45 ⫺ 45/2 22.5.
Notice that the Stackelberg equilibrium outcome (point S) differs from the
Cournot equilibrium outcome (point C ). Unlike the Cournot outcome, which was
symmetric, under the Stackelberg outcome, the leader produces more output than the
follower (exactly twice as much in fact). In fact, even though the market price is lower
under the Stackelberg outcome than under the Cournot outcome (compare the market price at point S to that at point C in Figure 13.6), the leader’s profit under the
Stackelberg outcome is higher than its profit at the Cournot equilibrium. This tells us
that an oligopolist benefits by choosing its output first. Where does this benefit come
from? Essentially, by choosing its output first, the leader, Samsung, can “manipulate”
LG’s output choice to its advantage. In particular, when Samsung chooses a quantity
that is greater than its Cournot equilibrium quantity, it forces LG into a position in
which LG’s optimal response is to choose a quantity that is less than its Cournot equilibrium quantity. (We can see this from the fact that LG’s reaction function is downward
sloping.) The intuition for why LG is forced into this position can be seen by imagining
how LG’s managers might react when they learn about Samsung’s decision to produce
the relatively large quantity of output at point S.
545
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CHAPTER 13
M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
Wow, Samsung has committed to produce 45 units of output; that’s a lot! With that much
output, the market price can’t be any higher than $55, and it would be that high only if we
didn’t produce anything! [P 100 45 55]. That puts us in a somewhat difficult position. If we produce the same quantity of output as Samsung, or anything even close to it,
the market price is going to be really low, which is bad news for us. Frankly, Samsung
hasn’t given us very much “wiggle room” to work with here. The best thing for us to do is
to be somewhat conservative in our output choice; sure, we don’t get as much market share
that Samsung has, but at least we keep the market price at a reasonably decent level. Those
guys over at Samsung, by getting the jump on us and moving first, have really boxed us in!
The Stackelberg model of oligopoly is a particular example of a sequential game,
one in which one player in the game moves before the other players do. We will study
sequential games in Chapter 14, and we will see that there can be a strategic value associated with the ability to be the first mover in the game.
13.3
DOMINANT
FIRM
MARKETS
I
n some industries, a single company with an overwhelming share of the market—
what economists call a dominant firm—competes against many small producers, each
of whom has a small market share. For example, in 2004, General Electric had 71.5
percent of the U.S. lightbulb market. The next largest competitor, Osram Sylvania,
had just 7.4 percent.17 During prior periods, U.S. Steel was the dominant firm in the
U.S. steel industry, and Alcoa was the dominant firm in the aluminum industry.
Figure 13.7 illustrates a model of price setting by a dominant firm. Market demand
is DM. The dominant firm sets the market price and splits the market demand with a
group of small firms that constitute the industry’s competitive fringe. Fringe firms produce identical products and act as perfect competitors: each chooses a quantity of output,
taking the market price as given. The curve SF is the competitive fringe’s supply curve.18
FIGURE 13.7 Dominant Firm Market
The dominant firm’s residual demand curve DR is the
horizontal difference between the fringe’s supply curve
SF and the market demand curve DM. The dominant
firm’s profit-maximizing quantity is 50 units, and its
profit-maximizing price is $50 per unit. At this price,
the fringe supplies 25 units.
17
Price (dollars per unit)
SF
$75
DR
$50
80 units
$35
A
MC
$25
MRR
0
10
25
DM
50
75 80 90
Quantity (units per year)
U.S. Business Reporter, Market Research, http://www.usbrn.com/mrktdetail2.asp?MarketID
MRK364105 (accessed January 1, 2007).
18
With a fixed number of fringe firms, SF is the horizontal sum of fringe marginal cost curves. The vertical
intercept of SF thus shows the minimum price at which a fringe firm would supply output.
547
13.3 DOMINANT FIRM MARKETS
The dominant firm’s problem is to find a price that maximizes its profits, taking
into account how that price affects the competitive fringe’s supply. To solve this problem,
we need to identify the dominant firm’s residual demand curve DR, which will tell us
how much the dominant firm can sell at different prices. We derive DR by subtracting
the fringe’s supply from the market demand at each price. For example, at a price of
$35, market demand is 90 units, and the price-taking fringe would supply 10 units.
The dominant firm’s residual demand at a price of $35 is thus 80 units. Point A is thus
one point on DR. By identifying the horizontal distance between DM and SF at every
price, we can trace out the full residual demand curve. At prices less than $25 per unit,
fringe firms will not supply output, and the dominant firm’s residual demand curve
coincides with the market demand curve. At $75, the dominant firm’s residual demand
shrinks to zero, and fringe firms satisfy the entire market demand.
The dominant firm finds its optimal quantity and price by equating the marginal
revenue MRR associated with the residual demand curve to its marginal cost MC ($25
in Figure 13.7). We see that the dominant firm’s optimal quantity is 50 units per year,
with the profit-maximizing price of $50 per unit. We use the residual demand curve
rather than the market demand curve to determine the price because it is the residual
demand curve that tells us how much the dominant firm can sell at various market prices.
At a price of $50, market demand is 75 units per year, and the competitive fringe
supplies 25 units. By setting a price of $50, which is twice as high as the minimum
price of $25 at which fringe firms would be willing to supply output, the dominant
firm creates a price umbrella that allows some fringe firms to operate profitably. And
of course, as we have just shown, this price maximizes profit for the dominant firm,
which earns a profit equal to ($50 $25) ⫻ 50, or $1,250 per year.
Figure 13.8 shows what happens when the size of the competitive fringe grows
because additional fringe producers enter the market. The fringe’s supply curve pivots
rightward, from SF to S¿F (the fringe supplies more at a given price). This causes the
dominant firm’s residual demand curve to pivot leftward from DR to D¿R (the dominant
FIGURE 13.8
Dominant Firm Market When the
Size of the Competitive Fringe Grows
When the size of the fringe grows, the fringe’s supply
curve rotates rightward to SF , causing the residual
demand curve to rotate leftward to DR. The new profitmaximizing quantity for the dominant firm is 50 units,
and the profit-maximizing price is $42. At this price, the
fringe supplies 33 units of the total market demand of
83 units.
Price (dollars per unit)
SF
S'F
DR
D'R
$42
DM
MC
$25
MR'R
0
33 50
83
Quantity (units per year)
548
CHAPTER 13
limit pricing A strategy
whereby the dominant firm
keeps its price below the
level that maximizes its
current profit in order to
reduce the rate of expansion
by the fringe.
firm supplies less at a given price). As a result, the dominant firm’s profit-maximizing
price becomes $42 per unit, rather than $50 per unit. Its optimal quantity continues
to be 50 units, but the fringe’s supply increases from 25 to 33.19 The dominant firm’s
market share falls from 67 percent to 60 percent, and its profit falls from $1,250
to $833.
Given this, why doesn’t the dominant firm do something to slow the rate of
entry of fringe firms? The prices $50 and $42 maximize the dominant firm’s profit at
a particular point in time (e.g., in a given year). But if the rate of entry by fringe firms
depends on the current market price, the dominant firm might want to follow a
strategy of limit pricing, whereby the dominant firm keeps its price below the level
S
E
M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 3 . 3
D
Computing the Equilibrium in the Dominant Firm Model
Suppose that the market demand curve in a
global mining industry is given by Q d 110 ⫺ 10P,
where Q d is measured in millions of units of product
mined per year and P is measured in dollars per unit.
The industry is dominated by a large firm with a constant marginal cost of $5 per unit. There also exists a
competitive fringe of 200 firms, each of whom has a
marginal cost given by MC 5 ⫹ 100q, where q is the
output of a typical fringe firm.
generality in assuming that the market price will exceed 5
and that a fringe firm’s supply curve is given by q (P ⫺
5)/100. This is because the marginal cost of the dominant
firm is 5, and the dominant will not operate at a point at
which its price is less than its marginal cost. Given this,
the fringe’s overall supply curve is found by multiplying
the individual fringe supply curve by the number of fringe
firms (200): Q s (200)(P ⫺ 5)/100 2P ⫺10. Thus, the
overall fringe supply curve is Q s 2P ⫺10.
Problem
(b) We find the residual demand curve by subtracting
the overall fringe supply from the market demand
curve. Letting Q r denote residual demand, we have:
Q r Q d ⫺ Q s (110 ⫺ 10P) ⫺ (2P ⫺ 10), which implies
Q r 120 ⫺ 12P.
(a) What is the equation of the supply curve for the competitive fringe?
(b) What is the equation of the dominant firm’s residual
demand curve?
(c) What is the profit-maximizing quantity of the dominant firm? What is the resulting market price? At this
price, how much does the competitive fringe produce,
and what is the fringe’s market share (i.e., the fringe
quantity divided by total industry quantity)? What is the
dominant firm’s market share?
Solution
(a) To find the supply curve of the competitive fringe, we
proceed as follows. Each price-taking fringe firm produces to the point at which the market price equals
marginal cost: P 5 ⫹ 100q, or q (P ⫺ 5)/100. This
equation is valid only if the market price is greater than or
equal to 5. If the price is less than 5, each fringe firm
would produce nothing. In this exercise, there is no loss of
19
(c) To find the profit-maximizing quantity of the dominant firm, we first invert the residual demand (dropping
the superscript r) to get: P 10 ⫺ (1/12)Q. The corresponding marginal revenue curve is MR 10 ⫺ (1/6)Q.
Equating marginal revenue to marginal cost gives us 10 ⫺
(1/6)Q 5, which implies Q 30 million units per year.
The resulting market price is P 10 ⫺ (1/12)(30)
$7.50 per unit. At this price, the fringe’s overall supply
is: 2(7.50) ⫺ 10 5 million units per year. The total industry output is thus 35 million units: 30 million units
produced by the dominant firm and 5 million units produced by the fringe. The fringe’s market share is thus
5/(5 ⫹ 30) 14.29 percent, while the dominant firm’s
market share is 85.71 percent.
Similar Problems:
13.20, 13.21, 13.23
The dominant firm’s profit-maximizing quantity stayed at 50 units per year because of the way we constructed the demand curve and fringe supply curve for this example. A shift in the fringe’s supply curve
could, in general, change the dominant firm’s profit-maximizing output.
549
1 3 . 4 O L I G O P O LY W I T H H O R I Z O N TA L LY D I F F E R E N T I AT E D P R O D U C T S
that maximizes its current profit, in order to reduce the rate of expansion by the
fringe.20 Under limit pricing, the dominant firm sacrifices profits today in order to
keep future profits higher than they would otherwise be.
A limit pricing strategy is most appealing when a high current price induces the
competitive fringe to expand rapidly.21 Limit pricing is also attractive when the dominant firm takes the “long view” and emphasizes future over current profits in making
decisions. Finally, the limit pricing strategy tends to be attractive when a dominant
firm has a significant cost advantage over its rivals. A cost advantage allows the dominant firm to keep its price low to slow the rate of entry without much sacrifice of
current profit.
I
n many markets, such as beer, ready-to-eat breakfast cereals, automobiles, and soft
drinks, firms sell products that consumers consider distinctive from each other. In
these markets, we say that firms produce differentiated products. In this section, following up on our brief discussion in Chapter 11, we take a deeper look at product differentiation and then explore how firms in a differentiated products oligopoly might
compete against each other.
13.4
O L I G O P O LY
WITH
HORIZONTALLY
DIFFERENTIATED
PRODUCTS
W H AT I S P R O D U C T D I F F E R E N T I AT I O N ?
Economists distinguish between two types of product differentiation: vertical and
horizontal. Vertical differentiation is about inferiority or superiority. Two products
are vertically differentiated when consumers consider one product better or worse
than the other. Duracell batteries are vertically differentiated from generic storebrand batteries because they last longer. This makes Duracell batteries unambiguously superior to store-brand batteries.
Horizontal differentiation is about substitutability. Two products, A and B, are
horizontally differentiated when, at equal prices, some consumers view B as a poor
substitute for A and thus will buy A even if A’s price is higher than B’s, while other
consumers view A as a poor substitute for B and thus will buy B even if B’s price is
higher than A’s. Diet Coke and Diet Pepsi are horizontally differentiated. Some consumers view Diet Pepsi as a poor substitute for Diet Coke, while others view Diet Coke
as a poor substitute for Diet Pepsi.
Horizontal differentiation and vertical differentiation are distinct forms of product differentiation. For example, all consumers might agree that Duracell batteries
are better than a store-brand battery because they last twice as long, but if all consumers also regard two store-brand batteries as equivalent to one Duracell battery,
then the two products, though vertically differentiated, would not be horizontally
20
It is an interesting question—beyond the scope of this book—why the rate of fringe expansion might
depend on the current industry price. One possibility is that existing fringe firms rely on current profits
to finance their expansion plans, and so a lower price will mean lower current profits and (for some) more
difficulty expanding their capacity. (Of course, if expansion is profitable, one might wonder why fringe
firms cannot go to their bankers and get a loan to fund their expansion plans.) This point is best explored
in advanced courses, such as industrial economics and finance.
21
These insights about the limit pricing problem come from D. Gaskins, “Dynamic Limit Pricing:
Optimal Pricing under the Threat of Entry,” Journal of Economic Theory 3 (September 1971): 306–322.
vertical differentiation
A situation involving two
products such that
consumers consider one
product better or worse
than the other.
horizontal differentiation
A situation involving two
products such that some
consumers view one as a
poor substitute for the
other and thus will buy the
one even if its price is
higher than the other’s.
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A P P L I C A T I O N
M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
13.3
U.S. Steel: The Price of Dominance22
With sales of over $6 billion, the U.S. Steel group of
the USX Corporation is one of America’s largest steel
companies. But while large in absolute terms, U.S.
Steel currently accounts for less than 15 percent of
U.S. domestic steel sales. At one time, though, U.S.
Steel was much more dominant. In fact, when it was
formed (by merger) in 1901, U.S. Steel produced 66
percent of the steel ingot sold in the United States. In
those days, U.S. Steel was a classic dominant firm.
However, as Table 13.6 shows, U.S. Steel’s market
share soon began to decline, and by the mid-1930s, it
had fallen to 33 percent of the market. According to
economic historians Thomas K. McCraw and Forest
Reinhardt:
For three decades [1900–1930], U.S. Steel followed
patterns of pricing and investment that guaranteed an erosion of its market share. Instead of
raising barriers to entry into the steel industry,
it lowered them. It neither tried vigorously to
retain its existing markets nor to take advantage
of new growth opportunities in structural and
rolled markets (p. 616).
Why didn’t U.S. Steel follow an aggressive strategy of limit pricing to slow the expansion by rival
firms? Our discussion of dominant firm pricing sheds
light on this question. Scholars who have studied the
history of U.S. Steel believe that before World War II
(1941–1945), U.S. Steel probably did not have an appreciable cost advantage over its competitors. F. M.
Scherer writes, “Although some of the Corporation’s
plants may have had lower costs, on average USS
could pour and shape steel at costs no lower than
22
TABLE 13.6
1901–1935
U.S. Steel’s Market Share,
Year
Market
Share
Year
Market
Share
1901
1905
1910
1915
66%
60%
54%
51%
1920
1925
1930
1935
46%
42%
41%
33%
those of its rivals, actual or potential.”23 In addition,
as Scherer notes, entry into the steel industry in the
early twentieth century took time. It required building an integrated steel mill, and in those days it was
not easy to secure either financial capital or reliable
sources of iron ore.
As a result, it probably made sense for U.S. Steel
to eschew an aggressive limit pricing strategy and
instead set prices at or close to the levels implied by
the dominant firm model. And as we saw from Figure
13.8, with an expanding fringe, this implied an erosion of the dominant firm’s share over time. Hideki
Yamawaki provides some statistical evidence that U.S.
Steel actually behaved this way.24 Using data on steel
prices and production (by U.S. Steel and rival firms)
from that era, Yamawaki shows that U.S. Steel’s pricing decisions were influenced by the market share of
fringe producers. He also shows that the price set by
U.S. Steel significantly influenced the fringe’s rate of
production and the rate at which the fringe expanded
over time. Based on this evidence, we can conclude
that the logic of the dominant firm model nicely fits
competitive dynamics in the U.S. steel industry from
1900 to 1940.
This example was inspired by a fuller and more detailed discussion of U.S. Steel’s history and dominant
firm pricing behavior by F. M. Scherer in Chapter 5 of his book Industry Structure, Strategy, and Public
Policy (New York: HarperCollins, 1996). The quotation below and the data in Table 13.6 come from
T. K. McCraw and F. Reinhardt, “Losing to Win: U.S. Steel’s Pricing, Investment Decisions, and Market
Share, 1901–1938,” Journal of Economic History 49 (September 1989): 593–619.
23
F. M. Scherer, Industry Structure, Strategy, and Public Policy (New York: HarperCollins, 1996), p. 155.
24
H. Yamawaki, “Dominant Firm Pricing and Fringe Expansion: The Case of the U.S. Iron and Steel
Industry, 1907–1930,” Review of Economics and Statistics 67 (August 1985): 429–437.
1 3 . 4 O L I G O P O LY W I T H H O R I Z O N TA L LY D I F F E R E N T I AT E D P R O D U C T S
differentiated.25 If the store-brand price were less than half the price of Duracell, all
consumers would choose the store brand. By contrast, although few people could
make a compelling case that Diet Coke has an unambiguously higher quality than
Diet Pepsi, some consumers have loyalties toward one brand over the other, and thus
do not regard the products as perfect substitutes. These brands are horizontally differentiated but not vertically differentiated.
Horizontal differentiation is an important concept for the theory of oligopoly and
monopolistic competition that we study in this chapter. Firms selling horizontally differentiated products have downward-sloping demand curves, as Figure 13.9 shows.
In Figure 13.9(a), where horizontal differentiation is weak, the firm’s demand is
quite sensitive to a change in its own price and the prices of its rivals. A relatively small
increase in the firm’s own price (from $30 to $35) results in a relatively large decrease
in quantity (from 40 to 20 units), and a small decrease in the price charged by a competitor also results in a large decrease in the quantity sold by the firm, illustrated by
the large leftward shift in the demand curve from D to D⬘.
In Figure 13.9(b), where horizontal differentiation is strong, the firm’s demand is
much less sensitive to a change in its own price and the prices of its rivals. A small
increase in the firm’s own price (from $30 to $35) results in only a small decrease in
quantity (from 40 to 38 units), and a small decrease in the price charged by a competitor also results in only a small decrease in the quantity sold by the firm, illustrated by
the small leftward shift in the demand curve from D to D.
$35
$30
20
40
Quantity (units per month)
(a) Weak horizontal differentiation
D
D'
Price (dollars per unit)
Price (dollars per unit)
D"
D
$35
$30
38 40
Quantity (units per month)
(b) Strong horizontal differentiation
FIGURE 13.9 Horizontal Differentiation and the Firm’s Demand Curve
In panel (a), horizontal differentiation is weak. The firm’s demand curve D is downward sloping, but the quantity demanded is sensitive to changes in the firm’s price. A given increase in
price, say from $30 per unit to $35 per unit, holding competitors’ prices fixed, leads to a large
reduction in the quantity demanded. Moreover, when competitors reduce their prices, the
firm’s demand curve shifts leftward, from D to D, by a large amount. By contrast, in panel (b),
horizontal differentiation is stronger. The firm’s demand is not as sensitive to a change in its
own price, and when competitors cut their prices, the firm’s demand curve shifts leftward, from
D to D, by a relatively smaller amount.
25
In the language of Chapters 4 and 5, consumer indifference curves for Duracell batteries and store-brand
batteries would be linear. In reality, consumers might not equate two store-brand batteries with one Duracell
battery because of the convenience factor. A battery that lasts longer takes up less space than two batteries
and does not have to be changed as often. For simplicity, here we ignore the convenience factor.
551
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13.4
Smartphone Wars26
For many years the cell phone market was dominated
by Nokia and Motorola, which sold inexpensive phones
with limited extra functionality. A distinctive competitor was the Blackberry made by Research in Motion
(RIM). The Blackberry was the first “smartphone”—
essentially a small hand-held computer that allowed
its users to not only make cellular calls, but also to receive and send e-mail messages, manage their appointment calendar, and so forth. For years, the Blackberry
was relatively unique, with high horizontal differentiation from standard cell phones. Most Blackberry users
were businesspeople, who often had their phones paid
for by their employers.
In recent years the capabilities of smartphones
have increased dramatically, and their prices have
fallen. Industry analysts estimate that roughly 40 percent of consumers now use or are planning to buy a
smartphone by the end of 2010. In January 2010, the
most popular consumer smartphone was Apple’s
iPhone, which leapfrogged the Blackberry in functionality. This phone continued Apple’s tradition of
elegant design and an easy-to-use interface. In addition
to a phone and camera, the iPhone could be used
to check e-mail, surf the Web, and store and play
music and videos exactly as in an iPod. The phone also
included a GPS (global positioning satellite) capability. It had a relatively large screen compared to other
phones, and it used an innovative touchscreen interface, while other phones used small keys for entry of
text and numbers. An important innovation and horizontal differentiator of the iPhone was the presence
of “apps”—small applications that iPhone users could
download (sometimes for free, sometimes for a few
dollars). Thousands of apps were developed by many
small companies, allowing iPhone users to add extensive functionality to their smartphones. These various
features of the iPhone enabled it to achieve horizontal
26
product differentiation vis-à-vis other smartphones,
such as the Blackberry.
The iPhone’s popularity grew rapidly, with sales
of over 5 million units in 2009 compared to less than 1
million in 2008. In response to the first iPhone models,
RIM introduced the Pre, a smartphone with features
somewhat similar to the iPhone. However, two days
after the Pre was launched, Apple announced its
newest model, the iPhone 3GS. This announcement
stalled sales of the Pre. By the beginning of 2010,
Apple’s unit sales were nearly as high as RIM’s.
The success of the iPhone spurred additional
competitive responses from Motorola and Google.
Google designed an operating system for smartphones called Android, hoping to open up the market for smartphones to many manufacturers. The
goal was to spur innovation in apps to compete with
those on the iPhone, and also innovation and cost
competition among phone manufacturers. Motorola
was an early entry in this initiative, announcing the
Droid phone in late 2009 to strong reviews.
If the Android initiative succeeds, Apple’s iPhone
may lose much of its horizontal differentiation. At
the time of this writing, it was too early to tell
whether or not that would happen. However, one
survey in January 2010 found that the percentage of
consumers planning to buy a new smartphone in
2010 who said that they would buy an Android
phone was 21 percent, compared to 6 percent a half
year earlier. Meanwhile the percentage planning to
buy an iPhone had fallen from 32 percent to 28 percent. Notably, the survey found that potential users
of iPhones and Android phones had very similar patterns for usage of smartphones (in terms of calls,
Internet browsing, apps, social networking, and instant messaging)—clear evidence of Apple’s potential
loss of differentiation. If the survey numbers turn
into actual sales, Android phones’ market share will
be nearly the same as Apple’s in its first year.
See Raven Zachary, “Who’s Winning the Smartphone Wars?” O’Reilly Radar, August 24, 2009,
http://radar.oreilly.com/raven (accessed March 11, 2010); “Smartphone Wars: Android Phones Close in
on iPhone,” ABC News, Ahead of the Curve, January 20, 2010, http://blogs.abcnews.com/aheadofthecurve/
2010/01/smartphone-wars-android-phones-close-in-on-iphone.html (accessed March 11, 2010).
1 3 . 4 O L I G O P O LY W I T H H O R I Z O N TA L LY D I F F E R E N T I AT E D P R O D U C T S
BERTRAND PRICE COMPETITION WITH
H O R I Z O N TA L LY D I F F E R E N T I AT E D P R O D U C T S
Let’s now study how firms in a differentiated products market would set their prices.
To do so, we return to the model of Bertrand price setting and adapt it to deal with
horizontally differentiated products.27 As a specific illustration of this model, let’s consider a market in which horizontal differentiation is significant: the U.S. cola market.
Farid Gasmi, Quang Vuong, and J. J. Laffont (GVL) have used statistical methods
to estimate residual demand curves for Coke (Firm 1) and Pepsi (Firm 2):28,29
Q1 64 ⫺ 4P1 ⫹ 2P2
(13.1)
Q2 50 ⫺ 5P2 ⫹ P1
(13.2)
GVL also estimated that Coca-Cola and Pepsi had marginal costs of $5 and $4, respectively.30 Given these demand curves and marginal costs, what price should each
firm charge?
As in the Cournot model, an equilibrium occurs when each firm is doing the best
it can given the actions of its rival. The logic of finding this equilibrium is similar to
the logic using the Cournot model, so we begin by deriving each firm’s price reaction
function—that is, its profit-maximizing price as a function of its rival’s price.
Consider Coca-Cola’s problem. Figure 13.10(a) shows Coke’s demand curve D8
when Pepsi sets a price of $8. This curve tells us how much Coke can sell at various
prices, given that Pepsi’s price remains fixed at $8 [note that D8 satisfies equation
(13.1)]. For example, if Coke sets a price of $7.50, it can sell 50 million units. Equating
Coca-Cola’s marginal revenue MR8 to its marginal cost MC tells us that its profitmaximizing output is 30 units. To sell this quantity, Coke must set a price of $12.50.
Thus, $12.50 is Coke’s best response to Pepsi’s price of $8. Figure 13.10(b) shows that
when Pepsi sets a price of $12, Coca-Cola’s best response is to charge $13.50.
These results provide data for plotting Coke’s price reaction function, and we
could derive similar data for Pepsi that would let us price Pepsi’s price reaction function. Figure 13.11 shows both these reaction functions: R1 shows how Coke’s profitmaximizing price varies with Pepsi’s price; R2 shows how Pepsi’s profit-maximizing
price varies with Coke’s price. Note that the profit-maximizing prices for Coke that
are shown in Figure 13.10, (P1 $12.50, P2 $8) and (P1 $13.50, P2 $12), are
on R1 (though not specifically labeled in Figure 13.11). Note, too, that the reaction
functions are upward sloping. Thus, the lower your rival’s price is, the lower your own
price should be.
27
We could also study a Cournot quantity-setting model of competition with differentiated products. Just
as the Cournot model with no product differentiation leads to a different equilibrium than the Bertrand
price model, the Cournot quantity-setting model with product differentiation leads to a different equilibrium price than the Bertrand model that we study in this section. You will get a chance to prove this
point for yourself in Problem 13.32.
28
The use of this example was inspired by our former colleague Matt Jackson, who used it in teaching his
microeconomics classes at the Kellogg Graduate School of Management.
29
F. Gasmi, Q. Vuong, and J. Laffont, “Econometric Analysis of Collusive Behavior in a Soft-Drink Market,”
Journal of Economics and Management Strategy (Summer 1992): 277–311. To keep the numbers simple, we have
rounded GVL’s estimates (which come from Model 10 in the paper) to the nearest whole number. In their
paper, prices are inflation-adjusted and are expressed in dollars per unit, while quantities are expressed in
millions of units of cola; a unit is defined as 10 cases, with twelve 24-ounce cans in each case.
30
These are also expressed in dollars per unit.
553
Coca-Cola's price
(dollars per unit)
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M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
$12.50
D8
$7.50
$5.00
MC
Coca-Cola's price
(dollars per unit)
554
$13.50
D12
MC
$5.00
MR8
0
30
Coca-Cola's quantity
(million of units per year)
(a) Coke's profit-maximization problem when
Pepsi's price = $8
MR12
50
0
34
Coca-Cola's quantity
(million of units per year)
(b) Coke's profit-maximization problem when
Pepsi's price = $12
FIGURE 13.10
Profit-Maximizing Price Setting by Coca-Cola
MC is Coca-Cola’s marginal cost curve. Panel (a): If Pepsi’s price is $8, Coke’s demand curve is D8,
and its corresponding marginal revenue curve is MR8. Coca-Cola will maximize profit at a quantity of 30 units and a price of $12.50. Panel (b): If Pepsi’s price is $12, Coke’s demand curve is
D12, and its corresponding marginal revenue curve is MR12. Coca-Cola will maximize profit at a
quantity of 34 units and a price of $13.50. These results can be used to plot Coca-Cola’s price
reaction function, shown in Figure 13.11.
R1 (Coke's reaction
FIGURE 13.11
Bertrand
Equilibrium for Coke and Pepsi
Coke’s reaction function is R1. Pepsi’s
reaction function is R2. The Bertrand
equilibrium occurs where the two
reaction functions intersect (point E,
where Coke charges a price of $12.56
and Pepsi charges a price of $8.26).
This differs from the monopoly equilibrium (point M, where Coke’s price
would be $13.80 and Pepsi’s price
would be $10.14).
P2 (Pepsi's price, dollars per unit)
function)
Bertrand
equilibrium
$10.14
$8.26
Monopoly
E
M prices
R2 (Pepsi's reaction
function)
$12.56 $13.80
P1 (Coca-Cola's price, dollars per unit)
At the Bertrand equilibrium (point E ), each firm chooses a price that maximizes
its profit given the other firm’s price.31 As shown in Figure 13.11, this occurs where
the two reaction functions intersect (P*1 $12.56, P*2 $8.26). By substituting these
prices back into the demand functions, we can compute the equilibrium quantities for
Coca-Cola and Pepsi: Q*1 30.28 million units and Q*2 21.26 million units. In fact,
31
You will see in Chapter 14 that the Bertrand equilibrium, like the Cournot equilibrium, is a particular
example of a Nash equilibrium. For this reason, some textbooks refer to the Bertrand equilibrium as the
Nash equilibrium in prices.
1 3 . 4 O L I G O P O LY W I T H H O R I Z O N TA L LY D I F F E R E N T I AT E D P R O D U C T S
the average (inflation-adjusted) prices over the time period of GVL’s study
(1968–1986) were actually $12.96 for Coca-Cola and $8.16 for Pepsi. The corresponding quantities were 30.22 million units and 22.72 million units. Thus, the Bertrand
model, when applied to the demand curves estimated by GVL, does a good job of
matching the actual pricing behavior of these two firms in the U.S. market.
Pepsi’s equilibrium price is so much lower than Coca-Cola’s price for two important reasons. First, Pepsi’s marginal cost is lower than Coke’s. Second, and more subtly,
Pepsi’s own price elasticity of demand is larger than Coke’s (as shown in Table 2.7).32
Since we know (from Chapter 11) that profit maximization along a downward-sloping
demand curve implies an inverse elasticity pricing rule (IEPR), applying the IEPR to
Coke and Pepsi’s pricing problem implies that Pepsi should have a smaller markup than
Coke. A smaller markup applied to a smaller marginal cost makes Pepsi’s price lower
than Coca-Cola’s.
Given the equilibrium prices, the percentage contribution margins (PCMs) for
Coke and Pepsi are
P*1 ⫺ MC1
12.56 ⫺ 5
0.60, or 60 percent
P*1
12.56
P*2 ⫺ MC2
8.26 ⫺ 4
0.52, or 52 percent
P*2
8.26
Coke’s PCM implies that for every dollar’s worth of Coke that Coca-Cola sells, it has
60 cents left over to cover marketing expenses, company overhead, interest, and taxes.
This PCM is higher than the average PCM of all U.S. manufacturing firms.33 This
example thus illustrates how product differentiation softens price competition. When
products are as strongly differentiated as Coke and Pepsi are, price cutting is less
effective for stealing a rival’s business than when products are perfect substitutes. Of
course, Coke and Pepsi incur a heavy cost to achieve this product differentiation. Both
companies spend hundreds of millions of dollars in the United States to advertise their
colas, and as discussed in the introduction to this chapter, the two companies compete
intensely for exclusive distribution deals on college campuses to develop brand loyalties among young people.
Even though horizontal differentiation softens price competition, the Bertrand
equilibrium prices do not correspond to the monopoly prices (i.e., the prices that
would maximize the joint profit of Pepsi and Coca-Cola). As Figure 13.11 shows
(point M ), these prices are about $13.80 for Coke and $10.14 for Pepsi. As in the
Cournot model, independent profit-maximizing oligopolists will typically not attain
the outcome that a profit-maximizing monopolist would, because neither firm takes
into account the adverse effect that a price cut or the beneficial effect that a price increase
would have on its rival.
32
Since GVL computed their elasticities at the actual average prices, your calculations based on the
computed equilibrium prices won’t exactly match those in Application 2.5, but they will be close.
33
A commonly used estimate of the PCM can be contructed from data from the U.S. Census of
Manufacturing:
Sales revenue ⫺ Materials cost ⫺ Factory payrolls
PCM ⬇
Sales revenue
This measure uses material and labor costs as a proxy for marginal cost. Historically, this measure of PCM
has been on the order of 23 to 25 percent for all U.S. manufacturing firms.
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E
M A R K E T S T RU C T U R E A N D C O M P E T I T I O N
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 3 . 4
D
Computing a Bertrand Equilibrium with Horizontally Differentiated Products
Suppose Coca-Cola’s and Pepsi’s demand
curves are given by Q1 (64 ⫹ 2P2) ⫺ 4P1 and Q2
(50 ⫹ P1) ⫺ 5P2, respectively. [These correspond to
equations (13.1) and (13.2) with terms rearranged and
with parentheses used to highlight terms that the firm
views as fixed.] Coca-Cola’s marginal cost is $5 per unit,
and Pepsi’s marginal cost is $4 per unit.
Problem
(a) What is Coca-Cola’s profit-maximizing price when
Pepsi’s price is $8?
(b) What is the equation of Coca-Cola’s price reaction
function (i.e., Coca-Cola’s profit-maximizing price when
Pepsi sets an arbitrary price P2)?
(c) What are Coca-Cola’s and Pepsi’s profit-maximizing
prices and quantities at the Bertrand equilibrium?
Solution
(a) Substitute P2 8 into Coke’s demand curve to get
Q1 (64 ⫹ 2(8)) ⫺ 4P1 80 ⫺ 4P1, or P1 20 ⫺
0.25Q1. The associated marginal revenue curve is MR
20 ⫺ 0.5Q1. Equating this to Coke’s marginal cost gives
20 ⫺ 0.5Q1 5, or Q1 30. Substituting this back
into Coke’s demand curve yields P1 20 ⫺ 0.25(30), or
A P P L I C A T I O N
(b) Solving Coke’s demand curve for P1 gives P1 (16 ⫹
P2/2) ⫺ Q1/4. The associated marginal revenue curve is
MR (16 ⫹ P2/2) ⫺ Q1/2. Equating marginal revenue
to marginal cost yields (16 ⫹ P2/2) ⫺ Q1/2 5, or Q1
22 ⫹ P2. Substituting this back into Coke’s demand curve
gives P1 (16 ⫹ P2/2) ⫺ (22 ⫹ P2)/4, or P1 10.5 ⫹
P2/4. This is the equation of Coca-Cola’s price reaction
function. (Note that we could find Pepsi’s price reaction
function in the same way, starting with Pepsi’s residual
demand curve. Doing so would give P2 7 ⫹ P1/10.)
(c) The Bertrand equilibrium is at the point where the
two reaction functions are equal (i.e., where the two
curves intersect). Thus, the Bertrand equilibrium prices
are the prices that simultaneously solve the two firms’
reaction functions: P1 ⫺ P2/4 10.5 (Coke’s reaction
function, rearranged) and P2 ⫺ P1/10 7 (Pepsi’s reaction
function, rearranged), or P*1 $12.56 and P*2 $8.26.
Substituting these prices back into each firm’s residual
demand curve yields the Bertrand equilibrium quantities:
Q*1 30.28 units and Q*2 21.26 units.
Similar Problems: 13.26, 13.28, 13.29, 13.30
13.5
Chunnel versus Ferry
One of the most impressive feats of modern engineering is the 32-mile-long Channel Tunnel (Chunnel) that
links Calais, France, to Dover, England. Eurotunnel
(ET), the company that owns and operates the
Chunnel, offers two main services: passenger service
and freight service. Under ET’s passenger service,
called Le Shuttle, you drive your car aboard one of the
specially designed rail cars at a terminus of the tunnel,
and a train then transports your car (with you inside)
through the tunnel to the other end.34 Under ET’s
freight services, trucks are driven aboard special rail
cars, and the train transports the trucks through the
34
P1 12.50. Thus, Coke’s profit-maximizing price is
$12.50 when Pepsi’s price is $8.
This service is also offered for buses.
tunnel. For both of these services, ET competes
against cross-channel ferries. When the Chunnel
opened, there were two major ferry companies:
Britain’s P&O and Sweden’s Stena Line. Together, they
carried about 80 percent of the cross-channel passenger and freight traffic. Since then, these two companies have merged their cross-channel operations and
compete as a duopolist against ET.
Before the Chunnel opened, John Kay, Alan
Manning, and Stefan Szymmanski (KMS) used the
Bertrand model of price competition to analyze the
likely outcome of price competition between ET and
the ferry operators (which they presciently treated as a
single firm) in the market for freight service. Using
1 3 . 4 O L I G O P O LY W I T H H O R I Z O N TA L LY D I F F E R E N T I AT E D P R O D U C T S
information from ET’s 1987 prospectus (a document
prepared for investors and lenders discussing its plan for
doing business) and some educated back-of-theenvelope conjectures, KMS estimated price reaction
functions for both ET and the ferry operators, as shown
in Figure 13.12. The Bertrand equilibrium that they predicted occurred at a price of £87 for the tunnel and
£150 for the ferry operators. The large difference between ET’s equilibrium price and the ferries’ equilibrium
price reflect KMS’s estimate that the marginal cost of
£ 160
Tunnel price (£)
ET’s freight shuttle service would be substantially less
than the marginal cost of freight shuttle service by ferry.
KMS’s analysis suggested that ET would become
a formidable competitor in the cross-channel freight
market. Two years after the opening of the Chunnel,
ET was capturing 44 percent of the cross-channel
truck traffic versus 40 percent for the ferries. By 2002,
ET’s share of the cross-channel truck market had
grown to over 50 percent, and the truck shuttle service
was probably ET’s most profitable business.
Ferry reaction function
£120
Tunnel reaction function
FIGURE 13.12
£ 80
£ 40
£0
£ 125
557
£ 150
£ 175
£ 200
Ferry price (£)
A P P L I C A T I O N
£ 225
£ 250
Bertrand
Equilibrium: The Chunnel and
the Ferries
The curves are the reaction functions for
the Channel Tunnel and the ferries. The
Bertrand equilibrium occurs at a price of
£87 for the Chunnel and £150 for the
ferries.
Source: Figure 18.7 from “Pricing the
Tunnel,” in J. Kay, The Business of
Economics (New York: Oxford University
Press), 1996.
13.6
Wireless Number Portability
November 24, 2003 was an important day for the U.S.
wireless telephone industry. Beginning then, subscribers were allowed to keep their phone numbers
when switching from one wireless provider to another. In other words, if you switched your cellular
phone service from, say, Verizon to Cingular, your
phone number would remain the same. Prior to wireless number portability, the need to change your
phone number if you changed wireless providers
created a potentially significant switching cost for consumers. You would need to inform your co-workers,
friends, and family members of your new phone
number, and you might miss important phone calls
while they were learning your new number.
By turning consumers who would otherwise have
been shoppers into “loyalists,” wireless number nonportability had the effect of strengthening the horizontal product differentiation between providers.
Since wireless phone service is essentially the same
product no matter who delivers it, any significant
horizontal differentiation went away on November
24, 2003. Economic theory predicts that the onset of
portability would move the industry from an equilibrium in which the percentage contribution margins
are moderate to large, to one in which price is much
closer to marginal cost, perhaps even approaching
558
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the outcome predicted by the Bertrand model of
oligopoly with homogeneous products, in which the
equilibrium is actually equal to marginal cost.
Did this happen? Since 2003, over 10 million individuals in the United States have taken their cell phone
numbers with them when they switched service
providers. Research by Minjung Park documents intensified price competition as a result.35 The average price
for a monthly plan with the fewest minutes of call time
fell only about 1 percent. However, call plans with more
minutes had larger price drops. Plans with intermediate
13.5
MONOPOLISTIC
COMPETITION
minutes fell by about 5 percent, and plans with large
numbers of minutes fell by about 7 percent. In addition,
the dispersion (variance) in prices fell dramatically.
What we learn from this example is that removing
conditions that create switching costs can intensify competition. Put another way, creation of switching costs
across sellers can reduce competition and keep prices
high. Firms understand this, of course, which is why we
see phenomena such as frequent flyer programs, loyalty
cards, and hundreds of millions of dollars spent every
year on advertising aimed at differentiating products.
A monopolistically competitive market has three distinguishing features.
36
First, the
market is fragmented—it consists of many buyers and sellers. Second, there is free
entry and exit—any firm can hire the inputs (labor, capital, and so forth) needed to
compete in the market, and they can release these inputs from employment when they
do not need them. Third, firms produce horizontally differentiated products—
consumers view firms’ products as imperfect substitutes for each other.
Local retail and service markets often have these characteristics. Consider, for
example, the restaurant market within the city of Evanston, Illinois. The market is
highly fragmented—the Evanston Yellow Pages, for example, has nearly five pages of
restaurant listings. The Evanston restaurant market also has free entry and exit.
Prospective restaurateurs can easily rent space, acquire cooking equipment, and hire
servers. A comparison of the Yellow Pages listings for 2004 with those for 2010 reveals
a remarkable turnover of establishments. When times are good, new restaurants are
opened. When a restaurant proves to be unprofitable, it is shut down.
Market fragmentation and free entry and exit are also characteristics of perfectly
competitive markets. But unlike perfectly competitive firms, Evanston restaurants are
characterized by significant product differentiation. There are many different types of
restaurants (Chinese, Thai, Italian, vegetarian) that cater to the wide variety of buyer
tastes in Evanston. Some restaurants are formal, while others are casual. And each
restaurant is conveniently located for people who live or work close to it but might be
inconvenient for people who have to drive several miles to get to it.
S H O R T- R U N A N D L O N G - R U N E Q U I L I B R I U M
I N M O N O P O L I S T I C A L LY C O M P E T I T I V E M A R K E T S
In choosing their prices, monopolistic competitors behave much like the differentiated products oligopolists that we studied in the previous section. Even though the
market is fragmented, each firm’s demand curve is downward sloping because of product
differentiation. Taking the prices of other firms as given, each firm maximizes its
profit at the point at which its marginal revenue equals marginal cost.
Figure 13.13 illustrates the profit-maximization problem facing a typical firm
under monopolistic competition. The firm faces a demand curve D. When the firm
maximizes its profit along this demand curve, it charges a price of $43 and produces
35
Minjung Park, “The Economic Impact of Wireless Number Portability,” Working Paper, University of
Minnesota, October 2009.
36
This model of monopolistic competition was developed by the economist Edward Chamberlin in his
book, The Theory of Monopolistic Competition (Cambridge, MA: Harvard University Press, 1933).
Price (dollars per unit)
13.5 MONOPOLISTIC COMPETITION
559
$43
MC
FIGURE 13.13
AC
MR
D
57
Quantity (units per month)
Profit Maximization and
Short-Run Equilibrium under Monopolistic
Competition
Each firm faces the demand curve D and maximizes
profit at the point where marginal revenue MR
equals marginal cost MC, at a quantity of 57 units
and a price of $43. This is a short-run equilibrium
but not long run, because the price exceeds the
firm’s average cost AC, indicating profit opportunities that will attract new entrants.
Price (dollars per unit)
an output of 57 units. The price of $43 is the firm’s best response to the prices charged
by other firms in the market. As in the Bertrand model of oligopoly with differentiated products, the market attains an equilibrium when every firm is charging a price
that is a best response to the set of prices charged by all other firms in the market. Let’s
suppose this condition holds when each firm in the market sets a price of $43 (i.e., we
will assume that all the firms in the market are identical).
What, then, makes monopolistic competition different from a differentiated
products oligopoly? The key difference is that monopolistically competitive markets
are characterized by free entry. If there are profit opportunities in the market, new entrants will appear to seize them. In Figure 13.13, note that the price of $43 exceeds
the firm’s average cost, which means the firm is earning positive economic profits.
The situation in Figure 13.13 constitutes a short-run equilibrium—a typical firm is
maximizing profits given the actions of rival firms—but it is not a long-run equilibrium because firms will enter the market to take advantage of the profit opportunity.
As more firms come into the market, each firm’s share of overall market demand
will fall—that is, the typical firm’s demand curve will shift leftward. Entry and the
resultant leftward shift in firms’ demand curves will cease when firms make zero economic profit. In Figure 13.14, this occurs at a price of $20, where each firm’s demand
$43
AC
$20
D'
47 57
Quantity (units per month)
D
FIGURE 13.14
Long-run Equilibrium under
Monopolistic Competition
As firms enter the monopolistically competitive
market, each firm’s demand curve shifts leftward
from D to D⬘. Long-run equilibrium occurs at a
price of $20 and a quantity of 47, where D⬘ is
just tangent to the average cost curve AC, and
the firm makes zero economic profit.
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curve D⬘ is tangent to its average cost curve AC. Put another way, the margin between
a firm’s price and its variable costs is just sufficient to cover its fixed costs of operation
and the up-front costs of entering the business. Given that this is so, entrants have no
incentive to come into the market.
P R I C E E L A S T I C I T Y O F D E M A N D, M A R G I N S,
AND NUMBER OF FIRMS IN THE MARKET
In monopolistically competitive markets, free entry and exit of firms determines how
many firms ultimately compete in the market. Figure 13.15 illustrates two possible
long-run equilibria.
In Market A, consumers are sensitive to price differences when they choose among
existing sellers. A seller in this market has a demand curve that is highly price elastic.
In a long-run equilibrium (where the demand curve D is tangent to the average cost
curve AC ), the margin (P * ⫺ MC ) between price and marginal cost is small, and firms
produce a large volume of output. By contrast, in Market B, consumers are not especially sensitive to price differences among competing sellers, so a firm’s demand is not
as sensitive to price as in Market A. In a long-run equilibrium, the margin between
price and marginal cost is large and each firm produces a small volume of output. If the
total number of units purchased in equilibrium is about the same in Markets A and B,
Market B would have more firms than Market A because each firm in market B sells a
smaller quantity than each firm in market A.
D O P R I C E S FA L L W H E N M O R E F I R M S E N T E R ?
P*
AC
MC
MR
Quantity (units per month)
(a) Market A firms face relatively elastic
demand
D
Price (dollars per unit)
Price (dollars per unit)
When we studied the Cournot model earlier in this chapter, we saw that the equilibrium price went down as more firms competed in the market. Figure 13.14 portrays a
similar phenomenon in a monopolistically competitive market. In that figure, the
entry of more firms resulted in a reduction in the market price.
P*
AC
MC
MR
D
Quantity (units per month)
(b) Market B firms face relatively inelastic
demand
FIGURE 13.15 Price Elasticity of Demand and Long-Run Equilibrium
In Market A, firms face relatively elastic demand. At a long-run equilibrium, the margin P* ⫺ MC
between price and marginal cost is small, and each firm produces a large volume of output. In
Market B, firms face relatively less elastic demand. At a long-run equilibrium, the margin between
price and marginal cost is large, and each firm produces a small volume of output.
13.5 MONOPOLISTIC COMPETITION
A P P L I C A T I O N
13.7
Wine or Roses?
If you look in your local Yellow Pages, you will probably
see that there are a lot more florists than liquor stores.
For example, in the 2010 Chicago Yellow Pages, there
are approximately 500 florists and 350 liquor stores. Why
is this? Do these numbers tell us that there is significantly
more demand for roses than wine? Probably not. In fact,
the typical U.S. household probably spends more per
year on wine, beer, and spirits than it does on flowers.
Instead, this pattern of local retail market structures probably reflects at least in part the logic of
Figure 13.15.37 The figure implies that when there is
free entry, markets in which firms can attain high
margins of price over marginal cost should contain
numerous small firms, while markets in which firms
have low margins of price over marginal cost should
have bigger but fewer firms. In a high-margin market
such as B in Figure 13.15, a firm does not need a large
volume of sales in order to cover the up-front costs of
entry and the fixed costs of doing business. Many
firms can fit into the market, and with free entry,
many firms will enter. In low-margin markets such as
A in Figure 13.15, a firm needs a larger volume in
order to cover costs. Fewer firms can fit into the market,
so even with free entry, fewer will enter.
In retailing, the margin between price and marginal cost is best approximated by what is called the
gross margin, which represents the difference between
a product’s price and its average cost to the retailer, expressed as a percentage of the price. Flower shops typically have gross margins that exceed 40 percent, while
liquor stores have gross margins closer to 20 percent.
The logic of Figure 13.15 tells us that, all else being
equal, local retail markets should have more florists
than liquor stores. It also suggests that we should see
more jewelry stores (gross margins around 50 percent)
then bakeries (gross margins around 40 percent), and
more bakeries than hardware stores (gross margins
around 20 to 30 percent). In Chicago’s Yellow Pages
there are approximately 540 jewelers, 420 bakeries, and
170 hardware stores. Page through your local Yellow
Pages to see whether this pattern holds in your town.
But this will not necessarily always happen. To see why, consider Figure 13.16,
which shows a monopolistically competitive industry that has attained a long-run
equilibrium at a price of $50. Suppose, now, that all firms experience a decrease in
their average cost (represented by a shift from AC to AC⬘ in the figure). At the current price of $50, firms now enjoy positive economic profit, which encourages the
entry of additional firms. When long-run equilibrium is restored, a typical firm again
earns zero profits, but this occurs at the higher price of $55 per unit. The entry of
more firms has driven the equilibrium price up!
Why could this happen? One possible reason is that the new entrants might lure
away the less loyal customers of existing firms—those that are more or less indifferent among competing sellers—leaving each existing firm with a small core of loyal
customers. In effect, the entry of additional firms into the market could push existing
firms into narrow niches of the market. For example, in the DVD rental market in an
urban area, the entry of new DVD rental stores might cause an existing store to lose
customers located far away from the store, leaving the store to operate in a niche that
consists of customers located in the few surrounding blocks. Another possible reason
is that as more firms enter the market, consumers might find it more difficult to learn
and compare the prices of all the sellers. With less efficient comparison shopping,
consumers could become less sensitive to price in choosing which seller to buy from.
37
561
It could also, of course, reflect other factors, such as differences in cost conditions across the two retail
trades, differences in the extent of product variety available in flower shops versus liquor stores, and the
fact that beer and wine (but not spirits) can also be purchased in grocery and convenience stores.
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FIGURE 13.16 Equilibrium Price under
Monopolistic Competition
Initially, the market is in long-run equilibrium at
a price of $50 and with each firm facing the demand curve D. If the average cost curve shifts
from AC to AC⬘, firms start earning positive economic profit. More firms enter the market, shifting each firm’s demand curve from D to D⬘. In
the new long-run equilibrium, the price ($55) is
higher than before, even with more firms in the
market.
Price (dollars per unit)
562
AC
$55
50
AC'
D'
D
Quantity (units per month)
Either or both of these factors would cause the typical firm’s demand curve to become
steeper as more firms enter, as Figure 13.16 shows. When demand shifts in this fashion due to new entry, each firm’s output could fall by such a large amount that it moves
to a higher point along its new average cost curve. At the new long-run equilibrium,
more firms are in the market, but each firm is smaller than before and charges a higher
price.
A P P L I C A T I O N
13.8
When a Good Doctor Is Hard to Find
Local markets for doctors are a good example of
monopolistic competition. Different doctors produce
differentiated products, and entry and exit are not
difficult. Mark Pauly and Mark Satterthwaite studied
the relationship between price and the number of
physicians in 92 metropolitan markets in the United
States.38 After controlling for demographic and market factors that might plausibly affect the average
price of a patient’s visit to a primary care doctor, Pauly
and Satterthwaite found that an increase in the number of primary care physicians per square mile (a
measure of the number of primary care doctors in the
local market) was associated with an increase in the
average price per office visit. In other words, markets
with more firms also had higher prices.
38
What would explain this finding? Pauly and
Satterthwaite observe that consumers search among
physicians mainly by asking friends, relatives, or coworkers for recommendations. In local markets with
a small number of doctors (three or four, for example), search is easy: Each doctor will probably have a
well-known reputation throughout the market. Most
consumers will probably have a pretty clear impression of each doctor and the prices he or she charges.
However, in markets with many physicians, it is probably harder for consumers to keep straight the various
pieces of information they might learn about different doctors in the market. As a result, consumer
search tends to be much less efficient. Because it is
harder for consumers to comparison shop, consumers
might become less sensitive to price in markets in
which there are many doctors. In such markets, an
M. Pauly and M. Satterthwaite, “The Pricing of Primary Care Physicians’ Services: A Test of the Role
of Consumer Information,” Bell Journal of Economics 12 (1982): 488–506.
C H A P T E R S U M M A RY
individual physician’s demand curve would be more
likely to resemble D⬘ in Figure 13.16 than D.
To explore whether the efficiency of consumer
search might have had something to do with the pattern of prices they observed, Pauly and Satterthwaite
looked at whether physicians’ prices in markets in which
a large proportion of the population had recently
563
moved (and thus had poorer information about local
doctors) were higher than in markets in which households were more settled. They were. This and other
evidence they collected suggests that the efficiency of
the consumer search process is an important determinant of prices in local physicians markets.
CHAPTER SUMMARY
• In a homogeneous products oligopoly, a small number
of firms sell virtually identical products. In a dominant
firm market, one firm has a large share of the market and
competes against numerous smaller firms, with all firms
offering virtually identical products. In a differentiated
products oligopoly, a small number of firms sell differentiated products. Under monopolistic competition, many
firms sell differentiated products.
• The four-firm concentration ratio (4CR) and the
Herfindahl-Hirschman Index (HHI) are two quantitative metrics used to describe market structures.
• The Cournot model of a homogeneous products
oligopoly presumes that each firm is a quantity taker—the
firm accepts its rivals’ outputs as given and then produces
an output that maximizes its profit. At a Cournot equilibrium, each firm’s output is a best response to all other
firms’ outputs, and no firm has any after-the-fact regrets
about its output choice. (LBD Exercises 13.1, 13.2)
• The Cournot model applies to firms that make a single, once-and-for-all decision on output. The Cournot
equilibrium is a natural outcome when firms simultaneously choose output on a once-and-for-all basis and have
full confidence in the rationality of their rivals.
• Cournot firms have market power. The Cournot
equilibrium price will be less than the monopoly price
but greater than the perfectly competitive price. (LBD
Exercise 13.2)
• We can reconcile the different predictions made
about industry equilibrium in the Cournot and Bertrand
models in two ways. First, the Cournot model can be
thought of as pertaining to long-run capacity competition, while the Bertrand model can be thought of as pertaining to short-run price competition. Second, the two
models make different assumptions about the expectations each firm has about its rivals’ reactions to its competitive moves.
• In the Stackelberg model of oligopoly, one firm (the
leader) makes its quantity choice first. The other firm
(the follower) observes that output and then makes its
quantity choice.
• In the Stackelberg model, the leader generally produces a higher quantity of output than it does in the
Cournot equilibrium, while the follower produces less
than its Cournot equilibrium output. By choosing
its quantity first, the leader can manipulate the follower’s
output choice to its advantage. As a result, the leader
earns a higher profit than it would have earned at the
Cournot equilibrium.
• In a dominant firm market, the dominant firm takes
the competitive fringe’s supply curve into account in setting a price. If the fringe’s supply is growing over time, the
dominant firm’s price will fall and its share of the market
might also fall. To prevent this, the dominant firm might
follow a strategy of limit pricing. (LBD Exercise 13.3)
• We can characterize the Cournot equilibrium using
a modified inverse elasticity pricing rule (IEPR).
• Two products are vertically differentiated when consumers view one product as unambiguously better or
worse than the other. Two products are horizontally differentiated when some consumers regard one as a poor
substitute for the other, while other consumers have the
opposite opinion.
• In the Bertrand model of a homogeneous products
oligopoly, each firm selects a price to maximize profits,
given the prices other firms set. If all firms have the same
constant marginal cost, the Bertrand equilibrium price is
equal to marginal cost.
• In a Bertrand equilibrium with differentiated products, equilibrium prices generally exceed marginal cost.
When horizontal product differentiation between the
firms is significant, the gap between prices and marginal
costs can be substantial. (LBD Exercise 13.4)
• With a larger number of firms in the industry, the
Cournot equilibrium industry output goes up and the
equilibrium market price goes down.
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• In a monopolistically competitive market, each firm
faces a downward-sloping demand curve. A short-run
equilibrium is attained when every firm chooses a profitmaximizing price, given the prices of all other firms. In a
long-run equilibrium, free entry drives firms’ economic
profits to zero.
• Under some conditions, the entry of more firms into
a monopolistically competitive market can result in a
long-run equilibrium with a higher price than before the
new entry.
REVIEW QUESTIONS
1. Explain why, at a Cournot equilibrium with two
firms, neither firm would have any regret about its output
choice after it observes the output choice of its rival.
2. What is a reaction function? Why does the Cournot
equilibrium occur at the point at which the reaction
functions intersect?
3. Why is the Cournot equilibrium price less than the
monopoly price? Why is the Cournot equilibrium price
greater than the perfectly competitive price?
4. Explain the difference between the Bertrand model
of oligopoly and the Cournot model of oligopoly. In a
homogeneous products oligopoly, what predictions do
these models make about the equilibrium price relative to
marginal cost?
5. What is the role played by the competitive fringe in
the dominant firm model of oligopoly? Why does an
increase in the size of the fringe result in a reduction in
the dominant firm’s profit-maximizing price?
6. What is the difference between vertical product differentiation and horizontal product differentiation?
7. Explain why, in the Bertrand model of oligopoly
with differentiated products, a greater degree of product
differentiation is likely to increase the markup between
price and marginal cost.
8. What are the characteristics of a monopolistically
competitive industry? Provide an example of a monopolistically competitive industry.
9. Why is it the case in a long-run monopolistically
competitive equilibrium that the firm’s demand curve is
tangent to its average cost curve? Why could it not be a
long-run equilibrium if the demand curve “cut through”
the average cost curve?
PROBLEMS
13.1. Beryllium oxide is a chemical compound used in
pharmaceutical applications. Beryllium oxide can only be
made in one particular way, and all firms produce their
version of beryllium oxide to the exact same standards of
purity and safety. The largest firms have market shares
given in the following table:
Firm
Mercury
Mars
Jupiter
Saturn
Market Share
80%
1%
1%
1%
a) What is the four-firm (4CR) concentration ratio for
this industry?
b) What is the Herfindahl-Hirschman Index (HHI) for
this industry?
c) Of the market structures described in Table 13.1,
which one best describes the beryllium oxide industry?
13.2. The cola industry in the country of Inner
Baldonia consists of five sellers: two global brands, Coke
and Pepsi, and three local competitors, Bright, Quite,
and Zight. Consumers view these products as similar, but
not identical. The market shares of the five sellers are as
follows:
Firm
Coca-Cola
Zight
Pepsi
Bright
Quite
Market Share
25%
24%
23%
20%
8%
a) What is the 4CR concentration ratio for this industry?
b) What is the HHI for this industry?
c) Of the market structures described in Table 13.1, which
one best describes the cola industry in Inner Baldonia?
13.3. Outer Baldonia is a largely rural country with
many small towns. Each town typically contains a retail
store selling livestock feed. In virtually all towns, there is
only one such store. The farmers who purchase feed
PROBLEMS
from these stores typically live outside of town. Often,
they will purchase from a store in the town closest to
them, but if farmers learn through word of mouth that
a feed retailer in a more distant town is selling feed
at a lower price, they will sometimes go to that store to
obtain feed.
The countrywide market shares of the largest feed
stores in Outer Baldonia are shown in the following
table:
Firm
Ben’s Feed and Supplies
Joe’s Hog and Cattle Supply
Hogwarts
Dave’s Livestock and Tools
Ron’s Supply Shed
Eddie’s Feed Coop
Market Share
2%
1%
1%
0.50%
0.25%
0.25%
a) What is the 4CR concentration ratio for the livestock
feed store market in Outer Baldonia?
b) What is the HHI for this industry?
c) Of the market structures described in Table 13.1,
which one best describes the livestock feed market in
Outer Baldonia?
13.4. In the following, let the market demand curve be
P 70 ⫺ 2Q, and assume all sellers can produce at a
constant marginal cost of c 10, with zero fixed costs.
a) If the market is perfectly competitive, what is the
equilibrium price and quantity?
b) If the market is controlled by a monopolist, what is
the equilibrium price and quantity? How much profit
does the monopolist earn?
c) Now suppose that Amy and Beau compete as Cournot
duopolists. What is the Cournot equilibrium price?
What is total market output, and how much profit does
each seller earn?
13.5. A homogeneous products duopoly faces a
market demand function given by P 300 ⫺ 3Q, where
Q Q1 ⫹ Q2. Both firms have a constant marginal cost
MC 100.
a) What is Firm 1’s profit-maximizing quantity, given
that Firm 2 produces an output of 50 units per year?
What is Firm 1’s profit-maximizing quantity when Firm
2 produces 20 units per year?
b) Derive the equation of each firm’s reaction curve and
then graph these curves.
c) What is the Cournot equilibrium quantity per firm
and price in this market?
d) What would the equilibrium price in this market be if
it were perfectly competitive?
e) What would the equilibrium price in this market be if
the two firms colluded to set the monopoly price?
565
f ) What is the Bertrand equilibrium price in this market?
g) What are the Cournot equilibrium quantities and industry price when one firm has a marginal cost of 100 but
the other firm has a marginal cost of 90?
13.6. Zack and Andon compete in the peanut market.
Zack is very efficient at producing nuts, with a low
marginal cost cZ 1; Andon, however, has a constant
marginal cost cA 10. If the market demand for nuts is
P 100 ⫺ Q, find the Cournot equilibrium price and the
quantity and profit level for each competitor.
13.7. Let’s consider a market in which two firms compete as quantity setters, and the market demand curve is
given by Q 4000 ⫺ 40P. Firm 1 has a constant marginal
cost equal to MC1 20, while Firm 2 has a constant marginal cost equal to MC2 40.
a) Find each firm’s reaction function.
b) Find the Cournot equilibrium quantities and the
Cournot equilibrium price.
13.8. In a homogeneous products duopoly, each firm
has a marginal cost curve MC 10 ⫹ Qi, i 1, 2. The
market demand curve is P 50 ⫺ Q, where Q Q1 ⫹ Q2.
a) What are the Cournot equilibrium quantities and
price in this market?
b) What would be the equilibrium price in this market if
the two firms acted as a profit-maximizing cartel?
c) What would be the equilibrium price in this market if
firms acted as price-taking firms?
13.9. Suppose that demand for cruise ship vacations is
given by P 1200 ⫺ 5Q, where Q is the total number of
passengers when the market price is P.
a) The market initially consists of only three sellers,
Alpha Travel, Beta Worldwide, and Chi Cruiseline. Each
seller has the same marginal cost of $300 per passenger.
Find the symmetric Cournot equilibrium price and output
for each seller.
b) Now suppose that Beta Worldwide and Chi
Cruiseline announce their intention to merge into a single
firm. They claim that their merger will allow them to
achieve cost savings so that their marginal cost is less
than $300 per passenger. Supposing that the merged
firm, BetaChi, has a marginal cost of c ⬍ $300, while
Alpha Travel’s marginal cost remains at $300, for what
values of c would the merger raise consumer surplus
relative to part (a)?
13.10. A homogeneous products oligopoly consists of
four firms, each of which has a constant marginal cost
MC 5. The market demand curve is given by P 15 ⫺ Q.
a) What are the Cournot equilibrium quantities and
price? Assuming that each firm has zero fixed costs, what
is the profit earned by each firm in equilibrium?
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b) Suppose Firms 1 and 2 merge, but their marginal cost
remains at 5. What are the new Cournot equilibrium quantities and price? Is the profit of the merged firm bigger or
smaller than the combined profits of Firms 1 and 2 in the
initial equilibrium in part (a)? Provide an explanation for
the effect of the merger on profit in this market.
13.11. An industry is known to face market price elasticity of demand ⑀Q, P ⫽ ⫺3. (Assume this elasticity as
constant as the industry moves along its demand curve.)
The marginal cost of each firm in this industry is $10 per
unit, and there are five firms in the industry. What would
the Lerner Index be at the Cournot equilibrium in this
industry?
13.12. Besanko, Inc., is one of two Cournot dupolists in
the market for gizmos. It and its main competitor
Schmedders Ltd. face a downward-sloping market
demand curve. Each firm has an identical marginal cost
that is independent of output. Please indicate how the following will affect Besanko’s and Schmedders’s reaction
functions, and the Cournot equilibrium quantities produced by Besanko and Schmedders.
a) Leading safety experts begin to recommend that all
home owners should replace their smoke detectors with
gizmos.
b) Besanko and Schmedders’s gizmos are made out of
platinum, with each gizmo requiring 1 kg of platinum.
The price of platinum goes up.
c) Besanko, Inc.’s total fixed cost increases.
d) The government imposes an excise tax on gizmos
produced by Schmedders, but not on those produced by
Besanko.
13.13. Suppose that firms in a two-firm industry choose
quantities every month, and each month the firms sell at
the market-clearing price determined by the quantities
they choose. Each firm has a constant marginal cost, and
the market demand curve is linear of the form P a ⫺
bQ, where Q is total industry quantity and P is the market
price. Suppose that initially each firm has the same constant marginal cost. Further suppose that each month the
firms attain the Cournot equilibrium in quantities.
a) Suppose that it is observed that from one month to the
next Firm 1’s quantity goes down, Firm 2’s quantity goes
up, and the market price goes up. A change in the demand
and/or cost conditions consistent with what we observe is:
i) The market demand curve shifted leftward in a parallel
fashion.
ii) The market demand curve shifted rightward in a parallel fashion.
iii) Firm 1’s marginal cost went up, while Firm 2’s marginal cost stayed the same.
iv) Firm 2’s marginal cost went up, while Firm 1’s marginal cost stayed the same.
v) All of the above are possible.
b) Suppose that it is observed that from one month to
the next, Firm 1’s quantity goes down, Firm 2’s quantity
goes down, and the market price goes down. A change in
the demand and/or cost conditions consistent with what
we observe is:
i) The market demand curve shifted leftward in a parallel fashion.
ii) The market demand curve shifted rightward in a parallel fashion.
iii) Firm 1’s marginal cost went up, while Firm 2’s marginal cost stayed the same.
iv) Firm 2’s marginal cost went down, while Firm 1’s
marginal cost stayed the same.
v) All of the above are possible.
c) Suppose that it is observed that from one month to
the next, Firm 1’s quantity goes up, Firm 2’s quantity
goes up, and the market price goes up. A change in the
demand and/or cost conditions consistent with what we
observe is:
i) The market demand curve shifted leftward in a parallel fashion.
ii) The market demand curve shifted rightward in a parallel fashion.
iii) Both firms’ marginal costs went up by the same
amount.
iv) Both firms’ marginal costs went down by the same
amount.
v) All of the above are possible.
d) Suppose that it is observed that from one month to the
next, Firm 1’s quantity goes up, Firm 2’s quantity goes
up, and the market price goes down. A change in the
demand and/or cost conditions consistent with what we
observe is:
i) The market demand curve shifted leftward in a parallel fashion.
ii) The market demand curve shifted rightward in a parallel fashion.
iii) Both firms’ marginal costs went up by the same
amount.
iv) Both firms’ marginal costs went down by the same
amount.
v) All of the above are possible.
13.14. An industry consists of two Cournot firms selling
a homogeneous product with a market demand curve
given by P ⫽ 100 ⫺ Q1 ⫺ Q2. Each firm has a marginal
cost of $10 per unit.
a) Find the Cournot equilibrium quantities and price.
b) Find the quantities and price that would prevail if the
firms acted “as if ” they were a monopolist (i.e., find the
collusive outcome).
PROBLEMS
c) Suppose Firms 1 and 2 sign the following contract.
Firm 1 agrees to pay Firm 2 an amount equal to T dollars for every unit of output it (Firm 1) produces.
Symmetrically, Firm 2 agrees to pay Firm 1 an amount
T dollars for every unit of output it (Firm 2) produces.
The payments are justified to the government as a crosslicensing agreement whereby Firm 1 pays a royalty for
the use of a patent developed by Firm 2, and similarly,
Firm 2 pays a royalty for the use of a patent developed
by Firm 1. What value of T results in the firms achieving the collusive outcome as a Cournot equilibrium?
d) Draw a picture involving reaction functions that
shows what is going on in this situation.
13.15. Consider an oligopoly in which firms choose
quantities. The inverse market demand curve is given by
P 280 ⫺ 2(X ⫹ Y ), where X is the quantity of Firm 1,
and Y is the quantity of Firm 2. Each firm has a marginal
cost equal to 40.
a) What is the Cournot equilibrium outputs for each
firm? What is the market price at the Cournot equilibrium? What is the profit of each firm?
b) What is the Stackelberg equilibrium, when Firm 1
acts as the leader? What is the market price at the
Stackelberg equilibrium? What is the profit of each firm?
13.16. The market demand curve in a commodity
chemical industry is given by Q 600 ⫺ 3P, where Q is
the quantity demanded per month and P is the market
price in dollars. Firms in this industry supply quantities
every month, and the resulting market price occurs at the
point at which the quantity demanded equals the total
quantity supplied. Suppose there are two firms in this
industry, Firm 1 and Firm 2. Each firm has an identical
constant marginal cost of $80 per unit.
a) Find the Cournot equilibrium quantities for each
firm. What is the Cournot equilibrium market price?
b) Assuming that Firm 1 is the Stackelberg leader, find
the Stackelberg equilibrium quantities for each firm.
What is the Stackelberg equilibrium price?
c) Calculate and compare the profit of each firm under the
Cournot and Stackelberg equilibria. Under which equilibrium is overall industry profit the greatest, and why?
13.17. Consider a market in which the market demand
curve is given by P 18 ⫺ X ⫺ Y, where X is Firm 1’s
output, and Y is Firm 2’s output. Firm 1 has a marginal
cost of 3, while Firm 2 has a marginal cost of 6.
a) Find the Cournot equilibrium outputs in this market.
How much profit does each firm make?
b) Find the Stackelberg equilibrium in which Firm 1 acts
as the leader. How much profit does each firm make?
13.18. Consider a market in which we have two firms,
one of which will act as the Stackelberg leader and the
other as the follower. As we know, this means that each
567
firm will choose a quantity, X (for the leader) and Y (for
the follower). Imagine that you have determined the
Stackelberg equilibrium for a particular linear demand
curve and set of marginal costs. Please indicate how X
and Y would change if we then “perturbed” the initial
situation in the following way:
a) The leader’s marginal cost goes down, but the follower’s marginal cost stays the same.
b) The follower’s marginal cost goes down, but the
leader’s marginal cost stays the same.
13.19. Suppose that the market demand for cobalt is
given by Q 200 ⫺ P. Suppose that the industry consists
of 10 firms, each with a marginal cost of $40 per unit.
What is the Cournot equilibrium quantity for each firm?
What is the equilibrium market price?
13.20. Consider the same setting as in the previous problem, but now suppose that the industry consists of a dominant firm, Braeutigam Cobalt (BC), which has a constant
marginal cost equal to $40 per unit. There are nine other
fringe producers, each of whom has a marginal cost curve
MC 40 ⫹ 10q, where q is the output of a typical fringe
producer. Assume there are no fixed costs for any producer.
a) What is the supply curve of the competitive fringe?
b) What is BC’s residual demand curve?
c) Find BC’s profit-maximizing output and price. At this
price, what is BC’s market share?
d) Repeat parts (a) to (c) under the assumption that the
competitive fringe consists of 18 firms.
13.21. Apple’s iPod has been the portable MP3-player
of choice among many gadget enthusiasts. Suppose that
Apple has a constant marginal cost of 4 and that market
demand is given by Q 200 ⫺ 2P.
a) If Apple is a monopolist, find its optimal price and
output. What are its profits?
b) Now suppose there is a competitive fringe of 12
price-taking firms, each of which has a total cost function
TC(q) 3q2 ⫹ 20q with corresponding marginal cost
curve MC 6q ⫹ 20. Find the supply function of the
fringe (Hint: A competitive firm supplies along its marginal cost curve above its shutdown point).
c) If Apple operates as the dominant firm facing competition from the fringe in this market, now what is its
optimal output? How many units will fringe providers
sell? What is the market price, and how much profit does
Apple earn?
d) Graph your answer to part (c).
13.22. Britney produces pop music albums with the
total cost function TC(Q) 8Q. Market demand for pop
music albums is P 56 ⫺ Q. Suppose there is a competitive fringe of price-taking pop music artists, with total
supply function Qfringe 2P ⫺ y, where y ⬎ 0 is some
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positive integer. If Britney behaves like a dominant firm
and maximizes her profit by selling at a price of P 16,
find (i) the value of y, (ii) Britney’s output level, and
(iii) the output level of the competitive fringe.
13.23. The market demand curve in the nickel industry
in Australia is given by Q d 400 ⫺ 8P. The industry is
dominated by a large firm with a constant marginal cost
of $10 per unit. There also exists a competitive fringe of
100 firms, each of which has a marginal cost given by
MC 10 ⫹ 50q, where q is the output of a typical fringe
firm.
a) What is the equation of the supply curve for the competitive fringe?
b) Restricting your attention to the range of prices that
exceed the dominant firm’s marginal cost, what is the
equation of the residual demand curve?
c) What is the profit-maximizing quantity of the dominant firm? What is the resulting market price? At this
price, how much does the competitive fringe produce,
and what is the fringe’s market share (i.e., the fringe
quantity divided by total industry quantity)? What is the
dominant firm’s market share?
d) Let’s consider a twist on the basic dominant firm
model. Suppose the Australia government, concerned
about the amount of dominance in the nickel industry
decides to break the dominant firm into two identical
firms, each with a constant marginal cost of $10 per unit.
Suppose further that these two firms act as Cournot
quantity setters, taking into account the supply curve of
the competitive fringe. What is the Cournot equilibrium
quantity produced by each dominant firm? What is the
equilibrium market price? At this price, how much does
the competitive fringe produce, and what is the fringe’s
market share?
13.24. Consider the Coke and Pepsi example discussed
in the chapter.
a) Explain why each firm’s reaction function slopes upward. That is, why does Coke’s profit-maximizing price
go up the higher is Pepsi’s price? Why does Pepsi’s
profit-maximizing price go up the higher Coke’s price is?
b) Explain why Pepsi’s profit-maximizing price seems to
be relatively insensitive to Coke’s price. That is, why is
Pepsi’s reaction function so flat?
13.25. Again consider the Coke and Pepsi example
discussed in the chapter. Use graphs of reaction functions to illustrate what would happen to equilibrium
prices if:
a) Coca-Cola’s marginal cost increased.
b) For any pair of prices for Coke and Pepsi, Pepsi’s demand went up.
13.26. Two firms, Alpha and Bravo, compete in the
European chewing gum industry. The products of the
two firms are differentiated, and each month the two
firms set their prices. The demand functions facing each
firm are:
QA 150 ⫺ 10PA ⫹ 9PB
QB 150 ⫺ 10PB ⫹ 9PA
where the subscript A denotes the firm Alpha and the
subscript B denotes the firm Bravo. Each firm has a constant marginal cost of $7 per unit.
a) Find the equation of the reaction function of each firm.
b) Find the Bertrand equilibrium price of each firm.
c) Sketch how each firm’s reaction function is affected by
each of the following changes:
i) Alpha’s marginal cost goes down (with Bravo’s marginal cost remaining the same).
ii) Alpha and Bravo’s marginal cost goes down by the
same amount.
iii) Demand conditions change so that the “150” term in
the demand function now becomes larger than 150.
iv) The “10” and “9” terms in each demand function
now become larger (e.g., they become “50” and “49,”
respectively).
d) Explain in words how the Bertrand equilibrium price
of each firm is affected by each of the following changes:
i) Alpha’s marginal cost goes down (with Bravo’s marginal cost remaining the same).
ii) Alpha and Bravo’s marginal cost goes down by the
same amount.
iii) Demand conditions change so that the “150” term in
the demand function for each firm now becomes larger
than 150.
iv) The “10” and “9” terms in each demand function
now become larger (e.g., they become “50” and “49,”
respectively).
13.27. When firms choose outputs, as in the Cournot
model, reaction functions slope downward. But when
firms choose prices, as in the Bertrand model with differentiated products, reaction functions slope upward. Why
do output reaction functions differ from price reaction
functions in this way?
13.28. Suppose that Jerry and Teddy are the only two
sellers of designer umbrellas, which consumers view as
differentiated products. For simplicity, assume each seller
has a constant marginal cost equal to zero. When Jerry
charges a price pJ and Teddy charges pT, consumers
would buy a total of
qJ 100 ⫺ 3pJ ⫹ pT
PROBLEMS
umbrellas from Jerry. In similar fashion, Teddy faces a
demand curve of
qT 100 ⫺ 3pT ⫹ pJ
Illustrate each seller’s best-response function on a graph.
What are the equilibrium prices? How much profit does
each seller earn?
13.29. United Airlines and American Airlines both fly
between Chicago and San Francisco. Their demand
curves are given by QA 1000 ⫺ 2PA ⫹ PU and QU
1000 ⫺ 2PU ⫹ PA.
QA and QU stand for the number of passengers per
day for American and United, respectively. The marginal
cost of each carrier is $10 per passenger.
a) If American sets a price of $200, what is the equation
of United’s demand curve and marginal revenue curve?
What is United’s profit-maximizing price when
American sets a price of $200?
b) Redo part (a) under the assumption that American
sets a price of $400.
c) Derive the equations for American’s and United’s
price reaction curves.
d) What is the Bertrand equilibrium in this market?
13.30. Three firms compete as Bertrand price competitors in a differentiated products market. Each of the
three firms has a marginal cost of 0. The demand curves
of each firm are as follows:
Q1 80 ⫺ 2P1 ⫹ P23
Q2 80 ⫺ 2P2 ⫹ P13
Q3 80 ⫺ 2P3 ⫹ P12
where P23 is the average of the prices charged by Firms 2
and 3, P13 is the average of the prices charged by Firms 1
and 3, and P12 is the average of the prices charged by
Firms 1 and 2 [e.g., P12 0.5(P1 ⫹ P2)]. What is the
Bertrand equilibrium price charged by each firm?
13.31. The Baldonian shoe market is served by a monopoly firm. The demand for shoes in Baldonia is given
by Q 10 ⫺ P, where Q is millions of pairs of shoes (a
right shoe and left shoe) per year, and P is the price of a
pair of shoes. The marginal cost of making shoes is constant and equal to $2 per pair.
a) At what price would the Baldonian monopolist sell
shoes? How many shoes are purchased?
b) Baldonian authorities have concluded that the shoe
sellers monopoly power is not a good thing. Inspired by
the U.S. government’s attempt several years ago to break
Microsoft into two pieces, Baldonia creates two firms:
one that sells right shoes and the other that sells left
shoes. Let P1 be the price charged by the right-shoe
569
producer and P2 be the price charged by the left-shoe
producer. Of course, consumers still want to buy a pair of
shoes (a right one and a left one), so the demand for pairs
of shoes continues to be 10 ⫺ P1 ⫺ P2. If you think about
it, this means that the right-shoe producer sells 10 ⫺ P1 ⫺
P2 right shoes, while the left-shoe producer sells 10 ⫺
P1 ⫺ P2 left shoes. Since the marginal cost of a pair of
shoes is $2 per pair, the marginal cost of the right-shoe
producer is $1 per shoe, and the marginal cost of the
left-shoe producer is $1 per shoe.
i) Derive the reaction function of the right-shoe producer (P1 in terms of P2). Do the same for the left-shoe
producer.
ii) What is the Bertrand equilibrium price of shoes?
How many pairs of shoes are purchased?
iii) Has the breakup of the shoe monopolist improved
consumer welfare?
Note: To see the potential relevance of this problem to
the Microsoft antitrust case, you might be interested in
reading Paul Krugman, “The Parable of Baron von
Gates,” New York Times (April 26, 2000).
13.32. Reconsider Problem 13.29, except suppose
American and United take each other’s quantity as given
rather than taking each other’s price as given. That is,
assume that American and United act as Cournot competitors rather than Bertrand competitors. The inverse
demand curves corresponding to the demand curves in
Problem 13.29 are39
2
1
QA ⫺ Q U
3
3
1
2
PU 1000 ⫺ QU ⫺ QA
3
3
a) Suppose that American chooses to carry 660 passengers per day (i.e., QA 660). What is United’s profitmaximizing quantity of passengers? Suppose American
carries 500 passengers per day. What is United’s profitmaximizing quantity of passengers?
b) Derive the quantity reaction function for each firm.
c) What is the Cournot equilibrium in quantities for
both firms? What are the corresponding equilibrium
prices for both firms?
d) Why does the Cournot equilibrium in this problem
differ from the Bertrand equilibrium in Problem 13.29?
PA 1000 ⫺
13.33. Let’s imagine that a local retail market is monopolistically competitive. Each firm (and potential entrant)
is identical and faces a marginal cost that is independent
39
We derived the inverse demand curves by solving the two demand curves simultaneously for the prices, PA and PU, in terms of
the quantities, QA and QU.
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of output and is equal to $100 per unit. Each firm has an
annual fixed cost of $300,000 per month. Because each
active firm perceives itself facing a price elasticity of demand equal to ⫺2, the inverse elasticity pricing condition
implies that the profit-maximizing price for each firm is
(P ⫺ 100)/P 1/2 or P 200. If each firm charges an
equal price, they will evenly split the overall market demand of 96,000 units per month.
a) How many firms will operate in this market at a longrun equilibrium?
b) How would your answer change if each firm faced a
price elasticity of demand of ⫺4/3 and charged a profitmaximixing price of $400 per unit?
13.34. The Thai food restaurant business in Evanston,
Illinois, is monopolistically competitive. Suppose that
each existing and potential restaurant has a total cost
function given by TC 10Q ⫹ 40,000, where Q is the
number of patrons per month and TC is total cost per
month. The fixed cost of $40,000 includes fixed operating expenses (such as the salary of the chef ), the lease on
the building space where the restaurant is located, and
interest expenses on the bank loan needed to start the
business in the first place.
Currently, there are 10 Thai restaurants in
Evanston. Each restaurant faces a demand function given
P ⫺5P 4 where P is the price of a typical enby Q 4,000,000
N
trée at the restaurant, P is the price of a typical entrée averaged over all the other Thai restaurants in Evanston,
and N is the total number of restaurants. Each restaurant
takes the prices of other Thai restaurants as given when
choosing its own price.
a) What is the own-price elasticity of demand facing a
typical restaurant?
b) For a typical restaurant, what is the profit-maximizing
price of a typical entrée?
c) At the profit-maximizing price, how many patrons
does a typical restaurant serve per month? Given this
number of patrons, what is the average total cost of a typical restaurant?
d) What is the long-run equilibrium number of Thai
restaurants in the Evanston market?
A P P E N D I X : The Cournot Equilibrium and the Inverse Elasticity Pricing Rule
At a Cournot equilibrium, each firm equates its marginal cost to the marginal revenue
corresponding to its residual demand curve:
P* ⫹
¢P
Q* MC,
¢Q i
for i 1, 2, . . . , N
(A.1)
where Q*i is firm i ’s equilibrium output. Rearranging condition (A.1) gives us
P* ⫺ MC
¢P Q*i
⫽⫺
P*
¢Q P*
(A.2)
Multiplying the top and bottom of the right-hand side of (A.2) by overall market output
Q* gives us
P* ⫺ MC
¢P Q* Q*i
⫽ ⫺a
b
P*
¢Q P * Q*
(A.3)
Now note that (¢P/¢Q)(Q*/P*) 1/⑀Q, P (i.e., the inverse of the price elasticity of demand). Moreover, note that Q*i /Q* firm i’s equilibrium market share. If all firms are
identical, then they will split the market evenly. Thus, Q*i /Q* 1/N. We can thus write
the Cournot equilibrium condition in (A.3) as a modified inverse elasticity pricing rule:
P * ⫺ MC
1
1
⫽⫺
⫻
⑀Q, P
P*
N
(A.4)
14
GAME THEORY AND
STRATEGIC BEHAVIOR
14.1
THE CONCEPT OF NASH
EQUILIBRIUM
APPLICATION 14.1
Everyone Loses Except
the Lawyers
Chicken in Orbit: Winning
the Battle for Satellite Radio in North America
APPLICATION 14.3 Bank Runs
APPLICATION 14.2
14.2
T H E R E P E AT E D P R I S O N E R S ’
DILEMMA
Shoot-to-Kill, Live-and-LetLive, or Tit-for-Tat?
APPLICATION 14.5
Collusion in Japanese Sumo
Wrestling
APPLICATION 14.6 The Cost of War
APPLICATION 14.4
14.3
S E Q U E N T I A L - M OV E G A M E S
A N D S T R AT E G I C M OV E S
Irreversibility and Credible
Strategies by Airlines
APPLICATION 14.7
What’s in a Game?
The market for automobiles in China experienced a boom during the first decade of the new millennium.
By 2004 the number of automobiles in Beijing alone was growing at 1,000 per week, and in some years the
demand nationally was growing at as much as 50 percent per year.1 A wave of investment in production
capacity transformed a country that had few private automobiles 25 years ago. By 2009 the number of light
vehicles sold in China had equaled the number sold in the United States, a remarkable benchmark achieved
several years earlier than experts had predicted before the unfolding of the worldwide economic crisis.2
1
“The Rich Hit the Road,” The Economist, June 17, 2004.
“Motoring Ahead: More Cars Are Now Sold in China than in America,” The Economist online, October 23,
2009, http://www.economist.com/daily/news/displaystory.cfm?story_id=14732026&fsrc=nwl (accessed
May 4, 2010).
2
571
Major automobile firms like Honda and Toyota often relish the opportunity to enter growing markets
around the world, and they, along with other producers, have entered the Chinese market. But they have
learned that they must think about more than the growth in demand when they make decisions about
adding production capacity to any market, even one that is growing rapidly. Automobile plants are expensive, and the profitability of a new plant depends on many factors, including decisions made by rival firms.
If production capacity grows too fast in China, “the milk and honey could dry up,” and China could “be a
much tougher market than many people think.”3
Honda and Toyota have faced similar decisions about entry in other markets at other times. For
example, in the late 1990s, both Honda and Toyota had to decide whether to build new auto assembly
plants in North America.4 By adding more production capacity, each firm would be able to sell more cars
in the United States and Canada. On the face of it, the decision to add capacity seemed sound. Both
Honda and Toyota were making money from the cars they sold in North America, and by selling more
cars each company would make even more money.5 But because demand in the North American automobile market was not growing that fast, a decision by both firms to build new plants and increase
production would probably make prices on competing models (e.g., Honda Civics and Toyota Corollas)
lower than they otherwise would be. It seemed possible that if both firms built new plants, both would
be worse off than if neither built new plants. Each firm’s decision making was thus complicated by the
interdependence between its decision and that of
its rival. Each firm would need to take into account
the probable behavior of the other.
Game theory is the branch of microeconomics
concerned with the analysis of optimal decision
making in competitive situations, in which the
actions of each decision maker have a significant
impact on the fortunes of rival decision makers.
Though the term game might sound frivolous, many
interesting situations can be studied as games. The
competitive interaction between Honda and Toyota
is one example. Other social interactions in which
game theory has been fruitfully applied include the
competition among buyers in auctions, races by
nations to accumulate nuclear weapons, and
competition between candidates in elections.
3
“Here Be Dragons,” The Economist (September 2, 2004).
See, for example, “Detroit Challenge: Japanese Car Makers Plan
Major Expansion of American Capacity,” The Wall Street Journal
(September 24, 1997), p. A1.
5
In addition, the Hondas and Toyotas built in the new plants in the
United States would be exempt from U.S. tariffs. Also, by building
U.S. plants, Honda and Toyota would insulate Japan from criticism by
U.S. politicians because the cars would be built by American workers.
4
572
573
14.1 THE CONCEPT OF NASH EQUILIBRIUM
Our goal in this chapter is to introduce you to the central ideas of game theory and to give you an
appreciation for the wide variety of competitive situations to which game theory can be applied. In
many ways, you began your study of game theory in Chapter 13. Most of the theories of oligopoly (e.g.,
Cournot, Bertrand) in that chapter are particular examples of game theory models. This chapter will
build on that foundation and equip you with basic game theory concepts and tools that will enable you
to analyze competitive interactions that arise in real life.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Explain the role of strategies and payoffs in a game.
• Identify dominant and dominated strategies in a game.
• Explain the difference between a pure strategy and a mixed strategy.
• Describe a Nash equilibrium.
• Solve for the Nash equilibria in one-shot games and repeated games.
• Solve for the Nash equilibria in simultaneous-move games and sequential games.
• Explain why some kinds of games can lead players to cooperate, while other kinds do not.
• Explain how limiting your options can have strategic value.
A SIMPLE GAME
14.1
To introduce the key ideas of game theory, we begin with the easiest kind of game to
analyze: a one-shot, simultaneous-move game. In this type of game, two or more players
make a single decision, at the same time. To illustrate, consider the competition between
Honda and Toyota described in the introduction. Recall that each firm faced the decision
of whether to build a new auto assembly plant. Table 14.1 shows the potential impact of
the two firms’ capacity expansion decisions. Each firm has two choices, or strategies—
build a new plant or do not build—and this gives rise to four capacity expansion scenarios. A player’s strategy in a game specifies the actions that the player might take under
every conceivable circumstance that the player might face. In a one-shot, simultaneousmove game, strategies are simple: they consist of a single decision.
In Table 14.1, the first entry in each cell is Honda’s annual economic profit (in
millions of dollars) under a scenario; the second entry is Toyota’s annual economic
profit (in millions of dollars).6 These profits represent the payoffs in the game: the
amount that each player can expect to get under different combinations of strategy
choices by the players. The payoffs in Table 14.1 show the extent to which the players
in this game are interdependent: Toyota’s payoff depends on what Honda does, and
vice versa. In game theory, a player will very rarely control its own fate. The payoffs
in Table 14.1 are fictitious but accurately reflect the dynamic that existed between
these two firms at the time.
THE CONCEPT
OF NASH
EQUILIBRIUM
6
In this, and in all subsequent tables in this chapter, we use the following convention. The first entry is
the payoff of the player listed on the side of the table—the so-called row player. The second entry is the
payoff of the player listed at the top of the table—the so-called column player.
game theory The
branch of microeconomics
concerned with the analysis
of optimal decision making
in competitive situations.
strategy A plan for the
actions that a player in a
game will take under every
conceivable circumstance
that the player might face.
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TABLE 14.1
Capacity Expansion Game between Toyota and Honda*
Toyota
Build a New Plant
Do Not Build
Honda
Build a New Plant
Do Not Build
16, 16
15, 20
20, 15
18, 18
*Payoffs are in millions of dollars.
THE NASH EQUILIBRIUM
Nash equilibrium A
situation in which each
player in a game chooses
the strategy that yields
the highest payoff, given
the strategies chosen by
the other players.
Game theory seeks to answer the question: What is the likely outcome of a game? To identify “likely outcomes” of games, game theorists use the concept of a Nash equilibrium.
At a Nash equilibrium, each player chooses a strategy that gives it the highest payoff, given
the strategies of the other players in the game. This is the same idea that we used in
Chapter 13 to define a Cournot equilibrium (in a quantity-setting oligopoly) and a
Bertrand equilibrium (in a price-setting oligopoly). In fact, both of these equilibria are
particular examples of the Nash equilibrium.
In this game, the Nash equilibrium strategy for each firm is “build a new plant.”
• Given that Toyota builds a new plant, Honda’s best response is also to build a
new plant: It gets a profit of $16 million if it builds but only $15 million if it
does not build. (Note: For the “row” player Honda, we compare payoffs between
the two rows.)
• Given that Honda builds a new plant, Toyota’s best response is to build: It gets a
profit of $16 million if it builds versus the $15 million it gets if it doesn’t expand
its capacity. (Note: For the “column” player Toyota, we compare payoffs between
the two columns.)
Why does the Nash equilibrium represent a plausible outcome of a game? Probably
its most compelling property is that the Nash equilibrium outcome is self-enforcing. If
each party expects the other party to choose its Nash equilibrium strategy, then both parties will, in fact, choose their Nash equilibrium strategies. At the Nash equilibrium, then,
expectation equals outcome—expected behavior and actual behavior converge. This
would not be true at non–Nash equilibrium outcomes, as the game in Table 14.1 illustrates. If Toyota (perhaps foolishly) expects Honda not to build a new plant but builds a
new plant of its own, then Honda—pursuing its own self-interest—would confound
Toyota’s expectations, build a new plant, and make Toyota worse off than it expected to be.
THE PRISONERS’ DILEMMA
prisoners’ dilemma
A game situation in which
there is a tension between
the collective interest of
all of the players and the
self-interest of individual
players.
The capacity-expansion game between Toyota and Honda illustrates a noteworthy aspect of a Nash equilibrium. The Nash equilibrium does not necessarily correspond to
the outcome that maximizes the aggregate profit of the players. Toyota and Honda
would be collectively better off by not building new plants. However, the rational pursuit of self-interest leads each party to take an action that is ultimately detrimental to
their collective interest.
This conflict between collective interest and self-interest is often referred to
as the prisoners’ dilemma. The game in Table 14.1, as well as both the Cournot
quantity-setting and Bertrand price-setting models from Chapter 13, are particular
14.1 THE CONCEPT OF NASH EQUILIBRIUM
TABLE 14.2
575
Prisoners’ Dilemma Game
Confess
Ron
Confess
Do Not Confess
5,
5
10, 0
David
Do Not Confess
0, 10
1, 1
examples of prisoners’ dilemma games—games in which the Nash equilibrium does not
coincide with the outcome that maximizes the collective payoffs of the players in the
game. The term prisoners’ dilemma is based on the following scenario: Two suspects in
a crime, David and Ron, are arrested and placed in separate cells. The police, who have
no real evidence against either, privately give each prisoner the chance to confess and
implicate the other suspect for the crime. They tell each prisoner that if neither confesses, both will be convicted on a minor charge and will serve just 1 year in jail. If both
confess, both will be convicted of the more serious crime but will be treated somewhat
leniently because they cooperated, and each will go to jail for 5 years. But if one suspect confesses and the other doesn’t, the one that confesses will go free, while the other
will be convicted of the crime and spend 10 years in jail. Table 14.2 shows the payoffs
for this game, with jail terms corresponding to negative payoffs.
The Nash equilibrium in this game is for each player to confess. Given that David
confesses, Ron gets a lighter jail term by confessing than by not confessing. And given
that Ron confesses, David gets a lighter jail term by confessing than by not confessing.
In equilibrium, both prisoners end up confessing and serving 5 years in jail, even though
collectively they would be better off not confessing and spending only 1 year in jail.
The prisoners’ dilemma is widely studied throughout the social sciences.
Psychologists, political scientists, sociologists, and economists find the prisoners’
dilemma a compelling scenario because the tension it portrays between an individual
player’s self-interest and a group’s collective interest shows up in many different ways
in the world around us. For example, business firms start price wars, even though all
firms in the industry get hurt as a result. Politicians run “attack ads” even though the
ill will and distrust they engender make it difficult for the winner of the election to
govern effectively. Analysis of the prisoners’ dilemma game can help us understand
why these apparently counterproductive outcomes can occur.
D O M I N A N T A N D D O M I N AT E D S T R AT E G I E S
Dominant Strategies
In the game between Toyota and Honda in Table 14.1, finding the Nash equilibrium
was easy because for each firm, the strategy “build a new plant” was better than “do
not build” no matter what strategy the other firm chose (e.g., if Toyota builds a new
plant, Honda gets $16 million instead of $15 million by building a new plant, too; if
Toyota doesn’t build, Honda gets $20 million instead of $18 million by building a new
plant). In this situation, we say that “build a new plant” is a dominant strategy. A
dominant strategy is a strategy that is better than any other strategy a player might
choose, no matter what strategy the other player follows. When a player has a dominant strategy, that strategy will be the player’s Nash equilibrium strategy.
Dominant strategies are not inevitable. In many games some or all players do not
have dominant strategies. Consider, for example, the capacity expansion game in
dominant strategy A
strategy that is better than
any other a player might
choose, no matter what
strategy the other player
follows.
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TABLE 14.3
Capacity Expansion Game between Marutti and Ambassador*
Ambassador
Build a New Plant
Do Not Build
Marutti
Build a New Plant
Do Not Build
12, 4
15, 6
20, 3
18, 5
*Payoffs are in millions of rupees.
Table 14.3 between Ambassador and Marutti in the automobile market in India. In
this market, Marutti is much bigger than Ambassador and makes better cars. It thus
gets far more profit than Ambassador does, no matter what capacity scenario occurs.
In this game, Marutti does not have a dominant strategy. It is better off not building a new plant if Ambassador builds one, but it prefers to build a new plant if
Ambassador doesn’t build. Despite the absence of a dominant strategy for Marutti,
there is still a Nash equilibrium: Ambassador builds a new plant, and Marutti doesn’t.
To see why, note that if Ambassador builds, Marutti’s best response is not to build:
Marutti gets 15 million rupees if it doesn’t build and only 12 million rupees if it builds.
And if Marutti does not build, Ambassador’s best response is to build: it gets 6 million
rupees if it builds but only 5 million rupees if it doesn’t build.
A P P L I C A T I O N
14.1
Everyone Loses Except the Lawyers
Modern American society has been criticized for being
excessively litigious. Individuals and firms seem increasingly willing to turn to lawyers to resolve their disputes.
But if, as is commonly argued, this dependence on litigation has significant social costs, why would a free
market system generate so much business for lawyers?
The research of two economists, Orley Ashenfelter
and David Bloom, suggests a possible answer.7 The decision to hire a lawyer to resolve a dispute is, they argue,
the result of a prisoners’ dilemma. Two parties in a dispute are collectively better off when they settle the dispute between themselves or hire a neutral arbitrator to
resolve their differences. But if a party believes that by
hiring a lawyer it will increase the odds of winning by a
sufficiently large amount to make hiring a lawyer worthwhile, it will be a dominant strategy to hire a lawyer. But
when both parties do this, the dispute is resolved no differently than if neither hired a lawyer, and each party is
worse off by the amount it pays its attorney.
To test this theory, Ashenfelter and Bloom analyzed public employee wage disputes from 1981 to
7
1984 in New Jersey. They also studied union grievance proceedings involving the rights of discharged
workers in Pennsylvania. In both cases, they found
strong evidence that hiring a lawyer is a dominant
strategy and that the decision to hire a lawyer reflects a prisoners’ dilemma type of situation. Based
on the New Jersey data, for example, they found that
when one party hired a lawyer, the chances of successfully persuading the arbitrator to accept its wage
proposal went up from roughly 50 to 75 percent.
When both sides hired lawyers, though, the odds of
winning remained roughly 50 percent, indicating
that the benefit of hiring a lawyer is canceled out
when the other party also hires a lawyer.
The possibility that hiring a lawyer is a Nash equilibrium outcome of a prisoners’ dilemma game suggests that making society less litigious is likely to
prove quite difficult. Lawyers clearly have no interest
in curbing the demand for their services, and the
logic of the prisoner’s dilemma suggests that a party
in a dispute has a strong individual incentive to hire
a lawyer, even though society as a whole would be
better off if he or she did not.
O. Ashenfelter and D. Bloom, “Lawyers as Agents of the Devil in a Prisoner’s Dilemma Game,” NBER
Working Paper No. W4447 (September 1993).
14.1 THE CONCEPT OF NASH EQUILIBRIUM
577
It is interesting to see how Marutti might figure out which strategy to choose in this
game. If it envisions this payoff matrix, it should realize that while it does not have a dominant strategy, Ambassador does (“build”). Thus, Marutti should reason that Ambassador
will choose this dominant strategy, and given this, Marutti should choose “do not build.”
The Nash equilibrium is a natural outcome of this game because Marutti’s executives—
putting themselves “inside the mind” of their rival—figure that their rival will choose its
dominant strategy, which then pins down what Marutti should do. Seeing the value of
placing yourself inside the mind of rival players in the game—seeing the world from their
perspective, not yours—is one of the most valuable lessons of game theory. Barry Nalebuff
and Adam Brandenberger call this allocentric reasoning, which should be contrasted with
egocentric reasoning, which views the world exclusively from one’s own perspective.8
Dominated Strategies
The opposite of a dominant strategy is a dominated strategy. A strategy is dominated
when the player has another strategy that gives it a higher payoff no matter what the other
player does. In Table 14.1, with just two strategies for each player, if one strategy is dominant then the other must be dominated. However, with more than two strategies available to each player, a player might have dominated strategies but no dominant strategy.
Identifying dominated strategies can sometimes help us deduce the Nash equilibrium in a game where neither player has a dominant strategy. To illustrate, let’s return
to the Honda–Toyota game, but now let’s suppose that each firm has three strategies:
Do not build, build a small plant, or build a large plant. Table 14.4 shows the payoffs
from each of these strategies.
Neither player in this game has a dominant strategy, and with three strategies
rather than two, the task of finding a Nash equilibrium seems rather daunting. But notice that for each player “build large” is a dominated strategy: No matter what Toyota
does, Honda is always better off by choosing “build small” rather than “build large.”
Similarly, no matter what Honda does, Toyota is always better off choosing “build
small” rather than “build large.” If each player thinks about the payoffs of the other—
that is, if each employs allocentric reasoning—each should conclude that its rival will
not choose “build large.” If each player assumes that the other will not choose “build
large” (and rules out choosing “build large” itself ), then the 3 ⫻ 3 game in Table 14.4
reduces to the 2 ⫻ 2 game in Table 14.5, which is the same game as in Table 14.1. In
this reduced game, each player now has a dominant strategy: “build small.” By eliminating a dominated strategy, we were able to find a dominant strategy for each player
that, in turn, enabled us to find the Nash equilibrium in the full game:9 for each firm
to build a small plant. (You can, by the way, verify this directly from Table 14.4: If either firm chooses “build small,” the other firm’s best response is also “build small.”)
TABLE 14.4
Modified Capacity Expansion Game between Toyota and Honda*
Honda
Build Large
Build Small
Do Not Build
Build Large
Toyota
Build Small
Do Not Build
0, 0
8, 12
9, 18
12, 8
16, 16
15, 20
18, 9
20, 15
18, 18
*Payoffs are in millions of dollars.
8
B. J. Nalebuff and A. M. Brandenberger, Coopetition (New York: Currency Doubleday, 1996).
This is the same logic that we employed in Chapter 13 when we argued that the Cournot equilibrium
was the natural outcome of the one-shot quantity game between Samsung and LG.
9
dominated strategy
A strategy such that the
player has another strategy
that gives a higher payoff
no matter what the other
player does.
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TABLE 14.5 Modified Capacity Expansion Game between Toyota
and Honda after Eliminating Dominated Strategies*
Toyota
Build Small Do Not Build
Build Small
Do Not Build
Honda
16, 16
15, 20
20, 15
18, 18
*Payoffs are in millions of dollars.
Summary: Finding a Nash Equilibrium by Identifying Dominant Strategies
and Eliminating Dominated Strategies
We can summarize the main conclusions of this section as follows:
• Whenever both players have a dominant strategy, those strategies will constitute
the Nash equilibrium in the game.
• If just one player has a dominant strategy, that strategy will be the player’s Nash
equilibrium strategy. We can find the other player’s Nash equilibrium strategy
by identifying that player’s best response to the first player’s dominant strategy.
• If neither player has a dominant strategy, but both have dominated strategies,
we can often deduce the Nash equilibrium by eliminating the dominated strategies and thereby simplifying the analysis of the game.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 4 . 1
D
Finding the Nash Equilibrium: Coke versus Pepsi
Table 14.6 shows Coke’s and Pepsi’s profits
for various combinations of prices that each firm might
charge.
Problem Find the Nash equilibrium in this game.
Solution We begin by searching for dominant
strategies. For Pepsi, a price of $8.25 is a dominant strategy because no matter which price Coke chooses, Pepsi’s
payoff is always higher in row 3—a price of $8.25—than
in any other row. Thus, the other three prices ($6.25,
$7.25, and $9.25) are dominated strategies for Pepsi. We
TABLE 14.6
note the elimination of these dominated strategies in
Table 14.6a by drawing a line through them.
If Coke assumes that Pepsi will follow its dominant
strategy, Coke’s best response is to set a price of $12.50
(the price that gives Coke its highest payoff in row 3).
The Nash equilibrium in this game is for Pepsi to set
a price of $8.25 and Coke to set a price of $12.50. (This
corresponds to the equilibrium we derived when discussing the Coke–Pepsi price competition in Chapter 13.)
Similar Problems: 14.3, 14.5, 14.6
Price Competition between Coke and Pepsi*
Pepsi
$6.25
$7.25
$8.25
$9.25
*Payoffs are in millions of dollars.
Coke
$12.50
$10.50
$11.50
66, 190
79, 201
82, 212
75, 223
68, 199
82, 211
86, 224
80, 237
70, 198
85, 214
90, 229
85, 244
$13.50
73, 191
89, 208
95, 225
91, 245
14.1 THE CONCEPT OF NASH EQUILIBRIUM
TABLE 14.6a Price Competition between Coke and Pepsi after Identifying
Pepsi’s Dominant Strategy and Dominated Strategies*
Pepsi
$6.25
$7.25
$8.25
$9.25
$10.50
Coke
$11.50
$12.50
$13.50
66, 190
79, 201
82, 212
75, 223
68, 199
82, 211
86, 224
80, 237
73, 191
89, 208
95, 225
91, 245
70, 198
85, 214
90, 229
85, 244
*Payoffs are in millions of dollars.
GAMES WITH MORE THAN ONE NASH EQUILIBRIUM
All of the games we have just studied had a unique Nash equilibrium. But some games
have more than one Nash equilibrium. A famous example is the game of Chicken:
Two teenage boys are going to prove their manhood to their friends. They each get
in their cars at opposite ends of a road and begin to drive toward each other at breakneck speed. If one car swerves before the other, the one that did not swerve (i.e., stays)
proves his manhood and becomes a hero to his friends, while the other loses face (he
is a “chicken”). If both swerve, nothing gets proven: Neither loses face, but neither
gains status either. If neither swerves, though, they crash into each other and are
either injured or killed.
Table 14.7 shows the payoffs for the game of Chicken between two teenagers,
Luke and Slick. There are two Nash equilibria in this game. The first is for Luke to
swerve and for Slick to stay. The other is for Luke to stay and Slick to swerve. To verify
that the first is a Nash equilibrium, note that if Luke swerves, Slick is better off staying
(payoff of 10) than swerving (payoff of 0). And if Luke stays, Slick is better off swerving
(payoff of 10), than staying (payoff of 100).
Do Chicken games occur in real life? In the 1950s and 1960s, many felt that a
Chicken game was a good description of how a nuclear showdown between the United
States and the Soviet Union would play out. The famous quote by John F. Kennedy’s
secretary of state, Dean Rusk, following the Cuban Missile Crisis, “We’re eyeball to
eyeball and the other fellow just blinked,” is an illustration of how one high-stakes
game of Chicken during the Cold War played out. Less dramatically, but perhaps
more pervasively, games of Chicken arise in economics when two firms compete in a
market that can profitably support only one firm. (In Chapter 11, we called these natural monopoly markets.) The Nash equilibrium in the Chicken game tells us that one
firm will eventually exit the market and one firm will survive.
TABLE 14.7
The Game of Chicken
Slick
Swerve
Luke
Swerve
Stay
0, 0
10, 10
Stay
10, 10
100, 100
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G A M E T H E O RY A N D S T R AT E G I C B E H AV I O R
14.2
Chicken in Orbit: Winning the Battle
for Satellite Radio in North America10
The satellite radio market in North America (United
States and Canada) resembled a high-stakes game of
Chicken in the mid- to late 2000s. Like satellite transmission, satellite radio involves the transmission of
radio signals using several satellites orbiting the Earth.
Satellite radio offers listeners near-perfect reception of
more than a hundred channels that appeal to all manner
of tastes. The service is thought to be particularly appealing to drivers (such as commercial truck drivers)
who have to travel long distances and traverse many
local radio markets. Beginning in 2001, two firms—XM
Satellite radio and Sirius Radio—fought to dominate
the emerging satellite radio market in North America.
The business of satellite radio involves high fixed
costs and low marginal costs because once a company
launches a satellite and acquires the rights to programming (e.g., the rights to carry sporting events),
the marginal cost of adding one more subscriber to its
subscription base is very low. A key implication of this
cost structure is that a satellite radio company needs a
critical mass of subscribers to break even financially.
Making the problem even more difficult for XM and
Sirius was the fact that the two companies used incompatible technologies, so that the receiver purchased to
receive one company’s service could not be used to receive the service of the other company. Even with the
expectations of rapid growth in the market, it was not
clear whether the market would be large enough to
allow two firms to coexist profitably in the market.
Given these realities, it was conceivable that the
satellite radio market in North America is a natural
monopoly. If so, the battle between XM and Sirius to
“win” this market can be understood as a game of
“Chicken.” Table 14.8 shows how we can use game
10
theory to predict the possible outcome of the battle
to dominate the North American satellite radio market. In the table, two firms, XM and Sirius, have the
choice of staying in the market or exiting. The payoffs in the table are hypothetical cumulative profits
that the firms would be expected to earn under various competitive scenarios.11 If (for the sake of illustration) we assume that the market can only support
one profitable firm and both firms choose to remain
in the market, each firm would be expected to incur
significant losses. However, if one firm were to exit
the market, the remaining firm would make a profit.
The game in Table 14.8 has two Nash equilibria:
In one, XM chooses “stay” and Sirius chooses “exit,”
while in the other, Sirius chooses “stay” and XM
chooses “exit.” Game theory, by itself, cannot tell us
which of these two Nash equilibria would be likely
to arise. We would need to know more about the
players and the particular circumstances they face in
order to make predictions about who would win.
In 2008 XM “swerved,” acquired by its rival to
form a new company in the United States, Sirius XM
Radio, Inc. After the merger, Sirius had over 18.5 million subscribers. In 2010 it offered subscriptions with
more than 140 channels of programming, including
applications for mobile devices such as the iPod and
iPhone and Blackberry phones.12
TABLE 14.8 The Game of Chicken between
XM and Sirius*
Sirius
Stay
XM
Stay
Exit
200, 200
0, 300
Exit
300, 0
0, 0
*Payoffs are in millions of dollars.
This example draws from “Satellite Radio: Winning the Competitive Skirmishes,” Satellite News, 27,
no. 21 (May 24, 2004) and “XM, Sirius Eye Pristine Radio Market in Canada,” Satellite News, 27, no. 15
(April 5, 2004).
11
Technically, the payoffs in Table 14.8 should be thought of as the present value of the profits (or losses) into
the future. As discussed in the Appendix to Chapter 4, a present value of a stream of profits involves adding
up the stream of profits over a period of years with the twist that we discount profits received in later years
to take into account the fact that a dollar of profit received 10 years from now is worth less than a dollar of
profit received today. The Appendix to Chapter 4 provides an introduction to the concept of present value.
12
“Sirius Completes Acquisition of XM Satellite,” Reuters, July 29. 2008, http://www.reuters.com/article/
idUSN2926292520080730?sp=true (accessed May 1, 2010).
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A P P L I C A T I O N
14.3
Bank Runs
If you have ever seen the movie It’s a Wonderful Life,
you probably remember the scene just after George
and Mary Bailey (Jimmy Stewart and Donna Reed) get
married. They are about to catch their train for their
honeymoon, when someone tells George: “There’s a
run on the bank!” In the ensuing scene, George goes
to his family’s business (the Bailey Brothers Building
and Loan) and is confronted with a mob of anxious depositors who are demanding to withdraw their money.
Rather than locking the doors as many real banks did
during the Great Depression of the 1930s, George does
his best to keep the Building and Loan open. He does
so by pleading with his depositors to not withdraw
their money, or at least, to withdraw only as much as
they need to pay their bills.
The financial events around the world in the past
decade have demonstrated that runs on banks and
other kinds of financial institutions are not a thing of
the past. Examples abounded in the financial crisis
surrounding the great recession at the end of the first
decade of the new millennium. During the subprime
mortgage crisis of 2007, the American firm Countrywide
Financial faced a run on its assets. In 2008 a run by the
bondholders of Bear Stearns, a global investment firm,
led the company to declare bankruptcy. Several other
institutions, including Washington Mutual, the largest
savings and loan in America, and Landsbanki, Iceland’s
second largest bank, failed in the wake of runs in 2008.
Why do runs occur? Are they the result of irrational
fear and hysteria, a sort of dysfunctional mass psychology? It might seem so. After all, if all depositors remained
clear-sighted and level-headed, they would realize that
everyone would be better off if there was no run on the
bank. The bank would remain open, and depositors
would eventually get their money. Or is something else
going on? Could bank runs be consistent with rational
maximizing behavior by depositors? Game theory suggests that the answer to the last question could be yes.
Table 14.9 presents a simple game theoretic analysis of a bank run. Two individuals have deposited $100
in the Bailey Building and Loan. The Building and Loan
has taken this money and invested it (perhaps lending
money for houses). If both depositors keep their
money in the bank (“don’t withdraw”), they will eventually get their deposit back with an interest payment
of $10, for a total payoff of $110. If both withdraw
their money at the same time (a bank run), though,
the bank must liquidate its investment and then close
its doors. In this case, each depositor gets 25 cents on
the dollar. If one depositor withdraws her money but
the other doesn’t, the bank again must liquidate its
investment and close. The depositor who withdraws
her money gets $50, but the unlucky depositor who
left her money in the bank loses everything.
Like the game of Chicken, the bank run game has
two Nash equilibria. The first is that both depositors
keep their money in the bank. If Depositor 2 chooses
“don’t withdraw,” Depositor 1 is better off choosing
“don’t withdraw” as well (a payoff of 110 versus a
payoff of 50). The same holds true for Depositor 1. The
second Nash equilibrium is for both players to withdraw their money. If Depositor 2 chooses “withdraw,”
Depositor 1’s best response is to choose “withdraw” as
well (and vice versa).
As in the game of Chicken, game theory cannot
tell us which equilibrium will occur, but it does teach
us that bank runs can occur. This is so even though we
assume that all depositors behave rationally and that
a bank run makes all depositors worse off. Thus, as
in the prisoners’ dilemma game, purposeful utilitymaximizing behavior by individuals will not necessarily result in an outcome that maximizes the collective
well-being of all the players in the game.
TABLE 14.9
The Bank Run Game*
Depositor 2
Don’t
Withdraw Withdraw
Depositor 1
Withdraw
Don’t Withdraw
*Payoffs are in dollars
Now that you have seen several games—some with a unique Nash equilibrium,
some with more than one Nash equilibrium—you might be wondering if there is a
systematic procedure for identifying the Nash equilibria in a game that is presented in
tabular form. That is what you will learn to do in Learning-By-Doing Exercise 14.2.
25, 25
0, 50
50, 0
110, 110
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L E A R N I N G - B Y- D O I N G E X E R C I S E 1 4 . 2
D
Finding All of the Nash Equilibria in a Game
Problem What are the Nash equilibria
in the game in Table 14.10?
Solution Generally speaking, the first step in finding
the Nash equilibria in a game should be to identify dominant or dominated strategies and attempt to simplify the
game, as we did in Learning-By-Doing Exercise 14.1.
But in this game, neither player has a dominant strategy
or any dominated strategies. (You should verify this before going further.) Thus, we cannot use this approach.
Instead, to find all the Nash equilibria in this game,
we proceed in three steps.
Step 1: Find Player 1’s best response to each of the
three possible strategies of Player 2. These are the strategies indicated by the circled payoffs in Table 14.10a.
Step 2: Find Player 2’s best response to each of the
three possible strategies of Player 1. These are the
strategies indicated by the boxed payoffs in Table 14.10a.
TABLE 14.10
Similar Problems: 14.1, 14.2, 14.4, 14.5, 14.6,
14.7, 14.8, 14.9, 14.22, 14.23, 14.24
What Are the Nash Equilibria?
Player 1
TABLE 14.10a
Step 3: Recall that at a Nash equilibrium every player
chooses a strategy that gives it the highest payoff, given
the strategies chosen by the other players in the game. In
Table 14.10a, this occurs in cells with both a circle and
a square. Thus, in this game, we have two Nash equilibria,
one where Player 1 chooses strategy A and Player 2
chooses strategy E, and another where Player 1 chooses
strategy C and Player 2 chooses strategy D.
The procedure we just used—first identifying Player
1’s best responses to each of Player 2’s strategies, then
identifying Player 2’s best responses to each of Player 1’s
strategies, then seeing where those best responses occur
together—is a surefire way to identify all the Nash equilibria in a game.
Strategy A
Strategy B
Strategy C
Strategy D
Player 2
Strategy E
Strategy F
4, 2
3, 10
12, 14
13, 6
0, 0
4, 11
1, 3
15, 2
5, 4
Player 1’s and Player 2’s Best Responses
Player 1
Strategy A
Strategy B
Strategy C
Strategy D
Player 2
Strategy E
Strategy F
4, 2
3, 10
12 , 14
13 , 6
0, 0
4, 11
1, 3
15 , 2
5, 4
M I X E D S T R AT E G I E S
In July 1999, the United States and the Chinese women’s soccer teams fought to a
0–0 tie in the final match of the Women’s World Cup. To decide the match, players on each team alternated in shooting penalty kicks, and the match eventually
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14.1 THE CONCEPT OF NASH EQUILIBRIUM
came down to a final penalty kick by the United States. If the U.S. player scored a
goal, the United States would win the match; if the Chinese goalie blocked the
kick, the game would continue, and the Chinese team would then have a chance to
win the match with a penalty kick of its own. Both the U.S. kicker and the Chinese
goalie had to make split-second decisions. Should the kicker aim left or right?
Should the goalie dive to the kicker’s left or right? If the Chinese goalie dove in the
direction in which the kicker aimed, the shot would be blocked, and the two teams
would remain tied and would move on to another penalty kick. If the goalie
guessed wrong, though, the U.S. team would score and win the match. (As you
might remember, the final U.S. kicker, Brandi Chastain, did make the final kick,
and the U.S. team won.)
Table 14.11 shows a payoff matrix that we might use to depict this final encounter
between the U.S. and Chinese teams. Winning the match gives the U.S. team a payoff of 10, while losing the match would give the Chinese team a payoff of ⫺10. If the
two teams remain tied, each receives (from this encounter) a payoff of 0.
This game does not appear to have a Nash equilibrium. If the Chinese goalie
believes the U.S. kicker will aim right, the goalie’s best strategy is to dive to the
kicker’s right. But if the U.S. kicker believes the Chinese goalie will dive to the kicker’s
right, the kicker’s best strategy is to aim left. And if the kicker aims left, the goalie’s
best response is to dive to the kicker’s left.
This game illustrates the contrast between a pure strategy and a mixed strategy. A pure strategy is a specific choice among the possible moves in the game. The
U.S. kicker has a choice between two pure strategies: “aim right” and “aim left.” By
contrast, under a mixed strategy, a player chooses among two or more pure strategies
according to prespecified probabilities.13 Even though some games might have no
Nash equilibrium in pure strategies, every game has at least one Nash equilibrium in
mixed strategies. The game in Table 14.11 illustrates this point: It does not have a
Nash equilibrium in pure strategies, but there is a Nash equilibrium in mixed strategies. The U.S. kicker should “aim right” with probability 1/2 and “aim left” with
probability 1/2. The Chinese goalie should “dive right” with probability 1/2 and
“dive left” with probability 1/2. If the U.S. kicker believes that the Chinese goalie
will dive right or left with probability 1/2, the U.S. kicker can do no better than to
choose to aim left or right with probability 1/2. Similarly, if the Chinese goalie believes that the U.S. kicker will aim right or left with probability 1/2, the goalie can
do no better than to choose to dive left or right with probability 1/2. Thus, when the
players choose these mixed strategies, each is doing the best it can given the actions
of the other.
TABLE 14.11
U.S. Kicker versus Chinese Goalie in the 1999 Women’s World Cup
U.S. Kicker
Aim Right Aim Left
Chinese Goalie
13
Dive to Kicker’s Right
Dive to Kicker’s Left
0, 0
10, 10
10, 10
0, 0
For this reason, mixed strategies are sometimes referred to as randomized strategies.
pure strategy A specific
choice of a strategy from the
player’s possible strategies
in a game.
mixed strategy A
choice among two or more
pure strategies according to
prespecified probabilities.
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The fact that games can have Nash equilibria in the form of mixed strategies
illustrates that unpredictability can have strategic value. When your opponent can
predict what you will do, you can leave yourself vulnerable to being exploited by your
opponent. Athletes in sports such as baseball, soccer, and tennis have long understood
this point, and the World Cup game illustrates it nicely. If the kicker knew which way
the goalie was going to dive, the kicker could simply aim the other way and score the
goal. There is value in being unpredictable, and mixed strategies illustrate how this
value is present in game theory.
S U M M A RY: H O W TO F I N D A L L T H E N A S H
E Q U I L I B R I A I N A S I M U LTA N E O U S - M OV E G A M E
W I T H T W O P L AY E R S
We can summarize the lessons of this section by outlining a five-step approach to
identifying the Nash equilibria in simultaneous-move games involving two players.
1. If both players have a dominant strategy, these constitute their Nash equilibrium
strategies.
2. If one player, say Player 1, has a dominant strategy, this is the player’s Nash
equilibrium strategy. We then find Player 2’s best response to Player 1’s dominant
strategy to identify Player 2’s Nash equilibrium strategy.
3. If neither player has a dominant strategy, we successively eliminate each player’s
dominated strategies in order to simplify the game, and then search for Nash
equilibrium strategies.
4. If neither player has dominated strategies, we identify Player 1’s best response to
each of Player 2’s strategies and then identify Player 2’s best response to each of
Player 1’s strategies. In a table representing the game, the Nash equilibria are the
cells where a Player 1 best response occurs together with a Player 2 best response.
(This approach, which is guaranteed to identify all the pure-strategy Nash equilibria in a game, was demonstrated in Learning-By-Doing Exercise 14.2.)
5. If the approach in Step 4 does not uncover any pure-strategy Nash equilibria—
that is, if the game does not have a Nash equilibrium in pure strategies, as in the
Womens’ World Cup game—we look for an equilibrium in mixed strategies.
14.2
THE
R E P E AT E D
PRISONERS’
DILEMMA
A key lesson of the prisoners’ dilemma is that the individual pursuit of profit max-
imization does not necessarily result in the maximization of the collective profit of a
group of players. But the prisoners’ dilemma is a one-shot game, and you might
wonder if the game would turn out differently if it was played over and over again by
the same players. When we allow the players to interact repeatedly, we open the possibility that each player can tie its current decisions to what its opponent has done in
previous stages of the game. This expands the array of strategies that the players can
follow and, as we will see, can dramatically alter the game’s outcome.
To illustrate the impact of repeated play, consider the prisoners’ dilemma game in
Table 14.12. For each player, “cheat” is a dominant strategy, but the players’ collective profit is maximized when both play “cooperate.” In a one-shot game, the Nash
equilibrium would be for both players to choose “cheat.”
1 4 . 2 T H E R E P E AT E D P R I S O N E R S ’ D I L E M M A
TABLE 14.12
585
Prisoners’ Dilemma Game
Player 1
Cheat
Cooperate
Player 2
Cheat
Cooperate
5, 5
1, 14
14, 1
10, 10
Payoff per period
But let’s now imagine that two players will be playing the game again and again,
into the foreseeable future. In this case, it is possible that the players might achieve an
equilibrium in which they play cooperatively. To see why, suppose that Player 1 believes that Player 2 will use the following strategy: “Start off choosing ‘cooperate’ and
continue to do so as long as Player 1 cooperates. The first time Player 1 chooses
‘cheat,’ Player 2 will choose ‘cheat’ in the next period and in all following periods.”
Of course, if Player 2 cheats in the ensuing periods, Player 1 might as well continue
to cheat as well. Player 2’s strategy is sometimes called the “grim trigger” strategy because one episode of cheating by one player triggers the grim prospect of a permanent
breakdown in cooperation for the remainder of the game.
Figure 14.1 illustrates that, by cooperating in every period, Player 1 can ensure
himself a stream of payoffs equal to 10 per period. By contrast, if Player 1 cheats, he receives a payoff of 14 in the current period and a payoff of 5 in all subsequent periods.
Which strategy is better? Without additional information about how Player 1 evaluates
current versus future payoffs we cannot say for sure. But if Player 1 places sufficiently
strong weight on future payoffs relative to current payoffs, Player 1 will prefer continued cooperation to cheating.14 This illustrates that in the repeated prisoners’ dilemma,
cooperation can, under certain circumstances, result from self-interested behavior on
the part of each player.
14
10
5
Now
A
B
Always cooperate
C
Cheat today
1
2
3
Number of periods from now
14
4
5
FIGURE 14.1 Payoffs in the Repeated
Prisoners’ Dilemma under the “Grim Trigger”
Strategy
If Player 1 cheats today, he receives a stream of
payoffs given by the light line. If he cooperates
today and in the future he can ensure himself
a stream of payoffs given by the dark line. The
distance of line segment AB represents the
one-time gain to Player 1 from cheating. The
distance of line segment BC represents the
reduction in each of Player 1’s subsequent
payoffs because Player 2 retaliates against
Player 1’s cheating.
We can formally represent the weight that players give to future versus current payoffs by the concept
of present value mentioned in footnote 11.
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The grim trigger strategy is not the only strategy that can induce cooperative
behavior in repeated prisoners’ dilemma games (we discuss another one, tit-for-tat,
in Application 14.4). The unifying feature of cooperation-inducing strategies is that
they punish the opposing player for cheating. For example, a player will voluntarily
cooperate in the repeated prisoners’ dilemma if the player anticipates that its rival
will eventually retaliate if the player cheats. The prospect of eventual retaliation and
the corresponding reduction in profit beyond the initial period (represented by the
length of line segment BC in Figure 14.1) is what provides an incentive for a player
to maintain cooperative behavior, even though cheating is the dominant strategy in
a one-shot game.
In light of this, it is possible to make some general statements about the likelihood
that players will be able to sustain cooperative behavior when they interact in a repeated prisoners’ dilemma game. Specifically, the likelihood of a cooperative outcome
increases under these conditions:
• The players are patient. That is, they value payoffs in future periods almost as
much as payoffs in the current period. For patient players, the adverse consequences of punishment loom large in comparison to the short-term gains from
cheating.
• Interactions between the players are frequent. This implies that the length of a
“period” is short and that the one-shot benefit to cheating accrues over a short
period of time.
• Cheating is easy to detect. This has the same effect, roughly, as shortening the
length of the period: A firm cannot get away with cheating for very long
and thus finds that the short-term benefit from noncooperative behavior is
fleeting.
• The one-time gain from cheating is relatively small. For example, the length of line
segment AB in Figure 14.1 is small in comparison to the eventual cost of cheating,
the length of line segment BC.
By contrast, the likelihood of a cooperative outcome diminishes under these conditions:
• The players are impatient. That is, they value current payoffs much more than
future payoffs.
• Interactions between the players are infrequent. This implies that the length of a
“period” is long and that the one-shot benefit to cheating accrues over a relatively long period of time.
• Cheating is hard to detect. When this is so, a firm can get away with cheating longer
and can enjoy the benefit from cheating over a relatively longer period of time.
• The one-time gain from cheating is large in comparison to the eventual cost of cheating.
Our analysis of the repeated prisoners’ dilemma game teaches an important lesson:
In competitive settings you must anticipate the reactions of your competitors. If you
are in a situation in which you will be interacting with the same group of competitors
over time, it is important to anticipate their likely responses to your moves. In particular, you need to understand how a competitor is likely to respond when you engage
in actions that could be construed as cheating. If, for example, you are a business
firm in a market and you cut price in order to increase your market share, you need
1 4 . 2 T H E R E P E AT E D P R I S O N E R S ’ D I L E M M A
A P P L I C A T I O N
14.4
Shoot-to-Kill, Live-and-Let-Live,
or Tit-for-Tat?15
Trench warfare is ugly and brutal. This was certainly
so along the Western front during World War I, where
the Allied army (France and Britain) faced the German
army. Still, as Robert Axelrod has written, despite the
grim circumstances, an unusual degree of cooperation emerged. Axelrod quotes a British staff officer
who wrote that he was:
astonished to observe German soldiers walking
about within rifle range behind their own line. Our
men appeared to take no notice. I privately made
up my mind to do away with that sort of thing
when we took over; such things should not be
allowed. These people evidently did not know
there was a war on. Both sides apparently believed
in the policy of “live and let live.”
Axelrod goes on to point out that these circumstances
were not isolated. “The live-and-let live system,” he
writes, “was endemic in trench warfare. It flourished
despite the best efforts of senior officers to stop it,
despite the passions aroused by combat, despite the
military logic of kill or be killed, and despite the ease
with which the high command was able to repress
any local efforts to arrange a direct truce.”
Axelrod interprets the “cooperative” trench warfare along the Western front as the outcome of a
repeated prisoners’ dilemma game. At any given point
along the line, the two players were Allied and German
battalions (military units consisting of roughly 1,000
men). On any given day, a battalion could “shoot-tokill,” a strategy corresponding to “cheat” in Table 14.12.
Or it could “Live-and-Let-Live,” a strategy that corresponds to “cooperate” in Table 14.12. Axelrod
argues that for each opposing battalion “shoot-to-kill”
was a dominant strategy. This is because each battalion
would occasionally be ordered by its army’s high command into a major battle in its area of the line (e.g., a
charge against the other side’s trenches). By shooting
to kill, a battalion would weaken its opponent, which
would increase the likelihood of survival should a major
engagement be ordered. At the same time, both sides
15
587
are better off when both “live-and-let-live” than when
both “shoot-to-kill.” The structure of the “game” between opposing battalions along the Western front
was thus a prisoners’ dilemma.
But if “shoot-to-kill” was a battalion’s dominant
strategy, why did cooperation emerge? The reason,
Axelrod argues, is that the prisoners’ dilemma game
between enemy battalions was a repeated game.
Trench warfare differs from other ways of fighting a
war because each side’s units face the same enemy
units for months at a time. Although cooperation between Allied and German battalions usually evolved
by accident (e.g., during periods of unusually rainy
weather during which fighting could not occur), the
close interaction between the same battalions allowed
them to follow strategies that tended to sustain the
cooperation once it had emerged.
A particularly valuable strategy for sustaining
cooperation between enemy battalions along the
Western front was “tit-for-tat.” Under this strategy,
you do to your opponent what your opponent did to
you last period. Along the Western front, it became
well understood that if one side exercised restraint,
the other would, too. If, by contrast, one side fired,
the other side would shoot back in a proportional
fashion. Wrote one soldier:
It would be child’s play to shell the road behind
the enemy’s trenches, crowded as it must be with
ration wagons and water carts, into a bloodstained
wilderness . . . but on the whole there is silence.
After all, if you prevent your enemy from drawing
his rations, his remedy is simple: he will prevent
you from drawing yours.
The “tit-for-tat” strategy was carried to strong numerical extremes. One soldier noted:
If the British shelled the Germans, the Germans
replied, and the damage was equal: if the Germans
bombed an advanced piece of trench and killed
five Englishmen, an answering fusillade killed five
Germans.
The use of tit-for-tat strategies meant that each
side realized that an aggressive act would be met by
an aggressive response. In choosing how to fight,
This example draws heavily from Chapter 4 of Robert Axelrod’s book, The Evolution of Cooperation
( New York: Basic Books, 1984), pp. 73–87.
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battalions on each side weighed the trade-off between
the short-term gain from shooting to kill against the
long-term cost from a breakdown in restraint. Facing
this trade-off, numerous battalions along the Western
front chose cooperation over noncooperation.
Eventually, as World War I came to a close, the
norm of cooperation along the Western front broke
down. The reason is that the high commands of both
the Allied and German armies took explicit steps to
end the tacit truces that had broken out along much
of the Western front. (In this sense, the high commands can be thought of as akin to antitrust enforcers
tit-for-tat A strategy in
which you do to your opponent in this period what
your opponent did to you in
the last period.
that attempt to break up tacitly collusive behavior
among business firms.) In particular, the armies’ commanders began to organize much more frequent and
larger raids in which the raiding parties were ordered
to kill enemy soldiers in their own trenches. This
changed the payoffs in the prisoners’ dilemma game
so that “shoot-to-kill” became a more attractive alternative to “live-and-let-live.” With larger and more
frequent raids, the traditional wartime norm of “kill
or be killed” took over, and by the time the war
ended, both sides had returned to an incessantly aggressive posture.
to anticipate whether your price cut will be detected, whether your competitor will
respond by matching the price, and if so, how long your competitor will take to
match. By ignoring the possibility of competitive responses, you run the risk of
overestimating the potential benefits that will accrue to you from various forms of
noncooperative behavior. You also run the risk of plunging your market into a costly
price war that will erase any temporary gains you might enjoy from having undercut
the prices of your competitors.
A P P L I C A T I O N
14.5
Collusion in Japanese Sumo
Wrestling16
Sumo is a uniquely Japanese form of wrestling in
which enormous men compete to wrestle each other
to the ground. Developed over 1,000 years ago as part
of a ritual to pay homage to the Shinto gods, the rules
of sumo are fairly simple: The first wrestler to touch
the floor with something other than the soles of his
feet, or the first wrestler to leave the ring, loses the
match. Sumo matches are very short, sometimes lasting just a few seconds, and rarely lasting more than a
minute. Every year, six major sumo tournaments involving over 60 wrestlers are held in Japan, with each
wrestler participating in 15 matches over 15 days.
In recent years, the sport has been roiled with
allegations that some sumo wrestlers may have colluded with each other to fix matches. Though no such
allegations have been formally proved, they cannot be
dismissed lightly; strong incentives to rig matches
do exist as a result of the way in which the ranking
16
system in Japan works. A wrestler who achieves a winning record in a 15-match tournament is guaranteed
to rise in the official rankings, and an increase in the
rankings can translate into significant financial rewards,
as well as enormous prestige. Given this incentive
structure, a wrestler who is “on the bubble” (close to
a winning record, e.g., one who has 7 wins and
7 losses) has a strong incentive to bribe a wrestler with
a clear winning record to deliberately lose.
Economists Mark Duggan and Steven Levitt have
studied the issue of collusion in sumo wrestling using
data on almost every official sumo match in Japan
between 1989 and 2000. They looked for the “footprints” of match rigging by, in effect, asking: If there
was match rigging, what would one expect to observe
in the data that wouldn’t be observed if there was no
systematic match rigging? And if one observed these
phenomena, can other plausible explanations be ruled
out? Duggan and Levitt find very strong evidence that
would be consistent with match rigging. For example,
they find that far more wrestlers finish with exactly
This example is based on M. Duggan and S. D. Levitt, “Winning Isn’t Everything: Corruption in Sumo
Wrestling,” American Economic Review, 92(4) (December 2002): 1594–1605.
1 4 . 2 T H E R E P E AT E D P R I S O N E R S ’ D I L E M M A
8 wins (the number needed to ensure a winning
record) than would be expected by chance. Further,
they find that winning percentages for wrestlers who
are on the bubble are particularly elevated on the last
day of the tournament as compared to other days.
The natural alternative hypothesis that would explain these findings is that sumo wrestlers who are on
the bubble try especially hard to win the eighth match
so as to guarantee a winning record; that is, they “step
it up a notch” and find a way to win. One way to discriminate between this hypothesis and the collusion
hypothesis is to use insights from the repeated prisoners’ dilemma model. That model tells us that the likelihood of wrestlers being able to sustain a collusive
deal should be positively related to the frequency with
which the wrestlers interact and the likelihood that
they will be paired again in the future. Duggan and
Levitt’s findings are consistent with this prediction.
A P P L I C A T I O N
They find, for example, that the unexpectedly large
number of wins by a wrestler on the bubble was
increased if the wrestler was engaged in a match with
another wrestler against whom he had wrestled frequently in the previous year. Further, they discovered
that a wrestler who is in the last year of his career (and
who therefore cannot participate in repeated play in
the future) is less likely to win an unexpectedly large
number of matches when he is on the bubble. These
patterns are consistent with the collusion hypothesis,
but there is no reason to expect to observe them if
wrestlers who were on the bubble were simply exerting extra special effort. Though Duggan and Levitt
have not uncovered a “smoking gun” showing that
collusion in sumo matches occurred, their indirect evidence is very powerful and suggests that the authorities who control sumo wrestling in Japan should be
alert to any signs that matches are being fixed.
14.6
The Cost of War17
An excellent illustration of what can happen when
one firm miscalculates competitor responses occurred
in the cigarette industry in Costa Rica in 1993. The
most famous cigarette price war of 1993 occurred in
the United States, when Philip Morris initiated its
“Marlboro Friday” price cuts. The lesser-known Costa
Rican price war, also initiated by Philip Morris, began
several months before and lasted a full year longer.
At the beginning of the 1990s, two firms dominated the Costa Rican cigarette market: Philip Morris,
with 30 percent of the market, and B.A.T., with 70
percent of the market. The market consisted of three
segments: premium, mid-priced, and value-for-money
(VFM). Philip Morris had the leading brands in the
premium and mid-priced segments (Marlboro and
Derby, respectively). B.A.T., by contrast, dominated
the VFM segment with its Delta brand.
Throughout the 1980s, a prosperous Costa Rican
economy fueled steady growth in the demand for cigarettes. As a result, both B.A.T. and Philip Morris were
able to sustain price increases that exceeded the rate
17
589
of inflation. However, in the late 1980s, the market
began to change. Health concerns slowed the demand for cigarettes in Costa Rica, a trend that hit the
premium and mid-priced segments much harder than
it did the VFM segment. In 1992, B.A.T. gained market
share from Philip Morris for the first time since the
early 1980s. Philip Morris faced the prospect of slow
demand growth and a declining market share.
On Saturday, January 16, 1993, Philip Morris reduced the prices of Marlboro and Derby cigarettes by
40 percent. The timing of the price reduction was not
by chance. Philip Morris reasoned that B.A.T.’s inventories would be low following the year-end holidays and
that B.A.T. would not have sufficient product to satisfy
an immediate increase in demand should it match or
undercut Philip Morris’s price cut. Philip Morris also initiated its price cut on a Saturday morning, expecting
that B.A.T.’s local management would be unable to respond without first undertaking lengthy consultations
with the home office in London.
However, B.A.T. surprised Philip Morris with the
speed of its response. Within hours, B.A.T. cut the
price of its Delta brand by 50 percent, a price that
We would like to thank Andrew Cherry (MBA 1998 Kellogg School of Management) for developing
this example.
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industry observers estimated barely exceeded Delta’s
marginal cost. Having been alerted to Morris’s move
on Saturday morning, B.A.T. had salespeople out selling at the new price by Saturday afternoon.
The ensuing price war lasted about two years.
Cigarette sales increased 17 percent as a result of the
lower prices, but market shares did not change. By
the time the war ended in late 1994, Philip Morris’s
share of the Costa Rican market was unchanged, and
it was $8 million worse off than it had been before the
war started. B.A.T. lost even more—$20 million—but it
had preserved the market share of its Delta brand and
was able to maintain the same price gaps that had
prevailed across market segments before the war.
14.3
Why did Philip Morris act as it did? In the early
1990s, Philip Morris had increased Marlboro’s market
share at B.A.T.’s expense in other Central American
countries, such as Guatemala. Perhaps it expected
that it could replicate that success in Costa Rica. Still,
had it anticipated B.A.T.’s quick response, Philip
Morris should have realized that its price cut would
not result in an increase in market share. Whatever
the motivation for Philip Morris’s actions, this example highlights how quickly retaliation by competitors
can nullify the advantages of a price cut. If firms understand that and take the long view, their incentive
to use price as a competitive weapon to gain market
share will be blunted.
S
SEQUENTIALM OV E G A M E S
AND
S T R AT E G I C
M OV E S
o far, we have studied games in which players make decisions simultaneously. In
many interesting games, however, one player can move before other players do. These
are called sequential-move games. In a sequential-move game, one player (the first
mover) takes an action before another player (the second mover). The second mover
observes the action taken by the first mover before it decides what action it should
take. (The Stackelberg model of oligopoly discussed in Chapter 13 is a particular
example of a sequential-move game.) We shall see that the ability to move first in a
sequential-move game can sometimes have significant strategic value.
sequential-move
games Games in which
A N A LY Z I N G S E Q U E N T I A L - M OV E G A M E S
one player (the first mover)
takes an action before
another player (the second
mover). The second mover
observes the action taken
by the first mover before
deciding what action it
should take.
game tree A diagram
that shows the different
strategies that each player
can follow in a game and
the order in which those
strategies get chosen.
To learn how to analyze sequential-move games, let’s return to the simultaneous-move
capacity expansion game between Toyota and Honda in Table 14.4. (To refresh your
memory of that game, Table 14.13 shows the payoff table.)
Recall that the Nash equilibrium in this game was for Toyota and Honda to
choose “build small.”
But now suppose that Honda can make its capacity decision before Toyota decides
what to do (perhaps because it has accelerated its decision-making process). We now
have a sequential-move game in which Honda is the first mover and Toyota is the
second mover. To analyze this sequential-move game, we use a game tree, which
shows the different strategies that each player can follow in the game and the order in
which those strategies get chosen. Figure 14.2 shows the game tree for our capacity
TABLE 14.13
Honda
Capacity Expansion Game between Toyota and Honda*
Build Large
Build Small
Do Not Build
*Payoffs are in millions of dollars.
Build Large
Toyota
Build Small
Do Not Build
0, 0
8, 12
9, 18
12, 8
16, 16
15, 20
18, 9
20, 15
18, 18
1 4 . 3 S E Q U E N T I A L - M OV E G A M E S A N D S T R AT E G I C M OV E S
Honda Payoff Toyota Payoff
Build large
Build large
T
Build small
Do not build
Build large
H
Build small
T
Build small
Do not build
Build large
Do not build
T
Build small
Do not build
0
0
12
8
18
9
8
12
16
16
20
15
9
18
15
20
18
18
591
FIGURE 14.2 Game Tree for
the Sequential-Move Capacity
Expansion Game between Toyota
and Honda
Honda moves first and can choose
among three strategies: Toyota moves
next (having observed Honda’s move),
also choosing among the same three
strategies. Assuming that Toyota will
always make its best (payoffmaximizing) response, Honda can
maximize its own payoff by choosing “build large,” as Toyota’s best
response will be “do not build.”
expansion game. In any game tree, the order of moves flows from left to right. Because
Honda moves first, it is in the leftmost position. For each of Honda’s possible actions,
the tree then shows the possible decisions for Toyota.
To analyze the game tree in Figure 14.2, it is convenient to use a thought process
called backward induction. When you solve a sequential-move game using backward
induction, you start at the end of the game tree, and for each decision point (represented by the shaded squares), you find the optimal decision for the player at that
point. You continue to do this until you reach the beginning of the game. The thought
process of backward induction has the attractive property that it allows us to break a
potentially complicated game into manageable pieces.
To apply backward induction in this example, we must find Toyota’s optimal decision
for each of the three choices Honda might make: “do not build,” “build small,” and
“build large” (in Figure 14.2, Toyota’s optimal choices are underlined):
• If Honda chooses “do not build,” Toyota’s optimal choice is “build small.”
• If Honda chooses “build small,” Toyota’s optimal choice is “build small.”
• If Honda chooses “build large,” Toyota’s optimal choice is “do not build.”
As we work backward in the tree, we assume that Honda anticipates that Toyota
will choose its best response to each of the three actions Honda might take. We can
then determine which of Honda’s three strategies gives it the highest profit, by identifying the profit that Honda gets from each option it might choose, given that Toyota
responds optimally:
• If Honda chooses “do not build,” then given Toyota’s optimal reaction, Honda’s
profit will be $15 million.
• If Honda chooses “build small,” then given Toyota’s optimal reaction, Honda’s
profit will be $16 million.
• If Honda chooses “build large,” then given Toyota’s optimal reaction, Honda’s
profit will be $18 million.
Thus, Honda attains the highest profit when it chooses “build large.” The Nash equilibrium in this game is for Honda to choose “build large” and for Toyota to choose
“do not build.” At this equilibrium, Honda’s profit is $18 million and Toyota’s profit
is $9 million.
backward induction A
procedure for solving a
sequential-move game by
starting at the end of the
game tree and finding the
optimal decision for the
player at each decision point.
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Notice that the Nash equilibrium of the sequential-move game differs significantly from that of the simultaneous-move game (both firms choose “build small”).
Indeed, in the sequential-move game, Honda’s equilibrium strategy (“build large”)
would be dominated if Toyota and Honda made their capacity choices simultaneously.
Why is Honda’s behavior so different when it can move first? Because in the sequentialmove game, the firm’s decision problems are linked through time: Toyota can see what
Honda has done, and Honda counts on a rational response by Toyota to whatever
action it chooses. This allows Honda to force Toyota into a corner. By committing to
a large-capacity expansion, Honda puts Toyota in a position where the best it can do
is not build. By contrast, in the simultaneous-move game, Toyota cannot observe
Honda’s decision beforehand, and therefore Honda cannot force Toyota’s hand.
Because of this, the choice of “build large” by Honda is not nearly as compelling as it
is in the sequential-move game.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 4 . 3
D
An Entry Game
scale or a small scale?
on a large scale and for Kodak to launch a price war. You
can see this most easily by noting that “large” is your
dominant strategy. Given that you choose this, Kodak
will respond by launching a price war. At this Nash equilibrium, your profit will be $2 million per year.
But you can do better if you can turn this into a
sequential-move game. Figure 14.3 shows the game tree
if you can commit to your scale of operation in advance,
before Kodak decides what to do. If you choose “large,”
Kodak’s best response, as we just saw, is to fight a price
war, and you get a payoff of $2 million per year. But if
you choose “small,” Kodak’s best response is “accommodate,” and you get a payoff of $4 million per year. Thus,
if you can move first, your optimal strategy is “small.”
The Nash equilibrium in the sequential-move game is
for you to enter on a small scale and for Kodak to respond by accommodating.
Solution If you and Kodak choose your strategies
Similar Problems: 14.10, 14.15, 14.17, 14.21,
Avinash Dixit and Barry Nalebuff, authors
of a delightful book on game theory, Thinking Strategically,
have written, “It takes a clever carpenter to turn a tree into
a table; a clever strategist knows how to turn a table into a
tree.”18 In this exercise, we illustrate their point in the context of a simple entry game.
Suppose you own a firm that is considering entry
into the digital camera business, where you will compete head to head with Kodak (which, let’s say, currently
has a monopoly). Kodak can react in one of two ways: It
can start a price war or it can be accommodating. You
can enter this business on a large scale or a small scale.
Table 14.14 shows the payoffs you and Kodak are likely
to get under the various scenarios that could unfold.
Problem Should you enter this business on a large
simultaneously, the Nash equilibrium is for you to enter
TABLE 14.14
14.22, 14.23
Entry into the Digital Camera Business*
Kodak
Accommodate
You
Small
Large
4, 20
8, 10
Price War
1, 16
2, 12
*Payoffs are in millions of dollars.
18
A. Dixit and B. Nalebuff, Thinking Strategically (New York: Norton, 1991), p. 122.
1 4 . 3 S E Q U E N T I A L - M OV E G A M E S A N D S T R AT E G I C M OV E S
Your Payoff
Accommodate
Small
K
Price war
Kodak's Payoff
4
20
1
16
8
10
2
12
Y
Accommodate
Large
K
Price war
FIGURE 14.3 Game Tree for Entry into the Digital Camera Business
You move first by deciding whether to enter on a small scale or a large scale. Kodak then responds
by accommodating your entry or launching a price war. Your best choice is to enter on a small
scale, to which Kodak will respond by accommodating.
T H E S T R AT E G I C VA L U E O F L I M I T I N G O N E ’ S O P T I O N S
In the sequential-move capacity expansion game, Honda committed in advance to a
particular course of action, whereas Toyota had the flexibility to respond to Honda.
Yet, Honda’s equilibrium profits were twice as large as Toyota’s. The firm that tied its
hands in advance fared better than the firm that maintained flexibility.
This illustrates a profound point. Strategic moves that seemingly limit options can
actually make a player better off, or, put another way, inflexibility can have value. This
is so because a firm’s commitments can alter its competitors’ expectations about how it
will compete, and this, in turn, will lead competitors to make decisions that benefit the
committed firm. In the Honda–Toyota game, when Honda commits itself in advance
to an apparently inferior strategy (“build large”), it alters Toyota’s expectations about
what it will do. Had Honda not made the commitment, Toyota would understand that
it would have been in Honda’s interest to choose “build small,” which in turn would
have led Toyota to choose “build small” as well. By committing in advance to the more
aggressive strategy of building a large plant, Honda makes it less appealing for Toyota
to expand its capacity, moving the industry to an equilibrium that makes Honda better
off than it would have been in the Nash equilibrium of the simultaneous-move game.
Generals throughout history have understood the value of inflexibility, as the
famous example of Hernan Cortes’s conquest of Montezuma’s Aztec empire in Mexico
illustrates. When he landed in Mexico, Cortes ordered his men to burn all but one of
his ships. Rather than an act of lunacy, Cortes’s move was purposeful and calculated:
By eliminating their only method of retreat, Cortes’s men had no choice but to fight
hard to win. According to Bernal Diaz del Castillo, who chronicled Cortes’s conquest
of the Aztecs, “Cortes said that we could look for no help or assistance except from
God for we now had no ships in which to return to Cuba. Therefore we must rely on
our own good swords and stout hearts.”19
19
This quotation comes from Chapter 2 of Richard Luecke’s book Scuttle Your Ships before Advancing:
And Other Lessons from History on Leadership and Change for Today’s Managers (New York: Oxford
University Press, 1994).
593
594
strategic moves
Actions that a player takes
in an early stage of a game
that alter the player’s
behavior and the other
players’ behavior later in
the game in a way that is
favorable to the first player.
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Honda’s preemptive capacity expansion and Cortes’s decision to scuttle his ships
are examples of strategic moves. A strategic move is an action you take in an early
stage of a game that alters your behavior and your competitors’ behavior later in the
game in a way that is favorable to you.20 In business life, there are many examples of
strategic moves. Decisions about how to position a product in the marketplace (“Do
we aim at a mass market or at a high-end niche?”), about how to compensate executives (“Do we reward our executives based on profitability or based on market
share?”), and about product compatibility (“Do we make our product compatible with
those of our competitors?”) are all examples of strategic moves because they can have
an important impact on how competition in the marketplace unfolds later on.21 For
example, a firm’s decision to position its product in a high-end niche might have
strategic value by reducing the fierceness of price competition with other competitors.
This is so even though the direct effect of a niche strategy would be to limit the size
of the product’s potential market.
Strategic moves are relevant in other domains besides business. For example, the
Israeli government has for many years maintained a policy that it will not—under any
circumstances—negotiate with terrorists. The objective of this commitment is to deter
terrorist organizations from using hostage-taking as a strategy to induce Israel to make
concessions, such as the release of prisoners. This policy ties Israel’s hands, and it is
possible to imagine particular circumstances in which an absolute stance against negotiation could be unwise. But if an unconditional refusal to negotiate alters the game by
deterring terrorist acts, then this sort of inflexibility can have enormous strategic value.
In order for a strategic move to work, it must be visible, understandable, and hard
to reverse. In our capacity expansion example, Toyota must observe and understand
that Honda has made the commitment to the “build large” strategy. Otherwise, this
move will not affect Toyota’s decision making. Irreversibility is necessary in order for
the strategic move to be credible. Toyota must believe that Honda will not back down
from its commitment to build a large plant. This is important because in our simple
example, Honda’s ideal course of action is to bluff Toyota into believing that it intends to
choose “build large,” thereby causing Toyota to choose “do not build,” but then to actually choose “build small.” For example, Honda might announce that it intends a large
capacity expansion project in the hope that Toyota will then abandon its decision to expand. Once this happens, Honda would then scale back its own decision to expand. If
Honda bluffs in this fashion and induces the outcome (“build small,” “do not build”),
Honda will enjoy a profit of $20 million, as opposed to the $18 million it would get if
it carried out its “build large” strategy. Of course, Toyota should understand this and
discount as bluster any claims that Honda makes regarding its intention to choose the
aggressive strategy, unless those claims are backed up with credible actions.
What makes a strategic move hard to reverse? One factor that contributes to irreversibility is the extent to which the strategic move involves the creation of specialized
assets—assets that cannot be easily redeployed to alternative uses. To illustrate, suppose
that Airbus, hoping to get a jump on arch-rival Boeing, decides to invest resources to
build next-generation superjumbo jets before Boeing decides whether it will offer a
20
This term was coined by Thomas Schelling in his book The Strategy of Conflict (Cambridge, MA.:
Harvard University Press, 1960).
21
See J. Tirole, Theory of Industrial Organization (Cambridge, MA.: MIT Press, 1988) for a careful analysis
of these and many other strategic moves. Chapter 7 of D. Besanko, D. Dranove, and M. Shanley,
Economics of Strategy, 3rd ed. (New York: Wiley, 2004) contains a less formal treatment of the economics
of strategic moves in a business setting.
1 4 . 3 S E Q U E N T I A L - M OV E G A M E S A N D S T R AT E G I C M OV E S
595
similar product.22 The multibillion-dollar investment in tooling and equipment that
Airbus must make to build superjumbo jets is very specialized. Once these investments
are made, the tooling and equipment have no good alternative uses. Given this, once
Airbus has built its capacity for manufacturing superjumbo jets, it will be unlikely to back
down by shutting down its factory unless competitive circumstances become so bad that
it cannot cover its nonsunk costs. The specialized nature of the assets implies that most of
Airbus’s cost are sunk, so average nonsunk costs are small. This creates a strong economic
incentive for Airbus not to reverse its strategic move. This irreversibility is especially important in Boeing’s and Airbus’s race to develop superjumbo jets because most observers
believe that market demand is insufficient to profitably support more than one firm.
Contracts can also facilitate irreversibility. One example of this is a most favored
customer clause (MFCC). If a seller includes such a clause in a sales contract with a
buyer, the seller is required to extend the same price terms to the buyer that it extends
to its other customers. For example, if the seller discounts below its list price to steal
a customer from a competitor, the buyer with an MFCC in its contract is entitled to
the same discount. The MFCC makes discounting “expensive,” and for this reason it
can create a credible commitment not to discount below the official list price.
Sometimes even public statements of intentions to take actions (“We plan to introduce a new and improved version of our existing product six months from now”) make
it hard for a firm to reverse course. For this to be true, however, the firm’s competitors
and customers must understand that the firm or its management would put something
at risk by failing to match words with actions; otherwise, they will recognize that talk
is cheap and discount the claims, promises, or threats the firm is making. The credibility of public announcements is enhanced when it is clear that the reputation of the firm
or its senior management would suffer if the firm failed to carry out what it has said it
will do. In the computer software industry, it is more common for established firms,
such as Microsoft, to make promises about new product performance and introduction
dates than it is for smaller firms or industry newcomers. This may, in part, be related
to the fact that a newcomer has far more to lose in terms of credibility with consumers
and opinion setters in personal computer magazines (an important forum for product
reviews) than an established firm has. For this reason, smaller firms may be more
reluctant to make claims than established firms that have had a past track record of success. Failure to match actions to words will result in a significant loss of face or diminution of reputation for the smaller firm and its senior management.
A P P L I C A T I O N
14.7
Irreversibility and Credible
Strategies by Airlines
How irreversible are the business decisions that real
companies actually make? Ming-Jer Chen and Ian
MacMillan set out to answer this question in the air-
22
line industry.23 They asked airline executives and
industry analysts (e.g., financial analysts and academic
experts) to rank the degree of irreversibility in various
competitive moves that airlines often make. They
learned that, in the opinion of industry participants
and observers, mergers/acquisitions, investments in
the creation of hub airports, and feeder alliances with
Superjumbo jets are ultralarge jets capable of carrying 500 or 600 passengers. The largest available
commercial jet, Boeing’s 747, can carry up to 400 passengers. Airbus has actually decided to develop a
superjumbo jet, the A380.
23
M. -J. Chen and I. C. MacMillan, “Nonresponse and Delayed Response to Competitive Moves: The Role
of Competitor Dependence and Action Irreversibility,” Academy of Management Journal, 35 (1992): 539–570.
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commuter airlines had the highest degree of irreversibility. Decisions to abandon a route, increases in
commission rates for travel agents, promotional advertising campaigns, and pricing decisions were seen
by industry participants and experts as being the easiest moves to reverse.
Chen and MacMillan hypothesized that competitors are less likely to match an airline’s competitive
move when the original move is hard to reverse. Their
logic is akin to that in the Honda–Toyota example in
this chapter. The more credible a firm’s commitment
to an aggressive strategic move is, the more likely it is
that its competitors will respond by choosing a less
aggressive strategy. This logic would suggest that a
preemptive move by one airline to expand its route
system by acquiring another airline is less likely to
provoke a matching response than is a decision to
engage in a short-term promotional or advertising
campaign. Chen and MacMillan tested this hypothesis
through an exhaustive study of competitive moves
and countermoves reported over an eight-year period
(1979–1986) in a leading trade publication of the airline industry, Aviation Daily. In general, their findings
support their hypothesis: Harder-to-reverse moves are
less frequently matched than easier-to-reverse moves.
The study suggests that price cuts are especially
provocative and thus likely to be matched frequently
and quickly. MacMillan and Chen found that rival airlines responded to price cuts more frequently than to
other moves they saw as having a similar, or even
higher, degree of irreversibility.
CHAPTER SUMMARY
• Game theory is the branch of economics concerned
with the analysis of optimal decision making when all
decision makers are presumed to be rational, and each is
attempting to anticipate the actions and reactions of its
competitors.
• A Nash equilibrium in a game occurs when each
player chooses a strategy that gives the highest payoff,
given the strategies chosen by the other players in the
game. (LBD Exercises 14.1, 14.2)
• Prisoners’ dilemma games illustrate the conflict
between self-interest and collective interest. In the Nash
equilibrium of a prisoners’ dilemma game, each player
chooses a “noncooperative” strategy, even though it is in
the players’ collective interest to pursue a cooperative
strategy.
• A dominant strategy gives a higher payoff than any
other strategy the player might follow, no matter what
the other player does. A dominated strategy gives a
lower payoff than another strategy, no matter what the
other player does.
• When both players in a game have a dominant strategy,
those strategies define the Nash equilibrium. If one
player has a dominant strategy, the Nash equilibrium is
defined by the other player’s best response to that strategy. If neither player has a dominant strategy, we can
often find the Nash equilibrium by eliminating dominated strategies.
• In many games, some or all players may have neither
a dominant strategy nor dominated strategies, and some
games, such as Chicken, have more than one Nash equilibrium. To find the Nash equilibria in any game, first
find Player 1’s best response to each of Player 2’s strategies, then find Player 2’s best response to each of Player
1’s strategies, and then see where these best responses
occur together.
• A pure strategy is a specific choice among the possible moves in a game. Under a mixed strategy, a player
chooses among two or more pure strategies according to
prespecified probabilities. Every game has at least one
Nash equilibrium in mixed strategies.
• In a repeated prisoners’ dilemma game, the players
might, in equilibrium, play cooperatively. The likelihood of a cooperative outcome is enhanced when the
players are patient, their interactions are frequent,
cheating is easy to detect, and the one-shot gain from
cheating is small.
• An analysis of sequential-move games reveals that
moving first in a game can have strategic value. (LBD
Exercise 14.3)
• A strategic move is an action you take in an early stage
of a game that alters your behavior and your competitors’
behavior later in the game in a way that is favorable to
you. Strategic moves can limit a player’s flexibility but in
so doing can have strategic value.
597
PROBLEMS
REVIEW QUESTIONS
1. What is a Nash equilibrium? Why would strategies
that do not constitute a Nash equilibrium be an unlikely
outcome of a game?
2. What is special about the prisoners’ dilemma game? Is
every game presented in this chapter a prisoners’ dilemma?
3. What is the difference between a dominant strategy
and a dominated strategy? Why would a player in a game
be unlikely to choose a dominated strategy?
4. What is special about the game of Chicken? How
does the game of Chicken differ from the prisoners’
dilemma game?
5. Can a game have a Nash equilibrium even though
neither player has a dominant strategy? Can a game have
a Nash equilibrium even though neither player has a
dominated strategy?
6. What is the difference between a pure strategy and a
mixed strategy?
7. How can cooperation emerge in the infinitely repeated prisoners’ dilemma game even though in a singleshot prisoners’ dilemma, noncooperation is a dominant
strategy?
8. What are the conditions that enhance the likelihood
of a cooperative outcome in a repeated prisoners’
dilemma game?
9. What is the difference between a simultaneousmove game and a sequential-move game?
10. What is a strategic move? Why must strategic
moves be hard to reverse in order to have strategic value?
PROBLEMS
14.1. What is the Nash equilibrium in the following
game?
Player 2
Left Right
Player 1
Up
Down
2, 6
0, 9
8, 5
12, 3
14.2. Ignoring mixed strategies, does the following
game have a Nash equilibrium? Does it have more than
one Nash equilibrium? If so, what are they?
Player 2
Player 1
North
South
West
East
2, 1
3, 2
1000, 900
2, 1
14.3. Does either player in the following game have a
dominant strategy? If so, identify it. Does either player
have a dominated strategy? If so, identify it. What is the
Nash equilibrium in this game?
Left
Player 1
Up
Down
15, 12
13, 11
whether to follow an aggressive advertising strategy, in
which the firm significantly increases its spending on
media and billboard advertising over last year’s level, or a
restrained strategy, in which the firm keeps its advertising spending equal to last year’s level. The profits associated with each strategy are as follows:
Pepsi
Aggressive Restrained
Coca-Cola
Aggressive
Restrained
8, 10
5, 14
14.4. Coca-Cola and Pepsi are competing in the
Brazilian soft-drink market. Each firm is deciding
$170, $40
$120, $100
What is the Nash equilibrium in this game? Is this game
an example of the prisoners’ dilemma?
14.5. In the Castorian Airline market there are only
two firms. Each firm is deciding whether to offer a frequent flyer program. The annual profits (in millions of
dollars) associated with each strategy are summarized in
the following table (where the first number is the payoff
to Airline A and the second to Airline B):
Airline B
With Frequent No Frequent
Flyer Program Flyer Program
Player 2
Middle Right
14, 8
12, 9
$100, $80
$80, $140
With Frequent
Flyer Program
Airline A
No Frequent
Flyer Program
200, 160
340, 80
160, 280
240, 200
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a) Does either player have a dominant strategy? Explain.
b) Is there a Nash equilibrium in this game? If so what is it?
c) Is this game an example of the prisoners’ dilemma?
Explain.
14.6. Asahi and Kirin are the two largest sellers of beer
in Japan. These two firms compete head to head in the
dry beer category in Japan. The following table shows
the profit (in millions of yen) that each firm earns when
it charges different prices for its beer:
Kirin
Asahi
¥630
¥660
¥690
¥720
Alcatel
Enter
Do Not Enter
1,000,
1,000
500, 0
0, 500
0, 0
Ignoring mixed strategies, find all of the Nash equilibria
in this game.
14.9. ABC and XYZ are the only two firms selling gizmos in Europe. The following table shows the profit (in
millions of euros) that each firm earns at different prices
(in euros per unit). ABC’s profit is the left number in each
cell; XYZ’s profit is the right number.
¥630
180, 180
184, 178
185, 175
186, 173
¥660
¥690
178, 184
175, 185
183, 183
182, 192
192, 182
191, 191
194, 180
198, 190
Price
20
24
28
32
¥720
173, 186
180, 194
190, 198
196, 196
20
60, 60
68, 56
70, 50
72, 46
24
28
56, 68
50, 70
66, 66
64, 84
84, 84
82, 82
88, 60
96, 80
32
46, 72
60, 88
80, 96
92, 92
a) Does Asahi have a dominant strategy? Does Kirin?
b) Both Asahi and Kirin have a dominated strategy: Find
and identify it.
c) Assume that Asahi and Kirin will not play the dominated strategy you identified in part (b) (i.e., cross out the
dominated strategy for each firm in the table). Having
eliminated the dominated strategy, show that Asahi and
Kirin now have another dominated strategy.
d) Assume that Asahi and Kirin will not play the dominated strategy you identified in part (c). Having eliminated this dominated strategy, determine whether Asahi
and Kirin now have a dominant strategy.
e) What is the Nash equilibrium in this game?
14.7.
Nokia
Do Not Enter
Enter
Consider the following game:
Player 2
Left
Right
Player 1
Up
1, 4
100, 3
Down
0, 3
0, 2
a) What is the Nash equilibrium in this game?
b) If you were Player 1, how would you play this game?
14.8. It is the year 2099, and the moon has been colonized by humans. Alcatel (the French telecom equipment
company) and Nokia (the Finnish telecom equipment
company) are trying to decide whether to invest in the
first cellular telecommunications system on the moon.
The market is big enough to support just one firm profitably. Both companies must make huge expenditures in
order to construct a cellular network on the moon. The
payoffs that each firm gets when it enters or does not
enter the moon market are as follows:
XYZ
ABC
Is there a unique Nash equilibrium in this game? If so,
what is it? If not, why not? Explain clearly how you arrive at your answer.
14.10. Two pipeline firms are contemplating entry into
a market delivering crude oil from a port to a refinery.
Pipeline 1, the larger of the two firms, is contemplating
its capacity strategy, which we might broadly characterize as “aggressive” and “passive.” The “aggressive” strategy involves a large increase in capacity aimed at increasing the firm’s market share, while the passive strategy
involves no change in the firm’s capacity. Pipeline 2, the
smaller competitor, is also pondering its capacity expansion strategy; it will also choose between an “aggressive
strategy” or a “passive strategy.” The following table
shows the present value of the profits associated with
each pair of choices made by the two firms:
Pipeline 2
Aggressive
Passive
Pipeline 1
Aggressive
Passive
75, 25
90, 45
100, 30
110, 40
a) If both firms decide their strategies simultaneously,
what is the Nash equilibrium?
b) If Pipeline 1 could move first and credibly commit to
its capacity expansion strategy, what is its optimal strategy? What will Pipeline 2 do?
14.11. Lucy and Ricky are making plans for Saturday
night. They can go to either a ballet or a boxing match.
Each will make the choice independently, although as
599
PROBLEMS
you can see from the following table, there are some benefits if they end up doing the same thing. Ignoring mixed
strategies, is there a Nash equilibrium in this game? If so,
what is it?
Ricky
Boxing Match
Ballet
Lucy
Ballet
Boxing Match
100, 30
90, 90
90, 90
30, 100
14.12. Suppose market demand is P ⫽ 130 ⫺ Q.
a) If two firms compete in this market with marginal cost
c ⫽ 10, find the Cournot equilibrium output and profit
per firm.
b) Find the monopoly output and profit if there is only
one firm with marginal cost c ⫽ 10.
c) Using the information from parts (a) and (b), construct
a 2 ⫻ 2 payoff matrix where the strategies available to
each of two players are to produce the Cournot equilibrium quantity or half the monopoly quantity.
d) What is the Nash equilibrium (or equilibria) of the
game you constructed in part (c)?
14.13.
Consider the following game, where x ⬎ 0:
Firm 2
Firm 1
High Price
Low Price
High Price
Low Price
140, 140
90 x, 90 x
20, 160
50, 50
a) For what values of x do both firms have a dominant
strategy? What is the Nash equilibrium (or equilibria) in
these cases?
b) For what values of x does only one firm have a dominant strategy? What is the Nash equilibrium (or equilibria) in these cases?
c) Are there any values of x such that neither firm has a
dominant strategy? Ignoring mixed strategies, is there a
Nash equilibrium in such cases?
14.14. Professor Nash announces that he will auction
off a $20 bill in a competition between Jack and Jill, two
students chosen randomly at the beginning of class. Each
student is to privately submit a bid on a piece of paper;
whoever places the highest bid wins the $20 bill. (In the
event of a tie, each student gets $10.) The catch, however,
is that each student must pay whatever he or she bid, regardless of who wins the auction. Suppose that each student
has only two $1 bills in his or her wallet that day, so the
available strategies to each student are to bid $0, $1, or $2.
a) Write down a 3 ⫻ 3 payoff matrix describing this
game.
b) Does either student have any dominated strategies?
c) What is the Nash equilibrium in this game?
d) Suppose that Jack and Jill each could borrow money
from the other students in the class, so that each of them
had a total of $11 to bid. Would ($11, $11) be a Nash
equilibrium?
14.15. Consider the following game between Sony, a
manufacturer of video cassette players, and Columbia
Pictures, a movie studio. Each firm must decide whether to
use the VHS or Beta format—Sony to make video players,
Columbia to release its movies for rental or purchase.
Columbia Pictures
Beta
VHS
Sony
Beta
VHS
20, 10
0, 0
0, 0
10, 20
a) Restrict attention to pure strategies. Does either firm
have a dominant strategy? What is (are) the Nash equilibrium (equilibria) of this game?
b) Is there a mixed strategy Nash equilibrium in this
game? If so, what is it?
c) Restrict attention again to pure strategies, but now
focus on a sequential-move game in which Sony chooses
its strategy first. What is (are) the Nash equilibrium
(equilibria) of this game?
14.16. In a World Series game, Tim Lincecum is pitching and Joe Mauer is batting. The count on Mauer is 3 balls
and 2 strikes. Lincecum has to decide whether to throw a
fastball or a curveball. Mauer has to decide whether to
swing or not swing. If Lincecum throws a fastball and
Mauer doesn’t swing, the pitch will almost certainly be a
strike, and Mauer will be out. If Mauer does swing, however, there is a strong likelihood that he will get a hit. If
Lincecum throws a curve and Mauer swings, there is a
strong likelihood that Mauer will strike out. But if
Lincecum throws a curve and Mauer doesn’t swing, there is
a good chance that it will be ball four and Mauer will walk
(assume that a walk is as good as a hit in this instance).
The following table shows the payoffs from each
pair of choices that the two players can make:
Joe Mauer
Swing
Do Not Swing
Tim Lincecum
Fastball
Curveball
100, 100
100, 100
100, 100
100, 100
a) Is there a Nash equilibrium in pure strategies in this
game?
b) Is there a mixed strategy Nash equilibrium in this
game? If so, what is it?
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14.17. In the mid-1990s, Value Jet wanted to enter the
market serving routes that would compete head to head
with Delta Airlines in Atlanta. Value Jet knew that Delta
might respond in one of two ways: Delta could start a
price war or it could be “accommodating,” keeping the
price at a high level. Value Jet had to decide whether it
would enter on a small scale or on a large scale. The
annual profits (in zillions of dollars) associated with each
strategy are summarized in the following table (where
the first number is the payoff to Value Jet and the second
the payoff to Delta):
Delta
Accommodate Price Low
(Price High) (Price War)
Value Jet
Enter on Small Scale
Enter on Large Scale
8, 40
16, 20
2, 32
4, 24
a) If Value Jet and Delta choose their strategies simultaneously, what strategies would the two firms choose at
the Nash equilibrium, and what would be the payoff for
Value Jet? Explain.
b) As it turned out, Value Jet decided to move first, entering on a small scale. It communicated this information
by issuing a public statement announcing that it had limited aspirations in this marketplace and had no plans to
grow beyond its initial small size. Analyze the sequential
game in which Value Jet chooses “small” or “large” in the
first stage and then Delta accommodates or starts a price
war in the second stage. Did Value Jet enhance its profit
by moving first and entering on a small scale? If so, how
much more did it earn with this strategy? If not, explain
why not? (Hint: Draw the game tree.)
14.18. Besanko, Inc. and Braeutigam, Ltd. compete in
the high-grade carbon fiber market. Both firms sell identical grades of carbon fiber, a commodity product that
will sell at a common market price. The challenge for
each firm is to decide upon a capacity expansion strategy.
The following problem pertains to this choice.
a) Suppose it is well known that long-run market demand
in this industry will be robust. In light of that, the payoffs
associated with various capacity expansion strategies that
Besanko and Braeutigam might pursue are shown in the
following table. What are the Nash equilibrium capacity
choices for each firm if both firms make their capacity
choices simultaneously?
b) Again, suppose that the table gives the payoffs to each
firm under various capacity scenarios, but now suppose
that Besanko can commit in advance to a capacity strategy. That is, it can choose no expansion, modest expansion, or major expansion. Braeutigam observes this
choice and makes a choice of its own (no expansion or
modest expansion). What is the equilibrium in this
sequential-move capacity game?
Braeutigam
No
Modest
Expansion
Expansion
Besanko
No Expansion
Modest Expansion
Major Expansion
$1,013, $1,013
$1,125, $844
$1,013, $506
$844, $1,125
$900, $900
$675, $450
14.19. Boeing and Airbus are competing to fill an
order of jets for Singapore Airlines. Each firm can offer a
price of $10 million per jet or $5 million per jet. If both
firms offer the same price, the airline will split the order
between the two firms, 50–50. If one firm offers a higher
price than the other, the lower-price competitor wins the
entire order. Here is the profit that Boeing and Airbus
expect they could earn from this transaction:
Boeing
Airbus
P ⴝ $5m
P ⴝ $10m
P ⴝ $5m
30, 30
270, 0
P ⴝ $10m
0, 270
50, 50
(payoffs are in millions of dollars)
a) What is the Nash equilibrium in this game?
b) Suppose that Boeing and Airbus anticipate that they
will be competing for orders like the one from Singapore
Airlines every quarter, from now to the foreseeable future.
Each quarter, each firm offers a price, and the payoffs are
determined according to the table above. The prices
offered by each airline are public information. Suppose
that Airbus has made the following public statement:
To shore up profit margins, in the upcoming quarter we
intend to be statesmanlike in the pricing of our aircraft
and will not cut price simply to win an order. However,
if the competition takes advantage of our statesmanlike
policy, we intend to abandon this policy and will compete
all out for orders in every subsequent quarter.
Boeing is considering its pricing strategy for the
upcoming quarter. What price would you recommend that
Boeing charge? Important note: To evaluate payoffs,
imagine that each quarter, Boeing and Airbus receive
their payoff right away. (Thus, if in the upcoming quarter, Boeing chooses $5 million and Airbus chooses
$10 million, Boeing will immediately receive its profit of
$270 million.) Furthermore, assume that Boeing and
Airbus evaluate future payoffs in the following way: a
stream of payoffs of $1 starting next quarter and received
in every quarter thereafter has exactly the same value as a
one-time payoff of $40 received immediately this quarter.
601
PROBLEMS
c) Suppose that aircraft orders are received once a year
rather than once a quarter. That is, Boeing and Airbus
will compete with each other for an order this year (with
payoffs given in the table above), but their next competitive encounter will not occur for another year. In terms
of evaluating present and future payoffs, suppose that
each firm views a stream of payoffs of $1 starting next
year and received every year thereafter as equivalent to
$10 received immediately this year. Again assuming that
Airbus will follow the policy in its public statement
above, what price would you recommend that Boeing
charge in this year and beyond?
14.20. Consider a buyer who, in the upcoming month,
will make a decision about whether to purchase a good
from a monopoly seller. The seller “advertises” that it offers a high-quality product (and the price that it has set is
based on that claim). However, by substituting low-quality
components for higher-quality ones, the seller can reduce the quality of the product it sells to the buyer, and
in so doing, the seller can lower the variable and fixed
costs of making the product. The product quality is not
observable to the buyer at the time of purchase, and so the
buyer cannot tell, at that point, whether he is getting a
high-quality or a low-quality good. Only after he begins
to use the product does the buyer learn the quality of the
good he has purchased.
The payoffs that accrue to the buyer and seller from
this encounter are as follows:
that if the buyer purchases the product and learns that he
has bought a high-quality good, he will return the next
month and buy again. Indeed, he will continue to purchase, month after month (potentially forever!), as long
as the quality of the product he purchased in the previous
month is high. However, if the buyer is ever unpleasantly
surprised—that is, if the seller sells him a low-quality
good in a particular month—he will refuse to purchase
from the seller forever after. Suppose that the seller
knows that the buyer is going to behave in this fashion.
Further, let’s imagine that the seller evaluates profits in
the following way: a stream of payoffs of $1 starting next
month and received in every month thereafter has exactly
the same value as a one-time payoff of $50 received immediately this month. Will the seller offer a low-quality
good or a high-quality good?
14.21. Two firms are competing in an oligopolistic
industry. Firm 1, the larger of the two firms, is contemplating its capacity strategy, which could be either “aggressive”
or “passive.” The aggressive strategy involves a large
increase in capacity aimed at increasing the firm’s market
share, while the passive strategy involves no change in the
firm’s capacity. Firm 2, the smaller competitor, is also
pondering its capacity expansion strategy; it will also
choose between an aggressive strategy and a passive strategy. The following table shows the profits associated with
each pair of choices:
Firm 2
Aggressive
Passive
Seller
Sell High-Quality Sell Low-Quality
Product
Product
Buyer
Purchase
Do Not
Purchase
$5, $6
$0,
$4
$4, $12
$0,
Firm 1
Aggressive
Passive
25, 9
33, 10
30, 13
36, 12
$1
The buyer’s payoff (consumer surplus) is listed first; the
seller’s payoff (profit) is listed second.
Answer each of the following questions, using the
preceding table.
a) What are the Nash equilibrium strategies for the buyer
and seller in this game under the assumption that it is
played just once?
b) Let’s again suppose that the game is played just once
(i.e., the buyer makes at most one purchase). But suppose
that before the game is played, the seller can commit to offering a warranty that gives the buyer a monetary payment
W in the event that he buys the product and is unhappy
with the product he purchases. What is the smallest value
of W such that the seller chooses to offer a high-quality
product and the buyer chooses to purchase?
c) Instead of the warranty, let’s now allow for the possibility of repeat purchases by the buyer. In particular, suppose
a) If both firms decide their strategies simultaneously,
what is the Nash equilibrium?
b) If Firm 1 could move first and credibly commit to its
capacity expansion strategy, what is its optimal strategy?
What will Firm 2 do?
14.22. The only two firms moving crude oil from an
oil-producing region to a port in Atlantis are pipelines:
Starline and Pipetran. The following table shows the annual profit (in millions of euros) that each firm would
earn at different capacities. Starline’s profit is the left
number in each cell; Pipetran’s profit is the right number.
At the current capacities (with no expansion) Starline is
earning 40 million euros, and Pipetran is earning 18 million euros annually. Each company is considering an
expansion of its capacity. Since Pipetran is a fairly small
company, it can consider only a small expansion to its
capacity. Starline has the ability to consider both a small
and a large expansion.
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ABC’s profit is the left number in each cell; XYZ’s profit
is the right number. For example, if ABC makes the
sweet cereal and XYZ produces the high-fiber cereal, annual profits will be 50 million pesos for ABC and 60 million pesos for XYZ.
Pipetran
No Expansion
Small
Starline
No Expansion
Small
Large
40, 18
48, 14
38, 10
28, 22
32, 16
24, 5
XYZ
High Fiber
Sweet
a) If the two firms make their decisions about expansion
simultaneously, is there a unique Nash equilibrium? If so,
what is it? If not, why not? Explain whether this game is
an example of a prisoners’ dilemma.
b) Would Starline have a first-mover advantage if capacities were chosen sequentially? If so, briefly explain how
it might credibly implement this strategy.
c) Suppose you were hired to advise Pipetran about its
choice of capacity. If Pipetran has the option of moving
first, should it do so? Explain.
14.23. ABC and XYZ are the two cereal manufacturers
contemplating entry into a South American market. Each
will be able to build one plant, and that plant can be used
to make either a cereal that is high in fiber and low in
calories (High Fiber) or a less healthy cereal with a sweet
taste (Sweet). Once a plant is chosen to produce one kind
of cereal, it will be prohibitively expensive to switch production to the other type. The following table shows the
annual profit (in millions of pesos) that each firm would
earn given the production choices of the two firms.
ABC
Sweet
High Fiber
50, 60
30, 40
20, 30
40, 60
a) If the two firms choose the type of plant simultaneously, is there a unique Nash equilibrium? If so, what is
it? If not, why not?
b) Would ABC have a first-mover advantage if capacities
were chosen sequentially? If so, briefly explain how it
might credibly implement this strategy.
c) Would XYZ have a first-mover advantage if capacities
were chosen sequentially? If so, briefly explain how it
might credibly implement this strategy.
14.24. Cities A, B, and C are located in different countries. The only airline serving the market between A and
B is Ajax Air. Its total cost is CAjax ⫽ 20QAB. The airfare
between A and B is PAB. Also, the only carrier serving the
market between B and C is Sky Air. Its total cost is CSky ⫽
20QBC. The airfare between B and C is PBC. The two airlines do not serve any other markets.
SKY
PBC
100
PAB
100
1600
1600
PAB
95
1875
2000
PAB
90
2100
2400
PAB
85
2275
2800
AJAX
PAB
80
2400
3200
PAB
70
2500
4000
PAB
65
2475
4400
PAB
60
2400
4800
PAB
55
2275
5200
PBC
95
PBC
90
2000
1875
2250
2250
2450
2625
2600
3000
2700
3375
2750
4125
2700
4500
2600
4875
2450
5250
2800
3200
2275
2400
2625
3000
3375
2450
2600
2700
2800
3150
3500
2800
2925
3000
2925
3250
3575
3150
3250
3300
3000
3300
3600
3500
3575
3600
3000
3250
3500
4200
4225
4200
2925
3150
3375
4550
4550
4500
2800
3000
3200
4900
4875
4800
2625
2800
2975
5250
5200
5100
PBC
85
PBC
80
PBC
70
PBC
65
PBC
60
PBC
55
4000
2500
4125
2750
4200
3000
4225
3250
4200
3500
4000
4000
3825
4250
3600
4500
3325
4750
4400
2475
4500
2700
4550
2925
4550
3150
4500
3375
4250
3825
4050
4050
3800
4275
3500
4500
4800
2400
4875
2600
4900
2800
4875
3000
4800
3200
4500
3600
4275
3800
4000
4000
3675
4200
5200
2275
5250
2450
5250
2625
5200
2800
5100
2975
4750
3325
4500
3500
4200
3675
3850
3850
PROBLEMS
All traffic on the network flows between A and C,
using B only as a point to interconnect with the other airline. (In other words, no traffic originates or terminates
at B.) The demand for passenger service between A and
C is QAC ⫽ 220 ⫺ PAC, where Q is the number of units
of passenger traffic demanded when PAC, the total airfare
between A and C, is PAB PBC.
A
B
Ajax Air: Air fare = PAB
Cost: CAB = 20QAB
C
Sky Air: Air fare = PBC
Cost: CBC = 20QBC
a) The preceding table shows the profits for each carrier for
various combinations of airfares. The upper left number in
a cell shows Ajax’s profit; the lower right number shows
603
Sky’s profit. Suppose Ajax charges PAB ⫽ 100 and Sky
charges PBC ⫽ 90. Determine the profit for each of the
two carriers, and enter your calculation in the table.
b) Currently, Ajax and Sky are not allowed to coordinate
prices. They must act noncooperatively when setting
their fares. Using the preceding table, find the Nash
equilibrium fares. Explain how you arrived at your
answer.
c) The two airlines have been lobbying antitrust authorities to allow them to merge, an act that would enable
them to price jointly as a monopolist. The merged airline
would still stop at B for refueling. The cost and demand
curves would not change if the carriers merged. Use the
table to determine what price the merged entity would
charge for a trip between A and C, and explain your reasoning clearly.
15
RISK AND
INFORMATION
15.1
DESCRIBING RISKY OUTCOMES
APPLICATION 15.1
Tumbling Dice and the Lucky
Number 7
15.2
E VA L UAT I N G R I S K Y O U T C O M E S
APPLICATION 15.2
Risk Premia for Employee
Stock Options
15.3
B E A R I N G A N D E L I M I N AT I N G R I S K
If AIG Can Collapse,
Why Would Anyone Supply Insurance?
APPLICATION 15.4 Obamacare and Adverse
Selection in the Health Insurance Market
APPLICATION 15.3
15.4
A N A LY Z I N G R I S K Y D E C I S I O N S
APPLICATION 15.5
Putting Money in a Hole
in the Ground?
15.5
AU C T I O N S
APPLICATION 15.6
The Winner’s Curse
in the Classroom
APPLICATION 15.7
Google AdWords
What Are My Chances of Winning?
No company better symbolizes the emergence of the Internet as a vehicle for commerce than Amazon.com.
Launched as “Earth’s Biggest Bookstore” in July 1995 by 32-year-old Jeff Bezos, Amazon.com now offers
DVDs, videos, toys, consumer electronics, clothing, tools, and even groceries. For some consumers,
Amazon.com is their first and only destination.
What would have happened if you had invested in Amazon.com? Suppose in September 1999, you
had bought $1,000 worth of Amazon’s stock. Figure 15.1 shows how the market value of that $1,000
investment would have changed over the next 11 years. In the first few months after your purchase, its value
would have grown, reaching about $1,060 by December 2001. However, over the next two years, the value
of your investment would have fallen substantially. By October 2001, in the wake of 9/11 and the bursting
604
$1,900
$1,800
$1,700
$1,600
$1,500
Value of $1,000 investment
$1,400
$1,300
$1,200
$1,100
$1,000
$900
$800
$700
$600
$500
$400
$300
$200
$100
9/
1/
1
3/ 999
1/
2
9/ 00
1/ 0
20
3/ 00
1/
2
9/ 001
1/
2
3/ 00
1/ 1
2
9/ 002
1/
2
3/ 00
1/ 2
20
9/ 03
1/
2
3/ 003
1/
2
9/ 004
1/
2
3/ 004
1/
2
9/ 005
1/
2
3/ 00
1/ 5
2
9/ 006
1/
2
3/ 006
1/
2
9/ 007
1/
2
3/ 007
1/
2
9/ 008
1/
2
3/ 008
1/
2
9/ 009
1/
2
3/ 009
1/
20
10
$50
FIGURE 15.1 `Value of $1,000 Invested in Amazon.com, September 1, 1999–April 1, 2010
The value of a $1,000 investment in Amazon.com stock in September 1999 has fluctuated considerably over the last eleven years. By April 2010, it was worth over $1,750.
Source: Yahoo Finance, http://finance.yahoo.com/q/hp?s=AMZN (accessed April 10, 2010).
of the technology bubble, your investment would have been worth just $87, a loss of value of nearly
93 percent. Five years after your initial investment, in September 2004, your investment would have been
worth about $500. Three years later, in September
2007, your investment would have rebounded in value
to $1,165, only to fall 14 months later in November
2008, in the midst of the financial crisis and the Great
Recession, to $534. But nearly a year and half after
that, your investment would have grown again,
cresting to over $1,700 as of April 2010.
The fate of Amazon.com’s stock provides an excellent example of risk. Investing in Amazon’s stock is like
riding a roller coaster in a fog bank. You know it will go
up and down, but you can’t predict when the ups and
downs will occur, nor how severe they will be. Economic
life is full of risk situations: entrepreneurs face a risk of
failure when they launch new businesses; sports teams
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R I S K A N D I N F O R M AT I O N
face a risk of sub-par performance when they sign a free agent to an expensive contract; households face
the risk of large medical bills if a person in the household becomes sick or experiences an accident; and
bidders face the risk of overpaying for items of unknown value when participating in auctions.
This chapter is about risk, imperfect information, and how we can employ tools from microeconomics
to analyze risky phenomena and decisions made in the face of risk.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Describe risky outcomes using the concepts of probability, expected value, and variance.
• Illustrate how the shape of an individual’s utility function describes his or her attitudes toward risk.
• Calculate expected utility as a way to evaluate risky outcomes.
• Compute the risk premium for a risk-averse decision maker.
• Explain why risk-averse individuals would purchase full insurance if it is fairly priced.
• Contrast two different types of asymmetric information in insurance markets: moral hazard and adverse
selection.
• Analyze risky decisions using a decision tree.
• Differentiate between different types of auctions.
• Explain the concept of the winner’s curse.
Our goal in introducing these tools and concepts is to help you better understand economic environments
such as insurance and auction markets in which risk and imperfect information play a central role. We also
hope that these tools and concepts will help you make better decisions in your own lives—decisions perhaps
about what job to accept, whether to buy stock in an Internet-based company, such as Amazon, or how
much to bid at an Internet auction site, such as eBay or Yahoo!.
15.1
DESCRIBING
RISKY
OUTCOMES
S
uppose you have just bought $100 worth of stock in a company such as
Amazon.com. You don’t know how the stock will perform over the next year—its
value could go up or down—so the stock is risky. But just how risky is it? How does
the riskiness of this stock compare to the riskiness of other investments you might
have made with this money? Answering this question involves describing a risky outcome. In this section, you will learn three concepts for describing risky outcomes:
probability distributions, expected value, and variance.
L OT T E R I E S A N D P R O BA B I L I T I E S
Even though you don’t know what the value of your stock will be next year, you can
still describe what it might be. In particular, suppose you know that over the next year,
one of three things will happen to your $100 investment:
• Its value could go up by 20 percent to $120 (outcome A).
• Its value could remain the same (outcome B).
• Its value could fall by 20 percent to $80 (outcome C ).
607
15.1 DESCRIBING RISKY OUTCOMES
1
0.90
0.80
Probability
0.70
0.60
0.50
FIGURE 15.2
0.40
0.30
0.20
0.10
0
C
B
A
80
100
120
Payoff (stock price in $)
Probability
Distribution of a Lottery
The probability of outcome A
(value of stock goes up
20 percent, to $120) is 0.30.
The probability of outcome B
(value of stock remains the
same, at $100) is 0.40. The
probability of outcome C
(value of stock goes down
20 percent, to $80) is 0.30.
Your investment in the stock is an example of a lottery. In real life, a lottery is a game
of chance. In microeconomics, we use the term lottery to describe any event—an investment in a stock, the outcome of a college football game, the spin of a roulette
wheel—for which the outcome is uncertain.
The lottery described above has three possible outcomes: A, B, and C. The
probability of a particular outcome of a lottery is the likelihood that this outcome will
occur. If there is a 3 in 10 chance that outcome A will occur, we say that the probability of A is 3/10, or 0.30. If outcome B has a 4 in 10 chance of occurring, we say that
the probability of B is 4/10, or 0.40. And if there is a 3 in 10 chance that outcome C
will occur, the probability of C is 0.30. The probability distribution of the lottery
depicts all possible outcomes in the lottery and their associated probabilities. The bar
graph in Figure 15.2 shows the probability distribution of our Internet company’s
stock price. Each bar represents a possible outcome, and the height of each bar measures the probability of that outcome. For any lottery, the probabilities of the possible
outcomes have two important properties:
lottery Any event for
which the outcome is
uncertain.
probability The likelihood that a particular
outcome of a lottery
will occur.
probability distribution
A depiction of all possible
payoffs in a lottery and their
associated probabilities.
• The probability of any particular outcome is between 0 and 1.
• The sum of the probabilities of all possible outcomes is equal to 1.
Where do probabilities and probability distributions come from? Some probabilities result from laws of nature. For example, if you toss a coin, the probability that it
will come up heads is 0.50. You can verify this by flipping a coin over and over again.
With a large enough number of flips (100 or 200), the proportion of heads will be
about 50 percent.
However, not all risky events are like coin flips. In many cases, it might be difficult to deduce the probabilities of particular outcomes. For example, how would you
really know whether your stock has a 0.30 chance of going up by 20 percent? Your
assessment reflects not immutable laws of nature but a subjective belief about how
events are likely to unfold. Probabilities that reflect subjective beliefs about risky
events are called subjective probabilities. Subjective probabilities must also obey the
subjective probabilities
Probabilities that reflect
subjective beliefs about
risky events.
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two properties of probability just described. However, different decision makers might
have different beliefs about the probabilities of possible outcomes of a given risky
event. For example, an investor more optimistic than you might believe the following:
• Probability of A 0.50 (there is a 5 in 10 chance that the stock’s value will go
up by 20 percent).
• Probability of B 0.30 (there is a 3 in 10 chance that the stock’s value will stay
the same).
• Probability of C 0.20 (there is a 2 in 10 chance that the stock’s value will go
down by 20 percent).
These subjective probabilities differ from yours, but they still obey the two basic laws
of probability: each is between 0 and 1, and they add up to 1.
E X P E C T E D VA L U E
expected value A
measure of the average
payoff that a lottery will
generate.
Given the probabilities associated with the possible outcomes of your risky investment, how much can you expect to make, that is, what is the expected value of the
investment? The expected value of a lottery is the average payoff that the lottery will
generate. We can illustrate this with our Internet stock example:
Expected value probability of A ⫻ payoff if A occurs
probability of B ⫻ payoff if B occurs
probability of C ⫻ payoff if C occurs
Applying this formula we get
Expected value ⫽ (0.30 ⫻ 120) (0.40 ⫻ 100) (0.30 ⫻ 80)
⫽ 100
The expected value of your Internet stock is a weighted average of the possible payoffs, where the weight associated with each payoff equals the probability that the
payoff will occur. More generally, if A, B, . . . , Z denote the set of possible outcomes
of a lottery, then the expected value of the lottery is as follows:
Expected value ⫽ probability of A ⫻ payoff if A occurs
probability of B ⫻ payoff if B occurs . . .
probability of Z ⫻ payoff of Z occurs
As in the coin tossing example, the expected value of a lottery is the average payoff you would get from the lottery if the lottery were repeated many times. If you made
the same investment over and over again and averaged the payoffs, that average would
be nearly indistinguishable from the lottery’s expected value of $100.
VA R I A N C E
Suppose you had a choice of two investments—$100 worth of stock in an Internet company or $100 worth of stock in a public utility (an electric company or a local waterworks).
609
1
1
0.90
0.90
0.80
0.80
0.70
0.70
0.60
0.60
Probability
Probability
15.1 DESCRIBING RISKY OUTCOMES
0.50
0.40
0.50
0.40
0.30
0.30
0.20
0.20
C
0.10
0
80
B
100
A
120
Payoff (stock price in $)
(a) Internet company
0.10
0
B
C
80
A
100
120
Payoff (stock price in $)
(b) Public utility company
FIGURE 15.3 Probability Distributions, Riskiness, and Variances
The riskiness of investing in the Internet company is much greater than the riskiness of investing in the public utility company. The probability that the actual outcome will differ from the
expected outcome (outcome B in both cases) is 6 in 10 for the Internet investment but only 2 in
10 for the public utility investment. This is reflected in the difference in the variances ($240 for
the Internet investment and $80 for the public utility investment).
Figure 15.3 depicts the probability distributions of the stock prices of these two companies. The expected values of the two stocks are the same: $100 (you should verify this).
However, the Internet stock is riskier than the public utility stock because the stock of
the public utility will probably remain at its current value of $100, but the Internet stock
has a greater likelihood of going up or down. In other words, with the Internet stock, an
investor stands to gain more or lose more than with a stock in a public utility.
We characterize the riskiness of a lottery by a measure known as the variance.
The variance of a lottery is the sum of the probability-weighted squared deviations of
the possible outcomes of the lottery. The squared deviation of a possible outcome
is the square of the difference between the lottery’s payoff for that outcome and the
expected value of the lottery. Here is how to compute the variance in the case of our
Internet investment, with the probable outcomes shown in Figure 15.3(a):
1. Find the expected value (EV ); in this case, as shown in the previous section,
EV $100.
2. Find the squared deviation of each outcome; then multiply it by the probability
of that outcome to find the probability-weighted squared deviation:
• Squared deviation of outcome A (payoff of $120) (payoff EV)2
($120 $100)2 $400.
Probability-weighted squared deviation of outcome A 0.30 ⫻ $400 ⫽ $120.
• Squared deviation of outcome B (payoff of $100) ⫽ (payoff ⫺ EV )2 ⫽
($100 ⫺ $100)2 ⫽ $0.
variance The sum of the
probability-weighted
squared deviations of the
possible outcomes of the
lottery.
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Probability-weighted squared deviation of outcome B 0.40 ⫻ $0 ⫽ $0.
• Squared deviation of outcome C (payoff of $80) ⫽ (payoff ⫺ EV )2 ⫽
($80 ⫺ $100)2 ⫽ $400.
Probability-weighted squared deviation of outcome C ⫽ 0.30 ⫻ $400 ⫽ $120.
3. Add up the probability-weighted squared deviations to find the variance:
Variance ⫽ $120 $0 $120 $240.
A P P L I C A T I O N
15.1
Tumbling Dice and the Lucky
Number 7
The number 7 is sometimes characterized as a lucky
number. Perhaps this is because in the Book of
Genesis in the Bible, the story of Creation unfolds
over six days, with the seventh day being sanctified as
the Sabbath. Or perhaps it is because there are seven
colors in a rainbow (red, orange, yellow, green, blue,
indigo, and violet). Or perhaps it is because, in number theory, seven is what is known as a lucky prime
number (a set that also includes primes such as 3, 13,
31, 37, 43, 67, 73, 79, 127 and 151). But perhaps, also,
it is because seven is the expected value of the sum of
a pair of two-sided dice, something of practical significance in gambling games such as craps or board
games such as Monopoly or Strat-O-Matic Baseball!
1
To illustrate, suppose that you throw a pair of
six-sided dice, and add the result. To calculate the
expected value, first write down all of the possible
outcomes. There are 36 possible pairs (6 times 6),
ranging from 2 to 12:
Value of a Pair of Dice
Die 2
Die 1
standard deviation
The square root of the
variance.
If we did the same computation for the investment in a public utility company, with the
probable outcomes shown in Figure 15.3(b), we would find that the variance $80.1
These results reflect what we can see intuitively by looking at Figure 15.3. The
public utility investment is much less risky than the Internet investment because the
probability that the outcome will equal the expected value (outcome B in both cases)
is 8 in 10 for the public utility investment but only 4 in 10 for the Internet investment.
An alternative measure of the riskiness of a lottery is the standard deviation,
which is simply the square root of the variance. Thus, the standard deviation of the
Internet stock is 1240 15.5, and the standard deviation of the public utility stock
is 180 8.9.
If the variance of one lottery is bigger than the variance of another lottery, it follows that the standard deviation of the first lottery will be bigger than the standard deviation of the second lottery. Thus, the standard deviation provides us with the same
information about the relative riskiness of lotteries as does the variance.
1
2
3
4
5
6
1
2
3
4
5
6
2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
5
6
7
8
9
10
6
7
8
9
10
11
7
8
9
10
11
12
The reason we square the difference (deviation) between payoff and EV is that, when the EV is greater
than the payoff (as in outcome C of both investments), the difference is a negative number. If we had
computed the variances of our two investments using deviations instead of squared deviations, the positive
and negative deviations would have canceled out, and the variance in both cases would have been zero
(you can verify this by doing the math). Thus, we would have obscured rather than revealed the very
different riskiness of the two investments.
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1 5 . 2 E VA L UAT I N G R I S K Y O U T C O M E S
The next table uses that information to calculate
the expected value. Column 2 shows the count of possible sums. For example, the most likely outcome is 7,
which can happen 6 ways out of 36 (as seen in the first
table). The bottom row of column 2 shows the total
of 36 outcomes. Column 3 then divides the count in
column 2 by 36 to give the probability for each possible sum of the dice. Note that these sum to 1.0 at the
bottom of column 3, as they should. Column 4 calculates the value of each pair, times the probability from
column 4. Summing those at the bottom gives the expected value, which exactly equals 7. Indeed, looking
at the counts in column 2, that makes sense. For example, values of 6 and 8 are equally likely, as are those
of 5 and 9, 4 and 10, and so on. The distribution of
outcomes is symmetric around its expected value of 7.
Calculating the Expected Value & Variance of a Pair of Dice
Probability
Probability ⴛ
Value of Dice
Deviation
(Value ⴚ
Expected Value)
Deviation2
Probability ⴛ
Deviation2
1
2
3
4
5
6
5
4
3
2
1
0.028
0.056
0.083
0.111
0.139
0.167
0.139
0.111
0.083
0.056
0.028
0.056
0.167
0.333
0.556
0.833
1.167
1.111
1.000
0.833
0.611
0.333
⫺5
⫺4
⫺3
⫺2
⫺1
0
1
2
3
4
5
25
16
9
4
1
0
1
4
9
16
25
0.694
0.889
0.750
0.444
0.139
0.000
0.139
0.444
0.750
0.889
0.694
7.000
5.833
36
1.000
(Expected Value)
(Variance)
Value
of Dice
Count
2
3
4
5
6
7
8
9
10
11
12
Total
We can also calculate the variance associated
with the throw of a pair of dice. To calculate the variance, we first need to calculate the deviation of each
value from the expected value, and then square that.
This is in columns 5–6. Column 7 then multiplies the
squared deviations by the probabilities. The total at
the bottom of column 7 is the variance, equal to
about 5.8.
I
n the previous section, we saw how to describe risky outcomes using probability distributions, expected values, and variances. In this section, we explore how a decision
maker might evaluate and compare alternatives whose payoffs have different probability distributions and thus different degrees of risk. In particular, we will show how we
can use the concept of a utility function that we studied in Chapter 3 to evaluate the
benefits that the decision maker would enjoy from alternatives with differing amounts
of risk.
UTILITY FUNCTIONS AND RISK PREFERENCES
Imagine that you are about to graduate and that you have two job offers. One offer is
to join a large, established company. At this company, you will earn an income of
$54,000 per year. The second offer is from a new start-up company. Because this company has been operating at a loss, you are offered a token salary of $4,000 (i.e., you
15.2
E VA L UAT I N G
RISKY
OUTCOMES
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CHAPTER 15
R I S K A N D I N F O R M AT I O N
Utility function
Utility
T
S
FIGURE 15.4 Utility Function with
Diminishing Marginal Utility
Marginal utility is diminishing because a
given increment to income increases utility
by much more when income is low than
when income is high: When income is low
($4,000), utility increases by the distance
from point Q to point R; when income is
high ($104,000), utility increases by the
distance from point S to point T.
R
Q
0 4
104
Income (thousands of dollars per year)
will work virtually for free). However, the company also promises you a bonus of
$100,000 if the company manages to become profitable during the upcoming year.
Based on your assessment of the company’s prospects, there is a 0.50 probability that
you will get the bonus and a 0.50 probability that you will not. Based on the salary offers
of the two companies, which job would you accept?2
You face an interesting decision. Your salary at the established company is a sure
thing—that is, the probability of receiving $54,000 is 1.0 (no other outcome is possible),
so the expected value is 1.0 ⫻ $54,000 ⫽ $54,000. Your salary at the start-up company
is a lottery—a 0.50 chance of receiving $4,000 and a 0.50 chance of receiving
$104,000, so the expected value is (0.50 ⫻ $4,000) (0.50 ⫻ $104,000) ⫽ 54,000.
Thus, the expected values of the two offers are equal. Even so, it seems unlikely that
you would view the offers as identical. After all, though you might get rich quick if
you receive your bonus, you also face a significant risk of ending up with only $4,000.
By contrast, the salary at the established company entails no risk.
How do we evaluate choices among alternatives that have different risks? One way
is to use the concept of a utility function. In Chapter 3, we saw that utility is a measure
of satisfaction from consuming a bundle of goods and services. Figure 15.4 depicts a
possible relationship between your utility U and your income I. This utility function is
increasing in income, so you prefer more income to less. It also exhibits diminishing
marginal utility (also discussed in Chapter 3) because the extra utility that you get from
an increment to your income gets smaller as your income increases. Thus, when your
income is low (say, $4,000), a small increase in income increases your utility by an
amount equal to the distance from point Q to point R. However, when your income is
high (say, $104,000), an equally small increase in income increases your utility by a much
smaller amount, equal to the distance from point S to point T.
2
In real life, you would decide between the two jobs based not only on the current salary offers, but also
on your long-term earning prospects at each company. And you would undoubtedly, consider various
nonmonetary aspects of the two jobs, such as the nature of the work, working hours, and location.
613
1 5 . 2 E VA L UAT I N G R I S K Y O U T C O M E S
C
320
Upside
of lottery
B
Utility
230
190
D
Downside
of lottery
60
Utility
function
A
0 4
54
104
Income (thousands of dollars per year)
FIGURE 15.5 Utility Function and Expected Utility
Your utility if you take the job with the established company will be 230 (point B). If you take
the job with the start-up, there is a 0.50 probability that your utility will be 320 (point C, if you
earn $104,000) and a 0.50 probability that your utility will be 60 (point A, if you earn $4,000),
yielding an expected utility of 190 (point D). Because your utility with the established company
is greater than your expected utility with the start-up company, you will prefer the offer from
the established company.
Figure 15.5 shows how we would use a utility function to evaluate your two job
offers:
• Your utility at the established company corresponds to point B, where you receive
an income of $54,000 and achieve a utility of 230—that is, U(54,000) 230.
• Your utility at the new company when you do not receive a bonus corresponds
to point A, where you receive an income of $4,000 and achieve a utility of 60—
that is, U(4,000) 60.
• Your utility at the new company when you receive a bonus corresponds to
point C, where you receive an income of $104,000 and achieve a utility of
320—that is, U(104,000) 320.
• Your expected utility at the start-up company (i.e., the expected value of your
utility levels if you worked there) [0.5 ⫻ U(4,000)] [0.5 ⫻ U(104,000)] ⫽
(0.5 ⫻ 60) (0.5 ⫻ 320) ⫽ 190. This corresponds to point D.
More generally, the expected utility of a lottery is the expected value of the utility levels
that the decision maker receives from the payoffs in the lottery. Thus, if A, B, . . . , Z
denote a set of possible payoffs of a lottery, then the expected utility of the lottery is
as follows:
Expected utility ⫽ probability of A ⫻ utility if A occurs
probability of B ⫻ utility if B occurs . . .
probability of Z ⫻ utility if Z occurs
(15.1)
expected utility The
expected value of the utility
levels that the decision
maker receives from the
payoffs in a lottery.
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CHAPTER 15
risk averse A characteristic of a decision maker
who prefers a sure thing to
a lottery of equal expected
value.
S
E
R I S K A N D I N F O R M AT I O N
The analysis in Figure 15.5 shows that although the expected values of the two offers
are equal, your expected utility at the new company is lower than the utility you will get
if you work for the established company. If you evaluate the offers according to the utility function in Figure 15.5, you will prefer the offer from the established company.
The utility function in Figures 15.4 and 15.5 depicts the preferences of a decision
maker who is risk averse, one who prefers a sure thing to a lottery of equal expected
value. In the example above, a risk-averse decision maker would prefer the certain
salary of the established company to the risky salary of the start-up company. In general, a utility function that exhibits diminishing marginal utility (like the one in
Figure 15.5) implies that the utility of a sure thing will exceed the expected utility of
a lottery with the same expected value. To see why this is the case, note that if you
go to work for the start-up company, the upside of the lottery is that you might have
$50,000 more income ($104,000 ⫺ $54,000) than if you worked at the established
company, while the downside is that you might have $50,000 less income ($54,000 ⫺
$4,000). Because of diminishing marginal utility, the reduction in utility from the
downside (230 ⫺ 60 ⫽ 170) is bigger than the gain in utility from the upside (320 ⫺
230 ⫽ 90), as Figure 15.5 shows. With diminishing marginal utility, the decision
maker is thus hurt more by the downside of a lottery than he or she is helped by the
upside. This tends to make the risk-averse decision maker prefer the sure thing.
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 5 . 1
D
Computing the Expected Utility for Two Lotteries
for a Risk-Averse Decision Maker
Consider the two lotteries depicted in Figure 15.3. They
have the same expected value, but the first (investing in
the Internet company’s stock) has a larger variance than
the second (investing in the public utility company’s
stock). This tells us that the first lottery is riskier than
the second lottery. Suppose that a risk-averse decision
maker has the utility function U(I ) 1100I, where I
denotes the payoff of the lottery.
Problem Which lottery does the decision maker
prefer—that is, which one has the bigger expected utility?
Solution Compute the expected utility of each lottery using equation (15.1):
Expected utility of investing in public utility stock
0.1018,000 0.80110,000 0.10112,000
0.10(89.4) 0.80(100) 0.10(109.5) 99.9
Since investing in the public utility company’s stock
has the higher expected utility, a risk-averse decision
maker prefers it to the Internet company’s stock. This
illustrates a general point: If lotteries L and M have the
same expected value, but lottery L has a lower variance
than lottery M, a risk-averse decision maker will prefer
L to M.
Similar Problems: 15.5, 15.6, 15.7, 15.8
Expected utility of investing in Internet stock
0.3018,000 0.40110,000 0.30112,000
⫽ 0.30(89.4) 0.40(100) 0.30(109.5) 99.7
risk neutral A characteristic of a decision maker
who compares lotteries
according to their expected
value and is therefore indifferent between a sure thing
and a lottery with the same
expected value.
R I S K - N E U T R A L A N D R I S K - L OV I N G P R E F E R E N C E S
Risk aversion is only one of the possible attitudes that decision makers might have
toward risk. A decision maker might also be risk neutral or risk loving. When a decision maker is risk neutral, he or she compares lotteries only according to their
1 5 . 2 E VA L UAT I N G R I S K Y O U T C O M E S
615
expected values and is therefore indifferent between a sure thing and a lottery with the
same expected value. To see why, note that a risk-neutral decision maker has a linear
utility function, U a bI, where a is a nonnegative constant and b is a positive constant. Consider a lottery with payoffs I1 and I2 and associated probabilities p and 1 ⫺ p.
The expected utility EU of the lottery is
EU ⫽ p(a bI1 ) (1 ⫺ p)(a bI2 )
a b[ pI1 (1 ⫺ p)I2 ]
The term in the square brackets is the expected value EV of the lottery, so EU ⫽ a bEV.
Thus, when the expected value equals the payoff of the sure thing (i.e., when EV I ),
the expected utility equals the utility of the sure thing (i.e., EU U).
Returning to our job offer example, we see that if you were risk neutral, you
would be indifferent between the sure $54,000 salary you would receive from the established company and the expected salary of $54,000 associated with the offer from
the start-up company. Figure 15.6 shows the utility function of a risk-neutral individual. Since the utility function is a straight line, the marginal utility of income is
constant—that is, the change in utility from any given increment to income is the
same, no matter what the decision maker’s income level.
When a decision maker is risk loving, he or she prefers a lottery to a sure thing
that is equal to the expected value of the lottery. In the job offer example, your
expected utility from accepting the offer from the start-up company would exceed the
utility that you get from accepting the offer from the established company. As shown
in Figure 15.7, a risk-loving decision maker has a utility function that exhibits increasing marginal utility—that is, the change in utility from any given increment to income
goes up as the decision maker’s income goes up.
risk loving A characteristic of a decision maker
who prefers a lottery to a
sure thing that is equal to
the expected value of the
lottery.
Utility function
Utility
T
S
R
Q
0 4
104
Income (thousands of dollars per year)
FIGURE 15.6 Utility Function for a RiskNeutral Decision Maker
The utility function is a straight line, so
marginal utility is constant. The change in
utility from any given increment to income is
the same, no matter what the decision maker’s
income level (e.g., the distance from point
Q to point R is the same as the distance from
point S to point T ).
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CHAPTER 15
R I S K A N D I N F O R M AT I O N
Utility function
T
Utility
S
FIGURE 15.7 Utility Function for a
Risk-Loving Decision Maker
The utility function exhibits increasing marginal utility. The change in utility from any
given increment to income goes up as the
decision maker’s income goes up (e.g., the
distance from point Q to point R is less than
the distance from point S to point T ).
S
E
R
Q
0 4
104
Income (thousands of dollars per year)
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 5 . 2
D
Computing the Expected Utility for Two Lotteries:
Risk-Neutral and Risk-Loving Decision Makers
Suppose two decision makers are each considering the
investments in the two lotteries depicted in Figure 15.3.
One decision maker is risk neutral, with the utility function U(I ) 100I, while the other is risk loving, with the
utility function U(I ) 100I 2, where I denotes the payoff
of the lottery.
Problem
(a) Which lottery does the risk-neutral decision maker
prefer?
(b) Which lottery does the risk-loving decision maker
prefer?
Solution
(a) For the risk-neutral decision maker
Expected utility of investing in Internet stock
0.30(8,000) 0.40(10,000) 0.30(12,000)
10,000
Expected utility of investing in public utility stock
0.10(8,000) 0.80(10,000) 0.10(12,000)
10,000
Since the two investments have the same expected utility,
the risk-neutral decision maker is indifferent between
them. Notice that the expected utility of each lottery is
equal to a hundred times the expected value of each lottery.
This illustrates a general point: For a risk-neutral decision
maker, the ranking of the expected utilities of lotteries will exactly
correspond to the ranking of the expected payoffs of the lotteries.
(b) For the risk-loving decision maker
Expected utility of investing in Internet stock
0.30(100)(802 ) 0.40(100)(1002 )
0.30(100)(1202 ) 1,024,000
Expected utility of investing in public utility stock
0.10(100)(802 ) 0.80(100)(1002 )
0.10(100)(1202 ) 1,008,000
The risk-loving decision maker will prefer investing in
the Internet stock, since the expected utility is higher
than it is for investing in the public utility stock. This
illustrates a general point. If lotteries L and M have the
same expected value, but lottery L has a higher variance than
lottery M, a risk-loving decision maker will prefer L to M.
Similar Problems: 15.7, 15.8
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1 5 . 3 B E A R I N G A N D E L I M I N AT I N G R I S K
We have now seen how to describe the riskiness of lotteries using the tools of 15.3
expected value and variance. We have also seen how we can compute the expected
utility of lotteries in order to determine an individual’s preferences among them.
Finally, we saw how we could use a utility function to characterize an individual’s
attitude toward risk (risk averse, risk neutral, or risk loving).
Although an individual could conceivably be risk neutral or risk loving, economists
believe that for big, important decisions, such as whether to purchase insurance coverage for an automobile or how much of one’s wealth to invest in the stock market, most
individuals tend to act as if they were risk averse. For example, why do most car owners willingly pay monthly premiums for coverage of collision damage on their cars even
though for most people the chance of having a costly automobile crash is relatively
small (certainly less than 50–50 within any given year)? The answer is that when it
comes to damage on our cars, most of us are risk averse. We believe that our insurance
premiums are a small price to pay for the peace of mind that comes from knowing that
if we ever did damage our vehicles, the cost of repairing or replacing the vehicle would
be covered by our insurance policy. However, individuals do not strive to completely
eliminate risk from their lives. Some motorists buy insurance policies with large
deductibles (i.e., policies in which damage up to a certain amount is not covered), and
many individuals invest at least a portion of their wealth in the stock market.
So when would risk-averse individuals choose to bear risk, and when would they
choose to eliminate it? In this section, we explore this question first by introducing
the concept of a risk premium and then by examining a risk-averse individual’s incentives to purchase insurance.
BEARING AND
E L I M I N AT I N G
RISK
RISK PREMIUM
In our job offer example, we saw that if you are risk averse, you prefer the certain income from the established company to the risky income from the start-up company.
However, we “cooked” this example to make your expected salary from the start-up
company equal to your certain salary from the established company. If your expected
salary had been sufficiently bigger than your certain salary, you might have preferred
the job at the start-up to the job at the established company, as shown in Figure 15.8.
C
Utility
320
190
60
F
D
A
0 4
29
54
104
Income (thousands of dollars per year)
FIGURE 15.8 A Risk-Averse Decision Maker
Might Prefer a Lottery to a Sure Thing
If the salary offer from the established company
were only $29,000 per year, your expected utility
from the start-up company’s offer (point D) would
exceed the utility from the established company’s
offer (point F ). In this case, you would prefer the
lottery to the sure thing. (Compare this figure to
Figure 15.5.)
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CHAPTER 15
risk premium The
necessary difference
between the expected
value of a lottery and the
payoff of a sure thing to
make the decision maker
indifferent between the
lottery and the sure thing.
The figure shows that when the expected salary of the start-up company is $54,000
and the established firm offers a certain salary of just $29,000, your expected utility at
the start-up company (point D) exceeds your utility at the established firm (point F ).
This illustrates an important point: A risk-averse decision maker might prefer a gamble to a sure thing if the expected payoff from the gamble is sufficiently larger than
the payoff from the sure thing. Put another way, a risk-averse decision maker will bear
risk if there is additional reward to compensate for the risk.
How big this reward must be is indicated by the risk premium of the lottery. The
risk premium is the minimum difference between the expected value of a lottery and
the payoff of a sure thing that would make the decision maker indifferent between
the lottery and the sure thing. To see what this means, consider again the situation
where a risk-averse decision maker chooses a sure thing over a lottery when the payoff
of the sure thing and the expected payoff of the lottery are equal. Suppose the payoff
of the sure thing were just a little less—the decision maker might still prefer it to the
risky lottery. But now suppose the payoff of the sure thing keeps decreasing in small
increments—at some point, the decision maker will equally prefer the sure thing and
the lottery (i.e., will become indifferent between the two). The risk premium tells us
the point at which this happens. It is the amount by which the payoff of the sure thing
must decrease to make the decision maker indifferent between it and the lottery. In a
lottery with two payoffs, I1 and I2, with probabilities p and 1⫺ p, respectively, we can
find the risk premium (RP) using the following formula3:
R I S K A N D I N F O R M AT I O N
pU(I1 ) (1 ⫺ p)U(I2 ) ⫽ U(pI1 (1 ⫺ p)I2 ⫺ RP)
The expression pI1 (1 ⫺ p)I2 is the expected value (EV ) of the lottery (as described
earlier in Section 15.2), so this formula becomes
pU(I1 ) (1 ⫺ p)U(I2 ) ⫽ U(EV ⫺ RP)
(15.2)
Returning to the job offer example, Figure 15.9 shows how to find the risk premium graphically. The expected value of the lottery (the job at the start-up company)
Utility function
Utility
320
FIGURE 15.9 The Risk Premium for a
Risk-Averse Decision Maker
If the salary offer from the established company were $37,000 per year, you would be
indifferent between the start-up company’s
offer and the established company’s offer
because the two offers would have the
same utility (190). The risk premium is given
by the length of line segment ED, which
equals $17,000.
3
E
190
60
A
0 4
C
D
Risk
premium
37
54
104
Income (thousands of dollars per year)
The derivation of this formula is too complex to present here.
1 5 . 3 B E A R I N G A N D E L I M I N AT I N G R I S K
619
is $54,000, corresponding to point D, where utility 190. You will be indifferent between the two jobs when they have equal utility. A utility of 190 (corresponding to
point E on the utility function) is attained when the salary offer from the established
company is $37,000. (We will show how to compute a risk premium in Learning-ByDoing Exercise 15.3.) Thus, the risk premium—the difference between the expected
payoff from the lottery and the payoff from the sure thing at the point where you are
indifferent between the two jobs—is $17,000 ($54,000 ⫺ $37,000).
This means that if the established company offered you a salary of $37,000, you
would prefer the job with the start-up company only if the expected salary at the startup exceeded the established company’s offer by more than the risk premium. (In other
words, the expected salary would have to be at least $54,001 to make you prefer the
start-up job and bear the risk.)
An important determinant of the risk premium is the variance of the lottery. If
two lotteries have the same expected value but different variances, the lottery with the
bigger variance will entail a higher risk premium. This implies that the reward a riskaverse individual requires for bearing risk becomes larger as the risk increases.
A P P L I C A T I O N
15.2
Risk Premia for Employee
Stock Options
Many companies use some form of pay for performance
as incentives for their employees. In the last 15 years,
the use of employee stock options as an incentive
has increased dramatically worldwide. Stock options
are particularly common in new ventures and hightechnology companies, but their use has also grown in
many other industries. An employee stock option gives
the employee the right, typically for a three-year period, to purchase one share of the company’s stock, at
an exercise price that is set at the time the option is
granted. Most employee options are granted with the
exercise price equal to the firm’s stock price on the day
the option is given to the employee. If the stock price
falls, there is no reason for the employee to exercise the
option, since it would be cheaper to buy a share of stock
on the open market. However, if the stock price rises,
the employee can profit from exercising the option.
Consider this example. Joe works for Apple
Computer. He is granted 100 employee stock options
with the exercise price equal to Apple’s stock price
today, $50. If the stock price rises to $75, he can exercise his option by paying $50 for each option, receiving a share of stock instead of the option. That share
4
will be worth $75, so he can make a before-tax profit
of $25 per option. Thus, Joe benefits if Apple’s stock
prices rises in the future, which is why firms sometimes use options to provide incentives for employees.
Options are a very risky form of compensation,
since they depend on the value of the firm’s stock price,
and stock prices are highly variable. For example, stock
prices often decline. In Joe’s example, if Apple’s stock
price falls, then his options are worthless. Even if the
stock price rises, its future value is highly uncertain, so
the value of Joe’s options is highly uncertain.
Do employees demand a risk premium for accepting options instead of salary in their compensation
packages? A report by the compensation consulting
firm Watson Wyatt suggests that they do. They surveyed employees at large companies to estimate how
much fixed salary employees would be willing to
exchange for stock options in their pay package. The
firm then compared those values to the expected
value of those options.4 Their estimate is that employees would discount stock options by 30 to 50 percent
compared to their expected values, and would discount stock grants (in effect, stock options with the
exercise price set to zero) by 15 to 20 percent. These
imply very large risk premia for both types of compensation, which is consistent with the fact that both options and stock are very risky forms of compensation.
The standard method to estimate the expected value of the cash flows from an option of this form is to
use the Black-Scholes Formula, which was developed by economists Fischer Black and Myron Scholes in
the 1970s. Scholes eventually won the Nobel Prize in Economics for this formula. Black was deceased, or
he presumably would have shared the prize.
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E
R I S K A N D I N F O R M AT I O N
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 5 . 3
D
Computing the Risk Premium from a Utility Function
Let’s return to the salary lottery that we
just discussed and suppose that your utility function is
given by U ⫽ 1I. (This generates a graph very similar
to that in Figure 15.9.)
Problem
(a) Find the risk premium associated with the start-up
company’s salary offer.
(b) Suppose that the start-up company offered you a
zero salary but a bonus of $108,000 if the company
meets its growth targets. (This has the same expected
value but a higher variance than the initial offer, as you
can easily verify.) What is the risk premium associated
with this offer?
Solution
(a) Recall equation (15.2): pU(I1) (1 ⫺ p)U(I2) ⫽
U(EV ⫺ RP ). Also recall that for the start-up job offer
lottery, one payoff (I1) is $104,000, the other payoff (I2)
is $4,000, the probability of each payoff is 0.50, and the
expected value is $54,000. Find the risk premium by
solving equation (15.2) for RP:
0.50 1104,000 0.5014,000 154,000 ⫺ RP
192.87 154,000 ⫺ RP
Squaring both sides of this equation and rounding to the
nearest whole number gives 37,199 ⫽ 54,000 ⫺ RP, or
RP ⫽ 16,801.
(b) In this case, I1 ⫽ $0 and I2 ⫽ $108,000, so
equation (15.2) becomes 0.5010 0.501108,000
154,000 ⫺ RP, or RP ⫽ 27,000. (This confirms that as
the variance of a lottery increases, holding the expected
value fixed, so does the risk premium.)
Similar Problems: 15.12, 15.14, 15.15
W H E N W O U L D A R I S K - AV E R S E P E R S O N C H O O S E
TO E L I M I N AT E R I S K ? T H E D E M A N D F O R I N S U R A N C E
fairly priced insurance
policy An insurance
policy in which the insurance
premium is equal to the
expected value of the promised insurance payment.
Our analysis of the risk premium tells us that a risk-averse individual will bear risk
only if there is a sufficiently big reward for doing so. The logic of risk aversion also
sheds light on the circumstances under which a risk-averse person would choose to
eliminate risk by buying insurance.
To illustrate, let’s imagine that you are risk averse and you have just purchased a
new car. If all goes well—if the car works as planned and if you don’t have an accident—
you will have $50,000 of income available for consumption of the goods and services
that you would typically purchase over the course of a year. If, however, you have an
accident and you are uninsured, you would expect to pay $10,000 for repairs. This
would leave just $40,000 available for consumption of other goods and services. Let’s
suppose that the probability of your having an accident is 0.05, so the probability of
your not having an accident is 0.95. Thus, if you remain uninsured, you face a lottery:
a 5 percent chance of $40,000 in disposable income and a 95 percent chance of
$50,000 in disposable income.
Let’s now suppose that you have the opportunity to buy $10,000 worth of annual
insurance coverage at a total cost of $500 per year ($500 is called the insurance
premium). Under this policy, the insurance company agrees to pay for up to $10,000
worth of repairs on your automobile in the event that you have an accident. This
insurance policy has two notable features. First, it provides full coverage (up to
$10,000) for any damage you might suffer if you have an accident.5 Second, it is a
fairly priced insurance policy. A fairly priced insurance policy is one in which the
5
In the language of the insurance business, we would say that this policy fully indemnifies you against your loss.
1 5 . 3 B E A R I N G A N D E L I M I N AT I N G R I S K
621
insurance premium is equal to the expected value of the promised insurance payment.
Because there is a 5 percent chance that the policy will pay $10,000 and a 95 percent
chance that it will pay nothing, the expected value of the promised insurance payment
is (0.05 ⫻ $10,000) (0.95 ⫻ 0) ⫽ $500.6 If the insurance company sold this policy
to many individuals with an accident risk that is similar to yours, it would expect to
break even on these policies.
We can use the logic of risk aversion to show that you should jump at the chance
to buy this policy. If you buy the policy, you get
• $50,000 ⫺ $500 ⫽ $49,500, if you do not have an accident
• $50,000 ⫺ $500 ⫺ $10,000 $10,000 $49,500, if you have an accident
The insurance policy thus eliminates all of your risk and allows you to consume $49,500
worth of goods and services no matter what. If you do not buy the policy, you get
• $50,000 if you do not have an accident
• $40,000 if you have an accident
The expected value of your consumption in this case is (0.95 ⫻ $50,000) (0.05 ⫻
$40,000) ⫽ $49,500. Thus, the expected value of your consumption if you do not buy
insurance is equal to the certain value of your consumption if you do buy insurance.
Because a risk-averse decision maker prefers a sure thing to a lottery with the same
expected value, you will prefer to buy a fair insurance policy that provides full coverage
against a loss rather than buy no insurance at all.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 5 . 4
D
The Willingness to Pay for Insurance
Your current disposable income is $90,000.
Suppose that there is a 1 percent chance that your house
may burn down, and if it does, the cost of repairing it will
be $80,000, reducing your disposable income to $10,000.
Suppose, too, that your utility function is U 1I.
Problem
(a) Would you be willing to spend $500 to purchase an
insurance policy that fully insures you against your loss?
(b) What is the highest price that you would be willing
to pay for an insurance policy that fully insures you in
the event that your house burns down?
Solution
(a) If you do not purchase insurance, your expected
utility is 0.99190,000 0.01110,000 298. If you do
purchase full insurance at a price of $500, your disposable
6
income is $89,500 whether or not your house burns
down. (Your insurance policy costs $500, but if your house
does burn down, the insurance company will compensate
you for the $80,000 cost of repairs.) Thus, your expected
utility from purchasing insurance is 189,500 ⫽ 299.17.
Since your expected utility is higher if you purchase the
insurance policy than if you do not, you would be willing
to purchase the insurance at a price of $500.
(b) Let P be the price of the insurance policy. If you
purchase the policy, your expected utility is 190,000 ⫺ P.
The highest price that you would be willing to pay is a
P such that you are just indifferent between purchasing insurance and not purchasing insurance: 190,000 ⫺ P ⫽
298, or 90,000 ⫺ P ⫽ 88,804, which implies that P
$1,196. Thus, the most you’d be willing to pay for the
insurance policy is $1,196.
Similar Problems: 15.17, 15.18, 15.19, 15.20
Another way to describe a fairly priced policy is that the insurance premium per dollar of insurance
coverage ($500/$10,000) is equal to the probability of an accident.
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R I S K A N D I N F O R M AT I O N
15.3
If AIG Can Collapse, Why Would
Anyone Supply Insurance?
We have just seen that a risk-averse consumer has an
incentive to demand insurance. But why would anyone
have an incentive to supply insurance? You might
guess that if risk-averse preferences explain insurance
demand, then risk-loving preferences explain insurance supply. After all, aren’t insurance suppliers really
taking a gamble that the insured party will not experience a loss? The dramatic collapse of the insurance
firm AIG in 2008 illustrates the consequences that
can arise when that gamble does not pay off. But the
answer to why insurance gets supplied is more subtle
than this and does not require that insurance suppliers
be risk lovers. A brief look at the history of insurance
will help clarify this point.
In his engaging history of the concept of risk,
Against the Gods, Peter Bernstein points out that the
insurance business had its roots in the ancient world.7 In
ancient Greece and Rome, for example, an early version
of life insurance was provided by occupational guilds.
These groups asked their members to contribute to
a pool that would then be used to provide financial support to a family if the head of the family unexpectedly
died. In medieval Italy, an early version of crop insurance
arose when farmers created cooperative organizations
that would insure one another against losses due to
bad weather. Under this arrangement, farmers in one
part of the country that experienced good weather
would compensate farmers in another part of the country whose crops had been impaired by bad weather.
And the most famous insurance company of all, Lloyds
of London, started in 1771 when a group of individuals
(the Society of Lloyds) who did business at Lloyds coffeehouse agreed to commit their personal wealth to
underwrite any losses incurred by group members and
their customers. The group that paid insurance premiums to the society included shipowners, merchants, and
building owners.
These historical examples illustrate the basic principle
of insurance: A group of people who have not sustained
losses provides money to compensate other people who
have sustained losses. In modern economies, insurance
companies such as Prudential and State Farm in effect
7
serve as intermediaries in this process. For example,
State Farm will use the cash that you paid last month for
your automobile insurance policy to compensate some
other car owner who had the misfortune to experience
an automobile accident this month.
Viewed in this way, insurance is fundamentally
about sharing risk among a group of individuals so
that no one in the group bears an undue amount of
risk. Because of this, insurance markets can arise even
when all parties are risk averse, as long as the risk the
parties bear are, to some degree, independent of each
other. That is, when one individual (or a group of individuals) suffers a loss, there must be other individuals
who do not suffer a loss. This is usually true of almost
all risks for which some form of insurance exists. A notable example of when independence does not hold
involved the housing mortgage industry in 2008–2009.
To understand this example, it is first necessary to describe mortgage securitization and credit default swaps.
When a bank loans money to purchase a home,
the home owner is promising to make monthly payments for the life of the mortgage (usually 30 years).
However, there is a risk that the home owner will stop
making those payments, for example, if the owner loses
a job and can no longer afford the mortgage. In typical
economic times, the risk that a home owner will default
on a mortgage in this way is independent of the risk of
default on mortgages issued to other home owners. If
one mortgage defaults, home owners with other mortgages typically keep making their payments to the
bank. In effect, the bank charges a small profit margin
to all mortgage holders, as a form of insurance for
when one mortgage goes into default. In fact, the
mortgage industry spreads the risks of mortgage even
more broadly through mortgage securitization. Banks
sell their mortgages to companies such as Fannie
Mae (Federal National Mortgage Association), a federally sponsored corporation. Fannie Mae then issues
mortgage-backed securities, similar to bonds, the
value of which depends on the monthly payments on
thousands of mortgages. Investors and mutual funds
can buy these securities as part of their portfolios. Thus,
the risks from thousands of individual mortgages (hopefully independent of each other) are combined, and
that joint risk is then spread over many investors. In the
See especially Chapter 5 of P. L. Bernstein, Against the Gods: The Remarkable Story of Risk ( New York:
John Wiley & Sons), 1996.
1 5 . 3 B E A R I N G A N D E L I M I N AT I N G R I S K
early 2000s, investors could also purchase collections of
mortgage-backed securities known as collateralized
debt obligations (CDOs), which were essentially
groupings of mortgage-backed securities, segmented
according to the riskiness of the underlying mortgages.
(In the language of Wall Street, these groupings were
known as tranches.)
Even with the spreading of risks, some investors in
mortgage-backed securities or CDOs sought to purchase
insurance on their investments. This insurance was
known as a credit-default swap. A credit default swap
protects the owner of a bond or a CDO against the risk
of default, that is, the possibility that the bond or CDO
stops generating flows of repayments and lose value. In
effect, a credit default swap is an insurance policy on
the bond or the CDO. An important issuer of credit
default swaps on CDOs was the insurance firm AIG.
Like any insurance supplier, suppliers of credit default swaps like AIG counted on the independence of
the risks it was insuring. Unfortunately, in the late
2000s, such independence was an illusion. Between
1997 and 2005, the U.S. housing market experienced
a dramatic increase in prices. By the early 2000s, the
market was in the midst of a speculative bubble in
which many individuals decided to invest a large part
of their personal wealth in their homes. Banks also
greatly increased the extent to which they were willing to issue “subprime” mortgages—which had much
higher risks of default. In 2006, the bubble began to
deflate, and housing prices began to fall, to the point
623
that many home owners owed more on their mortgage than the current market value of their home. At
the same time, interest rates on adjustable rate subprime mortgages began to “reset” from low “teaser
rates” (designed to attract borrowers in the first
place) to much higher rates. These developments
began to trigger a wave of mortgage defaults in
2006. As the rate of defaults rose dramatically in 2006
and 2007, not only did holders of mortgage-backed
securities and CDOs experience significant losses, so
too did insurers of those securities such as AIG.
Indeed, AIG failed—and was bailed out by the U.S.
government—because it had inadequate capital reserves to pay off the claims of those to whom it had
sold credit default swaps. These developments took
many by surprise—including apparently the ratings
agencies such as Moody’s and Standard and Poor’s
that had given AAA ratings to CDOs consisting of
mortgage bonds containing subprime mortgages.
Many people (including policymakers such as Alan
Greenspan and traders in Wall Street investment
banks such as Bear Stearns, Lehman Brothers, and
Merrill Lynch) evidently had not anticipated that
housing prices would decline and trigger massive
subprime defaults. The unusual and dramatic decline
in housing prices meant that there was far less independence in the default risks of individual mortgages
than many investors on Wall Street had believed.
Regrettably, the result was the massive financial crisis
in 2008 and the Great Recession of 2008–2010.
ASYMMETRIC INFORMATION IN INSURANCE MARKETS:
MORAL HAZARD AND ADVERSE SELECTION
If you own a car, take a look at your automobile insurance policy. You will probably
see that you have what is known as a deductible. A deductible makes the car owner
responsible for a portion (e.g., the first $1,000 worth) of the damage from an accident,
while the insurance company insures the rest. A deductible transforms an insurance
policy from one of full insurance to one of partial insurance.8
Why do insurance policies have deductibles? An important reason is the presence
of asymmetric information, which refers to situations in which one party knows
more about its own actions or personal characteristics than another party. In insurance
markets, there are two important forms of asymmetric information: moral hazard,
which arises when the insured party can take hidden actions that affect the likelihood
of an accident, and adverse selection, which arises when a party has hidden information about its risk of an accident or loss.
8
Co-payments in health insurance policies do the same thing. A co-payment makes the insured party
responsible for a prespecified portion (e.g., 10 percent or $10) of his or her medical bills.
asymmetric information
A situation in which one
party knows more about its
own actions or characteristics than another party.
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Hidden Action: Moral Hazard
moral hazard A phenomenon whereby an
insured party exercises less
care than he or she would
in the absence of insurance.
Suppose that you have just purchased a fairly priced insurance policy that completely
reimburses you for any damage that your car suffers as a result of an automobile accident. Now that you know that you are fully insured, how careful will you be? Perhaps
not as careful as you would have been had you not been fully insured. Perhaps you
drive faster or behave more recklessly under adverse weather conditions. Perhaps you
take less care to protect your car against vandals or thieves (e.g., by parking it on the
street rather than in a garage). The net effect of your exercising less care when you
are fully insured is that your probability of suffering damage goes up. Perhaps instead
of a 10 percent chance of a loss, it is now 15 or even 20 percent.
This illustrates the concept of moral hazard, whereby an insured party exercises
less care than he or she would in the absence of insurance. Since the insurance company cannot monitor the everyday actions of its policyholders—those actions are
hidden from its view—once it sells you the policy it can’t do much to affect your behavior. This is a problem for the insurance company because moral hazard can directly affect its profits. If the policy allowed the insurance company to just break even,
assuming a probability of damage equal to 10 percent, and if fully insured individuals
behave more recklessly because they are fully insured and the probability of damage
rises to 20 percent, the insurance company will lose money.
One way that the insurance company might deal with moral hazard would be to
pay for damage only in cases in which the insured party could demonstrate that his or
her recklessness or neglect was not the cause of the accident. But enforcing such contract provisions is often impractical. The insurance company would need to conduct
detailed investigations of every accident, and even if it did so, getting at the truth
would be very difficult—it would be easy for individuals to hide or shade the truth (“I
really was obeying the speed limit!”).
A better solution is for the insurance company to provide incentives for careful
driving. Deductibles are one way to provide such incentives. If you know that you will
have to pay a portion of the repair bill in the event of an accident, there is a good
chance that you will be more focused on driving carefully. This means that, in competing for the business of risk-averse customers, insurance companies face an interesting trade-off. The insurance has to be complete enough (i.e., it has to cover a large
enough portion of the expected damage) to make people buy it, while the deductible
has to be large enough to make people take care.
Hidden Information: Adverse Selection
adverse selection A
phenomenon whereby an
increase in the insurance
premium increases the
overall riskiness of the pool
of individuals who buy an
insurance policy.
Adverse selection is another reason that insurance policies often do not provide full
insurance. While moral hazard refers to the effect of an insurance policy on the
incentives of individual consumers to exercise care, adverse selection refers to how the
magnitude of the insurance premium affects the types of individuals who buy insurance. In particular, adverse selection means that an increase in the insurance premium
increases the overall riskiness of the pool of individuals who buy insurance.
The population consists of all sorts of individuals. Some individuals are skillful
or careful drivers, but some are not as skillful or careful and have a higher risk of an
accident. Insurance companies understand this, of course, which is why some classes
of drivers (young folks, for instance) face higher auto insurance premiums than other
classes of drivers (those over 30 years old).
But insurance companies can go only so far in distinguishing good risks from bad
risks. Even within broad risk classes, individuals might vary greatly in terms of their risk
1 5 . 3 B E A R I N G A N D E L I M I N AT I N G R I S K
625
characteristics, and information about the inherent riskiness of a prospective policyholder is often hidden. The inability to distinguish among the riskiness of individuals
who buy insurance gives rise to the adverse selection problem. Consider, for example, a
company that sells health insurance. For a given insurance premium, a policy that fully
insured the individual’s medical bills would be more attractive to an individual who faces
a high risk of illness (e.g., because of heredity or lifestyle) than one who faces a low risk
of illness. This makes such a policy costly for the insurance company to offer. You might
wonder whether raising the insurance premium would be a way for the insurance company to offset this high cost. But when the insurance company offers the same policy to
all potential consumers and cannot distinguish among individuals according to their risk
of illness, increasing the insurance premium makes matters even worse: High-risk individuals would continue to buy insurance (because it is so valuable to them), but some
low-risk individuals might conceivably choose to go without health insurance.9 The
increase in the insurance premium that is needed to offset the expected cost of the insurance adversely affects the pool of potential customers (hence the term adverse selection).
How could an insurance company make money in the face of adverse selection?
One way would be to offer consumers an array of different policies and allow potential consumers to select the one they most prefer. A policy with a large deductible and
low premium would appeal to someone who is convinced that his chances of illness
are low, whereas a policy with a smaller deductible but larger premium would be
relatively more attractive to someone who faces a more significant risk of illness. Another
way insurance companies deal with adverse selection is by selling insurance to groups
of individuals. For example, if all employees in a particular company participate in a
mandatory companywide group health insurance plan, the insurance company offering the group plan will face a mix of high- and low-risk individuals. Had an identical
insurance policy been offered on an individual-by-individual basis, low-risk individuals might opt not to purchase health insurance coverage, thus adversely affecting the
mix of individuals covered by the insurance policy.
A P P L I C A T I O N
15.4
Obamacare and Adverse Selection
in the Health Insurance Market
In March 2010, the U.S. Congress passed one of the
most significant pieces of domestic legislation in
decades, the Patient Protection and Affordable Care
Act (PPACA), typically referred to as the health care
reform bill, or even more casually, as Obamacare (because
the reform was strongly supported by President Barack
Obama). Even though the bill is described as health care
reform, the heart of the legislation is really about reform
of the health insurance market, and more specifically
the health insurance market for individuals. The case
for reform, and the approach that the PPACA takes to
9
reform, is directly related to the issue of adverse selection in health insurance markets.
In the United States, most people who have
health insurance coverage receive it from their employer. In 2008, 58.5 percent of the U.S. population
was covered by an insurance plan obtained through
the workplace; 8.9 percent of the public was covered
by a plan purchased in the individual health insurance
market; 29 percent was covered by a government
health insurance plan (either Medicare, Medicaid, or a
military health care plan), and 15.4 percent of the
population (or about 46.3 million people) did not
have health insurance coverage. (Note: These percentages add up to more than 100 percent because
Or, perhaps, low-risk individuals might seek out less expensive alternatives, such as joining a health
maintenance organization.
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an individual may have both individual coverage and
some form of government insurance as well.)10 As has
been noted in the text, employer-based health insurance coverage solves the adverse selection problem
by pooling risk across a large group of people, so that
the insurance rates paid by the company (or the workers) reflect the average health risk of the employees
in the company, not the risk of the high-risk workers.
However, the individual health insurance market
is different. This market provides insurance for those
who cannot obtain health insurance from an employer
or government health insurance plan. Unlike group
health plans in a company or government-provided
insurance, healthier individuals who might otherwise
purchase health insurance in the individual market
may instead decide to go without insurance. The result
of this behavior can lead to an adverse selection “death
spiral” that operates something like this: Insurance
companies set prices based on the average health risk of
the anticipated purchasers of insurance, but at these
prices, relatively healthy individuals choose to go without coverage, and the riskiness of the pool of insured is
worse than anticipated. This, in turn, induces insurance
companies to increase premiums to cover their now
higher-than-expected insurance expenses. But if insurance premiums are increased, even more individuals
will opt out of the market, leaving an even higher risk
pool. If insurance companies are still unable to cover
their expenses, they may raise rates even more, leading
to more individuals opting out of the insurance market,
and an even higher risk pool still. The end result might
be a very thin market with very high premiums that
only the highest risk individuals are willing to pay.
Broadly, this describes the individual health insurance
market in the United States. The significant number of
individuals who go without health insurance is, in part,
a reflection of adverse selection in the health insurance
market.
In practice, insurance companies do take steps to
sell insurance policies based on differences in individuals’ health risks. And setting prices based on different
risk profiles is a possible antidote to adverse selection
death spirals and may help health insurance markets
operate more efficiently. This is why, for example, individuals may not be able to obtain health insurance
coverage in the individual insurance market if they
10
have a preexisting condition. The preexisting condition
is a signal of the individual’s intrinsic health risk. But
denial of coverage based on preexisting conditions is
unpopular and seen by some as fundamentally unfair,
since preexisting conditions may arise through no fault
of the individual. Further, denial of coverage based on
preexisting conditions adds to the population of the
uninsured. And from an economic efficiency perspective, a large uninsured population may be problematic.
Uninsured parties may lack the access to the health
care system that would otherwise induce them to engage in preventive care (e.g., annual checkups) or seek
care when a medical condition is treatable. Without
health insurance, individuals may wait until the problem is so severe that high-cost emergency care is the
only option. Distortions in medical decisions stemming
from lack of health insurance may raise the overall cost
of medical care in the United States.
A key goal of Obamacare is to reduce the number
of uninsured, while at the same time eliminating denial
of coverage based on preexisting conditions (or in the
parlance of insurance, providing “guaranteed issue” of
insurance). By itself, adopting guaranteed issue could
actually worsen the adverse selection problem.
Knowing that you cannot be turned away for health
insurance, you might wait until you need health care to
purchase insurance. Thus, to prevent the system from
being “gamed” in this way (which is really an extreme
form of adverse selection) and to deal with adverse
selection and the thinness of the individual health insurance market more generally, the PPACA mandates that
all individuals must have health insurance (the so-called
individual mandate). For those who do not have health
insurance through an employer or do not qualify for
government insurance programs such as Medicaid, individuals will have the ability to purchase health insurance policies on state (or multistate) exchanges, which
are intended to create competitive markets that include
a broad pool of individuals with diversified health risks.
But an individual mandate creates another problem: It
forces individuals to purchase insurance who do not
want it (i.e., their maximum willingness to pay for insurance is less than the premium) or who cannot afford it.
To deal with this issue, the PPACA provides meanstested subsidies to individuals who purchase insurance
in the exchanges.
U.S. Bureau of the Census, Income, Poverty, and Health Insurance Coverage in the United States: 2008,
September 2009, http://www.census.gov/prod/2009pubs/p60-236.pdf (accessed April 30, 2010).
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Most of these provisions do not go into effect
until 2014. Still, the individual mandate to purchase
insurance has become a lightning rod for criticism of
Obamacare. Almost immediately after the PPACA was
signed into law, attorneys general from several states
(e.g., Virginia, Florida) initiated lawsuits to block the
individual mandate. Whether these efforts will succeed
remains to be seen. It is important to note, though,
that an individual mandate is not necessarily needed
to deal with gaming of the system under guarantee
issue or to address the more general issue of adverse
selection in the health insurance market. To deal with
adverse selection, sufficiently attractive subsidies, or
low rates through a “public insurance option,” might
be enough to entice a broad base of individuals with
diversified health risks into the individual insurance
market. And to counteract gaming, regulations could
be enacted that require that those who have dropped
their insurance coverage, and attempt later on to
restart it, to pay penalties equal to 2 or 3 months of
“back” insurance coverage.
A key point, though, is that dealing with adverse
selection in the health insurance market is not easy.
Obamacare is complex because a set of interlocking
pieces—guaranteed issue, individual mandate, subsidies, and insurance exchanges—must work together to
provide a mechanism to move the United States closer
to universal health insurance coverage. How these reforms work in practice will be one of the most interesting ongoing economic stories of the next decade.
L
et’s now analyze how a decision maker might choose a plan of action in the face of
risk. We do so by introducing you to the concept of a decision tree, a diagram that
describes the options and risks faced by a decision maker. It is a valuable tool for identifying the optimal plan of action when a decision maker faces risk.
15.4
A N A LY Z I N G
RISKY
DECISIONS
D E C I S I O N T R E E BA S I C S
To illustrate how a decision tree can be used to choose among risky alternatives, we
begin with a simple example. Suppose an oil company has just discovered a new reserve
of oil offshore in the North Sea. It can construct either of two types of offshore drilling
platforms: a large-capacity facility or a small-capacity facility. The size of the facility the
firm would want to construct depends on the amount of oil in the reservoir:
• If the reservoir is large, and the firm builds . . .
—a large facility, the firm’s profit is $50 million.
—a small facility, the firm’s profit is $30 million.
• If the reservoir is small, and the firm builds . . .
—a large facility, the firm’s profit is $10 million.
—a small facility, the firm’s profit is $20 million.
In this example, if the firm knew for sure that the reservoir was large, then it would
build a large facility, and if it knew for sure that the reservoir was small, it would build a
small facility. But the oil company doesn’t know the size of the reservoir. It believes that
the reservoir will be large with a probability of 0.50 and small with a probability of 0.50.
Figure 15.10 illustrates the oil company’s decision tree. A decision tree has four
basic parts:
• Decision nodes. A decision node, represented by n in the tree drawing, indicates
a particular decision that the decision maker faces. Each branch from a decision
node corresponds to a possible decision.
decision tree A diagram
that describes the options
available to a decision
maker as well as the risky
events that can occur at
each point in time.
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FIGURE 15.10
Decision
Tree for Oil Company’s
Facility Size Decision
At decision node A, the company has two choices—build
a large facility or build a small
facility. At chance nodes B
and C, the company faces lotteries with two possible outcomes (the reservoir is large
or the reservoir is small, each
with a probability of 0.50).
The company’s payoff (i.e., its
profit) depends on its decision
at node A and the actual
outcome.
Reservoir is large
(probability = 0.5)
Build large
facility
Oil company's payoff (millions)
$50
B
Reservoir is small
(probability = 0.5)
$10
A
Reservoir is large
(probability = 0.5)
Build small
facility
$30
C
Reservoir is small
(probability = 0.5)
$20
• Chance nodes. A chance node, represented by ~ in the tree drawing, indicates a
particular lottery that the decision maker faces. Each branch from a chance
node corresponds to a possible outcome of the lottery.
• Probabilities. Each possible outcome has a probability. The sum of the probabilities of all possible outcomes from a chance node must add up to 1.
• Payoffs. Each branch at the right-hand end of the tree has a payoff associated
with it. The payoff is the value of the result from each possible combination of
choices and risky outcomes. If the decision maker is risk neutral, payoffs are
monetary values. If the decision maker is risk averse or risk loving, payoffs are
the utilities associated with the monetary values of the payoffs.
Now let’s apply these concepts to the oil company’s decision tree in Figure 15.10.
First, let’s assume that the company is risk neutral, so the payoffs represent monetary
values (i.e., the company’s actual profit at each outcome).11 Decision node A represents the company’s facility size decision, with the two possible choices shown on the
branches extending from the decision node (“build large facility” and “build small
facility”). Chance nodes B and C represent the lotteries the company faces depending
on its decision at node A. Each lottery has two possible outcomes, shown on the
branches extending from the chance nodes (“reservoir is large” and “reservoir is
small”), and the probability of each outcome is 0.50. The company’s profit depends on
the decision made at node A and the actual outcome of the corresponding lottery.
That is, if the company decides at node A to build a large facility, profit will be either
$50 million (if the reservoir is large) or $10 million (if the reservoir is small); if the
company decides at node A to build a small facility, profit will be either $30 million
(if the reservoir is large) or $20 million (if the reservoir is small).
To choose its optimal plan of action, the oil company would begin by calculating the
expected value of each lottery12 and then choose the decision at node A that leads to the
lottery with the higher expected value. Thus, the company would evaluate the tree by
working backward, from right to left. This is called folding the tree back and is identical to
the process of backward induction that we used to analyze game trees in Chapter 14.
11
If we had assumed that the firm was risk averse, we would need to specify a utility function for the firm
and evaluate the utility of the profit of each outcome.
12
If the firm were risk averse, it would evaluate the expected utility of the payoffs using the firm’s utility
function.
1 5 . 4 A N A LY Z I N G R I S K Y D E C I S I O N S
Build large
facility
Oil company's expected payoff (millions)
0.5($50) + 0.5($10) = $30
A
Build small
facility
0.5($30) + 0.5($20) = $25
FIGURE 15.11 Folded Back Decision
Tree for Oil Company’s Facility Size Decision
Compare this figure to Figure 15.10. We have
(1) replaced the payoffs for each outcome
with the expected payoff for each lottery and
then (2) folded the expected payoffs back
over the lotteries. Now it is easy to see that
the oil company’s best decision is to build a
large facility. (That decision leads to the
higher expected payoff.)
The expected value of the lottery at chance node B is (0.5 ⫻ 50 million) (0.5 ⫻
10 million) ⫽ $30 million. The expected value of the lottery at chance node C is
(0.5 ⫻ $30 million) (0.5 ⫻ $20 million) ⫽ $25 million. This is shown in Figure 15.11,
where we have simplified the decision tree by replacing the payoffs for each outcome
with the expected payoff for each lottery and then folding the expected payoffs back
over the lotteries. Hiding the chance nodes in this way lets us see immediately that the
company’s best bet (its optimal decision) is to build a large facility.
DECISION TREES WITH A SEQUENCE OF DECISIONS
The decision trees in Figures 15.10 and 15.11 were easy to analyze because the decision maker faced just one decision. But sometimes decision makers face a sequence of
decisions or must make a decision following the outcome of a chance event. To illustrate decision tree analysis in this more complicated setting, let’s add an additional
twist to our oil company example. The firm can still build a large facility or a small facility, but suppose that it can also conduct a seismic test to determine the size of the
reservoir before it makes the decision about the size of the facility. Suppose, for a moment, that the test is costless and 100 percent accurate.13 Should the firm conduct the
test, and if so, how much better off would the firm be by doing so?
To answer these questions, consider the firm’s decision tree in Figure 15.12. The top
two decision branches coming out of decision node A are the same as in Figures 15.10
and 15.11, while the third branch represents the new alternative: conduct a seismic test
before building the facility. If the firm conducts the test, it will learn whether the reservoir is large or small, as depicted by chance node D. The decision to conduct a test leads
to a chance node because, before the firm conducts the test, it does not know what its
outcome will be.
In our example, the test has two possible outcomes, each with a probability of 0.50
and each leading to another decision:
• If the test says that the reservoir is large, the firm would face the decision represented by decision node E, where it could choose to build a large facility (with a
payoff of $50 million) or a small facility (with a payoff of $30 million).
• If the test says that the reservoir is small, the firm would face the decision represented by decision node F, where it could again choose to build a large facility
(with a payoff of $10 million) or a small facility (with a payoff of $20 million).
13
629
In the next section, we will discuss what happens when (as is the case in reality) the test is costly.
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CHAPTER 15
Build large
facility
(no test)
R I S K A N D I N F O R M AT I O N
Reservoir is large
(probability = 0.5)
$50
B
Reservoir is small
(probability = 0.5)
$10
Reservoir is large
(probability = 0.5)
Build small
facility
(no test)
A
Oil company's payoff (millions)
$30
C
Reservoir is small
(probability = 0.5)
$20
Test says reservoir is large
(probability = 0.5)
Build large facility
Payoff
(millions)
$50
E Build small facility
Conduct seismic
test first
$30
D
Test says reservoir is small
(probability = 0.5)
Build large facility
$10
F Build small facility
$20
FIGURE 15.12 Decision Tree for Oil Company’s Facility Size Decision with an Option to Test
Compare this figure to Figure 15.10. Now the company has an option to conduct a seismic test
at no cost. This option leads to the new chance node D, whose outcomes lead to decision
nodes E and F. If we compare the payoffs associated with the choices at these decision nodes,
we can cross out the inferior choices. Then we can calculate the expected payoffs of the lotteries, fold back the tree, and find the company’s optimal decision (see Figure 15.13).
Decision nodes E and F (unlike decision node A) do not lead to lotteries but directly
to outcomes with payoffs. Thus, in the process of folding back the tree (working
from right to left), we need not calculate expected payoffs from these decisions, but
instead will simply compare the actual payoffs. Clearly, the preferred decision at
node E (where the test says the reservoir is large) is to build a large facility, while
the preferred decision at node F (where the test says the reservoir is small) is to build
a small facility. We represent this by crossing out the inferior decisions as shown in
Figure 15.12. Doing so turns chance node D into a simple lottery with two possible
outcomes and payoffs, each with a probability of 0.50. If the test says the reservoir
is large and the firm builds a large facility, the payoff is $50 million; if the test says
the reservoir is small and the firm builds a small facility, the payoff is $20 million.
The expected payoff of this lottery is (0.5 ⫻ $50 million) (0.5 ⫻ $20 million) ⫽
$35 million.
Now we can simplify the tree as shown in Figure 15.13, where we have again replaced the payoffs for each outcome with the expected payoff for each lottery and then
folded the expected payoffs back over the lotteries. Once again, it is easy to evaluate
the decision tree: the optimal decision at node A is to conduct the seismic test, since
that decision leads to the highest expected payoff ($35 million, versus a $30 million
expected payoff for building a large facility without testing and a $25 million expected
1 5 . 4 A N A LY Z I N G R I S K Y D E C I S I O N S
Build large
facility
(no test)
631
Oil company's expected payoff (millions)
0.5($50) + 0.5($10) = $30
Build small
facility
(no test)
FIGURE 15.13
0.5($30) + 0.5($20) = $25
A
Conduct seismic
test first
0.5($50) + 0.5($20) = $35
Folded Back Decision
Tree for Oil Company’s Facility Size Decision
with an Option to Test
Compare this figure to Figure 15.12. The folded
back decision tree makes it clear that the oil
company’s best plan of action is to conduct the
seismic test and then decide whether to build a
small facility or a large one.
payoff for building a small facility without testing). Thus, the firm’s optimal plan of
action can be summarized as follows:
• Conduct the seismic test.
• If the test says the reservoir is large, build a large facility.
• If the test says the reservoir is small, build a small facility.
This example illustrates the basic steps involved in constructing and analyzing a
decision tree.
Begin by mapping out the sequence of decisions and risky events.
For each decision, identify the alternative choices the decision maker can make.
For each risky event, identify the possible outcomes.
Assign probabilities to the risky events.
Identify payoffs for all possible combinations of decision alternatives and risky
outcomes.
6. Finally, find the optimal sequence of decisions by folding back the tree. In so
doing, you identify the expected value of the lotteries at each chance node and
determine the highest expected payoff option at each decision node. The payoff
corresponding to that highest expected payoff option then becomes the value you
assign to that decision node.
1.
2.
3.
4.
5.
T H E VA L U E O F I N F O R M AT I O N
When faced with risky decisions, decision makers benefit from information that helps
them reduce or even eliminate the risk. The value of information is reflected in the
fact that oil companies spend money to perform seismic tests before drilling oil wells,
that consumer products companies spend money to test market new products before
they roll them out on a national scale, and that prospective presidential candidates
spend money taking polls and establishing exploratory committees before throwing
their hats into the ring. The decision tree analysis that we just went through can help
us identify the economic value of information.
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Let’s summarize the results of the decision tree analysis of the oil company example
in the previous section:
• When the oil company cannot conduct a seismic test, its optimal course of action is
to build a large facility. Its expected payoff from this course of action is $30 million.
• When the oil company can conduct a seismic test at no cost, its optimal course of
action is to conduct a test. If the test indicates a large reservoir, the company should
build a large facility. If the test indicates a small reservoir, the company should build
a small facility. Its expected payoff from this course of action is $35 million.
• Thus, when the firm can conduct a seismic test at no cost, its expected payoff is
$5 million higher than when it cannot conduct a test.
value of perfect information The increase in
a decision maker’s expected
payoff when the decision
maker can—at no cost—
conduct a test that will
reveal the outcome of a
risky event.
This example illustrates the value of perfect information (VPI), the increase in
a decision maker’s expected payoff when the decision maker can—at no cost—conduct
a test that will reveal the outcome of a risky event. In our oil-drilling example, the VPI
is $5 million, the difference between the expected payoff when the decision maker can
conduct a costless seismic test and the expected payoff when the decision maker makes
the optimal decision with no test.
Why does perfect information have value? It is not, as you might initially guess,
because individuals are risk averse. We can see this in two ways. First, even though the
seismic test revealed the true size of the oil reservoir, it did not eliminate the decision
maker’s risk: Before the test is taken, its outcome is uncertain and thus represents a
risk for the decision maker. Second, risk aversion by itself cannot account for the value
of perfect information because there was a positive VPI, even though we assumed that
the firm is risk neutral.
Perfect information has value because it allows the decision maker to tailor its decisions to the actual situation. In our example, the oil company fares best when it can match
the size of the drilling facility to the size of the oil reservoir (a small facility maximizes profits from a small reservoir, and a large facility maximizes profits from a large reservoir).
The VPI tells us the maximum amount of money the firm would be willing to pay
for a test that revealed perfect information. It is, in short, the firm’s willingness to pay
for a crystal ball. In this case, if the seismic test costs $4 million, the firm should conduct it: It would be paying $4 million for a test that is actually worth $5 million. If, by
contrast, the test costs $7 million, it would not be worth doing. The firm would be
better off making a choice without the results of the seismic test.
A P P L I C A T I O N
15.5
Putting Money in a Hole
in the Ground?14
Oil prices have risen dramatically in recent years, driven
in part by economic growth in emerging markets such
as China and India. Concerns continue to be raised
14
about existing supplies of oil. These two factors—
potentially limited supply plus increasing demand—
mean that oil prices may remain high in the future.
During periods of high and rising prices, the
stakes from oil exploration are enormous. Oil companies are running out of oil fields with low costs of
We thank Jason Sheridan for suggesting this application and providing important background information. Also see “New Oil Field Deep in the Gulf a Potential Giant,” Houston Chronicle (September 6, 2006);
and “Deep Oil, Deep Unknowns,” Forbes (October 2, 2009).
633
1 5 . 5 AU C T I O N S
access and extraction of oil. Increasingly, they are
being forced to search for oil in remote locations, including below the ocean floor. When oil is located in
these locations, it is very expensive to extract, since oil
wells must be drilled at great depths under extreme
conditions. The costs of building an oil well under the
ocean, of pumping the oil out, and of building a
pipeline to transport any oil found to refineries often
run into the hundreds of millions of dollars. (As the BP
Deepwater Horizon catastrophe illustrated in the
spring and summer of 2010, drilling for oil in the
ocean can also entail significant environmental costs,
some of which the oil company may be liable for.) At
the same time, a successful oil well may produce billions of dollars in revenue given current high prices.
This is precisely the set of conditions in which information has very high value for decision makers.
The costs of committing to drilling a new oil well on
the ocean floor are extremely high. The benefits of a
successful well are even higher. Unfortunately, the
odds that a well will end up with disappointing yield
are also reasonably high. Therefore, mistakes can be
highly costly, while correct decisions can be highly
profitable. These are some of the highest-stake decisions made in the world today. For these reasons oil
companies spend enormous sums trying to improve
the quality of their information before committing to
drilling a new oil well in remote locations.
A recent example is the Jack oil field in the Gulf
of Mexico, about 270 miles southwest of New
Orleans. Petroleum geologists have suspected that
the area might hold oil reserves that could be profitably exploited, but had little concrete information
to justify a well. They have therefore invested large
sums of money in obtaining better information on
A
this question. For years they have used oil exploration
ships to conduct preliminary geological tests. Based
on those first tests, Chevron and its partners drilled an
initial test well, the Jack 1, in 2004, to a depth of
29,000 feet below sea level. This test well suggested
that there might be more than 350 feet of oil sands.
A test well costs on the order of $150–$200 million.
Based on the promising results of Jack 1, the company
drilled Jack 2 to further test the potential for the oil
field, and later a third test well. Geologists had been
concerned that the oil would be difficult to pump out
of the ground since it was under such high pressure.
However, Chevron found enough positive pressure in
the oil in the ground (in other words, the pressure of
the oil to be pumped out of the ground was high
enough) that the field might have potential for extraction of 6,000 barrels of oil a day. To open the oil
field, it costs $2 to $4 billion to establish an oil platform in deep ocean. Each well—and a large field like
Jack will have 10 to 20 wells—costs about $100 million. Total capital investment can top $6 billion. Based
on these sets of tests, estimates are that the field may
ultimately produce $200 billion to $1 trillion in revenues. Chevron and its partners are proceeding with
development of the Jack field.
These tests have not resolved all of the uncertainty about the Jack oil field. However, they have
increased the odds that investing in the field may be
highly profitable. The high expenses that Chevron
and its partners incurred to obtain this information
attest to the value of improved information for decision making when the stakes are high. While these
costs have run into the billions of dollars, they are still
below the value of perfect information, since a great
deal of uncertainty remains.
uctions are a prominent part of the economic landscape. Since the mid-1990s,
several countries (e.g., the United States, the United Kingdom, and Germany) have
used auctions to sell portions of the airwaves for communications services such as mobile
telephones and wireless Internet access. Other countries, such as Mexico, have used
auctions to privatize state-owned companies such as railroads and telephone companies. And now, of course, auctions are available to anyone with an Internet connection, as companies such as eBay have helped make online auctions one of the fastest
growing areas of commerce on the World Wide Web.
15.5
AU C T I O N S
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Economists have been studying auctions for years, and a well-developed body of
microeconomic theory pertains to them. Auctions typically involve relatively few
players that make decisions under uncertainty. The analysis of auctions thus combines
the game theory we discussed in Chapter 14 with concepts relating to information and
decision making under uncertainty that we have discussed in this chapter. For this reason, a discussion of auctions provides a nice way of capping and integrating ideas from
both chapters.
T Y P E S O F AU C T I O N S A N D B I D D I N G E N V I R O N M E N T S
Auction Formats
English auction An
auction in which participants
cry out their bids and each
participant can increase his or
her bid until the auction ends
with the highest bidder winning the object being sold.
first-price sealed-bid
auction An auction in
which each bidder submits
one bid, not knowing the
other bids. The highest bidder
wins the object and pays a
price equal to his or her bid.
second-price sealedbid auction An auction
in which each bidder submits one bid, not knowing
the other bids. The highest
bidder wins the object but
pays an amount equal to
the second-highest bid.
Dutch descending
auction An auction in
which the seller of the object
announces a price, which is
then lowered until a buyer
announces a desire to buy
the item at that price.
private values A situation in which each bidder
in an auction has his or her
own personalized valuation
of the object.
common values A situation in which an item being
sold in an auction has the
same intrinsic value to all
buyers, but no buyer knows
exactly what that value is.
There are many different types of auctions. Perhaps the most familiar format (probably because it is often depicted in movies or on television) is the English auction.
Under this format, participants cry out their bids, and each participant can increase
his or her bid until the auction ends with the highest bidder winning the object.
Another common auction type is the first-price sealed-bid auction in which each
bidder submits one bid, not knowing the other bids. The highest bidder wins the
object and pays a price equal to his or her bid. Many auctions on eBay are, in effect,
sealed-bid auctions. Still another type of auction is the second-price sealed-bid
auction, which was used to sell airwave licenses in New Zealand. As in the first-price
sealed-bid auction, each bidder submits a bid and the high bidder wins. However,
the winning bidder pays an amount equal to the second-highest bid. Lastly, under the
Dutch descending auction format, often used to sell agricultural commodities such
as tobacco and flowers (including tulips in Holland, which explains the name), the
seller of the object announces a price, which is then lowered until a buyer announces
a desire to buy the item at that price.
Private Values versus Common Values
Auctions can also be classified as involving private values or common values. When
buyers have private values, each bidder has his or her own personalized valuation of
an object. You know how much the item is worth to you, but you are not sure how
much it is worth to other potential bidders. A setting in which bidders have private
values is the sale of antiques or art. For such items, individuals are likely to have idiosyncratic assessments of an item’s value and are probably not going to change their
minds if they find out that someone else assesses the item differently. In a private
values setting your attitude would be, “I don’t care what you think, I love that painting.”
When buyers have common values, the item has the same intrinsic value to all
buyers, but no buyer knows exactly what it is. To illustrate, imagine that your economics professor comes to class with a briefcase full of dollar bills that he or she intends
to auction. The monetary value of the dollars inside is the same to everyone, but no
one knows how many bills are actually inside. The assumption of common values
nicely characterizes the sale of items such as oil leases or U.S. treasury bills. In a common values setting, we usually assume that bidders have the opportunity to obtain
estimates of the value of the object (e.g., you can look inside the briefcase for 30 seconds). Your estimate would be your best guess about the value of the object. In this
situation, you might change your mind about the object’s value if you knew the estimates of other bidders. In particular, if you later learned that every other bidder had
a lower estimate of the object’s true value than you did, you would probably revise
your estimate of the object’s value downward.
1 5 . 5 AU C T I O N S
635
AU C T I O N S W H E N B I D D E R S H AV E P R I VAT E VA L U E S
To study bidding behavior in auctions, let’s first consider a setting in which bidders
have private values. We will explore three different auction formats: the first-price
sealed-bid auction, the English auction, and the second-price sealed-bid auction. Our
goals are to see how the rules of an auction affect the behavior of bidders and to see
how much revenue auctions raise for sellers.
First-Price Sealed-Bid Auctions
Suppose you and other bidders are competing to purchase an antique dining room
table that is being offered for sale on eBay. Also suppose (1) that this table is worth
$1,000 to you—that is, the most you are willing to spend to buy this table is $1,000,
(2) that you do not know the valuations of other potential bidders, and (3) that you
believe that some bidders could have valuations above or below $1,000.
In deciding on a bidding strategy, it might seem natural to submit a bid of $1,000.
After all, that is what the table is worth to you, and by bidding as high as possible, you
maximize your chances of winning. However, this is generally not your best strategy.
In a first-price sealed-bid auction, a bidder’s optimal strategy is to submit a bid that is
less than the bidder’s maximum willingness to pay.
To see why, let’s explore what happens when you reduce your bid from $1,000 to
$900. Not knowing the valuations of the other bidders, you can’t say for sure what the
consequences of this move will be. However, it’s likely that your probability of winning the auction will go down. Suppose that curve S in Figure 15.14 describes the relationship between your bid and the probability of winning. (In a moment, we’ll talk
about where S comes from.) If you bid $1,000, the expected value of your payment—
your bid multiplied by the probability of winning—is areas A B C D E F.
(Throughout this section, we will assume that bidders are risk neutral—they evaluate
benefits and costs according to their expected value.) If, by contrast, you bid $900,
your expected payment is areas E F. (Table 15.1 keeps track of these areas for you.)
Thus, with a bid of $900, your expected payment goes down by areas A B C D,
1
Probability of winning auction
S
0.7
A
B
C
0.4
FIGURE 15.14
F
E
0
D
900
Your bid ($)
1,000
Optimal Bidding in a First-Price
Sealed-Bid Auction
The curve S shows the relationship between your bid
and the probability of winning. If you bid $1,000, your
expected payment and your expected benefit are both
equal to A B C D E F, so your expected
profit is zero. If you bid $900, your expected payment
is E F and your expected benefit is D E F, so
your expected profit is D. You are better off bidding
$900 than $1,000.
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TABLE 15.1
Comparison of Different Bids in a First-Price Sealed-Bid Auction
Bid
$1,000
Expected benefit
Expected payment
Expected profit
A
A
0
B
B
C
C
D
D
$900
E
E
F
F
D
E
D
E
F
F
for two reasons: First, you pay less if you win; second, your probability of winning is
lower. Reducing your expected payment is good, but when you lower your bid, you
also reduce your expected benefit from winning the auction. Your expected benefit is
your $1,000 value times the probability of winning. When you bid $1,000, your expected benefit is areas A B C D E F, but when you bid $900 your expected
benefit is areas D E F. Thus, your expected benefit goes down by areas A B C.
So is it worth shading your bid? The answer is yes, because when you shade your bid,
your expected payment goes down by more than your expected benefit, and your net
gain (expected profit) from shading your bid is area D, compared with an expected
profit of zero if you bid $1,000. By shading your bid below your true valuation, you
reduce your probability of winning, but you more than make up for it by increasing
your net gain if you win the auction.
By how much should you shade your bid? This depends on the shape of S, which
depends on your beliefs about the bidding strategies of the other bidders, and that, in
turn, depends on your beliefs about their valuations. In the Nash equilibrium of the
bidding game, each player forms an assessment of the relationship between a bid and
the probability of winning—the S curve in Figure 15.14—by conjecturing a relationship between the valuations of each rival bidder and that bidder’s equilibrium bidding
behavior.15 In equilibrium, these conjectures must be consistent with bidders’ actual
behavior (we illustrate Nash equilibrium bidding strategies for a first-price sealed-bid
auction in Learning-By-Doing Exercise 15.5).
With N bidders, the Nash equilibrium strategy for each bidder is to submit a bid
equal to (N ⫺ 1) /N times the bidder’s true valuation. Note that no matter how many
bidders there are, the bidder with the highest valuation wins the auction and pays a
price that is less than the bidder’s maximum willingness to pay. Moreover, equilibrium
bids go up as more bidders participate in the auction.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 5 . 5
D
Verifying the Nash Equilibrium in a First-Price Sealed-Bid
Auction with Private Values
Two women (Bidder 1 and Bidder 2) are competing to
buy an object in a first-price sealed-bid auction with
private values. Each believes that the other’s valuation is
equally likely to be anywhere in the interval between $0
and $200. (In other words, they believe that a $0 valua15
tion is as likely as a $1 valuation or a $2 valuation or a $3
valuation, and so on up to $200. It’s like spinning a wheel
numbered 0–200, with 0 and 200 in the same spot at the
top of the wheel: the wheel is as likely to stop at one
number as at any other number.
Remember from Chapter 14 that at a Nash equilibrium, each player in a game is doing the best it can
given the strategies of the other players.
1 5 . 5 AU C T I O N S
Problem Verify that each bidder’s Nash equilibrium
bid is half of her own valuation.
Solution Since each bidder has the same belief about
the other’s valuation, their optimal bidding strategies
will be the same. Therefore, we only need to verify that
Bidder 1’s Nash equilibrium bid is half of her valuation—
that is, we need to show that if Bidder 1 expects Bidder 2
to submit a bid equal to half of Bidder 2’s valuation, then
Bidder 1 will submit a bid equal to half of Bidder 1’s
valuation. We can show this by reasoning as follows.
If Bidder 1 expects Bidder 2 to submit a bid equal to
half of Bidder 2’s valuation, then Bidder 1 believes that
Bidder 2’s bid is equally likely to be anywhere in the interval between $0 and $100 (now the wheel has only 100
numbers).
Thus, if Bidder 1 submits a bid equal to Q, where
Q ⱕ 100, the probability that Bidder 1 will win the auction is Q/100. We can illustrate this by first assuming
that Bidder 2 bids as expected—that is, submits a bid
between $0 and $100—and then considering some of
Bidder 1’s possible bids. If, for example, Bidder 1 submits a bid of $50, her probability of winning 0.50
(i.e., there is a 0.50 probability that Bidder 2 will submit
a higher bid and a 0.50 probability that Bidder 2 will
submit a lower bid), and Q/100 0.50. If Bidder 1 submits a bid of $30, her probability of winning 0.30
(i.e., there is a 0.70 probability that Bidder 2 will submit
a higher bid and a 0.30 probability that Bidder 2 will
submit a lower bid), and Q/100 ⫽ 0.30. And so on. (In
the analogy of the wheel, the probability that the wheel
will stop at a number less than or equal to, say, 20, is
20/100, or 0.20.)
Now suppose Bidder 1’s valuation of the object is
$60. (Any number would work as well for the sake of this
argument.) In that case, Bidder 1’s profit from winning
the auction will be her expected benefit minus her expected payment. Her expected benefit is her valuation
times her probability of winning ⫽ (60 ⫻ Q/100), while
her expected payment is her bid times her probability of
winning ⫽ (Q ⫻ Q/100). Thus, Bidder 1’s profit ⫽ (60 ⫻
Q/100) ⫺ (Q ⫻ Q/100) ⫽ (0.60 ⫺ 0.01Q)Q.
This formula for Bidder 1’s profit is analogous to
the formula we saw in Chapter 11 for total revenue
along a linear demand curve [i.e., for a linear demand
curve P ⫽ a ⫺ bQ, total revenue ⫽ (a ⫺ bQ)Q]. Thus, the
formula for Bidder 1’s marginal profit is 0.60 ⫺ 0.02Q
(analogous to the formula we derived in Chapter 11 for
marginal revenue along a linear demand curve, a ⫺
2bQ). At Bidder 1’s profit-maximizing optimal bid, marginal profit is zero: 0.60 ⫺ 0.02Q ⫽ 0, or Q ⫽ 30.
Thus, for an arbitrary valuation (in this case, $60),
we have shown what we set out to show: if Bidder 1 expects Bidder 2 to submit a bid equal to half of Bidder 2’s
valuation, then Bidder 1 will submit a bid equal to half
of Bidder 1’s valuation.
Similar Problem: 15.26
English Auctions
Let’s now consider an English auction. Suppose that you and another bidder are competing to purchase an antique table that is worth $1,000 to you. Unknown to you,
your rival’s valuation of the table is $800. If the auctioneer opens the bidding at $300,
what should you do?
When buyers have private values, the dominant strategy in an English auction is
to continue bidding only as long as the high bid is less than the bidder’s maximum willingness to pay.16 To see why, suppose that your rival has just shouted out a bid of $450
and that the auctioneer will accept increases in bids in increments of $1. Clearly, you
should raise your bid to $451: The worst that can happen is that your bid will be topped
by the other bidder, in which case you are no worse off than you are now. The best that
can happen is that the other bidder will drop out, and you will get the table at a price
($451) that is below your willingness to pay.
If both players follow a strategy of bidding until the high bid reaches their maximum willingness to pay, it follows that the person who values the item the most (in
this example, that’s you) will win the item, paying a price that is just a shade higher
than the valuation of the bidder with the second-highest valuation. In this example, your
16
See Chapter 14 for a discussion of dominant strategies.
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rival drops out when you raise the bid to $801. As a result, you are able to buy a table
that is worth $1,000 to you for a price of $801.
Second-Price Sealed-Bid Auctions
Now suppose that the seller uses a second-price sealed bid auction to sell the antique
table. What bid should you submit? This auction seems much more complicated than
the English auction or the first-price sealed-bid auction. Interestingly, though, game
theory again yields a clear statement of ideal bidding behavior: Each bidder’s dominant strategy is to submit a bid equal to the bidder’s maximum willingness to pay. That
is, if your valuation of the table is $1,000, then submitting a bid of $1,000 is at least as
good as—and sometimes better than—submitting, any other bid, no matter what bids
you think rival bidders will submit. To see why, consider your options:
• If you bid less than your maximum willingness to pay of $1,000, you might win
or you might not, depending on the valuation of the other player. But no matter
what, you cannot hurt yourself by increasing your bid to $1,000 because, if you
win, you don’t pay your own bid but instead pay a price equal to the secondhighest bid. And by increasing your bid you might even help your chances of
winning. Thus, any bid less than your maximum willingness to pay is dominated
by a bid exactly equal to your maximum willingness to pay.
• What about bidding more than your maximum willingness to pay of $1,000, say
$1,050? This might seem appealing because you don’t actually pay your bid.
The problem is that this strategy can never help you, and it can sometimes hurt
you. If your rival bids more than $1,050, raising your bid from $1,000 to $1,050
doesn’t help you; you will lose the auction either way. If your rival bids less
$1,000, you would have won anyway if you had kept your bid at $1,000, and so
again, raising your bid doesn’t help you. And if the rival bids between $1,000
and $1,050, you win the table, but you’ve paid a price that is more than it is
worth to you. You would have been better off bidding $1,000 and not winning
the table. Thus, any bid that is greater than your maximum willingness to pay is
never better and sometimes worse than a bid that is exactly equal to your maximum
willingness to pay.
If each bidder follows the dominant strategy and submits a bid equal to the
bidder’s maximum willingness to pay, you will submit a bid equal to $1,000, while your
rival (whose valuation we have assumed to equal $800) will submit a bid of $800. As
in the English auction, you win the item, and the price that you pay—$800—is virtually identical to the $801 that you would have paid in an English auction. Remarkably,
the second-price sealed-bid auction, even though it entails different rules than the
English auction, generates virtually the same result. (The difference arises because in
the English auction we restricted the bid increment to $1. In general, the difference
between the payment made by the winning bidder in an English auction and a secondprice-sealed-bid auction depends entirely on the size of the bidding increment. If, in
the theoretical extreme, the bidding increment were vanishingly small, the payments
in the two auction formats would be equal.)
Revenue Equivalence
We have seen that under the three auction formats we have considered (first-price
sealed-bid auction, English auction, and second-price sealed-bid auction), when
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1 5 . 5 AU C T I O N S
bidders have private values and each bidder follows the Nash equilibrium strategy, the
bidder with the highest willingness to pay wins the auction. We have also seen that:
• In a first-price sealed-bid auction, the winning bidder pays a price that is less
than his or her maximum willingness to pay.
• In an English auction and in a second-price sealed-bid auction, the winning
bidder pays a price that is equal to the second-highest private valuation among
all the bidders in the auction.
Thus, each format successfully identifies the bidder with the highest valuation, but the
seller’s revenue (the winning bid) is less than that highest valuation. Remarkably, the
seller’s revenue in English and second-price sealed-bid auctions—the second-highest
private valuation among all the bidders in the auction—is also the seller’s revenue in
first-price sealed-bid auctions and in all other types of auctions when bidders have private
values and follow Nash equilibrium strategies. This surprising result (which is too complex to derive here) is called the revenue equivalence theorem: When bidders have
private values, all auction formats generate the same revenue for the seller, equal on
average to the second-highest private valuation among all the bidders in the auction.
AU C T I O N S W H E N B I D D E R S H AV E C O M M O N VA L U E S :
THE WINNER’S CURSE
Frequency of estimates and bids
When bidders have common values, a complication arises that does not occur when
bidders have private values, the winner’s curse: The winning bidder might bid an
amount that exceeds the item’s intrinsic value. To see how this can happen, suppose
your economics professor brings a briefcase full of dollar bills to class and auctions it
off. Every student is given a peek inside the briefcase to estimate how much it contains. You estimate that it contains $150, which represents the most you would be willing to bid. Of course, your classmates develop their own estimates, and these might
differ from yours. Let’s suppose that these estimates are distributed according to the
dashed bell-shaped curve shown in Figure 15.15. The height of this curve indicates
Distribution of bids
Distribution of estimates
80 100
Estimates and bids in dollars
150
revenue equivalence
theorem When participants in an auction have
private values, any auction
format will, on average,
generate the same revenue
for the seller.
winner’s curse A
phenomenon whereby
the winning bidder in a
common-values auction
might bid an amount
that exceeds the item’s
intrinsic value.
FIGURE 15.15
The Winner’s Curse
in an Auction with Common Values
The dashed bell-shaped curve shows the
distribution of bidders’ estimates, centered on the item’s intrinsic value of
$80. The solid bell-shaped curve shows
the distribution of bids, assuming that
bidders shade their bids as they would
in an auction with private values. The
winning bid will be in the right-hand
half of the distribution of bids and
might be in the shaded region, where
bids are greater than the item’s intrinsic
value. If so, the winning bidder will
have suffered the winner’s curse.
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the relative frequency of different estimates. The curve is centered on the true intrinsic value of the item (i.e., the actual amount of money in the briefcase, which is $80)
because it seems natural to assume that underestimates and overestimates balance out.
Suppose your professor uses a first-price sealed-bid auction to sell the money in
the briefcase. If you and your classmates shade your bids as you would in an auction
with private values, the distribution of bids will be another bell-shaped curve (the solid
curve in Figure 15.15), shifted to the left of the curve describing the distribution of
estimates. Now suppose that you submit a bid of $100, which is two-thirds of your
estimate. To your initial delight, yours is the high bid and you win the briefcase. But
when you count the money, you realize you have spent $100 to win $80. You’ve just
experienced the winner’s curse.
Figure 15.15 helps explain the winner’s curse phenomenon.17 The bid that wins
the auction will be drawn from the right-hand half of the distribution of bids. If, as in
Figure 15.15, the winning bidder has overestimated the value of the object being sold,
then even if the bidder shades his or her bid, that winning bid could still fall within a
region (like the shaded region in Figure 15.15) where winning bids exceed the true
value of the object.
How can you avoid the winner’s curse? A key lesson from our discussion of game
theory in Chapter 14 is that you should think ahead. You should anticipate that if you
win the auction, it will be because you had the highest estimate of the object’s value,
and you should adjust your bidding behavior accordingly. For example, in the briefcase auction you should reason as follows:
• I estimate that the value of the money in the briefcase is $150.
• But if I win the auction, it will mean that my estimate was higher than everybody
else’s, which means that the true value of the item is probably less than $150.
• Because my goal is to win the auction but not pay more than the item is actually
worth, I should act as if my estimate is not $150, but something less than $150,
say, a ⫻ $150, where a 6 1.
The amount by which you should discount your estimate, a, depends on how many
other bidders there are. Suppose the class has 29 other students. To determine how
much to shade your bid, you should ask yourself: “If I knew that my estimate of $150
was the largest of 30 estimates, what would be my best guess about the intrinsic value
of the item?” The answer is that the intrinsic value must be significantly less than
$150—for example, $85.18 This modified estimate of the value of the item ought to
be your starting point in devising a bid strategy. We say starting point because you
might want to scale down your bid even more (as you did in a private-values auction)
as you consider the possible bidding behavior of other bidders. The key point, though,
is that the possibility of the winner’s curse should make you even more conservative
in your bidding behavior than you would have been in an auction in which bidders
have private values.
The winner’s curse implies that if bidders adjust their bidding strategies to avoid
it, adding more bidders to the auction can actually make bidders behave more conservatively. This contrasts with the private-values case in which the addition of bidders
17
This diagram is based on a similar diagram in M. Bazerman and W. F. Samuelson, “I Won the Auction
But Don’t Want the Prize,” Journal of Conflict Resolution, 27, no. 4 (December 1983): 618–634.
18
Figuring out precisely the most probable intrinsic value of the item would require the application of
advanced probability theory.
1 5 . 5 AU C T I O N S
A P P L I C A T I O N
641
15.6
The Winner’s Curse
in the Classroom19
What do you think would happen if your economics
professor really did bring in a briefcase of dollar bills?
Do you think the class would suffer from the winner’s
curse? Two professors, Max Bazerman and William
Samuelson, did this experiment in a number of MBA
classes at Boston University, using jars of pennies and
nickels rather than a briefcase full of dollar bills. But
the experiment was essentially the one we just described. Students were asked to guess the amount of
money in the jar (the jar contained $8 worth of coins;
to motivate accurate guesses, a special prize was
awarded to the student whose guess came closest to
the actual amount of money in the jar). Students then
participated in a first-price sealed-bid auction in which
they submitted an amount they were willing to pay
for the money in the jar.
Bazerman and Samuelson found that students
systematically succumbed to the winner’s curse. In the
48 auctions they conducted, the average winning bid
was $10.01, resulting in an average loss of $2.01 for
the winning bidder. This finding is even more remarkable because students’ estimates of the amount of
money in the jar tended to be on the low side. The
average estimate was $5.13, $2.87 below the true
value. Thus, the winner’s curse in these auctions operated with special force. Despite underestimating
the value of the item, students still overbid relative to
its true value! Had subjects been unbiased in their
estimates—that is, had the true value of the item
been $5.13—the winning bidders’ average loss would
have been $4.88 ($10.01 – $5.13).
The lesson: Beware of the winner’s curse! The
temptation to bid aggressively in an auction is strong.
If you fall prey to it, you may well regret winning.
tends to inflate the Nash equilibrium bids in the auction. Why might you want to bid
more conservatively when more bidders participate in the auction? Think about it this
way: When are you more likely to have an overly optimistic estimate of the value of
an object—when you are the winning bidder in an auction with three bidders or in an
auction with 300 bidders? In the first case, if you win the auction, your estimate must
have exceeded just two others. In the second case, your estimate must have exceeded
299 others. You are much more likely to have an inflated estimate when yours is the
highest of 300 than when it is the highest of just three.
A P P L I C A T I O N
15.7
Google AdWords
Google is not only the world’s most popular search
engine, but it is also one of the world’s largest sellers
of advertising. The “Sponsored Links” that appear on
a Google search page—known as Google AdWords—
were paid for by the sponsoring companies. The sponsored links that appear depend on the keywords
entered by a Google user. Google uses an algorithm
(the details of which the company does not reveal) to
19
decide which sponsored links to show, and in what
order, on the search results page. The order of sponsored links is important, since the first link is most
likely to be clicked on. It is interesting to note that
Google’s interests are aligned with both advertisers
and its search engine customers. Customers get more
value from using Google if the sponsored links are
more relevant to their keyword search. Advertisers
get higher advertising elasticity of demand (the term
we used when discussing advertising in Chapter 12)
This example is based on Bazerman and Samuelson, “I Won the Auction But Don’t Want the Prize.”
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and thus greater benefits from their spending on ads.
Google gets more search-engine users, and higher
demand and prices for its sponsored links, if the algorithm it uses better achieves those objectives for its
customers and advertisers.
The price and order of a sponsored link depends
largely on two factors. One is the likelihood that a
user’s search will lead to a click on a sponsored link.
For example, some users are likely to be searching for
a product to buy, while others are looking for information or a weather report. Users who are searching
for something to buy are more likely to click on a
sponsored link and are thus more valuable to
Google’s advertising business. To assess this, Google
assigns a “quality score” to each keyword, based on
factors such as how often the user clicks on a link
after using that keyword in their search.
The second factor used in pricing and ordering a
sponsored link is how much advertisers are willing to
pay for an ad. Advertisers on Google’s search engine
page choose keywords that they think are relevant to
the product they are advertising. They then bid the
maximum amount they are willing to pay for each click
on their sponsored link from searches using that keyword. In other words, Google auctions off sponsored
links. An auction is an effective way to determine pricing in this case because there is enormous variation in
the types of keywords and in demand for advertising
associated with each ad. While Google could try to set
prices itself, the market it faces is so complex and everchanging that such an approach would be costly and
probably not very effective. Instead, an auction is a way
to assess the amount that advertisers are willing to pay
separately and automatically for each unique user
search. In effect, Google is running a new auction with
every search. Note that this approach is only possible
because of high-speed computers.
Google uses a generalized second-price sealedbid auction to sell AdWords. This format is similar to
the second-price sealed-bid auction described in this
chapter, but somewhat more complicated. In the standard second-price auction, there is a single winner of
the auction, who pays the second highest bid. In
Google’s auction, multiple advertisers “win” by having their ads placed on the page as sponsored links.
Each ad pays an amount equal to the next highest bid
that is below its bid. Thus, links that are higher on the
page pay more. Google developed this auction
method on its own, but before launching it they
asked economist Hal Varian to analyze its properties.
His conclusion was that the method would be quite
effective at efficiently allocating Internet advertising.
At that point, Google launched AdWords to great
success, and hired Varian to be the company’s chief
economist. AdWords are highly profitable not only to
Google, but also to its customers, since Google’s algorithms increase the likelihood that relevant advertisements are presented to users of its search engine. A
recent analysis by Varian estimates that the value of
the ads to advertising firms are approximately 2 to
2.5 times the cost of the ads.20 Thus, firms earn substantial producer surplus from AdWords.
If bidders respond to the possibility of the winner’s curse by shading their bids in
a sealed-bid auction, one might wonder whether a first-price sealed-bid auction is best
from the auctioneer’s perspective. It turns out that when bidders have common values, a better auction format for the seller is the English auction, in which bidders can
see the bids of the other players and can revise their opinion of the item’s value as the
bidding progresses. In particular, if you initially have a low estimate of an item’s value,
the fact that other players continue to bid aggressively on it will lead you to revise
your estimate upward. This, in turn, reduces your incentive and the incentives of
other bidders to shade bids downward in fear of the winner’s curse. Game theory
analysis can show that the auctioneer’s average revenue over many auctions will be
higher under an English auction than under a first-price sealed-bid auction, a secondprice sealed-bid auction, or a Dutch auction.21 This might partly explain why English
auctions are so prevalent in the real world.
20
21
Hal Varian, “Online Ad Auctions.” Working paper, University of California at Berkeley, 2009.
This is true both when bidders are risk neutral and when they are risk averse.
C H A P T E R S U M M A RY
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CHAPTER SUMMARY
• A lottery is any event whose outcome is uncertain.
We describe this uncertainty by assigning a probability
to each possible outcome of the lottery. These probabilities are each between zero and one, and the probabilities of all possible outcomes add up to one.
• Some probabilities are objective, resulting from laws
of nature (such as the 0.50 probability that a coin will
come up heads), while other probabilities are subjective,
reflecting someone’s beliefs (such as a belief about the
probability that a stock will go up or down in value).
• The expected value of a lottery is a measure of the
average payoff the lottery will generate.
• The variance of a lottery is a measure of the lottery’s
riskiness—the average deviation between the possible outcomes of the lottery and the expected value of the lottery.
• Utility functions can be used to assess decision makers’ preferences among alternatives with different
amounts of risk. Decision makers may be risk averse, risk
neutral, or risk loving.
• A risk-averse decision maker prefers a sure thing to a
lottery of equal expected value, evaluates lotteries
according to their expected utility, and has a utility function that exhibits diminishing marginal utility. (LBD
Exercise 15.1)
• A risk-neutral decision maker is indifferent between
a sure thing and a lottery of equal expected value, evaluates lotteries according to their expected value, and has
a utility function that exhibits constant marginal utility.
(LBD Exercise 15.2)
• A risk-loving decision maker prefers a lottery to a
sure thing of equal expected value, evaluates lotteries according to their expected utility, and has a utility function that exhibits increasing marginal utility. (LBD
Exercise 15.2)
• A risk premium is the minimum difference between
the expected value of a lottery and the payoff from a sure
thing that would make the decision maker indifferent between the lottery and the sure thing. (LBD Exercise 15.3)
• A fair insurance policy is one in which the price of
the insurance is equal to the expected value of the damage being covered. A risk-averse individual will always
prefer to purchase a fair insurance policy that provides
full insurance against a loss.
• Insurance companies must deal with the risks arising
from asymmetric information (e.g., by including deductibles in insurance policies). Asymmetric information
can take two forms: moral hazard (insured people may,
unbeknownst to the insurance company, behave in ways
that increase risk) and adverse selection (an increase in
insurance premiums may, unbeknownst to the insurance
company, increase the overall riskiness of the pool of insured people).
• A decision tree is a diagram that describes the options
and risks faced by a decision maker. We analyze decision
trees by starting at the right end of the tree and working
backwards, in a process called folding the tree back.
• The value of perfect information (VPI) is the increase in the decision maker’s expected payoff when the
decision maker can—at no cost—conduct a test that will
reveal the outcome of a risky event.
• Auctions are important in economics. There are different types of auction formats, including the English
auction, the first-price sealed-bid auction, the secondprice sealed-bid auction, and the Dutch descending
auction. Auctions can also be classified according to
whether bidders have private valuations of the item
being sold or common valuations.
• In a first-price sealed-bid auction with private values,
the bidder’s best strategy is to bid less than his or her
maximum willingness to pay (by an amount that depends
on the number of other bidders). (LBD Exercise 15.4)
• In an English auction with private values, the bidder’s
dominant strategy is to continue bidding as long as the
high bid is less than his or her maximum willingness to pay.
• In a second-price sealed-bid auction with private
values, the bidder’s dominant strategy is to submit a bid
equal to his or her maximum willingness to pay.
• In each of these three auction formats, the bidder with
the highest willingness to pay wins the auction, and the
seller’s revenue is always less than the highest valuation
among all bidders. The revenue equivalence theorem
shows that, in all types of auctions with private values
where bidders follow their Nash equilibrium strategies,
the seller’s revenue will, on average, be equal to the
second-highest private valuation among all bidders.
• In auctions with common values, bidders must worry
about the winner’s curse—bidding more than the item is
worth. The bidder’s best strategy is to discount his or
her estimate of the item’s value (by an amount that depends on the number of other bidders). The seller’s best
choice of format for an auction with common values is
the English auction, which generates a higher average
revenue than other formats.
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REVIEW QUESTIONS
1. Why must the probabilities of the possible outcomes
of a lottery add up to 1?
7. What is fair insurance? Why will a risk-averse consumer always be willing to buy full insurance that is fair?
2. What is the expected value of a lottery? What is the
variance?
8. What is the difference between a chance node and a
decision node in a decision tree?
3. What is the difference between the expected value
of a lottery and the expected utility of a lottery?
9. Why does perfect information have value, even for
a risk-neutral decision maker?
4. Explain why diminishing marginal utility implies
that a decision maker will be risk averse.
10. What is the difference between an auction in
which bidders have private values and one in which they
have common values?
5. Suppose that a risk-averse decision maker faces a
choice of two lotteries, 1 and 2. The lotteries have the
same expected value, but Lottery 1 has a higher variance
than Lottery 2. What lottery would a risk-averse decision maker prefer?
6. What is a risk premium? What determines the magnitude of the risk premium?
11. What is the winner’s curse? Why can the winner’s
curse arise in a common-values auction but not in a
private-values auction?
12. Why is it wise to bid conservatively in a commonvalues auction?
PROBLEMS
15.1. Consider a lottery with three possible outcomes: a payoff of ⫺10, a payoff of 0, and a payoff of
20. The probability of each outcome is 0.2, 0.5, and
0.3, respectively.
a) Sketch the probability distribution of this lottery.
b) Compute the expected value of the lottery.
c) Compute the variance and the standard deviation of
the lottery.
15.2. Suppose that you flip a coin. If it comes up
heads, you win $10; if it comes up tails, you lose $10.
a) Compute the expected value and variance of this lottery.
b) Now consider a modification of this lottery: You flip
two fair coins. If both coins come up heads, you win $10.
If one coin comes up heads and the other comes up tails,
you neither win nor lose—your payoff is $0. If both coins
come up tails, you lose $10. Verify that this lottery has
the same expected value but a smaller variance than the
lottery with a single coin flip. (Hint: The probability that
two fair coins both come up heads is 0.25, and the probability that two fair coins both come up tails is 0.25.)
Why does the second lottery have a smaller variance?
15.3. Consider two lotteries. The outcome of each
lottery is the same: 1, 2, 3, 4, 5, or 6. In the first lottery
each outcome is equally likely. In the second lottery,
there is a 0.40 probability that the outcome is 3, and a
0.40 probability that the outcome is 4. Each of the other
outcomes has a probability 0.05. Which lottery has the
higher variance?
15.4. Consider a lottery in which there are five possible payoffs: $9, $16, $25, $36, and $49, each occurring
with equal probability. Suppose that a decision maker
has a utility function given by the formula U 1I.
What is the expected utility of this lottery?
15.5. Suppose that you have a utility function given by
the equation U 150I. Consider a lottery that provides a payoff of $0 with probability 0.75 and $200 with
probability 0.25.
a) Sketch a graph of this utility function, letting I vary
over the range 0 to 200.
b) Verify that the expected value of this lottery is $50.
c) What is the expected utility of this lottery?
d) What is your utility if you receive a sure payoff of $50?
Is it bigger or smaller than your expected utility from the
lottery? Based on your answers to these questions, are
you risk averse?
15.6. You have a utility function given by U
2I 10 1I. You are considering two job opportunities.
The first pays a salary of $40,000 for sure. The other
pays a base salary of $20,000, but offers the possibility of
a $40,000 bonus on top of your base salary. You believe
PROBLEMS
that there is a 0.50 probability that you will earn the
bonus.
a) What is the expected salary under each offer?
b) Which offer gives you the higher expected utility?
c) Based on your answer to (a) and (b), are you risk averse?
15.7. Consider two lotteries, A and B. With lottery A,
there is a 0.90 chance that you receive a payoff of $0 and
a 0.10 chance that you receive a payoff of $400. With lottery B, there is a 0.50 chance that you receive a payoff of
$30 and a 0.50 chance that you receive a payoff of $50.
a) Verify that these two lotteries have the same expected
value but that lottery A has a bigger variance than lottery B.
b) Suppose that your utility function is U 1I 500.
Compute the expected utility of each lottery. Which lottery has the higher expected utility? Why?
c) Suppose that your utility function is U I 500.
Compute the expected utility of each lottery. If you have
this utility function, are you risk averse, risk neutral, or
risk loving?
d) Suppose that your utility function is U (I 500)2.
Compute the expected utility of each lottery. If you have
this utility function, are you risk averse, risk neutral, or
risk loving?
15.8. Consider two lotteries A and B. With lottery A,
there is a 0.8 probability that you receive a payoff of
$10,000 and a 0.2 chance that you receive a payoff of
$4,000. With lottery B, you will receive a payoff of
$8,800 for certain. You should verify for yourself that
these two lotteries have the same expected value, but
that lottery A has a higher variance. For each of the utility functions below, please fill in the table below:
Utility Function
U ⫽ 100 1I
U⫽I
I2
U⫽
10000
Expected Utility
Lottery A
Expected Utility
Lottery B
15.9. Sketch the graphs of the following utility functions as I varies over the range $0 to $100. Based on
these graphs, indicate whether the decision maker is risk
averse, risk neutral, or risk loving:
a) U 10I ⫺ (1/8)I 2
b) U (1/8)I2
c) U ln (I 1)
d) U 5I
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15.10. a) Write down the equation of a utility function
that corresponds to a risk-neutral decision maker. (Note:
there are many possible answers to this part and the next
two parts.)
b) Write down the equation of a utility function that corresponds to a risk-averse decision maker.
c) Write down the equation of a utility function that corresponds to a risk-loving decision maker.
15.11. Suppose that I represents income. Your utility
function is given by the formula U ⫽ 10I as long as I is
less than or equal to 300. If I is greater than 300, your
utility is a constant equal to 3,000. Suppose you have a
choice between having an income of 300 with certainty
and a lottery that makes your income equal to 400 with
probability 0.5 and equal to 200 with probability 0.5.
a) Sketch this utility function.
b) What is the expected value of each lottery?
c) Which lottery do you prefer?
d) Are you risk averse, risk neutral, or risk loving?
15.12. Suppose that your utility function is U ⫽ 1I.
Compute the risk premium of the two lotteries described
in Problem 15.7.
15.13. Suppose you are a risk-averse decision maker
with a utility function given by U(I ) ⫽ 1 ⫺ 10I ⫺2,
where I denotes your monetary payoff from an investment in thousands. You are considering an investment
that will give you a payoff of $10,000 (thus, I ⫽ 10) with
probability 0.6 and a payoff of $5,000 (I ⫽ 5) with probability 0.4. It will cost you $8,000 to make the investment. Should you make the investment? Why or why
not?
Which Lottery
Gives the Highest
Expected Utility?
Does the Utility
Function Exhibit Risk
Aversion, Risk Neutrality,
or Risk Loving?
15.14. You have a utility function given by U ⫽ 10 lnI.
where I represents the monetary payoff from an investment. You are considering making an investment which,
if it pays off, will give you a payoff of $100,000, but if
it fails, it will give you a payoff of $20,000. Each outcome is equally likely. What is the risk premium for this
lottery?
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R I S K A N D I N F O R M AT I O N
15.15. In the upcoming year, the income from your
current job will be $90,000. There is a 0.8 chance that
you will keep your job and earn this income. However,
there is 0.2 chance that you will be laid off, putting you
out of work for a time and forcing you to accept a lower
paying job. In this case, your income is $10,000. The expected value of your income is thus $74,000.
a) If your utility function has the formula 100I ⫺ 0.0001I 2,
determine the risk premium associated with this lottery.
b) Provide an interpretation of the risk premium in this
particular example.
15.16. Consider a household that possesses $100,000
worth of valuables (computers, stereo equipment, jewelry, and so forth). This household faces a 0.10 probability
of a burglary. If a burglary were to occur, the household
would have to spend $20,000 to replace the stolen items.
Suppose it can buy an insurance policy for $500 that
would fully reimburse it for the amount of the loss.
a) Should the household buy this insurance policy?
b) Should it buy the insurance policy if it cost $1,500?
$3,000?
c) What is the most the household would be willing to
pay for this insurance policy? How does your answer relate to the concept of risk premium discussed in the text?
15.17. If you remain healthy, you expect to earn an income of $100,000. If, by contrast, you become disabled,
you will only be able to work part time, and your average income will drop to $20,000. Suppose that you believe
that there is a 5 percent chance that you could become
disabled. Furthermore, your utility function is U ⫽ 1I.
What is the most that you would be willing to pay for an
insurance policy that fully insures you in the event that
you are disabled?
15.18. You are a risk-averse decision maker with a utility function U(I ) ⫽ 1 ⫺ 3200I ⫺2, where I denotes your
income expressed in thousands. Your income is $100,000
(thus, I ⫽100). However, there is a 0.2 chance that you
will have an accident that results in a loss of $20,000.
Now, suppose you have the opportunity to purchase an
insurance policy that fully insures you against this loss
(i.e., that pays you $20,000 in the event that you incur
the loss). What is the highest premium that you would
be willing to pay for this insurance policy?
15.19. You are a relatively safe driver. The probability
that you will have an accident is only 1 percent. If you do
have an accident, the cost of repairs and alternative
transportation would reduce your disposable income
from $120,000 to $60,000. Auto collision insurance that
will fully insure you against your loss is being sold at a
price of $0.10 for every $1 of coverage. Finally, suppose
that your utility function is U ⫽ 1I.
You are considering two alternatives: buying a policy
with a $1,000 deductible that essentially provides just
$59,000 worth of coverage, or buying a policy that fully
insures you against damage. The price of the first policy
is $5,900. The price of the second policy is $6,000.
Which policy do you prefer?
15.20. Consider a market of risk-averse decision makers, each with a utility function U ⫽ 1I. Each decision
maker has an income of $90,000, but faces the possibility
of a catastrophic loss of $50,000 in income. Each decision
maker can purchase an insurance policy that fully compensates her for her loss. This insurance policy has a cost
of $5,900. Suppose each decision maker potentially has a
different probability q of experiencing the loss.
a) What is the smallest value of q so that a decision maker
purchases insurance?
b) What would happen to this smallest value of q if the
insurance company were to raise the insurance premium
from $5,900 to $27,500?
15.21. An insurance company is considering offering
a policy to railroads that will insure a railroad against
damage or deaths due to the spillage of hazardous chemicals from freight cars. Different railroads face difference
risks from hazardous spills. For example, railroads operating on relatively new tracks face less risk than railroads
with relatively older right of ways. (This is because a key
cause of chemical spills is derailment of the train, and
derailments are more likely on older, poorer tracks.)
Discuss the difficulties that the insurance company
might face in offering this type of policy; that is, why
might it be difficult for the insurance company to make
a profit from this type of policy?
15.22. A firm is considering launching a new product.
Launching the product will require an investment of
$10 million (including marketing expenses and the costs
of new facilities). The launch is risky because demand
could either turn out to be low or high. If the firm does
not launch the product, its payoff is 0. Here are its possible payoffs if it launches the product.
Outcome
Demand is high
Demand is low
Probability
Payoff if Firm
Launches Product
0.5
0.5
$20 million
⫺$10 million
a) Draw a decision tree showing the decisions that the
company can make and the payoffs from following those
decisions. Carefully distinguish between chance nodes
and decision nodes in the tree.
b) Assuming that the firm acts as a risk-neutral decision
maker, what action should it choose? What is the expected payoff associated with this action?
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PROBLEMS
15.23. A large defense contractor is considering making
a specialized investment in a facility to make helicopters.
The firm currently has a contract with the government,
which, over the lifetime of the contract, is worth $100 million to the firm. It is considering building a new production plant for these helicopters; doing so will reduce the
production costs to the company, increasing the value of
the contract from $100 million to $200 million. The
cost of the plant will be $60 million. However, there is
the possibility that the government will cancel the contract. If that happens, the value of the contract will fall
to zero. The problem (from the company’s point of
view) is that it will only find out about the cancellation
after it completes the new plant. At this point, it appears
that the probability that the government will cancel the
contract is 0.45.
a) Draw a decision tree reflecting the decisions the firm
can make and the payoffs from those decisions. Carefully
distinguish between chance nodes and decision nodes in
the tree.
b) Assuming that the firm is a risk-neutral decision
maker, should the firm build a new plant? What is the
expected value associated with the optimal decision?
c) Suppose instead of finding out about contract cancellation after it builds the plant, the firm finds out about
cancellation before it builds the plant. Draw a new decision tree corresponding to this new sequence of decisions
and events. Again assuming that the firm is a risk-neutral
decision maker, should the firm build the new plant?
15.24. A small biotechnology company has developed
a burn treatment that has commercial potential. The
company has to decide whether to produce the new
compound itself or sell the rights to the compound to a
large drug company. The payoffs from each of these
courses of action depend on whether the treatment is
approved by the Food and Drug Administration (FDA),
the regulatory body in the United States that approves
all new drug treatments. (The FDA bases its decision on
the outcome of tests of the drug’s effectiveness on
human subjects.) The company must make its decision
before the FDA decides. Here are the payoffs the drug
company can expect to get under the two options it
faces:
Decision
Outcome
FDA approves
FDA does not
approve
Probability
Sell the
Rights
Produce
Yourself
0.20
0.80
$10
$ 2
$50
⫺$10
(payoffs are in millions of dollars)
a) Draw a decision tree showing the decisions that the
company can make and the payoffs from following those
decisions. Carefully distinguish between chance nodes
and decision nodes in the tree.
b) Assuming that the biotechnology company acts as a
risk-neutral decision maker, what action should it choose?
What is the expected payoff associated with this action?
15.25. Consider the same problem as in Problem 15.24,
but suppose that the biotech company can conduct its
own test—at no cost—that will reveal whether the new
drug will be approved by the FDA. What is the biotech
company’s VPI?
15.26. You are bidding against one other bidder in a
first-price sealed-bid auction with private values. You
believe that the other bidder’s valuation is equally likely
to lie anywhere in the interval between $0 and $500.
Your own valuation is $200. Suppose you expect your
rival to submit a bid that is exactly one half of its valuation. Thus, you believe that your rival’s bids are equally
likely to fall anywhere between 0 and $250. Given this,
if you submit a bid of Q, the probability that you win the
auction is the probability that your bid Q will exceed
your rival’s bid. It turns out that this probability is equal
to Q/250. (Don’t worry about where this formula comes
from, but you probably should plug in several different
values of Q to convince yourself that this makes sense.)
Your profit from winning the auction is profit ⫽ (200 ⫺
bid) ⫻ probability of winning. Show that your profitmaximizing strategy is bidding half of your valuation.
16
GENERAL EQUILIBRIUM
THEORY
16.1
GENERAL EQUILIBRIUM
A N A LYS I S : T WO M A R K E T S
APPLICATION 16.1
Net after Taxes?
16.2
GENERAL EQUILIBRIUM
A N A LYS I S : M A N Y M A R K E T S
Causes and Effects
of the 2007–2008 Oil Price Rise
APPLICATION 16.2
16.3
GENERAL EQUILIBRIUM
A N A LYS I S : C O M PA R AT I V E S TAT I C S
APPLICATION 16.3
Who Likes the Gas Tax Least?
16.4
THE EFFICIENCY OF
COMPETITIVE MARKETS
Experimental Economics
Looks at Pareto Efficiency
APPLICATION 16.4
16.5
GAINS FROM FREE TRADE
APPLICATION 16.5
Gains from Free Trade
APPENDIX
DERIVING THE DEMAND AND
S U P P LY C U RV E S F O R T H E G E N E R A L
EQUILIBRIUM IN FIGURE 16.9 AND
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 6 . 2
How Do Gasoline Taxes Affect the Economy?
Gasoline prices have often made the front pages of newspapers in the last decade. As the price of crude oil
soared to record levels in 2008, gasoline prices peaked at about $4.00 per gallon, well above the average
price of about $1.30 that prevailed at the turn of the millennium.
648
Figure 16.1 shows why the prices of crude oil and gasoline are strongly related to one another. The
cost of crude oil is the most significant determinant of the price you pay for gasoline at the pump. But the
figure shows that other factors are important in the retail price. Federal, state and local government taxes
are the second largest component. Federal and state taxes alone have averaged around $0.40 per gallon
throughout the first decade of the new millennium. In addition, some states impose sales taxes, and some
cities and counties impose further taxes.
While taxes on gasoline do vary from state to state, on average consumers in the United States pay
much lower taxes than consumers in many other countries. For example, during the last decade taxes have
often exceeded $4.00 per gallon in the United Kingdom, the Netherlands, and Germany.
Whom do you think is hurt more by gasoline taxes: lower-income households or higher-income households? The most straightforward answer is that lower-income households are hurt more. Gasoline taxes
make the price of gasoline higher than it would otherwise be, and lower-income households spend a
higher fraction of their income on gasoline than higher-income households.
But this straightforward answer might not be correct. Governments use the proceeds of gasoline taxes
to purchase goods and services. How these proceeds get spent can have an important impact on economic
activity in a variety of industries, which in turn can
affect the prices of the finished goods produced in
these industries and the prices of inputs employed by
these industries. As we will see in this chapter, once we
take into account the full effect of the tax as its impact
ripples through the economy, we might find that
higher-income households are hurt more by increases
in gasoline excise taxes than lower-income households.
General equilibrium theory is the part of microeconomics that studies how prices of finished goods
and inputs are determined in many markets simultaneously. Because the gasoline tax affects several
markets at the same time (e.g., the market for
gasoline, the market for construction services, and
the market for manual labor employed in the
construction trades), a general equilibrium analysis
would be appropriate for analyzing its impact on
the well-being of different kinds of households in
the economy.
CHAPTER PREVIEW
After reading and studying
this chapter, you will be able to:
• Distinguish between partial equilibrium analysis
and general equilibrium analysis.
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G E N E R A L E Q U I L I B R I U M T H E O RY
What We Pay for in a Gallon of Regular Gasoline
(March 2010)
Retail Price: $2.77/gallon
Taxes
14%
Distribution & Marketing
8%
Refining
9%
Crude Oil
69%
FIGURE 16.1 Cost
Components of Retail Gasoline
in the United States
The most important component
in the retail price of gasoline in
the United States is the cost of
crude oil. The figure shows that
federal and state taxes amounted
to about 14 percent of the price
at the pump in March of 2010.
Source: U.S. Department of
Energy, U.S. Energy Information
Administration http://tonto.eia.doe
.gov/oog/info/gdu/gasdiesel.asp
(accessed May 18, 2010).
• Explain how one can use general equilibrium analysis to explore the total impact of government
interventions with policies like an excise tax.
• Explain why Walras’ Law tells us that prices of goods and services are determined relative to the price
of one good or input, and not determined absolutely.
• Analyze the general equilibrium effects of an excise tax on a particular good.
• Apply general equilibrium theory to explore the efficiency of resource allocation in an economy consisting of many competitive markets, all of which are interrelated and reach equilibrium at the same time.
• Explain how countries benefit from free trade combined with specialization in the production of
goods for which a country has a comparative advantage.
16.1
GENERAL
EQUILIBRIUM
ANALYSIS:
TWO MARKETS
When we studied supply and demand analysis in Chapters 2, 9, and 10, we used what
is known as partial equilibrium analysis. A partial equilibrium analysis studies the
determination of price and output in a single market, taking as given the prices in all
other markets. In this section we introduce general equilibrium analysis, the study of
how price and output are determined in more than one market at the same time.
To see how the two types of analysis differ, let’s consider a simple example with
two markets: coffee and tea, as illustrated in Figure 16.2. Panel (a) shows supply and
demand in the market for coffee, while panel (b) shows supply and demand in the
market for tea.
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1 6 . 1 G E N E R A L E Q U I L I B R I U M A N A LYS I S : T WO M A R K E T S
SC
ST
Price (dollars per pound)
Price (dollars per pound)
S'C
$1.59
$1.50
$0.93
$0.79
$0.63
D'C
DC
0
(a) Coffee market
Quantity (millions of pounds)
D'T
DT
0
Quantity (millions of pounds)
(b) Tea market
FIGURE 16.2 Supply and Demand in the Coffee and Tea Markets
Coffee and tea are substitutes. Initially, the equilibrium prices are $0.93 per pound for coffee
and $0.63 per pound for tea. Then a severe frost damages the coffee crop, shifting the supply
curve for coffee leftward, from SC to S¿C. The effects of this shift eventually result in a new equilibrium: The demand curve for coffee has shifted from DC to D¿C , the demand curve for tea has
shifted from DT to D¿T, the equilibrium price of coffee is now $1.59 per pound, and the equilibrium price of tea is now $0.79 per pound.
General equilibrium analysis is applicable only if something links these two markets. In this example, we will assume (plausibly, we think) that consumers view coffee
and tea as substitute goods. Thus, an increase or decrease in the price of one good
(holding the price of the other good fixed) will cause a corresponding increase or
decrease in the demand for the other good. (For example, an increase in the price of
coffee—holding the price of tea fixed—will cause an increase in the demand for tea.)
Suppose that both markets are initially in equilibrium. The equilibrium price of
coffee is $0.93 per pound, where the demand curve for coffee DC intersects the supply
curve for coffee SC. The equilibrium price of tea is $0.63 per pound, where the demand
curve for tea DT intersects the supply curve for tea ST.
Now imagine that a severe frost in South America destroys a significant portion
of the coffee crop. As a result, the coffee supply curve shifts leftward, from SC to S¿C.
The initial impact is to increase the price of coffee from $0.93 to $1.50 per pound. But
because coffee and tea are substitutes, the increase in the price of coffee increases the
demand for tea. This shifts the demand curve for tea to the right. As a result, the equilibrium price of tea goes up. But things don’t stop here. Because coffee and tea are
substitutes, the increase in the price of tea increases the demand for coffee, which
shifts the demand curve for coffee to the right, which drives the price of coffee up
some more. This in turn increases the demand for tea, shifting the demand curve for
tea even further to the right. When all of these effects have played out, the demand
curve for tea has shifted from DT to D¿T, driving up the price of tea from $0.63 to
$0.79 per pound. The demand curve for coffee is now D¿C, and the equilibrium price
is $1.59. (In Learning-By-Doing Exercise 16.1, we show how to determine these equilibrium prices.)
partial equilibrium
analysis An analysis that
studies the determination of
equilibrium price and output
in a single market, taking as
given the prices in all other
markets.
general equilibrium
analysis An analysis
that determines the equilibrium prices and quantities
in more than one market
simultaneously.
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G E N E R A L E Q U I L I B R I U M T H E O RY
We have just gone through a simple general equilibrium analysis. This analysis is
significant for two reasons. First, we see that events in the coffee market cannot necessarily be viewed in isolation: The decrease in coffee supply had a significant impact on
the price of tea. Second, because coffee and tea are substitutes, an exogenous event in
the coffee market, for example, bad weather, that tends to increase the price of coffee,
will also tend to increase the price of tea; similarly, an exogenous event that tends to
decrease the price of coffee will also tend to decrease the price of tea. This tells us that
the prices of substitute goods will tend to be positively correlated.
A P P L I C A T I O N
16.1
Net after Taxes?
The last time you purchased a product on the Internet—
a book, a CD, or even a personal computer—you probably did not pay a sales tax on the transaction. This is not
because such transactions do not involve taxes; they
usually do. Rather, the burden is on you, the buyer, to
calculate and pay the state and local sales taxes on the
items that you buy. (If you don’t believe us, read the
fine print on the invoice for your purchase. It will probably say something like, “The purchaser is responsible
for remitting any additional taxes to the taxing authority.”) This is in contrast to sales in traditional retail outlets.
When you buy a CD at your local music store, for example, the store owner is responsible for paying the tax to
the relevant tax authority, not you. Of course, with millions of individual consumer transactions on the Web
every day, it is nearly impossible for state and local governments to force consumers to pay the sales taxes that
they owe.
The most straightforward way around this problem would be to treat Internet transactions like traditional retail transactions and require sellers to remit
the sales taxes, not consumers. However, states are not
legally allowed to assess sales taxes on goods sold by
companies outside of their own state. In order to assess
taxes on an Internet purchase, the company must have
a “physical presence” (such as a store, office, or distribution warehouse) in the state to which the goods are
shipped. For example, Amazon.com is the world’s
largest Internet retailer. It has a physical presence in the
states of Kansas, Kentucky, New York, North Dakota,
and Washington, so consumers who live in those states
are assessed state sales taxes by Amazon. Consumers
living in other states are not assessed taxes on Amazon
1
purchases. Technically, those consumers are supposed
to calculate their sales tax and send it to their state’s
tax agency, but few consumers actually do so. Thus, a
large percentage of Internet sales across state lines are
effectively tax free, which provides a competitive advantage for online retailers.
This tax advantage for online retailers may not
last for long. In 2008, New York State passed a law
requiring online retailers to collect sales taxes from
residents of New York if that retailer provides sales
referrals to any New York-based website. Amazon
not only sells products directly, but the site also acts
as an aggregator that in effect channels buyers to
other sellers, all over the United States. Therefore,
Amazon will be subject to the New York tax rule if it
is ultimately upheld in court. North Carolina and
Rhode Island have now passed similar laws; California,
Virginia, Illinois, Colorado and Hawaii are considering doing the same.1
What would happen if states were allowed to
collect sales taxes directly from sellers? Let’s use general equilibrium analysis to explore this question. In
particular, we want to examine the impact of this
policy not only on the prices of products such as CDs
and books that are purchased online, but also on the
prices of services, such as the provision of Internet
access—subscription to online services that allow you
to connect to the Web.
Figure 16.3 analyzes what might happen. In a
typical e-tail market such as the market for CDs, the
imposition of a requirement that sellers pay the sales
tax would raise the marginal cost of a typical CD seller,
which, as shown in Figure 16.3(a), would shift the
supply curve for online CD sales leftward, from SCD to
S⬘CD. As a result, the price of CDs sold online would go
Evan Halper, “Lawmakers Want to Tax Amazon Sales in California.” Los Angeles Times (February 20, 2010).
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SCD
New
price
Initial
price
DCD
SIA
Price (dollars per month)
Price (dollars per CD)
S'CD
Initial
price
New
price
D'IA
0
New
Initial
quantity quantity
Quantity (CDs per year)
(a) Online CD market
0
DIA
New
Initial
number of number of
subscribers subscribers
Quantity (number of subscribers)
(b) Market for Internet access
FIGURE 16.3 Effects of Internet Sales Taxes
Online merchandise (e.g., CDs) and Internet access are demand complements. If sellers were required to charge sales taxes for CDs (and other merchandise) sold online, the supply curve for
CDs would shift leftward from SCD to S¿CD , causing the online price of CDs to rise and the quantity sold to fall. The effect of this (plus the effect of charging sales taxes on other merchandise
sold online) would diminish the value of Internet access for consumers. The demand curve for
Internet access would shift leftward from DIA to D¿IA , driving the price of Internet access services
down and reducing the number of subscribers to those services.
up. Similar price increases would occur in other online
retail markets such as the markets for books, toys,
flowers, and personal computers, and the volume of
online sales of these products would go down. In fact,
research by economist Austan Goolsbee suggests that
this impact would be quite dramatic.2 He estimates
that applying existing sales taxes to Internet commerce
would reduce the number of online buyers by 24 percent. This large impact is explained by the fact that
consumers can easily get a product like a CD elsewhere
(e.g., at a local music store or Wal-Mart).
But the effect of collecting sales taxes from
Internet sellers would not stop there. As shopping on
the Internet became more expensive and consumers
did less of it, the benefits that consumers get from
being connected to the Internet would go down. As
Figure 16.3(b) shows, the demand curve for Internet
access services would shift leftward, from DIA to D⬘IA.
This leftward shift would result in a decrease in the
2
price of Internet access. Thus, if online merchants
were forced to collect sales taxes, the price of online
merchandise such as CDs would go up, and the price
of Internet access would go down. This reduction in
price would benefit consumers but would reduce the
profitability of Internet providers such as MSN. This
might explain why high-profile technology companies such as MSN have been vocal opponents of making it easier for states to assess Internet sales taxes.
Note the contrast between this analysis and our
earlier analysis of the coffee and tea markets. In that
analysis, the goods were demand substitutes, and as a
result their prices were positively correlated. In this
example, Internet access and online merchandise are
demand complements. As a result, exogenous events
in the online retailing market that tend to increase
the prices of online merchandise will tend to decrease
the price of the complementary good, Internet access
services.
Austan Goolsbee, “In a World without Borders: The Impact of Taxes on Internet Commerce.” Quarterly
Journal of Economics 115, no. 1(2000): 561–576.
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S
E
G E N E R A L E Q U I L I B R I U M T H E O RY
D
Finding the Prices at a General Equilibrium with Two Markets
The following table shows the equations of some of the demand and supply curves depicted in
Figure 16.2.
Coffee
Tea
Initial Demand Curve
Initial Supply Curve
Supply Curve after Frost
QdC 120 ⫺ 50PC 40PT
QdT 80 ⫺ 75PT 20PC
QsC 80 20PC
QsT 45 10PT
QsC 40 20PC
QsT 45 10PT
Problem
(a) What are the general equilibrium prices of coffee and
tea initially?
(b) What are the general equilibrium prices after a frost
damages the coffee crop?
Solution General equilibrium in the two markets
occurs at prices at which supply equals demand in both
markets simultaneously.
Q Cd
QCs
(a) Initially, general equilibrium occurs when
and Q Td QTs . Using the equations in the table above,
we can rewrite these equilibrium conditions as
120 ⫺ 50PC 40PT 80 20PC
80 ⫺ 75PT 20PC 45 10PT
16.2
GENERAL
EQUILIBRIUM
A N A LYS I S :
MANY
MARKETS
This is a system of two equations in two unknowns, PC
and PT. Solving these equations simultaneously gives us
PC $0.93 and PT $0.63. These are the prices at the
initial equilibrium.
(b) After the frost, the equilibrium conditions are
QCd QCs and Q Td QTs . Again using the equations in the
table above, we can rewrite these equilibrium conditions as
120 ⫺ 50PC 40PT 40 20PC
80 ⫺ 75PT 20PC 45 10PT
Again, this is a system of two equations in the two unknown prices. Solving this system gives us PC $1.59 and
PT $0.79. These are the prices at the equilibrium after
the frost.
Similar Problems:
16.1, 16.2, 16.3, 16.4
T
he previous section illustrated a simplified general equilibrium analysis focused on
just two markets at the same time. However, we sometimes need to study more than
two markets simultaneously. For example, to understand the effects of a gasoline
excise tax on low- and high-income households, we need to explore several markets
simultaneously, including markets for inputs. In this section, we see how to do this
kind of analysis.
T H E O R I G I N S O F S U P P LY A N D D E M A N D
IN A SIMPLE ECONOMY
Let’s consider an economy consisting of two types of households, white-collar households and blue-collar households. Each type of household purchases two goods, energy (e.g., electricity, heating fuel, motor fuel) and food. And each of these goods is
produced with two input services, labor and capital.
Figure 16.4 outlines the interactions between households and business firms in
this economy. Households, in their role as consumers of finished goods, purchase the
energy and food supplied by firms. Firms, in their role as consumers of input services,
purchase the services of labor and capital supplied by households. Households supply
labor as employees in business firms that need their services. Households supply capital by renting the land or the physical assets that they own to business firms or by selling their intellectual capital to these firms.
1 6 . 2 G E N E R A L E Q U I L I B R I U M A N A LYS I S : M A N Y M A R K E T S
655
Households demand
energy and food
Businesses supply
energy and food
Suppliers of
finished goods
Consumers of
finished goods
Business Firms
Households
Consumers
of inputs
Suppliers
of inputs
Businesses demand
capital and labor
Households supply
capital and labor
FIGURE 16.4 Interactions
between Firms and Households in a
General Equilibrium
Households, in their role as consumers of
finished goods, purchase the energy and
food that are supplied by business firms.
Firms, in their role as consumers of
inputs, purchase the services of labor and
capital that are supplied by households.
As Figure 16.4 illustrates, this economy thus has four major components:
•
•
•
•
Household demand for energy and food
Firm demand for labor and capital
Firm supply of energy and food
Household supply of labor and capital
Where do the demand and supply curves for these components come from?
The Demand Curves for Energy and Food Come from Utility Maximization
by Households
To derive the demand curves for energy and food, we need to consider the utilitymaximization problems of individual households. The quantity of energy a household
purchases is denoted by x, and the quantity of food a household purchases is denoted
by y. The label W denotes white-collar households, and B denotes blue-collar households. A white-collar household has a utility function UW (x, y), and a blue-collar
household has a utility function UB (x, y).
Each household derives income from supplying labor and capital inputs to business firms. We’ll assume that each household has a fixed endowment of labor and
capital. Let’s suppose that blue-collar households are the primary suppliers of labor
in our economy, while white-collar households are the primary suppliers of capital.
Let’s also suppose that the aggregate supply of labor is greater than the aggregate
supply of capital. This could be because there are more blue-collar households than
white-collar households or because the amount of labor supplied by each blue-collar
household is greater than the amount of capital supplied by each white-collar household. If the price received for a unit of labor is w and the price received for a unit of
capital is r, then the income of each type of household, IW and IB, will depend on w
and r.
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Suppose, now, that the price of energy is Px per unit, while the price of food is Py.
When a household maximizes its utility, it takes these prices and input prices as fixed.
The utility-maximization problems for households are thus:
max UW (x, y), subject to: Px x Py y IW (w, r)
(x, y)
max UB (x, y), subject to: Px x Py y IB (w, r)
(x, y)
where IW (w, r) and IB (w, r) signify that household incomes depend on the returns that
households receive from selling their labor and their capital and that these returns
depend on the prices of labor and capital, w and r.
The solutions to these utility-maximization problems yield the optimality conditions that we discussed in Chapter 4:
MRSW
x, y
Px
Py
and
MRSBx, y
Px
Py
(16.1)
That is, each household maximizes utility by equating its marginal rate of substitution
of x for y with the ratio of the price of x to the price of y. These optimality conditions,
along with the budget constraints, can be solved for the demand curves for each
household, which depend on the prices and household income.
Figure 16.5 shows the aggregate demand curves for energy and food for each type
of household. For example, DW
x in panel (a) is the aggregate demand for energy by all
white-collar households, while DBx is the demand for energy by all blue-collar households. (In this section and throughout the rest of this chapter, subscripts on demand
and supply curves refer to the commodity being demanded or supplied, and superscripts refer to the people or firms doing the demanding or supplying.) We find these
demand curves by summing the demand curves of all the individual households. The
$1
Dx
0.80
0.60
DxW
0.40
DxB
0.20
0
5,000
10,000
15,000
20,000
Quantity of energy (units per year)
(a) Energy market
Price of food (dollars per unit)
Price of energy (dollars per unit)
$1
Dy
0.80
0.60
DyW
0.40
0.20
DyB
0
5,000
10,000
15,000
Quantity of food (units per year)
(b) Food market
FIGURE 16.5 Demand Curves for Energy and Food
Panel (a): The aggregate demand curves for energy for white-collar households and blue-collar houseB
W
B
holds are DW
x and Dx . The market demand curve for energy (Dx) is the horizontal sum of Dx and Dx .
Panel (b): The aggregate demand curves for food for white-collar households and blue-collar houseB
W
B
holds are DW
y and Dy . The market demand curve for food (Dy) is the horizontal sum of Dy and Dy .
20,000
1 6 . 2 G E N E R A L E Q U I L I B R I U M A N A LYS I S : M A N Y M A R K E T S
B
overall market demand curve for energy, Dx, is the horizontal sum of DW
x and Dx . The
position of these demand curves will, in general, depend on the income levels of
households, the price of good y, and the particular tastes of each household as embodied by its utility function. That is, changes in household income or in the price of
B
good y will cause DW
x , Dx , and Dx to shift.
To summarize, the demand curves for energy and food in our simple economy come
from utility maximization by households. Summing the energy and food demand curves
of all individual households generates the market demand curves for each commodity.
The Demand Curves for Labor and Capital Come from Cost Minimization
by Firms
To derive the demand curves for labor and capital in the economy, we need to consider
the cost-minimization problems (i.e., the input choice decisions) faced by individual firms.
Assume that some firms produce energy while others produce food, that all energyproducing firms are identical and all food-producing firms are identical, and that each
market is perfectly competitive. Each individual energy producer has a production function x f (l, k), where l and k denote the amount of labor and capital used by an individual producer (uppercase L and K will refer to the aggregate amounts of labor and capital
in the market). Also assume that this production function is characterized by constant returns to scale (recall from Chapter 6 that this means that doubling the amount of labor
and capital exactly doubles the quantity of energy a typical producer can make). For an
energy producer that produces x units of energy, the cost-minimization problem is
min wl rk, subject to: x f (l, k)
(l,k)
Similarly, each food producer has a production function y g (l, k), which is also
characterized by constant returns to scale. The cost-minimization problem for a food
producer is
min wl rk, subject to: y g(l, k)
(l, k)
The solutions to these cost-minimization problems yield the optimality conditions that we discussed in Chapter 7:
MRTS l,xk
w
r
and
MRTS ly, k
w
r
(16.2)
That is, each firm chooses its cost-minimizing input combination by equating its marginal rate of technical substitution of labor for capital, MRTSl,k, to the ratio of the price
of labor to the price of capital. These optimality conditions, along with the production
constraints for energy and food, can be solved to determine the demand curves for
labor and capital for individual energy and food producers. These demand curves depend on the input prices w and r and on the total amount of output produced by a firm.
Figure 16.6 shows the aggregate demand curves for labor and capital for each industry, energy and food. We find these demand curves by summing the demand curves
of all the individual firms in each industry. For example, DLx in panel (a) is the aggrey
gate demand for labor by firms in the energy industry, while DL is the aggregate
demand for labor by firms in the food industry. The overall market demand curve for
y
labor, DL, is the horizontal sum of DLx and DL. The position of these demand curves
depends on the total amount of output produced in each industry, the price of the
other input, and the nature of the technology embodied in the production functions.
For example, an increase in the amount of output in the energy industry would
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G E N E R A L E Q U I L I B R I U M T H E O RY
$1
0.5
DL
DLx
y
DL
0
2000 4000 6000 8000
Quantity of labor (units)
(a) Labor market
Price of capital (dollars per unit)
CHAPTER 16
Price of labor (dollars per unit)
658
$1
0.5
DK
y
DK
0
DKx
2000 4000 6000 8000
Quantity of capital (units)
(b) Capital market
FIGURE 16.6 Demand Curves for Labor and Capital
Panel (a): The aggregate demand curves for labor for energy producers and food producers
y
are DLx and DL . The market demand curve for labor (DL) is the horizontal sum of DLx and
y
.
DL Panel (b): The aggregate demand curves for capital for energy producers and food
y
producers are Dkx and Dk. The market demand curve for capital (DK) is the horizontal sum
y
x
of Dk and Dk.
increase the demand for labor in that industry and would thus shift DLx (and thus DL)
rightward. By contrast, a decrease in the price of capital, r, would encourage firms to
y
substitute capital for labor and would shift both DLx and DL (and thus DL) to the left.
To summarize, the demand curves for labor and capital in each industry in our
simple economy come from cost minimization by individual firms. Summing the labor
and capital demand curves of all individual firms in both industries generates the market demand curves for both inputs.
The Supply Curves for Energy and Food Come from Profit Maximization
by Firms
We saw in Chapter 8 that the cost-minimization problem of each firm yields a total
cost curve and a marginal cost curve. Because each firm has a production function characterized by constant returns to scale, the marginal cost curve for an energy producer
is a constant, MCx, and the marginal cost curve for a food producer is also a constant,
MCy. Both of these curves are shown in Figure 16.7. The height of each curve depends
on the input prices w and r. Because the production function for food differs from the
production function for energy, the curves may depend on the input prices in different
ways. For example, if food production is labor-intensive (if it involves a high ratio of
labor to capital), then MCy might be more sensitive to the price of labor than MCx is.
Since the energy and food industries are assumed to be perfectly competitive,
firms in these industries act as price takers. Because a firm in the energy industry faces
a constant marginal cost, energy producers are willing to supply any positive amount
of output at a price Px equal to marginal cost MCx. This means that the industry supply curve for energy is perfectly elastic at that price. Thus, the industry supply curve
for energy Sx coincides with the marginal cost curve for energy production MCx, as
659
$1
0.8
Sx = MCx
0.6
0.4
0.2
0
5,000
10,000
15,000 20,000
Quantity of energy (units per year)
(a) Energy market
Price of food (dollars per unit)
Price of energy (dollars per unit)
1 6 . 2 G E N E R A L E Q U I L I B R I U M A N A LYS I S : M A N Y M A R K E T S
$1
0.8
Sy = MCy
0.6
0.4
0.2
0
5,000
10,000
15,000 20,000
Quantity of food (units per year)
(b) Food market
FIGURE 16.7 Supply Curves for Energy and Food
Panel (a): The marginal cost curve for energy MCx is also the market supply curve for energy Sx.
Panel (b): The marginal cost curve for food MCy is also the market supply curve for food Sy.
shown in Figure 16.7(a). Similarly, the industry supply curve for food Sy coincides
with the marginal cost curve for food production MCy, as shown in Figure 16.7(b).
Because the supply curves coincide with the marginal cost curves, the equilibrium
prices must equal the marginal costs:
Px MCx
and
Py MCy
(16.3)
Since we have constant returns to scale, marginal cost and average cost are equal, so
at these prices each producer earns zero profit. At this point, we still cannot say what
these equilibrium prices are, since the marginal costs in each market, MCx and MCy,
depend on the input prices w and r. And these input prices, in turn, depend on supply
and demand in the input markets. Thus, each of the markets in this economy is interdependent.
To summarize, the supply curves in each industry in our economy arise from
profit maximization by firms. Because production in both the energy and food industries is characterized by constant returns to scale, the supply curves in each industry
are horizontal lines corresponding to the industry’s marginal cost of production.
The Supply Curves for Labor and Capital Come from Profit Maximization
by Households
The final components of our economy are the supply curves for labor and capital.
Labor and capital in this economy are provided by households. As already mentioned,
each household can offer a fixed supply of labor and capital. Assume that there is no
opportunity cost to offering this supply of labor or capital. (This simplifies the presentation without affecting the main conclusions.) Profit maximization by individual
households thus implies that a household will supply its labor and capital as long as
those services can fetch a positive price in the marketplace. Also assume that households are indifferent between selling their labor to the energy or food industries as
long as the wage w that they get from either industry is the same. Similarly, households will supply capital to either industry as long as the price of capital services r is
the same in each industry.
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SL
Supply Curves for Labor
and Capital
Panel (a): The market
supply curve for labor
SL is a vertical line corresponding to the total
amount of labor that
households are willing
to supply. Panel (b): The
market supply curve for
capital SK is a vertical
line corresponding to
the total amount of
capital that households
are willing to supply.
1
0.5
0
2000 4000 6000 8000
Quantity of labor (units)
(a) Labor market
SK
$1.5
Price of capital (dollars per unit)
FIGURE 16.8
Price of labor (dollars per unit)
$1.5
1
0.5
0
2000 4000 6000 8000
Quantity of capital (units)
(b) Capital market
Figure 16.8 shows the implications of these assumptions. The market supply
curve for labor, SL, is a vertical line corresponding to the overall supply of labor, which
is predominantly provided by blue-collar households. Similarly, the market supply
curve for capital, SK, is a vertical line corresponding to the overall supply of capital,
which predominantly comes from white-collar households.
To summarize, the supply curves for labor and capital in our economy come from
profit maximization by households. Because we have assumed that each household has a
fixed supply of labor and capital that it can offer, these supply curves will be vertical lines.
THE GENERAL EQUILIBRIUM IN OUR SIMPLE ECONOMY
In our simple economy, four prices are simultaneously determined in a general equilibrium: a price Px for energy, a price Py for food, a price w for labor services, and a
price r for capital services. These latter two prices, in turn, determine household income, which is derived from their sales of labor and capital services to firms. The four
prices in our economy are interdependent. For example, the price of energy is determined by the marginal cost of energy, but the marginal cost of energy depends on the
prices of labor and capital. These prices are pinned down by market-clearing conditions in each of our four markets:
Household demand for energy Industry supply of energy
Household demand for food Industry supply of food
Industry demand for labor Household supply of labor
Industry demand for capital Household supply of capital
Figure 16.9 illustrates our simple economy when it is in a general equilibrium—that
is, when supply equals demand in all four markets simultaneously. Panels (a) and
(b) show that when the prices of labor and capital are $0.48 and $1.00, respectively,
the marginal costs of energy and food production are $0.79 and $0.70, respectively.
661
Dx
0
6202
Quantity of energy (units)
Price of labor (dollars per unit)
SL
$0.48
DL(r = $1.00,
X = 6202, Y = 4943)
7000
Quantity of labor (units)
(c) Labor market
MCy = Sy
(w = $0.48,
r = $1.00)
$0.70
Dy
0
4943
Quantity of food (units)
(b) Food market
(a) Energy market
0
Price of food (dollars per unit)
MCx = Sx
(w = $0.48,
r = $1.00)
$0.79
Price of capital (dollars per unit)
Price of energy (dollars per unit)
1 6 . 2 G E N E R A L E Q U I L I B R I U M A N A LYS I S : M A N Y M A R K E T S
SK
$1.00
DK (w = $0.48,
X = 6202, Y = 4943)
0
5000
Quantity of capital (units)
(d) Capital market
FIGURE 16.9 General Equilibrium
At the general equilibrium in our simple economy, all four markets (energy, food, labor, and
capital) are simultaneously in equilibrium. Panels (a) and (b) show that when the prices of labor
and capital are $0.48 and $1.00, the equilibrium prices of energy and food are $0.79 and $0.70,
and the equilibrium quantities of energy and food are 6202 and 4943 units. Panels (c) and
(d) show that when the quantities of energy and food demanded are 6202 and 4943 units,
the equilibrium prices of labor and capital are $0.48 and $1.00 per unit.
The equilibrium input prices thus determine the height of the industry supply curves,
Sx and Sy. These input prices also determine household incomes, IW (w, r) and IB (w, r),
which determines the positions of the demand curves for energy and food (Dx and Dy).
The intersection of demand and supply in the energy and food markets determines
the total output in these industries: 6202 units in the energy industry and 4943 units
in the food industry. These outputs, in turn, determine the positions of the labor and
capital demand curves in panels (c) and (d). And it is the intersection of these input
demand curves with the input supply curves, SL and SK, that determines the equilibrium prices of labor and capital ($0.48 and $1.00). This explanation of Figure 16.9
began and ended with the prices of labor and capital. Figure 16.9 illustrates the same
cycle of interdependence at a general equilibrium that is pictured in Figure 16.4.
Thus, to summarize, we have seen the following:
• The equilibrium input prices in the labor and capital markets determine the
positions of the supply and demand curves in the energy and food markets.
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• These supply and demand curves determine the equilibrium prices and quantities
in the energy and food markets.
• The equilibrium quantities of energy and food determine the positions of the
demand curves in the labor and capital markets, and the point where these
curves cross the supply curves of labor and capital determines the equilibrium
prices of labor and capital.
From this analysis we can see that, even in our simple economy, we cannot analyze events in one market without taking into account how those events affect the
other markets. Application 16.2 illustrates how a change in the price of one good, like
oil, can affect the equilibrium in many other markets.
A P P L I C A T I O N
16.2
Causes and Effects of 2007–2008
Oil Price Rise
In 2007–2008, oil prices experienced a dramatic increase. The price had been rising gradually in the last
few years, from about $20 per barrel in 2001 to $70
per barrel in 2006. In late 2007 and continuing in
2008, the price rose suddenly, to a high of $145 per
barrel. This was followed by a remarkable fall in the
price to about $40 per barrel. A study of this incident
by economist James Hamilton reveals that these
events in the oil industry had important effects on
other parts of the U.S. economy.3
To begin, what caused the dramatic spike in oil
prices? While the media suggested at the time that it
might have been caused by “speculators,” Hamilton
concluded that the rise is explained by shifts in the
supply and demand curves. The world supply of oil
had risen gradually to 2005, but did not change much
from 2005 through 2008. One reason for this was declining production in some oil fields such as the North
Sea. While the supply did not change, demand was
increasing. Importantly, India’s and China’s emerging
economies were shifting out their demand for oil rapidly. For example, China’s economy had annual growth
rates of 7 percent, with similar increases in oil consumption. Prior studies had estimated that worldwide
demand for oil has relatively inelastic demand, with
price elasticity of demand of approximately –0.06.
Based on the quantity of oil sold at that time, this
elasticity is indeed consistent with a price increase
3
from $55 to $145 barrels per day, exactly what occurred. Static supply combined with an inelastic demand curve that shifted out with economic growth
appears to have caused the price to rise.
Hamilton argues that the subsequent fall in price
resulted from two factors. First, the world had entered what would become the largest recession since
the Great Depression. This had the effect of initially
shifting the demand curve for oil strongly to the left
and leading to a reversal of the recent price rise.
Second, demand became more elastic throughout
2008 as manufacturers and consumers made further
adjustments to their production and consumption in
response to the severe recession. In other words, elasticity of demand was larger in absolute value in the
longer run, exactly what we would expect.
Hamilton then went on to analyze some of the
effects of the oil price shock on the U.S. economy. Oil
is an important factor of production because of the
fundamental roles of energy and fuel in any economy. Every industry uses energy as an input, and consumers spend a nontrivial fraction of their budgets
on transportation. The increase in oil prices raised
manufacturing and fuel costs, which shifted supply
curves to the left for many goods. In addition, the
higher cost of gasoline reduced consumer demand
due to income effects. He estimates that perhaps half
of the reduced growth of GDP in 2008 may have been
caused by the oil price shock (with the steep fall in
the housing market being the other major cause).
James Hamilton, “Causes and Consequences of the Oil Shock of 2007–2008.” Brookings Papers on Economic
Activity, 2009.
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1 6 . 2 G E N E R A L E Q U I L I B R I U M A N A LYS I S : M A N Y M A R K E T S
An industry that was particularly affected by the
rise in oil prices was automobiles. In 2008, sales of
sport-utility vehicles (SUVs) declined 25 percent. At the
same time, domestic car sales fell 7 percent, imported
car sales rose 10 percent, but sales of imported light
trucks fell 22 percent. These results strongly suggest
that substitution effects were more important than
income effects. Consumers were shifting from vehicles
with low- fuel efficiency, such as SUVs and trucks, and
toward smaller and more fuel-efficient cars (imported
cars tend to fit into those categories). However, by the
end of 2008 the severity of the recession meant that
income effects were becoming important to consumer
demand, as sales of cars, both imported and domestic,
began to decline considerably as well. In addition, the
fall in demand for SUVs, trucks, and cars affected the
labor market. Seasonally adjusted employment in the
motor vehicle and parts industries declined by 125,000
over this period. Of course, that unemployment further reduced demand for many consumer goods, and
increasing problems in the housing market (especially
in cities such as Detroit in which vehicle manufacturing
is an important part of the local economy) contributed
further to the recession.
The general equilibrium analysis in Figure 16.9 highlights the relationship between
the scarcity of factors of production, the relative prices of those factors, and the distribution of income in the economy. In the economy in Figure 16.9, the aggregate supply of
capital is much less than the aggregate supply of labor (i.e., SK is closer to its vertical axis
than is SL). As a result, the price of capital services exceeds the price of labor (i.e., capital
services trade at a price premium compared to labor services). This, in turn, allows the
providers of capital inputs—the white-collar households in our economy—to earn higher
incomes than the providers of labor inputs—primarily blue-collar households.
Learning-By-Doing Exercise 16.2 shows how to write the supply-equals-demand
conditions that determine a general equilibrium for our simple economy.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 6 . 2
D
Finding the Conditions for a General Equilibrium with Four Markets
Suppose that the households in the simple economy depicted in Figure 16.9 have the characteristics
given in the following table:
Blue Collar
White Collar
Number of
Households
Labor Supplied
per Household
100
100
60 units
10 units
Also suppose that the supply and demand curves for the
markets in this economy are as shown in the following
Energy
Supply
Demand
1
3
Px w r
Food
2
3
50IW 75IB
Px
X
1
2
Py w r
Capital Supplied
per Household
Household Income
IB(w, r) 60w
IW (w, r) 10w 50r
0 units
50 units
table, where X is the overall quantity of energy demanded and Y is the overall quantity of food demanded:4
Labor
1
2
50IW 25IB
PY
Y
Capital
L 7000*
X r
Y r
L a b a b
3 w
2 w
2
3
1
2
K 5000*
2X w 3
Y w 2
K
a b a b
3 r
2 r
1
*Based on supply per household, as shown in the table showing the number of households above [L (100 ⫻ 10) ⫹
(100 ⫻ 60) ⫽ 7000; K ⫽ (100 ⫻ 50) ⫹ (100 ⫻ 0) ⫽ 5000].
4
In the Appendix, we show how these curves are derived from the cost-minimization problems of individual firms and the utility-maximization problems of individual households.
1
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Problem
(a) What are the supply-equals-demand conditions for
the energy and food markets?
(b) What are the supply-equals-demand conditions for
the labor and capital markets?
1
(c) How would we find the general equilibrium for this
economy?
Solution
(a) The supply-equals-demand condition in the energy
market is
50(10w 50r) 75(60w)
50IW 75IB
5000w 2500r
X
X
X
2
w3r 3
(16.4)
The supply-equals-demand condition in the food market is
1
1
w2r 2
50(10w 50r) 25(60w)
50IW 25IB
2000w 2500r
Y
Y
Y
Equations (16.4) and (16.5) identify the points at which
Sx Dx and Sy Dy in Figure 16.9.
(b) The supply-equals-demand condition in the labor
market is
Y r 2
X r 3
a b a b
3 w
2 w
2
7000
1
(16.6)
The supply-equals-demand condition in the capital
market is
Y w
2X w
a b a b
3 r
2 r
1
3
5000
1
2
(16.5)
(c) To find the general equilibrium we would solve the
four equations (16.4) through (16.7) for the four unknowns (w, r, X, and Y ). (We will not show the algebra
here.) We could then determine the equilibrium in each
market (and, thus, the general equilibrium) by plugging
the values of these unknowns back into equations (16.4)
through (16.7). This is how the equilibrium shown in
Figure 16.9 was actually determined.
Similar Problems:
16.5, 16.6
(16.7)
WA L R A S ’ L AW
Walras’ Law The law
that states that in a general
competitive equilibrium
with a total of N markets,
if supply equals demand in
the first N ⫺1 markets, then
supply will equal demand in
the N th market as well.
If you tried to solve the four equations in four unknowns in Learning-By-Doing
Exercise 16.2, you would discover something surprising: instead of having four distinct equations in four unknowns, you would really have three equations in four
unknowns. That is, one of our four supply-equals-demand equations is redundant.
This is an example of Walras’ Law, named after the Swiss economist Leon Walras,
who discovered it. Walras’ Law states that in a general competitive equilibrium with a
total of N markets (N 4 in our simple example), if supply equals demand in the first
N ⫺ 1 markets, then supply will necessarily equal demand in the Nth market as well.
The reason that Walras’ Law holds is straightforward. We saw earlier that a household’s income is equal to the payments made by firms for the labor and capital services
provided by the household. We also know that when households maximize their utilities, their budget constraints hold: A household’s expenditure on goods and services
equals the household’s income. Putting these two observations together implies that
total household expenditures on goods and services in the economy must therefore
equal total payments by firms to purchase inputs. This last condition, coupled with
supply-equals-demand in the first N ⫺ 1 markets in the economy, will ensure that supplyequals-demand in the Nth market as well.
Because of Walras’ Law, in the simple economy we analyzed above, we have three
market-clearing conditions but four unknowns. This implies that an equilibrium in
our economy will determine the prices in just three of our four markets. In the fourth
665
1 6 . 3 G E N E R A L E Q U I L I B R I U M A N A LYS I S : C O M PA R AT I V E S TAT I C S
market—which in our example we took to be the capital market—we can set the price
equal to any number we want. In our analysis we set that price equal to $1.
What is the significance of Walras’ Law? Walras’ Law tells us that our general
equilibrium analysis determines the prices of labor, energy, and food relative to the
price of capital, rather than determining the absolute levels of all of these prices. We
could, of course, have set the price of capital equal to a number other than $1, perhaps $2 or even $200. Had we done so, all of the other prices in our economy would
have changed. However, their ratio to the prespecified price of capital would remain
the same. For example, the ratio of the price of labor to the price of capital would remain at 0.48, no matter what our prespecified price of capital.
N
ow that we have seen how to determine the general competitive equilibrium for a
simple economy, how can we apply this approach? Economists use general equilibrium models to explore the effects of taxes or public policy interventions. Most of
these applications involve performing some kind of comparative statics analysis. For
example, economists might explore how changes in exogenous variables such as
household endowments of labor or capital or tax rates would affect the endogenous
variables—prices and quantities—that are determined in equilibrium. The models
that economists use for this purpose are much more complex than the simple model
we have presented here. In one analysis, economists looked at the effects of motor fuel
taxes using a model with more than 30 industries, seven different types of households,
and five inputs (capital and four different types of labor).5 In this section, we will
illustrate general equilibrium comparative statics analysis using the model we developed
in the previous section. Specifically, we will consider the general equilibrium impact of
an excise tax.
Suppose that the government in our simple economy imposes an excise tax of
$0.20 per unit in the market for energy. Also suppose that the proceeds are used to
buy goods from the food industry, which are then shipped outside the economy (e.g.,
distributed to countries experiencing famines). How does this tax affect prices and
quantities in the economy? Also, who is harmed more by this tax: blue-collar households or white-collar households?
You might think that blue-collar households are likely to be harmed more. As we
can see from Figure 16.10, blue-collar households tend to spend much more on energy
than on food in the initial equilibrium. By contrast, white-collar households spend
almost equally on both goods. However, when we work through the general equilibrium effects of the energy tax, we will see that the impact of the tax is not necessarily
greater on blue-collar households.
In performing our comparative statics analysis, we can take advantage of Walras’
Law and focus our attention on changes in the prices of energy, food, and labor, keeping the price of capital equal to $1.00 per unit. The most obvious impact of the tax,
as shown in panel (a) of Figure 16.11, is that it shifts the supply curve for energy upward by the amount of the tax ($0.20 per unit) from Sx to Sx 0.20. This results in a
$0.20 increase in the price of energy, from $0.79 to $0.99. This, in turn, means that
the equilibrium quantity of energy demanded goes down; thus, the demand for labor
5
A. Wiese, A. Rose, and G. Shluter, “Motor-Fuel Taxes and Household Welfare: An Applied General
Equilibrium Analysis,” Land Economics (May 1995): 229–243.
16.3
GENERAL
EQUILIBRIUM
A N A LYS I S :
COMPARATIVE
S TAT I C S
Price of energy (dollars per unit)
CHAPTER 16
G E N E R A L E Q U I L I B R I U M T H E O RY
$0.79
Dx
DxW
DxB
Price of food (dollars per unit)
666
Dy
DyW
DyB
0 1029 3914 4943
Quantity of food (units per year)
0 2734 3468 6202
Quantity of energy (units per year)
(a) Energy market
$0.70
(b) Food market
Household
Expenditures on Energy
Expenditures on Food
Blue collar
2734 units @ $0.79 per unit = $2,160
1029 units @ $0.70 per unit = $720
White collar
3468 units @ $0.79 per unit = $2,740
3914 units @ $0.70 per unit = $2,740
FIGURE 16.10
Purchases by Blue-Collar and White-Collar Households at the Initial
Equilibrium
Panel (a) shows the demand curves for energy for blue-collar and white-collar households (DW
x
and DBx ) and the overall demand curve for energy (Dx). Panel (b) shows the demand curves for
B
food by these households (DW
y and Dy ) and the overall demand curve for food (Dy). The table
shows the amount of money each type of household spends on each good.
by the energy industry also goes down. However, because the government spends the
proceeds of the tax on food, the aggregate demand for food, which now includes government demand as well as household demand, goes up, which results in an increase
in the demand for labor by food producers.
With labor demand by energy producers falling and labor demand by food producers rising, what happens to the overall demand for labor? In other words, does the
overall labor demand curve shift to the right or the left? In general, it could shift in
either direction. In Figure 16.11 we examine the case in which the labor demand curve
DL shifts rightward. This case would arise if the food industry uses more labor to produce a given unit of output than the energy industry does.6 Panel (c) of Figure 16.11
shows that when DL shifts to the right, the equilibrium price of labor w goes up. This
feeds back to increase the marginal costs of both energy and food, which increases
prices in these markets. But this increase in w also increases household incomes, particularly among the blue-collar households that derive most of their income from
labor. This works to shift demand rightward in both the energy and food markets.
6
In the last section of the Appendix, we show that when we compute the equilibrium using the production
functions that generated the supply curves for energy and food in Learning-By-Doing Exercise 16.2,
firms in the food industry do, in fact, use more labor to produce a given unit of output than do firms in
the energy industry.
667
Sx (excise tax)
Sx + $0.20
Sx (no tax)
$1.02
0.99
0.79
Dx (excise tax)
Dx (no tax)
0
6202
Quantity of energy (units)
(a) Energy market
Price of food (dollars per unit)
Price of energy (dollars per unit)
1 6 . 3 G E N E R A L E Q U I L I B R I U M A N A LYS I S : C O M PA R AT I V E S TAT I C S
Sy (excise tax)
Sy (no tax)
$0.74
0.70
Dy (excise tax)
Dy (no tax)
0
4943
Quantity of food (units)
Price of labor (dollars per unit)
(b) Food market
SL
$0.55
0.48
0
DL (excise tax)
DL (no tax)
7000
Quantity of labor (units)
(c) Labor market
FIGURE 16.11
General Equilibrium Effects of an Excise Tax: Comparative Statics Analysis
An excise tax of $0.20 per unit is imposed on energy, and the proceeds are used to purchase
food (which is then distributed outside the economy). This tax ultimately results in a new general
equilibrium: the price of energy rises from $0.79 to $1.02, the price of food rises from $0.70 to
$0.74, and the price of labor rises from $0.48 to $0.55.
Figure 16.11 shows that when we account for all of the equilibrium effects, the
new equilibrium involves a slightly higher price of labor (w $0.55 versus $0.48 initially) and higher prices for both energy and food (Px $1.02 versus $0.79 initially,
and Py 0.74 versus $0.70 initially). Figure 16.12 summarizes these effects. Because
the price of labor has gone up, blue-collar households enjoy a significant increase in
income, while white-collar households enjoy a modest increase in income. Both types
of households are hurt by the tax because of the higher prices. However, blue-collar
households are hurt less by the tax than white-collar households because of the greater
boost in income enjoyed by blue-collar households.
We deliberately constructed this example to show that it is not always obvious
who will be affected most by a public policy intervention such as a tax. Even though
the tax in our example is on energy, and blue-collar households spend a higher proportion of their income on energy, white-collar households are actually hurt more by
the tax. This became clear only as we worked through all the effects of the tax on the
way to a new general equilibrium. The example illustrates why economists often use
general equilibrium models to analyze public policy proposals.
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CHAPTER 16
G E N E R A L E Q U I L I B R I U M T H E O RY
Excise tax of $0.20
per unit imposed on
energy. Government
uses proceeds to buy
food
Effect on
energy market
Effect on
food market
Supply curve shifts
up; price goes up
and quantitly goes
down
Demand curve shifts
rightward; price
goes up and quantity
goes up
Effect on
labor market
Effect on
labor market
Demand curve shifts
leftward
Overall effect on
labor market
(assuming
that food
industry uses
more labor per
unit of output
than energy
industry does)
Demand curve shifts
rightward
Demand curve shifts
rightward, so price
goes up and
quantity goes up
Effect on
energy market
Supply curve shifts
up; price goes up
and quantity goes
down
Effect on
food market
Supply curve shifts
up; price goes up
and quantitly goes
up
Overall Effect
(new general equilibrium)
Price
Quantity
FIGURE 16.12
Energy
Food
Labor
Up
Down
Up
Up
Up
Unchanged
General Equilibrium Effects of an Excise Tax on Energy: Flowchart
The effects on the general equilibrium of a $0.20 per unit excise tax on energy—explained
in the text and represented as a comparative statics analysis in Figure 16.11—are shown
here in the form of a flowchart.
669
16.4 THE EFFICIENCY OF COMPETITIVE MARKETS
A P P L I C A T I O N
16.3
Who Likes the Gas Tax Least?
We have just analyzed a simple economy with only
four markets (energy, food, labor, and capital) and
two types of households (blue collar and white collar).
We constructed a reasonable example in which an excise tax on energy hurts white-collar households more
than blue-collar households, despite our initial expectation that the opposite would be true. Could this
happen in the real world?
Arthur Wiese, Adam Rose, and Gerald Shluter used
a general equilibrium analysis to address this question.7
The proceeds of state gasoline taxes have historically
been used to finance highway construction. When a
state collects more gasoline tax revenue and spends it
on road construction, this increases the demand for
road construction firms. This, in turn, increases the construction firms’ demand for labor, driving up wages of
manual labor in construction trades. The increase in
wages increases the marginal cost of production in
other industries that also employ manual labor.
As in our simple economy in Figure 16.11, the increase in wages for manual labor feeds through to
increase the prices of finished goods in industries that
employ manual labor. The increase in the prices of
these manufactured goods results in a reduction in the
quantity of these goods demanded by households,
which means that output in manufacturing industries
goes down. As these industries produce less output,
they employ smaller quantities of all types of labor,
including white-collar and professional labor. Some
of these industries might even lay off managers and
professionals.
The study by Wiese, Rose, and Shluter shows that
when all is said and done, the effects of an increase in
the gasoline tax are rather complicated. All consumers
are hurt by higher prices of gasoline and finished
goods. Moreover, households of white-collar and professional labor are hurt by the reduced demand for
their labor services. On the other hand, households
that supply manual labor benefit from higher wages.
Because lower-income households tend to supply a
disproportionately high share of manual labor, they
find that lower-income households are hurt less by
the tax than higher-income households. They also
find that if state governments spend a smaller proportion of the proceeds of gasoline taxes on construction
programs and use more of the proceeds for general
state spending (e.g., education), then an increase in
gasoline taxes would hurt the highest-income and the
lowest-income households the most, while hurting
middle-income households the least.
I
n Chapter 10, we saw that the competitive equilibrium in a single competitive market maximizes the net economic benefit that can be generated in that market. This
makes the competitive market outcome economically efficient. In this section, we explore whether economic efficiency arises in an economy in which many competitive
markets simultaneously achieve a general equilibrium. But before we begin our analysis, we need to refine our definition of economic efficiency and see how it applies at a
general competitive equilibrium.
16.4
THE
EFFICIENCY
OF
COMPETITIVE
MARKETS
W H AT I S E C O N O M I C E F F I C I E N C Y ?
At the general competitive equilibrium shown in Figure 16.9, energy and food are
consumed by households, and labor and capital are used by industries. We call such a
pattern of consumption and input usage an allocation of goods and inputs. We say
7
Arthur Wiese, Adam Rose, and Gerald Shluter, “Motor-Fuel Taxes and Household Welfare: An Applied
General Equilibrium Analysis.” Land Economics (1995): 229–243.
allocation of goods
and inputs A pattern
of consumption and input
usage that might arise in a
general equilibrium in an
economy.
670
CHAPTER 16
economically efficient
that an allocation of goods and inputs is economically efficient if there is no other
feasible allocation of goods and inputs that would make some consumers better off
without hurting other consumers (some books refer to this as Pareto efficient). By contrast, an allocation of goods and inputs is economically inefficient (or Pareto inefficient) if there is an alternative feasible allocation of goods and inputs that would make
all consumers better off than the initial allocation does. Put another way, for any inefficient allocation we can always find at least one efficient allocation that consumers
would unanimously prefer to the inefficient one. At an inefficient allocation of goods
and inputs, the economy is not getting all that it can get from its resources.
Given this definition of efficiency, a competitive equilibrium such as the one
shown in Figure 16.9 needs to satisfy three conditions if it is to be efficient:
(Pareto efficient) Characteristic of an allocation of
goods and inputs in an
economy if there is no other
feasible allocation of goods
and inputs that would make
some consumers better off
without hurting other
consumers.
economically inefficient
(Pareto inefficient) Characteristic of an allocation of
goods and inputs if there is
an alternative feasible allocation of goods and inputs that
would make all consumers
better off as compared with
the initial allocation.
exchange efficiency A
characteristic of resource
allocation in which a fixed
stock of consumption goods
cannot be reallocated among
consumers in an economy
without making at least
some consumers worse off.
input efficiency A
characteristic of resource
allocation in which a fixed
stock of inputs cannot be
reallocated among firms in
an economy without reducing the output of at least
one of the goods that is
produced in the economy.
substitution efficiency
A characteristic of resource
allocation in which, given
the total amounts of capital
and labor that are available
in the economy, there is no
way to make all consumers
better off by producing
more of one product and
less of another.
G E N E R A L E Q U I L I B R I U M T H E O RY
1. Given the total amounts of energy and food (goods) that are consumed by the two
types of households, white collar and blue collar, there is no way that we can reallocate these amounts among the households to make all households better off
than they are at the competitive equilibrium. That is, the allocation of goods must
satisfy the condition of exchange efficiency. Generally, we have efficiency in exchange when a fixed stock of consumption goods cannot be reallocated among
consumers in an economy without making at least some consumers worse off. We
have inefficiency in exchange when we can reallocate a fixed basket of consumption goods among consumers in a way that makes all consumers better off.
2. Given the total amounts of capital and labor (inputs) that are used by the two
types of firms, energy producers and food producers, there is no way that we
can reallocate these amounts among the firms so that they produce more energy
and more food than they do when they are at the competitive equilibrium. That
is, the allocation of inputs must satisfy the condition of input efficiency. Generally,
we have input efficiency when a fixed stock of inputs cannot be reallocated among
firms in an economy without reducing the output of at least one of the goods
that is produced in the economy. In other words, we have input efficiency when
an expansion of output in one industry (e.g., food) necessitates a reduction in
output in another industry (e.g., energy). We have input inefficiency when we
can reallocate a fixed stock of inputs among firms in a way that simultaneously
expands the output of all of the goods produced in the economy.
3. Given the total amounts of capital and labor that are available in the economy,
there is no way that we can make all consumers better off by producing more of
one product (e.g., energy) and less of the other (e.g., food). That is, the allocation
of goods and inputs in the economy must satisfy the condition of substitution
efficiency. By contrast, an allocation of goods and inputs is substitution inefficient
if we can make all consumers better off by producing more of one product and
less of another.
In the next three sections, we explore each of these notions of efficiency in greater
detail and show that the general competitive equilibrium in Figure 16.9 satisfies all
three efficiency conditions.
EXCHANGE EFFICIENCY
To see whether the competitive equilibrium satisfies the condition of exchange
efficiency, we will need to develop a graphical tool called the Edgeworth box, used
to describe exchange efficiency and inefficiency.
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16.4 THE EFFICIENCY OF COMPETITIVE MARKETS
Energy consumed by blue-collar household
5
Food consumed by white-collar household
0
G
1
0
5
Food consumed by blue-collar household
Total amount of food available
10
9
10
Energy consumed by white-collar household
Total amount of energy available
FIGURE 16.13
An Edgeworth Box
For an economy with two goods (energy and food) and two consumers (a blue-collar household
and a white-collar household), this Edgeworth box shows all possible allocations of the two goods
(each point in the box, such as point G, represents a possible allocation). The width of the box
shows the total amount of energy available (10 units); white-collar energy consumption (bottom
axis) increases from left to right, while blue-collar energy consumption (top axis) increases from
right to left. The height of the box shows the total amount of food available (10 units); white-collar
food consumption (left axis) increases from bottom to top, while blue-collar food consumption (right
axis) increases from top to bottom. At point G, a white-collar household consumes 5 units of energy
and 1 unit of food, while a blue-collar household consumes 5 units of energy and 9 units of food.
What Is the Edgeworth Box?
Imagine that a given amount of energy and food has been produced—10 units of each
product—and is going to be divided between two households in our economy, a whitecollar household and a blue-collar household. The diagram in Figure 16.13, called an
Edgeworth box, shows all of the possible allocations of the two goods. The width of
the Edgeworth box shows the total amount of energy available (10 units), while the
height of the box shows the total amount of food available (also 10 units). Each point
in the Edgeworth box represents one way to allocate the available energy and food. For
example, at point G, a white-collar household consumes 5 units of energy and 1 unit of
food, while a blue-collar household consumes 5 units of energy and 9 units of food.
Describing Exchange Efficiency Using the Edgeworth Box
Does the allocation represented by point G satisfy the condition of exchange efficiency?
The answer depends on the preferences of the households (i.e., on their utility functions).
In Figure 16.14, indifference curves for the white-collar household and indifference curves
for the blue-collar household are superimposed on the Edgeworth box from Figure 16.13.
Edgeworth box A
graph showing all the possible allocations of goods in
a two-good economy, given
the total available supply of
each good.
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G E N E R A L E Q U I L I B R I U M T H E O RY
Energy consumed by blue-collar household
10
8.5
5
0
Food consumed by white-collar household
Total amount of food available
Direction
in which
utility
Blue-collar increases
household
I
4
H
6
J
1
0
1.5
G
5
Food consumed by blue-collar household
White-collar
household
9
10
Energy consumed by white-collar household
Total amount of energy available
FIGURE 16.14
Trading to Reach an Allocation That Is Economically Efficient in Exchange
Indifference curves for the white-collar household and the blue-collar household cross at point
G and point J and are tangent at point H and point I. Points G and J (and all other points
where indifference curves cross) do not represent allocations that are economically efficient in
exchange, because at either point households could make trades that would let both households reach higher indifference curves. For example, the trade represented in the figure—the
white-collar household gives the blue-collar household 3.5 units of energy in exchange for
3 units of food—moves the allocation from point G to point H, where both are on higher indifference curves. Points H and I (and all other points where indifference curves are tangent) do
represent allocations that are economically efficient in exchange, because any trade at such a
point would put at least one household on a lower indifference curve.
White-collar consumption is represented on the left and bottom axes, while blue-collar
consumption is represented on the right and top axes, with opposite directions of increasing consumption for each good. This means that white-collar utility increases in a northeast direction, while blue-collar utility increases in a southwest direction.
Point G is on both a white-collar and a blue-collar indifference curve which cross
at point G. Compare point H—it is on two indifference curves also, but the two curves
are tangent at that point, rather than crossing. All points in the Edgeworth box are either like point G or like point H (e.g., point J is like point G, where two indifference
curves cross, while point I is like point H, where two curves are tangent).
Now note that point G cannot represent an exchange efficient allocation of the
two goods, because there are points, such as point H, where both households would
be on higher indifference curves. Thus, if the two households started at point G, they
could gain by exchanging (trading). For example, the white-collar household could
give the blue-collar household 3.5 units of energy in exchange for 3 units of food,
thereby reaching the allocation represented by point H, and both households would
be better off. At an allocation that is economically inefficient in exchange, there are potential
673
16.4 THE EFFICIENCY OF COMPETITIVE MARKETS
exchanges (trades) among consumers that would benefit all consumers. (The inefficiency
corresponds to the fact that these potential benefits are not being realized.)
We have seen that point G does not represent an exchange efficient allocation
(nor, by the same argument, does point J or any other point where indifference curves
cross). Which points, then, do represent exchange efficient allocations? As you might
suspect, exchange efficient allocations are represented by points (such as point H and
point I ) where indifference curves are tangent. Why? Because moving from such a
point would make at least one household worse off (i.e., would move at least one
household to a lower indifference curve). Thus, if the two households had traded as
described above to move from point G to point H, any further trade would hurt at
least one household. At an allocation that is economically efficient in exchange, there are no
potential trades among consumers that would benefit all consumers.
The Contract Curve
Consider the curve that connects all the exchange efficient allocations (i.e., all the
points of tangency) in the Edgeworth box, as shown in Figure 16.15. Such a curve is
called a contract curve. If the two households were free to bargain and make trades
of the two goods, and if all their trades were mutually beneficial, they would bargain
their way to an allocation that was economically efficient in exchange—that is, to
some point on the contract curve. The exact point they would reach would depend on
their starting point (i.e., on the initial allocation of goods). For example, if they started
Food consumed by white-collar household
Total amount of food available
I
H
4
6
K
1
G
1.5
5
Energy consumed by white-collar household
Total amount of energy available
The Contract Curve
The contract curve connects all the allocations in the Edgeworth box that are economically
efficient in exchange—that is, all the points where an indifference curve for the white-collar
household is tangent to an indifference curve for the blue-collar household. (The blue-collar
indifference curve tangent at point K is not shown.)
Food consumed by blue-collar household
Contract curve
0
FIGURE 16.15
that shows all the allocations
of goods in an Edgeworth
box that are economically
efficient.
Energy consumed by blue-collar household
5
0
8.5
10
contract curve A curve
9
10
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CHAPTER 16
G E N E R A L E Q U I L I B R I U M T H E O RY
at point G, they would reach a point on the contract curve between points I and K. It
is easy to see why: Between points I and K, both households are at least as well off as
they are at point G; but below point K on the contact curve, white-collar households
are worse off, while above point I, blue-collar households are worse off.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 6 . 3
D
Checking the Conditions for Exchange Efficiency
Two individuals, Sonia and Anne, together
have 6 apples and 10 pears. Let xS denote the quantity of
apples possessed by Sonia and yS denote the quantity of
pears possessed by Sonia. Similarly, let xA denote the quantity of apples that Anne has and yA denote the quantity of
pears that Anne has. Suppose, further, that for Sonia,
MRS Sonia
x, y
2yS
xS
while for Anne
MRS Anne
x, y
yA
xA
Finally, suppose that Sonia has 4 apples and 2 pears,
while Anne has 2 apples and 8 pears.
Problem
(a) Does the allocation of apples and pears between
Anne and Sonia satisfy the condition of exchange
efficiency?
(b) Can you find an exchange between Sonia and Anne
that makes both parties better off ?
Solution
(a) For this allocation to satisfy the condition of exchange efficiency, the indifference curves of Anne and
Sonia must be tangent to one another. To check whether
the tangency condition holds, we need to compute the
marginal rates of substitution for Sonia and Anne.
When Sonia has 4 apples and 2 pears, her marginal
rate of substitution of apples for pears is
MRS Sonia
x, y
2(2)
1
4
This tells us that Sonia is willing to give up one pear in
order to get one additional apple. Put another way, this
also tells us that Sonia is willing to give up one apple to
get one additional pear.
When Anne has 2 apples and 8 pears, her marginal
rate of substitution of apples for pears is
MRS Anne
x, y
8
4
2
This tells us that Anne is willing to give up 4 pears to get
1 additional apple.
We can see from these calculations that for Sonia
and Anne the marginal rates of substitution of apples for
pears are not equal. Therefore, their indifference curves
are not tangent, and the condition of exchange efficiency
does not hold.
(b) The fact that the existing allocation of apples and
pears is inefficient means that Anne and Sonia can both
be made better off by trading with each other. To see
why, suppose that Anne gives 2 of her pears to Sonia in
exchange for 1 of Sonia’s apples. This makes both individuals better off. To see why, recall that Anne was willing to give up four pears to get one additional apple.
Because she only gives up two pears to get that extra
apple, Anne is better off. What about Sonia? She was
willing to give up one apple to get one additional pear.
Under the proposed deal, Sonia gives up one apple to get
two extra pears. Thus, Sonia is better off as well. There
are other possible trades between Anne and Sonia that
would have made both better off. The key point is that
whenever the condition of exchange efficiency does not
hold, there is always the possibility of a beneficial gain
from trade between individuals in the economy.
Similar Problems:
16.10, 16.11, 16.12, 16.13, 16.14
Does the General Competitive Equilibrium Satisfy Exchange Efficiency?
Consider again the general equilibrium shown in Figure 16.9, where firms supply about
62 units of energy per household and about 49 units of food per household. At the
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16.4 THE EFFICIENCY OF COMPETITIVE MARKETS
equilibrium, a typical white-collar household consumes about 35 units of energy and
39 units of food, a typical blue-collar household consumes about 27 units of energy and
10 units of food, the equilibrium price of energy is $0.79 per unit, and the equilibrium
price of food is $0.70 per unit. Since this is a competitive equilibrium, the marginal rates
of substitution of the two types of households are equal, and each type of household maximizes its utility by setting its marginal rate of substitution equal to the ratio of the equilibrium prices (in the following equations, x denotes energy and y denotes food):
B
MRS W
x, y MRS x, y
Px
$0.79
1.13
Py
$0.70
Since the marginal rate of substitution equals the slope of the household’s indifference
curve, the indifference curves of the two types of households are tangent to one another
and tangent to a line whose slope (in absolute value) equals the ratio of the equilibrium
prices of energy and food. Finally, since the indifference curves are tangent, the allocation of energy and food at the equilibrium must be on the contract curve and must,
therefore, satisfy exchange efficiency. All this is depicted in Figure 16.16, where point E
in the Edgeworth box represents the allocation at the general equilibrium.
Energy consumed by typical blue-collar household
62
27
0
Food consumed by typical
white-collar household
Total amount of food available = 49 units
E
39
0
Contract curve
10
$0.79
$0.70
= – 1.13
slope = –
49
35
Energy consumed by typical white-collar household
62
Total amount of energy available = 62 units
FIGURE 16.16
Exchange Efficiency at the General Competitive Equilibrium
In this Edgeworth box, point E represents the allocation between the typical white-collar household
and the typical blue-collar household at the general equilibrium. At point E, the indifference curves
of the two types of households are tangent to one another and to a line whose slope (in absolute
value) equals the ratio of the equilibrium prices ($0.79 per unit for energy and $0.70 per unit for
food). Since the indifference curves are tangent, point E lies on the contract curve. Thus, at the general competitive equilibrium, there are no unexploited gains from exchanges between households.
Food consumed by typical
blue-collar household
49
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CHAPTER 16
G E N E R A L E Q U I L I B R I U M T H E O RY
Given that point E is on the contract curve, no exchanges between households are
possible that would benefit both types of households. This condition exists despite the
fact that households in this economy did not bargain with each other directly—all
transactions were between households and firms. This shows that, in a competitive
market, the outcome (the general equilibrium) is the same whether consumers bargain
freely and directly or not.
INPUT EFFICIENCY
We have just seen that the general competitive equilibrium results in an allocation of
consumption goods—energy and food—that is economically efficient in exchange.
But what about the allocation of labor and capital that emerges in equilibrium? Does
it satisfy the condition of input efficiency? As in the case of exchange efficiency, we
can draw an Edgeworth box (in this case, for inputs rather than goods) that will help
us answer this question.
Describing Input Efficiency Using the Edgeworth Box
Edgeworth box for
inputs A graph showing
all the possible allocations
of fixed quantities of labor
and capital between the
producers of two different
goods.
input contract curve
A curve that shows all the
input allocations in an
Edgeworth box for inputs
that are input efficienct.
An Edgeworth box for inputs, shown in Figure 16.17, illustrates how fixed quantities of the two inputs, labor and capital, can be allocated between producers of two
different goods—an energy producer and a food producer. The width of the box
shows the total amount of labor available (10 units), while the height of the box shows
the total amount of capital available (also 10 units). Input usage by the energy producer is represented on the left and bottom axes, while input usage by the food producer is represented on the right and top axes, with opposite directions of increasing
use of each input. This means that output by the energy producer increases in a northeast direction, while output by the food producer increases in a southwest direction.
Each point in the box represents one way to allocate all the available labor and capital. For example, at point G, the energy producer uses 1 unit of labor and 6 units of
capital, while the food producer uses 9 units of labor and 4 units of capital. The curves
shown in the box are isoquants for each producer (each isoquant represents the combinations of labor and capital that let the firm produce a given level of output).
An Edgeworth box for inputs has characteristics that are exactly parallel to those
of an Edgeworth box for goods. Thus, every point in the Edgeworth box for inputs in
Figure 16.17 is on two isoquants, an energy producer’s isoquant and a food producer’s
isoquant. At some points (e.g., point G ), the two isoquants cross, while at other points
(e.g., point H ), the two isoquants are tangent to one another. Points where isoquants
cross represent economically inefficient allocations of inputs because at such points it
is possible to reallocate inputs so as to increase output in both industries simultaneously
(e.g., at point G; we could reallocate the inputs to achieve the allocation represented by
point H, where outputs of both energy and food are higher). Points where isoquants
are tangent represent economically efficient allocations of inputs, because no such reallocations are possible (e.g., at point H, any reallocation of inputs that raises output in
one industry will lower it in the other). The input contract curve shown in Figure 16.17
(like the contract curve in Figure 16.15) connects all the economically efficient allocations of inputs (i.e., all the points where isoquants are tangent).
Does the General Competitive Equilibrium Satisfy Input Efficiency?
At a competitive equilibrium, given the prices of labor and capital, firms in each industry use a combination of inputs that minimizes the cost of production. As we saw
in Chapter 7, this implies that the marginal rates of technical substitution for energy
16.4 THE EFFICIENCY OF COMPETITIVE MARKETS
Labor used by food producer
9
10
0
6
4
G
Input contract curve
Capital used by energy producer
Total amount of capital available
Food
producer
Direction
in which
output
increases
Capital used by food producer
Energy
producer
H
I
0
10
1
Labor used by energy producer
Total amount of labor available
FIGURE 16.17 Input Efficiency in the Edgeworth Box
Isoquants for the food producer and the energy producer cross at point G and are tangent
at points H and point I. Point G (and any other point where isoquants cross) does not
represent an allocation of inputs that is economically efficient, because at either point
inputs could be reallocated in a way that would simultaneously increase outputs in both
industries. Points H and I (and all other points where isoquants are tangent) do represent
allocations of inputs that are economically efficient, because reallocation at such points
would decrease output in at least one industry. The input contract curve connects all the
allocations that satisfy input efficiency.
producers (denoted by x) and food producers (denoted by y) are both equal to the ratio
of the price of labor (w) to the price of capital (r):
y
MRTSl,xk MRTSl, k
w
r
Since the marginal rates of technical substitution are the absolute values of the slopes
of the isoquants in energy and food production, and since these slopes are equal at a
competitive equilibrium (where isoquants are tangent), it follows that a general competitive equilibrium satisfies input efficiency. That is, there is no reallocation of inputs
across industries that would allow one industry to increase its output without reducing output in the other industry.
677
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G E N E R A L E Q U I L I B R I U M T H E O RY
Production Possibilities Frontier
The production possibilities frontier shows all the
possible combinations of goods x and y that can
be produced using all the available inputs. Any
point inside the frontier (e.g., point H) is inefficient
because there must be at least one point on the
frontier representing larger quantities of both
goods (e.g., point I). At any given point on the
frontier, the absolute value of the slope is the
marginal rate of transformation of x for y (MRTx,y).
For example, at point I, the slope of the frontier
is –2, so MRTx,y 2, which tells us that producing
one additional unit of good x would mean producing
two fewer units of good y.
Quantity of good y
FIGURE 16.18
I
H
slope = –2, so MRTx,y = 2
0
Quantity of good x
SUBSTITUTION EFFICIENCY
We have seen that a general competitive equilibrium satisfies the conditions of exchange efficiency and input efficiency. Does it also satisfy substitution efficiency?
The Production Possibilities Frontier and the Marginal Rate of Transformation
production possibilities
frontier A curve that
shows all possible combinations of consumption goods
that can be produced in
an economy given the
economy’s available supply
of inputs.
marginal rate of transformation The absolute
value of the slope of the
production possibilities
frontier.
To determine whether the general competitive equilibrium satisfies substitution efficiency, we need to introduce the concept of the production possibilities frontier,
the possible combinations of consumption goods that can be produced in an economy
given the economy’s available supply of inputs. Figure 16.18 shows a production possibilities frontier for an economy with two goods, x and y. When the allocation of inputs across industries satisfies the condition of input efficiency, if more of good x is
produced, less of good y is produced. This is why the production possibilities frontier
is downward sloping. A point such as H, which lies beneath the production possibilities frontier, is inefficient. Indeed, such a combination of outputs could not arise in a
general competitive equilibrium because the equilibrium satisfies input efficiency (i.e.,
with input efficiency, firms producing good x are producing as much output as they
can, given the resources that are devoted to the production of good y, and vice versa).
The slope of the production possibilities frontier shows the amount of good y that
the economy must give up in order to gain one additional unit of good x. We call the
absolute value of the slope of the production possibilities frontier the marginal rate
of transformation of x for y, or MRTx,y. For example, at point I, the slope of the line
tangent to the production possibilities frontier is –2, so the MRTx,y is equal to 2. At
this point, the economy can get one additional unit of good x only by sacrificing two
units of good y. In this sense, the MRTx,y tells us the marginal opportunity cost of
good x in terms of forgone units of good y.
The marginal rate of transformation is equal to the ratio of the marginal costs of
goods x and y: MRTx, y MCx /MCy. To see why, imagine that we want to produce one
additional unit of good x. The incremental cost of the additional resources (capital and
labor) that are needed to produce this extra unit would equal MCx (let’s suppose that
this equals $6). Since the supply of resources in our economy is fixed, we need to take
away $6 worth of resources from the production of good y. If the marginal cost of good
y is currently $3, we would need to reduce our production of good y by two units in
order to free up the $6 worth of resources we need to produce one more unit of good x.
Quantity of food (y)
16.4 THE EFFICIENCY OF COMPETITIVE MARKETS
FIGURE 16.19
slope = –0.79/0.70 = –1.13,
so MRTx,y = 1.13
0
Quantity of energy (x)
679
Production Possibilities Frontier
for Our Simple Economy
In the simple economy whose equilibrium is described
in Figure 16.9, production functions had constant returns to scale, so the production possibilities frontier is
a straight line. The absolute value of its slope equals the
marginal rate of transformation MRTx,y, which equals
the ratio of the marginal costs (MCx /MCy) that arises in
a general equilibrium.
Thus, if the ratio of the marginal costs is MCx /MCy $6/$3 2, the marginal rate of
transformation of x for y will also be 2. This confirms that the marginal rate of transformation equals the ratio of the marginal costs.
In the simple economy whose equilibrium we described in Figure 16.9, every producer had a production function with constant returns to scale, and thus marginal cost
was independent of output. When this is the case, the production possibilities frontier
is a straight line, as shown in Figure 16.19, where MRTx, y 0.79/0.70 1.13, which
is the ratio of the marginal costs that arises in a general equilibrium.
Does the General Competitive Equilibrium Satisfy Substitution Efficiency?
Now we can use the concept of marginal rate of transformation to determine if we have
substitution efficiency at a general competitive equilibrium. Suppose that MRTx, y 1
but for each household in the economy MRSx, y 2. If that were the case, then each
additional unit of energy produced (good x) would require that one fewer unit of food
be produced (good y). However, because MRSx, y 2, each household would be willing to give up 2 units of food to get 1 additional unit of energy. In this case, household
utility would go up if more resources were devoted to energy production and fewer resources were devoted to food production. We can use similar reasoning to show that if
MRTx, y 7 MRSx, y, household utility would go up if fewer resources were devoted to
energy production and more resources were devoted to food production. What we
learn from this analysis is that in order for the competitive equilibrium to satisfy subB
stitution efficiency, it must be the case that MRTx, y MRSW
x, y MRSx, y. Is this condition satisfied at a competitive equilibrium? The answer is yes. Here’s why:
B
• We know that household utility maximization implies that MRSW
x, y MRSx, y
Px /Py.
• We also know that profit maximization by competitive firms implies that price
equals marginal cost in both the energy and food industries—that is, Px MCx
and Py MCy, which therefore means that Px /Py MCx /MCy.
• And as we have just seen, MRTx, y MCx /MCy.
B
Putting these three points together implies MRTx, y MRS W
x, y MRS x, y. That is,
substitution efficiency is satisfied at the general competitive equilibrium.
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A P P L I C A T I O N
G E N E R A L E Q U I L I B R I U M T H E O RY
16.4
Experimental Economics Looks
at Pareto Efficiency
In 2002, Vernon Smith shared the Nobel Prize in Economics with Daniel Kahneman. The Nobel Committee
cited Smith “for having established laboratory experiments as a tool in empirical economic analysis, especially in the study of alternative market mechanisms.”
Smith was a pioneer in the use of laboratory experiments to study economic questions that are difficult
to study in the real world. Because of Smith’s early
studies, experimental economics has grown into an
important subfield of economics, and lab experiments
are now regularly conducted in most areas of economics. In addition, experimental economics has had
important effects on practices in public policy and
business. For example, the economic field of auction
theory is highly abstract, using sophisticated mathematical modeling of advanced game theory. Modeling
the effects of specific rules for designing an auction
can be challenging. However, it can be quite easy to
test how an auction design mechanism works in practice by using lab experiments. The ability to do such
experiments has added to the practical application of
auction theory, enabling economists to provide far
better advice on the design and effects of different
types of auctions.
An example of an economic theory that is difficult to test in the real world is the Pareto efficiency of
markets. Real markets are more complicated than the
simple models used in economic theory. As an alternative to theoretical analysis, one can design experimental markets, letting the participants act in those markets
to see if the results accord with efficiency. Smith became
famous for a series of economic experiments that did
just that.
Smith built on what may have been the first economic experiments, designed by his teacher Edward
Chamberlin.8 Chamberlin set up simple markets for
his students to trade. In his market, each student was
a firm or consumer. Each firm was given a single unit
to sell at a given cost. Each consumer was assigned a
value for one unit of the good. Students went around
the room looking for another student to trade with.
8
When they agreed on a trade, they were removed
from the market. The process continued until no
more students were willing to trade with each other.
That might happen, for example, if all remaining
buyers had values below the costs of all remaining
sellers. Chamberlin found that that kind of market
did not tend to result in Pareto efficiency.
Smith later decided to change the rules of the
experiment in two ways. First, he allowed students to
call out bids to buy or sell their good (what we called
a double auction earlier in the book). This bidding
continued until there were no more students willing
to make new trades. Second, he repeated the experiment the next day. Trading on the second day took
place under the same rules as on the first day.
However, an important difference is that students had
observed the trades that were made on the first day.
Smith wrote, “I am still recovering from the shock of the
experimental results. The outcome was unbelievably
consistent with competitive price theory.” His students
converged to the Pareto efficient outcome predicted by
economic theory. He tried variations on this experiment
and concluded that this finding was robust. The experimental markets converged to competitive equilibrium
with as few as six participants.
In subsequent experiments Smith studied many
other market designs, and found that a wide range of
designs, including ones with several markets that
were linked to each other, tended to reach competitive equilibrium. He also found that if participants in
the experiment had actual financial incentives (so that
they could win or lose money)—as is of course true in
real markets—then a market performed even better.
A comparison of Smith’s results to Chamberlin’s
earlier findings highlights two structural features of
models that allow markets to come to equilibrium
more quickly than in the simpler theoretical models
economists often use. The first feature is information.
Participants in Smith’s market had much more information than those in Chamberlin’s because all bids were
announced publicly. With more information about
their options (the market value of the good they were
trying to sell or the competition they faced for the
good they were trying to buy), buyers and sellers were
more likely to find someone with whom they could
Theodore Bergstrom, “Vernon Smith’s Insomnia and the Dawn of Economics as Experimental Science,”
Scandinavian Journal of Economics 105, no. 1 (2003): 181–205.
16.4 THE EFFICIENCY OF COMPETITIVE MARKETS
profitably transact. Second, Smith’s market was dynamic
because participants made repeated trades. This
allowed them to change their behavior and gave them
additional information about supply and demand.
Experimental economics suggests that simple
markets do tend to come to Pareto efficient equilibrium
681
just as suggested by the theory in this section. The
field also shows that market imperfections, such as
transactions costs and externalities, can cause inefficient market outcomes. Those are the kinds of market
settings that we have discussed in other parts of this
text, such as oligopolistic markets.
P U L L I N G T H E A N A LYS I S TO G E T H E R : T H E
F U N DA M E N TA L T H E O R E M S O F W E L FA R E E C O N O M I C S
In the preceding sections, we saw that the allocation of goods and inputs at a competitive equilibrium satisfies our three criteria for economic efficiency: exchange efficiency, input efficiency, and substitution efficiency. This means that we have just
proven the First Fundamental Theorem of Welfare Economics:
The allocation of goods and inputs that arises in a general competitive equilibrium is
economically efficient. That is, given the resources available to the economy, there is no
other feasible allocation of goods and inputs that could simultaneously make all consumers
better off.
This theorem is remarkable. It tells us that, even though households and firms in
our economy behave independently and each pursues its own self-interest, the resulting equilibrium is efficient in the sense that it exploits all possible mutually beneficial
gains from trade or from the reallocation of inputs. This is the essence of the
“Invisible Hand” argument made by Adam Smith in his famous 1776 treatise, An
Inquiry into the Nature and Causes of the Wealth of Nations.9
Of course, even though the competitive equilibrium outcome is efficient, there is
no guarantee that all consumers fare equally well under the equilibrium. The wellbeing of an individual consumer depends on his or her endowment of scarce economic
resources. For example, we saw that in the equilibrium in Figure 16.9, white-collar
households (which supply capital) fared better than blue-collar households (which supply labor) because white-collar households owned the factor of production—capital—
that was scarcer and in more demand by producers. Had the pattern of ownership of
scarce inputs in the economy been different, the equilibrium distribution of income
and utility would have been different.
Figure 16.20 illustrates this point with a curve called the utility possibilities frontier,
which connects all the possible combinations of utilities that could arise at the various
economically efficient allocations of goods and inputs in a simple two-consumer economy. At point E, for example, the typical white-collar household enjoys greater utility
than the typical blue-collar household, while at point F, the distribution of utility is
more equal.
Could a social planner with the power to redistribute ownership of scarce resources do so in such a way as to create a general competitive equilibrium corresponding to any arbitrary point along the utility possibilities frontier? For example, could a
social planner in our two-consumer economy redistribute the available stock of labor
and capital in order to create a general equilibrium with the equal distribution of utility
9
Adam Smith, An Inquiry into the Nature and Causes of the Wealth of Nations, printed for W. Strahan and
T. Cadell, London, 1776.
First Fundamental
Theorem of Welfare
Economics The allocation of goods and inputs
that arises in a general
competitive equilibrium is
economically efficient—that
is, given the resources available to the economy, there
is no other feasible allocation
of goods and inputs that
could simultaneously make
all consumers better off.
utility possibilities
frontier A curve that
connects all the possible
combinations of utilities
that could arise at the
various economically
efficient allocations of
goods and inputs in a
two-consumer economy.
CHAPTER 16
G E N E R A L E Q U I L I B R I U M T H E O RY
FIGURE 16.20 The Utility
Possibilities Frontier
The utility possibilities frontier connects all the possible combinations
of utilities at economically efficient
allocations of goods and inputs.
Point F represents a more equitable
distribution of utility than point
E does.
Second Fundamental
Theorem of Welfare
Economics Any economically efficient allocation
of goods and inputs can be
attained as a general competitive equilibrium through
a judicious allocation of the
economy’s scarce supplies of
resources.
16.5
GAINS FROM
FREE TRADE
Utility of typical
blue-collar household
682
F
E
0
Utility of typical
white-collar household
corresponding to point F in Figure 16.20? The Second Fundamental Theorem of
Welfare Economics says that the answer to these questions—at least, in theory—
is yes:
Any economically efficient allocation of goods and inputs can be attained as a general
competitive equilibrium through a judicious allocation of the economy’s scarce supplies of
resources.
The significance of this theorem is that it reveals the possibility that an economy
could simultaneously attain an efficient allocation and one in which the resulting distribution of utility is in some sense equitable, or fair. However, this is by no means
easy to accomplish. As we saw in Chapter 10, most of the feasible mechanisms for redistributing wealth in a democratic society (e.g., taxes and subsidies) are themselves
costly—that is, they usually distort economic decisions and impair efficiency. Thus,
even though the goals of equity and efficiency are compatible with each other in theory,
in practice many public policy choices entail a trade-off between equity and efficiency,
as we saw in our analysis of public policy interventions in Chapter 10.
In our analysis of exchange efficiency in the previous section, we saw how trade
among individuals can make all individuals better off. In this section, we will see that
trade among countries can make all countries better off. This is the case even when
one country is unambiguously more efficient in producing everything than another
country.
F R E E T R A D E I S M U T UA L LY B E N E F I C I A L
To show that unrestricted free trade can benefit two countries, let’s consider a simple
example in which two countries—the United States and Mexico—can each produce
two goods: computers and clothing. For simplicity, let’s assume that each country produces these products with a single input: labor. Table 16.1 shows how many hours of
labor are required to produce each good.
683
16.5 GAINS FROM FREE TRADE
TABLE 16.1
Labor Requirements in the United States and Mexico
Computers (labor-hours per unit)
Clothing (labor-hours per unit)
10
60
5
10
United States
Mexico
20
6
0
(a) United States
FIGURE 16.21
Quantity of clothing (units per week)
Quantity of clothing (units per week)
For example, Table 16.1 says that in the United States it takes 10 labor-hours to
produce 1 computer, while in Mexico it takes 60 labor hours to produce that computer. Similarly, in the United States it takes 5 labor-hours to produce 1 unit of clothing, while in Mexico it takes 10 labor-hours to do so. Notice that Table 16.1 implies
that U.S. workers are more productive in both computer and clothing production
than their Mexican counterparts since it takes fewer U.S. labor-hours to make a unit
of either product.
Let’s assume that, in each country, there are 100 available labor-hours each week.
With the numbers in Table 16.1, we can draw the production possibilities frontiers for
the United States and Mexico. These are shown in Figure 16.21. For the United
States, the marginal rate of transformation of computers for clothing is 10/5, or 2.
This is because for every additional computer that is produced, 10 additional laborhours are required. With the supply of labor fixed, these 10 labor-hours would have
to be diverted from clothing production, which then means that 2 fewer units of
clothing can be produced. Put another way, in the United States, the opportunity cost
slope = –2, so
MRTcomputers, clothing = 2
H
7
Quantity of computers
(units per week)
10
20
slope = –6, so
MRTcomputers, clothing = 6
10
I
4
0
1
10
Quantity of computers
(units per week)
(b) Mexico
Production and Consumption in the United States and Mexico:
No Trade Situation
The straight line in panel (a) is the production possibilities frontier for the United States, while
the straight line in panel (b) is the production possibilities frontier for Mexico. If the countries
do not trade, U.S. consumers consume as many computers and units of clothing as U.S. producers
produce. Point H depicts this outcome. Similarly, without trade, Mexican consumers consume
as many computers and units of clothing as Mexican producers produce. Point I depicts this
outcome.
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CHAPTER 16
G E N E R A L E Q U I L I B R I U M T H E O RY
TABLE 16.2
Production and Consumption under No Trade
United States
Mexico
Total
Computers (units)
Clothing (units)
7
1
8
6
4
10
of one additional unit of clothing is 1/2 computer. By contrast, for Mexico, the marginal
rate of transformation of computers for clothing is 60/10 6. The opportunity cost
of one additional computer is 6 units of clothing, while the opportunity cost of one
additional unit of clothing is 10/60 or 1/6 of a computer.
Now suppose initially that there is no trade between the United States and Mexico.
Suppose, further, that 70 U.S. labor-hours are devoted to computer production, while
the remaining 30 are devoted to clothing production. As shown in Figure 16.21(a), this
implies that the U.S. economy operates at point H on its production possibilities
frontier: The U.S. economy produces—and U.S. consumers consume—7 computers and
6 units of clothing per week.10 We will assume that this combination of computers and
clothing is efficient for the U.S. economy.
Let’s suppose that in Mexico, 60 out of the 100 available labor-hours are devoted
to computer production, while the remaining 40 labor-hours are devoted to clothing
production. As Figure 16.21(b) shows, this means that the Mexican economy operates
at point I on its production possibilities frontier. At this point, the Mexican economy
produces—and Mexican consumers consume—1 computer and 4 units of clothing.
Let’s suppose that this outcome is efficient for Mexican consumers. Table 16.2 summarizes the situation for consumers in the United States and Mexico.
We will now see that the two countries can do better by trading with each other.
Suppose that the United States specializes in computer production, devoting all 100
hours of its available labor to that activity. Suppose, too, that Mexico specializes in the
production of clothing by devoting all 100 of its labor-hours to clothing production.
Table 16.3 shows the total production of the two countries under this situation, and
these outcomes are depicted by points J and K in Figure 16.22.
TABLE 16.3
Production under Free Trade
United States
Mexico
Total
10
Computers (units)
Clothing (units)
10
0
10
0
10
10
To see why, note that since each computer requires 10 hours of labor, the United States can produce
70 hours per week/10 hours per units 7 units per week if it devotes 70 hours a week to computer
production. Further, since each unit of clothing requires 5 hours of labor, the United States can produce
30 hours per week/5 hours per units 6 units per week if it devotes 30 hours a week to clothing
production.
685
20
L
6
H
J
0
7
8
Quantity of computers
(units per week)
(a) United States
Quantity of clothing (units per week)
Quantity of clothing (units per week)
16.5 GAINS FROM FREE TRADE
20
10
K
4
I
0
10
M
10
2
Quantity of computers
(units per week)
(b) Mexico
FIGURE 16.22
Production and Consumption in the United States and Mexico: Free Trade
Under free trade, the United States produces 10 computers and no units of clothing (point J),
while Mexico specializes in the production of clothing, making no computers and 10 units of
clothing (point K). The United States then trades 2 computers for 6 units of clothing. This
allows U.S. consumers to consume 8 computers and 6 units of clothing (point L), while Mexican
consumers consume 2 computers and 4 units of clothing (point M). Free trade makes consumers
in both countries better off than they were before.
Now suppose that the United States ships 2 computers per week to Mexico in
exchange for 6 units of clothing per week. This means that total consumption in both
countries is as shown in Table 16.4.
Trade makes both countries better off. Both countries consume just as many units
of clothing as before, but each country now has more computers. As Figure 16.22
shows, the specialization of production coupled with free trade allows each country to
consume “outside” its production possibilities frontier. Thus, when trade between two
countries is allowed, both countries can expand their consumption of some goods
without reducing their consumption of other goods.
Of course, in practice, not all consumers in the economy benefit equally from the
increased consumption opportunities made possible by free trade. In our example, the
United States produces less clothing under the free trade regime than it did in the
absence of trade. Workers whose skills are specialized to the textile industry might
experience reduced wages or even job losses if trade with Mexico were to commence.
Thus, even though the U.S. economy benefits in the aggregate from free trade, those
gains are not shared equally, at least in the short run, by all consumers in the economy.
TABLE 16.4
Consumption under Free Trade
United States
Mexico
Total
Computers (units)
Clothing (units)
8
2
10
6
4
10
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G E N E R A L E Q U I L I B R I U M T H E O RY
C O M PA R AT I V E A DVA N TAG E
comparative
advantage One country
has a comparative advantage over another country
in the production of good x
if the opportunity cost of
producing an additional
unit of good x —expressed
in terms of forgone units
of some other good y —is
lower in the first country
than in the second country.
absolute advantage
One country has an
absolute advantage over
another country in the
production of a good x if
production of one unit of x
in the first country requires
fewer units of a scarce
input (e.g., labor) than it
does in the second country.
The beneficial effect of free trade is a consequence of a very important idea in microeconomics: comparative advantage. One country (say Mexico) has a comparative
advantage over another (say, the United States) in the production of good x if the
opportunity cost of producing an additional unit of good x (e.g., clothing)—expressed
in terms of forgone units of good y (e.g., computers)—is lower in the first country
than it is in the second country. In our example, Mexico has a comparative advantage
over the United States in the production of clothing because, as we saw above, the
opportunity cost of 1 additional unit of clothing produced in Mexico is 1/6 of a computer, while the opportunity cost of 1 additional unit of clothing produced in the
United States is 1/2 of a computer.
By the same token, the United States has a comparative advantage over Mexico in
the production of computers because producing 1 additional computer in the United
States requires a sacrifice of 2 units of clothing, while 1 additional computer produced
in Mexico requires a sacrifice of 6 units of clothing.
Comparative advantage should be contrasted with absolute advantage. One
country has an absolute advantage over another country in the production of good x
if production of x in the first country requires fewer units of a scarce input (e.g., labor)
than it does in the second country. In our example, the United States has an absolute
advantage over Mexico in the production of both computers and clothing.
Nevertheless, the United States benefits from free trade with Mexico, because the benefits from free trade are determined by comparative advantage rather than absolute advantage. In general, starting from a situation in which two countries are not trading
with each other, two countries can make themselves better off by trading when each
country specializes in the production of goods for which it has a comparative advantage. Thus, in the previous section, we saw that when Mexico specializes in clothing
production (its comparative advantage) while the United States specializes in computer
production (its comparative advantage), both countries can end up strictly better off
through free trade.
A P P L I C A T I O N
16.5
Gains from Free Trade
Since the end of World War II, there has been a longterm trend toward reduction in barriers to international trade. In 1948 a General Agreement on Tariffs
and Trade (GATT) was signed by many nations. Over
the years the GATT process was used to negotiate and
implement treaties between nations to reduce tariffs
and quotas on imports, subsidies to domestic industries, and other barriers to free trade. GATT was
replaced in 1995 by the World Trade Organization
(WTO), which continued this process. The latest round
of these negotiations is the Doha Round, named after
the original meeting in Doha, Qatar in 2001. The
Doha Round represents an attempt to further reduce
barriers to trade, including services and labor.
However, these negotiations broke down in 2008 over
disagreements on issues such as reductions of agricultural subsidies and finding ways to make pharmaceuticals more available in developing nations (e.g., by
allowing production of generics in those nations).
Developing nations are often reluctant to eliminate
import protections for their domestic manufacturing
industries. As of 2010 it is not clear whether negotiations will move forward substantially any time soon.
Nevertheless, over the last 60 years there has been a
gradual, but significant, reduction in barriers to trade
worldwide.
C H A P T E R S U M M A RY
The movement toward free trade continues to be
controversial. Many protests have erupted in cities
conducting meetings of the Doha Round. Protestors
are concerned that reducing barriers to trade may increase poverty in developing nations and harm the
environment. Many fear that decreased protectionism
could cause unemployment as jobs move to other
countries. The evidence from previous free trade
agreements, however, suggests that many of these
fears might be unfounded. For example, many people
expected the North Atlantic Free Trade Agreement
(NAFTA) to cause unemployment in the United States
as manufacturing jobs moved to Mexico, where labor
is much less expensive. However, aggregate unemployment did not increase after implementation of NAFTA.
World Bank economist Kym Anderson employed
a general equilibrium model of the world economy to
estimate the effects of implementation of the proposed Doha Round reductions in barriers to trade.11
He concluded that the gains from freer trade would
be enormous. Under his pessimistic scenario (only a
25 percent reduction in trade barriers and agricultural
subsidies), the present value of the benefits net of
costs from 2010 through 2050 is estimated to be
about $13.4 trillion (calculated in 2010 dollars). In a
more optimistic scenario (50 percent reduction in
barriers and subsidies), Anderson estimates the present
value of benefits net of costs to be about $26.8 trillion. Roughly half of the benefits of trade liberalization would accrue to developing nations. A large part
of the gains to developing nations would be realized
in agricultural and textile industries because those
tend to have high barriers to trade. The poorest workers
687
in developing nations, who are employed disproportionately in those industries, would probably enjoy disproportionate gain. Implementation of the Doha
Round might therefore lead to significant reductions in
poverty and hunger, with corresponding improvements
in nutrition, health, medical care, and education.
The effects of reducing barriers to trade on the
environment are difficult to measure. However, there
are good reasons to expect that the net effect might
be positive in the long run. Many environmental
problems are caused by poverty (e.g., slash-and-burn
agriculture) and by industrialization of poor nations.
However, the experience of the twentieth century has
shown that as nations become richer, their citizens demand cleaner environments and tend to adopt policies designed to alleviate environmental problems. In
many developed nations, environmental qualities indicators (for example, for air or water pollution) are
now improving. Reduced trade barriers may significantly increase wealth, creating resources that could
be used to address environmental problems.
In one recent evaluation of proposed solutions to
various world problems, the Doha Round was ranked
as the second best policy for improving welfare, behind policies that would provide vitamin supplements
to malnourished children worldwide.12 Estimates
from general equilibrium analysis suggest that essentially all countries involved might benefit from reducing trade barriers. Why, then, do nations resist negotiations agreements such as the Doha Round? One
concern is that the elimination of trade barriers
would cause short-term adjustments, leaving some
workers worse off as their industries lose protection.
CHAPTER SUMMARY
• Partial equilibrium analysis studies the determination of price and output in a single market, taking as
given the prices in all other markets. By contrast, general equilibrium analysis studies the determination of
price and output in more than one market at the same
time. (LBD Exercise 16.1)
11
• An exogenous event that tends to decrease the price
of one good will also tend to decrease the prices of substitute goods. Thus, the prices of substitute goods will
tend to be positively correlated. By contrast, an exogenous event that tends to decrease the price of one good
will tend to increase the prices of complementary goods.
Kym Anderson, “Subsidies and Trade Barriers.” Copenhagen Consensus Report, 2004.
Copenhagen Consensus, 2008. The Copenhagen Consensus Center is a think-tank in Denmark that offers
suggestions about the best ways for governments and philanthropists to fund aid and development.
12
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CHAPTER 16
G E N E R A L E Q U I L I B R I U M T H E O RY
Thus, the prices of complementary goods will tend to be
negatively correlated.
• In a general equilibrium, demand for finished products comes from utility maximization by households,
while demand for inputs comes from cost minimization
by firms. The supply of finished products comes from
profit maximization by firms, while the supply of inputs
comes from profit maximization by households.
• In a general equilibrium, the prices of all goods are
determined simultaneously by supply-equals-demand
conditions in every market. (LBD Exercise 16.2)
• Walras’ Law tells us that a general equilibrium determines the prices of goods and inputs relative to the price
of one of the goods or inputs, rather than determining
the absolute levels of all prices.
• To determine the general equilibrium effects of an
excise tax on a particular good, we need to analyze the
impact of the tax on all markets in the economy, taking
into account the interdependencies that exist among
those markets.
• An allocation of goods and inputs is economically
efficient if there is no other feasible allocation of goods and
inputs that would make some consumers better off without hurting other consumers. By contrast, an allocation of
goods and inputs is economically inefficient if there is an
alternative feasible allocation of goods and inputs that
would make all consumers better off as compared with the
initial allocation.
• Economic efficiency requires exchange efficiency,
input efficiency, and substitution efficiency. (LBD
Exercise 16.3)
• All three efficiency conditions are satisfied at a general competitive equilibrium. This result is known as the
First Fundamental Theorem of Welfare Economics.
• The Second Fundamental Theorem of Welfare
Economics says that any economically efficient allocation of goods and inputs can be attained as a general
competitive equilibrium through a judicious allocation
of the economy’s scarce supplies of resources.
• Free trade between two countries can make both countries better off than they would be in the absence of trade.
• A country has a comparative advantage over another
in the production of a good if the opportunity cost of producing an additional unit of that good, expressed in terms
of forgone units of another good, is lower in the first
country than in the second country. Gains from free trade
are realized when countries specialize in the production of
goods for which they have a comparative advantage.
REVIEW QUESTIONS
1. What is the difference between a partial equilibrium
analysis and a general equilibrium analysis? When analyzing the determination of prices in a market, under what
circumstances would a general equilibrium analysis be
more appropriate than a partial equilibrium analysis?
2. In a general equilibrium analysis with two substitute goods, X and Y, explain what would happen to the
price in market X if the supply of good Y increased (i.e.,
if the supply curve for good Y shifted to the right).
How would your answer differ if X and Y were complements?
3. What role does consumer utility maximization play in
a general equilibrium analysis? What is the role played by
firm cost minimization in a general equilibrium analysis?
6. What is exchange efficiency? In an Edgeworth box
diagram, how do efficient allocations and inefficient
allocations differ?
7. How does exchange efficiency differ from input
efficiency? Could an economy satisfy the conditions for
exchange efficiency but not the conditions for input
efficiency?
8. Suppose an economy has just two goods, X and Y.
True or False: If the condition of input efficiency prevails,
we can increase the production of X without decreasing
the production of Y. Explain your answer.
What is Walras’ Law? What is its significance?
9. What is the production possibilities frontier? What
is the marginal rate of transformation? How does the
marginal rate of transformation relate to the production
possibilities frontier?
5. What is an economically efficient allocation? How
does an economically efficient allocation differ from an
inefficient allocation?
10. Explain how consumers in an economy can be made
better off if the marginal rate of transformation does not
equal consumers’ marginal rates of substitution.
4.
689
PROBLEMS
11. Explain how the conditions of utility maximization,
cost minimization, and profit maximization in competitive markets imply that the allocation arising in a general
competitive equilibrium is economically efficient.
12. What is comparative advantage? What is absolute
advantage? Which of these two concepts is more important in determining the benefits from free trade?
PROBLEMS
16.1. Consider the markets for butter (B) and mard
garine (M ), where the demand curves are Q M
⫽ 20 ⫺
d
2PM ⫹ PB and Q B ⫽ 60 ⫺ 6PB ⫹ 4PM and the supply
s
⫽ 2PM and Q Bs ⫽ 3PB.
curves are Q M
a) Find the equilibrium prices and quantities for butter
and margarine.
b) Suppose that an increase in the price of vegetable oil
s
shifts the supply curve of margarine to Q M
⫽ PM. How
does this change affect the equilibrium prices and quantities for butter and margarine? Using words and graphs,
explain why a shift in the supply curve for margarine
would change the price of butter.
16.2. Suppose that the demand curve for new automobiles is given by Q Ad ⫽ 20 ⫺ 0.7PA ⫺ PG, where QA and
PA are the quantity (millions of vehicles) and average
price (thousands of dollars per vehicle), respectively, of
automobiles in the United States, and PG is the price
of gasoline (dollars per gallon). The supply of automobiles is given by Q sA ⫽ 0.3PA. Suppose that the demand
and supply curves for gasoline are Q dG ⫽ 3 ⫺ PG and
Q sG ⫽ PG.
a) Find the equilibrium prices of gasoline and automobiles.
b) Sketch a graph that shows how an exogenous increase
in the supply of gasoline affects the prices of new cars in
the United States.
16.3. Studies indicate that the supply and demand
schedules for ties (t) and jackets ( j ) in a market are as
follows:
Demand for ties:
Supply of ties:
Demand for jackets:
Supply of jackets:
Q dt ⫽ 410 ⫺ 5Pt ⫺ 2Pj
Q st ⫽ ⫺60 ⫹ 3Pt
Q dj ⫽ 295 ⫺ Pt ⫺ 3Pj
Q sj ⫽ ⫺120 ⫹ 2Pj
The estimates of the schedules are valid only for prices at
which quantities are positive.
a) Find the equilibrium prices and quantities for ties and
jackets.
b) Do the demand schedules indicate that jackets and ties
are substitute goods, complementary goods, or independent goods in consumption? How do you know?
16.4. Suppose that the demand for steel in Japan is
given by the equation Q dS ⫽ 1200 ⫺ 4PS ⫹ PA ⫹ PT,
where QS is the quantity of steel purchased (millions of
tons per year), PS is the price of steel (yen per ton), PA
is the price of aluminum (yen per ton), and PT is the
price of titanium (yen per ton). The supply curve for
steel is given by Q sS ⫽ 4PS. Similarly, the demand and
supply curves for aluminum and for titanium are given
by Q dA ⫽ 1200 ⫺ 4PA ⫹ PS ⫹ PT (demand curve for
aluminum), Q sA ⫽ 4PA (supply curve for aluminum),
Q dT ⫽ 1200 ⫺ 4PT ⫹ PS ⫹ PA (demand curve for titanium), and Q sT ⫽ 4PT (supply curve for aluminum).
a) Find the equilibrium prices of steel, aluminum, and
titanium in Japan.
b) Suppose that a strike in the Japanese steel industry
shifts the supply curve for steel to Q sS ⫽ PS. What does
this do to the prices of steel, aluminum, and titanium?
c) Suppose that growth in the Japanese beer industry,
a big buyer of aluminum cans, fuels an increase in the
demand for aluminum so that the demand curve for
aluminum becomes Q dA ⫽ 1500 ⫺ 4PA ⫹ PS ⫹ PT.
How does this affect the prices of steel, aluminum, and
titanium?
16.5. Consider a simple economy that produces two
goods, beer (denoted by x) and quiche (denoted by y),
using labor and capital (denoted by L and K, respectively) that are supplied by two types of households,
those consisting of wimps (denoted by W ) and those
consisting of hunks (denoted by H ). Each household of
hunks supplies 100 units of labor and no units of capital.
Each household of wimps supplies 10 units of capital and
no units of labor. There are 100 households of each type.
Both beer and quiche are produced with technologies
exhibiting constant returns to scale. The market supply
curves for beer and quiche are
1
5
3
1
Px ⫽ w6r 6
Py ⫽ w4r 4
where w denotes the price of labor and r denotes the
price of capital. The market demand curves for beer and
quiche are given by
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CHAPTER 16
Px
20IW 90IH
X
Py
80IW 10IH
Y
G E N E R A L E Q U I L I B R I U M T H E O RY
d) In equilibrium how will the income of each whitecollar family compare with the income of each blue-collar
family?
where X and Y denote the aggregate quantities of beer
and quiche demanded in this economy and IW and IH are
the household incomes of wimps and hunks, respectively. Finally, the market demand curves for labor and
capital are given by
3Y r 4
X r 6
L a b
a b
4 w
6 w
5
1
5X w 6
Y w 4
a b a b
4 r
6 r
1
K
3
There are four unknowns in our simple economy:
the prices of beer and quiche, Px and Py, and the prices
of labor and capital, w and r. Write the four equations
that determine the equilibrium values of these unknowns.
16.6. In an economy, there are 40 “white-collar”
households, each producing 10 units of capital (and no
labor); the income from each unit of capital is r. There
are also 50 “blue-collar” households, each producing
20 units of labor (and no capital); the income from each
unit of labor is w.
Each white-collar household’s demand for energy is
XW 0.8MW /PX, where MW is income in the household. Each white-collar household’s demand for food is
YW 0.2MW /PY.
Each blue-collar household’s demand for energy is
XB 0.5MB /PX, where MB is income in the household.
Each blue-collar household’s demand for food is YB
0.5MB /PY.
Energy is produced using only capital. Each unit
of capital produces one unit of energy, so r is the marginal cost of energy. The supply curve for energy is
described by PX r, where PX is the price of a unit of
energy. Food is produced using only labor. Each unit
of labor produces one unit of food, so w is the marginal cost of food. The supply curve for labor is described by PY w, where PY is the price of a unit of
food.
a) In this economy, show that the amount of labor demanded and supplied will be 1,000 units. Show also that
the amount of capital demanded and supplied will be
400 units.
b) Write down the supply-equals-demand conditions for
the energy and food markets.
c) In equilibrium how will the price of a unit of energy
compare with the price of a unit of food?
16.7. One of the implications of Walras’ Law is that
the ratios of prices (rather than the absolute levels of
prices) are determined in general equilibrium. In
Learning-By-Doing Exercise 16.2, show that price labor
will be 25
52 ⬇ 0.48 of the price of capital, as illustrated in
Figure 16.9.
16.8. One of the implications of Walras’ Law is that
the ratios of prices (rather than the absolute levels of
prices) are determined in general equilibrium. In
Learning-By-Doing Exercise 16.2, show that the ratio of
the price of energy to the price of capital is about 0.79,
as illustrated in Figure 16.9.
16.9. One of the implications of Walras’ Law is that
the ratios of prices (rather than the absolute levels of
prices) are determined in general equilibrium. In
Learning-By-Doing Exercise 16.2, show that the ratio of
the price of food to the price of capital is about 0.7, as
illustrated in Figure 16.9.
16.10. Two consumers, Josh and Mary, together have
10 apples and 4 oranges.
a) Draw the Edgeworth box that shows the set of feasible
allocations that are available in this simple economy.
b) Suppose Josh has 5 apples and 1 orange, while Mary
has 5 apples and 3 oranges. Identify this allocation in the
Edgeworth box.
c) Suppose Josh and Mary have identical utility functions,
and assume that this utility function exhibits positive
marginal utilities for both apples and oranges and a diminishing marginal rate of substitution of apples for
oranges. Could the allocation in part (b)—5 apples and 1
orange for Josh; 5 apples and 3 oranges for Mary—be
economically efficient?
16.11. Ted and Joe each consume peaches, x, and
plums, y. The consumers have identical utility functions,
with MRS x,Joey 10yJ/xJ, MRS Ted
x, y 10yT/xT. Together,
they have 10 peaches and 10 plums. Verify whether each
of the following allocations is on the contract curve:
a) Ted: 8 plums and 9 peaches; Joe: 2 plums and 1 peach.
b) Ted: 1 plum and 1 peach; Joe: 9 plums and 9 peaches.
c) Ted: 4 plums and 3 peaches; Joe: 6 plums and 7
peaches.
d) Ted: 8 plums and 2 peaches; Joe: 2 plums and 8
peaches.
16.12. Two consumers, Ron and David, together own
1,000 baseball cards and 5,000 Pokémon cards. Let xR
denote the quantity of baseball cards owned by Ron and
PROBLEMS
yR denote the quantity of Pokémon cards owned by Ron.
Similarly, let xD denote the quantity of baseball cards
owned by David and yD denote the quantity of Pokémon
cards owned by David. Suppose, further, that for Ron,
D
MRSRx, y yR/xR, while for David, MRS x,
y yD /2xD.
Finally, suppose xR 800, yR 800, xD 200, and yD
4,200.
a) Draw an Edgeworth box that shows the set of feasible
allocations in this simple economy.
b) Show that the current allocation of cards is not economically efficient.
c) Identify a trade of cards between David and Ron that
makes both better off. (Note: There are many possible
answers to this problem.)
16.13. There are two individuals in an economy, Joe
and Mary. Each of them is currently consuming positive
amounts of two goods, food and clothing. Their preferences are characterized by diminishing marginal rate of
substitution of food for clothing. At the current consumption baskets, Joe’s marginal rate of substitution of
food for clothing is 2, while Mary’s marginal rate of substitution of food for clothing is 0.5. Do the currently
consumed baskets satisfy the condition of exchange efficiency? If not, describe an exchange that would make
both of them better off.
16.14. Consider an economy that consists of three individuals: Maureen (M), David (D), and Suvarna (S). Two
goods are available in the economy, x and y. The marginal rates of substitution for the three consumers are
2yM /xM, MRS David
2yD /xD,
given by MRS Maureen
x, y
x, y
Suvarna
yS /xS. Maureen and David are both
and MRS x, y
consuming twice as much of good x as good y, while
Suvarna is consuming equal amounts of goods x and y.
Are these consumption patterns economically efficient?
16.15. Two firms together employ 100 units of labor
and 100 units of capital. Firm 1 employs 20 units of
labor and 80 units of capital. Firm 2 employs 80 units of
labor and 20 units of capital. The marginal products of
the firms are as follows: Firm 1: MP1l 50, MP1k 50;
Firm 2: MP2l 10, MP2k 20. Is this allocation of inputs economically efficient?
16.16. There are two firms in an economy. Each of
them currently employs positive amounts of two inputs,
capital and labor. Their technologies are characterized
by diminishing marginal rate of technical substitution of
labor for capital. At the current operating basket, Firm
A’s marginal rate of technical substitution of labor for
capital is 3, while Firm B’s marginal rate of technical
substitution of labor for capital is 1. Do the current production baskets satisfy the condition of input efficiency?
If not, describe an exchange of inputs that would improve efficiency.
691
16.17. Two firms together employ 10 units of labor (l)
and 10 units of capital (k). The marginal rate of technical substitution of each firm is given by: MRTS1lk k1/l1
2
4k2/l2. Which of the following input
and MRTS lk
allocations satisfy the condition of input efficiency?
a) Firm 1 uses 5 units of labor, 5 units of capital; Firm 2
uses 5 units of labor, 5 units of capital.
b) Firm 1 uses 5 unit of labor, 8 units of capital; Firm 2
uses 5 units of labor; 2 units of capital.
c) Firm 1 uses 9 units of labor, 9 units of capital; Firm 2
uses 1 unit of labor; 1 unit of capital.
d) Firm 1 uses 2 units of labor; 5 units of capital; Firm 2
uses 8 units of labor; 5 units of capital.
16.18. Two firms together employ 20 units of labor
and 12 units of capital. For Firm 1, which uses 5 units of
labor and 8 units of capital, the marginal products of
labor and capital are MPl1 20 and MP1k 40. For Firm 2,
which uses 15 units of labor and 4 units of capital, the
marginal products are MP l2 60 and MP 2k 30.
a) Draw an Edgeworth box for inputs that shows the
allocation of inputs across these two firms.
b) Is this allocation of inputs economically efficient?
Why or why not? If it is not, identify a reallocation of
inputs that would allow both firms to increase their
outputs.
16.19. Consider an economy that produces two
goods: food, x, and clothing, y. Production of both
goods is characterized by constant returns to scale.
Given current input prices, the marginal cost of producing clothing is $10 per unit, while the marginal cost of
producing food is $20 per unit. What is the marginal
rate of transformation of x for y? How much clothing
must the economy give up in order to get one additional
unit of food?
16.20. An economy consists of two consumers ( Julie
and Carina), each consuming positive amounts of two
goods, food and clothing. Food and clothing are both
produced with two inputs, capital and labor, using
technologies exhibiting constant returns to scale. The
following information is known about the current consumption and production baskets: The marginal cost
of producing food is $2, and the price of clothing is $4.
The wage rate is 2/3 the rental price of capital, and
the marginal product of capital in producing clothing is
3. In a general competitive equilibrium, what must be
a) The price of food?
b) The marginal rate of transformation of food for
clothing?
c) The shape of the production possibilities frontier for
the economy?
d) The marginal product of labor in producing clothing?
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G E N E R A L E Q U I L I B R I U M T H E O RY
16.21. Consider an economy that uses labor and capital
to produce two goods, beer (x) and peanuts ( y), subject to
technologies that exhibit constant returns to scale. The
marginal cost of a 12-ounce can of beer is $0.50. The
marginal cost of a 12-ounce tin of peanuts is $1.00.
Currently, the economy is producing 1 million 12-ounce
cans of beer and 2 million 12-ounce tins of peanuts. The
marginal rates of technical substitution of labor for capital
in the beer and peanut industries are the same. Moreover,
there are 1 million identical consumers in the economy,
each with a marginal rate of substitution of beer for
peanuts given by MRSx, y 3y/x.
a) Sketch a graph of the economy’s production possibilities frontier. Identify the economy’s current output on
this graph.
b) Does the existing allocation satisfy substitution efficiency? Why or why not?
16.22. The United States and Switzerland both produce
automobiles and watches. The labor required to produce a
unit of each product is shown in the following table:
a) Which country has an absolute advantage in the production of watches? In the production of automobiles?
b) Which country has a comparative advantage in the
production of watches? in the production of automobiles?
16.23. Brazil and China can produce cotton and soybeans. The labor required to produce a unit of each
product is shown in the following table:
Brazilian and Chinese Labor Requirements
Cotton (labor-hours Soybeans (labor-hours
per unit)
per unit)
China
Brazil
20
10
100
80
a) Which country has an absolute advantage in the production of cotton? In the production of soybeans?
b) Which country has a comparative advantage in the
production of cotton? In the production of soybeans?
U.S. and Swiss Labor Requirements
Automobiles (labor- Watches (laborhours per unit)
hours per unit)
United States
Switzerland
5
20
50
60
A P P E N D I X : Deriving the Demand and Supply Curves for General Equilibrium in
Figure 16.9 and Learning-By-Doing Exercise 16.2
Recall that the simple economy in Figure 16.9 and Learning-By-Doing Exercise 16.2
has the following characteristics: There are 100 blue-collar households (B) and 100
white-collar households (W ); two goods—energy (x) and food ( y), each produced by
100 firms that specialize in that good (i.e., 100 energy producers and 100 food producers); and two inputs—labor (l ) and capital (k). The total amount of energy produced by all energy producers together is X, and the total amount of food produced
by all food producers together is Y.
In this appendix we will derive the demand and supply curves for this economy, as
depicted in Figure 16.9 and given in equation form in Learning-By-Doing Exercise 16.2.
These derivations are based on the following utility functions and production functions:
1
1
Utility function for white-collar household:
U W (x, y) x 2y 2
Utility function for blue-collar household:
U B (x, y) x 4 y 4
Production function for energy producer:
x 1.89l 3k 3
Production function for food producer:
3
1
1
1
y 2l 2k 2
2
1
693
A P P E N D I X : D E R I V I N G T H E D E M A N D A N D S U P P LY C U RV E S
DERIVING THE HOUSEHOLD AND MARKET DEMAND
C U RV E S F O R E N E R G Y A N D F O O D
We begin by deriving the demand curves for each household type in our economy, and
we then sum these demand curves to derive the market demand curves. To do this, we
use the techniques developed in Chapter 5.
Given the utility function for a white-collar household, the marginal utilities of
energy and food are
1 y 2
a b
2 x
1
1 x 2
W
MU y a b
2 y
1
MU W
x
The marginal rate of substitution of energy for food is the ratio of the marginal
W
W
utilities: MRS W
x, y MU x /MU y . Using the above expressions for marginal utility,
W
this ratio reduces to MRS x, y y/x. Assuming that the household maximizes its
utility subject to its budget constraint, it will equate the marginal rate of substitution to the ratio of the prices: MRS W
x, y Px /Py. In addition, the budget constraint
is satisfied. Thus, utility maximization gives us two equations in two unknowns, x
W
and y. First, y/x Px /Py (which follows from MRS W
x, y y/x and MRS x, y Px /Py).
Second, xPx yPy IW (which follows from the budget constraint), where IW denotes the household’s income level (which, recall, depends on the input prices, w
and r). When we solve these two equations for x and y (treating Px, Py, and IW as
constants), we get x (1/2)(IW /Px) and y (1/2)(IW /Py). These are a typical whitecollar household’s demand curves for energy and food.
Let’s suppose that our economy contains 100 such households. We can find the
aggregate demand curves for energy and food from white-collar households by mulW
tiplying the above expressions by 100. This yields the DW
x and Dy demand curves in
W
W
Figure 16.5: x 50IW/Px and y 50IW/Py.
Let’s now turn to the blue-collar households. Given the utility function for a bluecollar household, the marginal utilities of energy and food are
3 y 4
a b
4 x
1
MU Bx
1 x 4
a b
4 y
3
and
MUyB
Proceeding in the same way we did for white-collar households. We find that the
demand curves for a typical blue-collar household are x (3/4)(IB / Px) and y
(1/4)(IB /Py). Multiplying these by 100 gives us the aggregate demand curves for bluecollar households D Bx and D By in Figure 16.5: x B 75IB /Px and y B 25IB /Py.
We can now find the market demand curves for energy and food by horizontally
summing the demand curves for both types of household. Let X be the aggregate
amount of energy demanded in the economy. The market demand curve for energy is
thus X x W x B, or X (50IW/Px) (75IB /Px). In Learning-By-Doing Exercise
16.2 we expressed this as Px (50IW 75IB)/X. Similarly, the market demand curve
for food is Y yW yB, which we expressed as Py (50IW 25IB)/Y. Notice
that these market demand curves depend on the income levels of each individual
household.
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CHAPTER 16
G E N E R A L E Q U I L I B R I U M T H E O RY
D E R I V I N G T H E M A R K E T D E M A N D C U RV E S
F O R L A B O R A N D C A P I TA L
Given the production function for a typical energy producer, the marginal products
of labor and capital are
1 2
1
MPl a b1.89l 3k 3l⫺1,
3
and
1 2
2
MPk a b 1.89l 3 k 3k⫺1
3
Recall from Chapter 7 that the marginal rate of technical substitution MRTS l,xk is
the ratio of the marginal product of labor to the marginal product of capital:
MRTS l,x k MPl /MPk. Using the above expressions for marginal product, we find that
this ratio reduces to MRTS xl, k (1/2)(k/l ).
An energy producer minimizes its cost of production by equating the marginal rate
of technical substitution to the ratio of the input prices: MRTS l,xk w/r. In addition,
the quantity of labor and capital must be sufficient to produce the desired amount of
output x (i.e., the production function must be satisfied). Thus, cost minimization gives
us two equations in two unknowns, k and l. First, (1/2)(k/l ) w/r [which follows
from MRTS l,x k (1/2)(k/l ) and the requirement that MRTS l,xk w/r冥. Second,
1 2
x 1.89l 3k 3.
To solve these equations for k and l (treating w, r, and x as constants), we solve the
first equation for k and substitute the result into the second equation, which we then
solve for l. Solving the first equation for k gives us k (2wl )/r. Substituting this into
the second equation and solving for l gives us13
x r 3
l a b
3 w
2
This is the labor demand curve for a typical energy producer. To find the firm’s demand curve for capital, we substitute the above expression back into the expression for
k (2wl )/r. Doing this and simplifying gives us
2x w 3
a b
3 r
1
k
This is the capital demand curve for a typical energy producer.
13
Here are the details on how to simplify this expression. When we substitute k (2wl )/r into the production
function, we get
x 1.89l 3 a
1
2
2
2wl 3
w 321
b 1.89(2) 3 a b l 3 l 3
r
r
2
2
2
1
2
1
Using a calculator, we find that 1.89(2) 3 3. Also, l 3 l 3 l (3 3 ) l. Thus,
w
x 3a b l
r
2
3
or, rearranging terms,
x r 3
a b
3 w
2
l
This is the labor demand curve stated in the text.
695
A P P E N D I X : D E R I V I N G T H E D E M A N D A N D S U P P LY C U RV E S
Now consider the food industry. Given the production function for a typical food
producer, the marginal products of labor and capital are
1 1 1
MPl a b2l 2k2l⫺1
2
and
1 1 1
MPx a b2l 2k2k⫺1
2
Proceeding in the same way we did for a typical energy producer (but omitting the
actual computations), we find that the labor and capital demand curves for a typical food
producer are
y r 2
a b
2 w
1
l
y w 2
a b
2 r
1
and
k
Now we can find the overall market demand curves for labor and capital. The energy industry consists of 100 identical firms, each producing x units
of energy and
2
each with the labor demand curve derived above: l (x/3)(r/w) 3. The overall labor
demand curve for
energy producers l x is 100 times this expression:
2
x
3
x units of energy, total
l 100(x/3)(r/w) . Since there are 100 firms, each producing
2
energy production X 100x. Thus, l x (X/3)(r/w) 3. This is the equation for the
labor demand curve D Lx in Figure 16.6(a).
By similar logic, we can determine
that the overall labor demand curve for food
1
y
producers is l y (Y/2)(r/w) 2. This is the equation for the labor demand curve DL in
Figure 16.6(a).
The overall market demand curve L for labor is the sum of labor demands in the
energy and food industries:
X r 3
Y r 2
L a b a b
3 w
2 w
2
1
This is the equation for the labor demand curve DL in Figure 16.6(a).
We can use similar logic to derive the equation for the market demand for
capital:
2X w 3
a b (energy industry demand for capital)
3 r
3
Y w 2
y
K a b (food industry demand for capital)
2 r
1
Kx
y
These are the equations for the capital demand curves D kx and D k in Figure 16.6(b). The
sum of these equations is the overall market demand for capital, which we denote by K:
Y w 2
2X w 3
K
a b a b
3 r
2 r
1
1
This is the equation for the capital demand curve DK in Figure 16.6(b). Notice that
the economywide demands for labor and capital depend on the ratio of the input
prices and on the total output produced in each industry.
D E R I V I N G T H E M A R K E T S U P P LY C U RV E S
FOR ENERGY AND FOOD
Now let’s see how to derive the market supply curves for energy and food shown
in Figure 16.7. As we saw earlier in this chapter, the market supply curves are the
696
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G E N E R A L E Q U I L I B R I U M T H E O RY
marginal cost curves for energy and food production. We will derive these marginal cost curves in two steps.
First, let’s derive total cost curves for a typical energy producer and a typical food
producer (recall that we saw problems like this in Chapter 8). The total cost for a typical
energy producer is the sum of the producer’s costs for labor and capital,
TC wl rk. In the previous section, we derived the cost-minimizing quantities of labor
and capital for a typical energy producer. If we substitute these equations for l and k into
the TC equation, we get
x r 3
2x w 3
TC w c a b d r c a b d
3 w
3 r
2
1
which simplifies to14
2
1
TCx (w 3 r 3 ) x
Using similar logic, we can derive the total cost curve for a typical food producer:
1
1
TCy (w 2 r 2) y
Recall that marginal cost is the rate of change of total cost with respect to a
change in output. The total cost curve for an energy producer, which we just derived,
goes up at a constant rate as the firm’s output x goes up.1This
constant rate is the co2
efficient of x in the equation for the total cost curve, (w 3 r 3 ) . Thus, the marginal cost
curve for an energy producer is
1
2
MCx w 3 r 3
By the same token, the marginal cost curve for a food producer is the coefficient of y
in the equation for the total cost curve:
1
1
MCy w 2 r 2
Note that the marginal cost curves for energy and food producers depend on the
input prices for labor and capital. Until we know what these input prices are, we won’t
know the exact level of marginal cost. Also note that the marginal cost curves for
energy and food depend on the input prices in different ways. For example, the marginal
cost for energy depends more strongly on the price of capital than on the price of
labor. Ultimately, this is because of differences in the production functions for energy
and capital. Given these production functions, an energy producer uses a higher ratio
of capital to labor than does a typical food producer. That is, energy production is
more capital-intensive than is food production.
14
Here are the details of the simplification of this expression. Begin by rearranging terms:
2x w 3
x r 3
TC w c a b d r c a b d
3 w
3 r
2
2
2
2
1
1
1
x wr 3
2x r w3
2
3 w3
3 r 13
x 1 ⫺23 32
2x 1 ⫺31 13
w w r
rr w
3
3
1
1
2
Now, note that w1w⫺3 w1⫺ 3 w 3 and that r1r⫺3 r1⫺ 3 r 3 . Substituting these into the above expression gives
1 2
2x 31 32
x 1 2
w r xw 3 r 3
TCx w 3 r 3
3
3
17
EXTERNALITIES AND
PUBLIC GOODS
17.1
INTRODUCTION
APPLICATION 17.1
How to Avoid “Collapse”
of a Fish Species
17.2
EXTERNALITIES
APPLICATION 17.2
APPLICATION 17.3
Gone Surfing?
Congestion Pricing
in California
APPLICATION 17.4
APPLICATION 17.5
London’s Congestion Charge
Knowledge Spillovers and
Innovation
17.3
PUBLIC GOODS
APPLICATION 17.6
Free Riding on the Public
Airwaves
When Does the Invisible Hand Fail?
Economist Herbert Mohring has described a situation familiar to all of us: “The users of road and other
transportation networks not only experience congestion, they create it. In deciding how and when to travel,
most travelers take into account the congestion they expect to experience; few consider the costs their trips
impose on others by adding to congestion.”1 This scenario involves an externality that arises because each
driver bears only part of the costs that he or she imposes on society when making a trip. To see why, note
that, as a driver on the highway, your costs (i.e., the price of driving) include gas and oil, wear and tear on
your car, and any tolls, as well as the cost of your time spent driving (you could have spent that time doing
something productive). These are the costs you are likely to take into account when deciding whether to
drive, but there are other costs that you are much less likely to consider because you do not bear them yourself—for instance, adding to traffic congestion and thereby increasing the travel time (and associated cost)
for other drivers. The costs that you as a driver impose on society include both these kinds of costs—the
ones you bear yourself (internal costs) and the ones borne by others (external costs).
1
See H. Mohring, “Congestion,” Chapter 6 in Essays in Transportation Economics and Policy: A Handbook
in Honor of John R. Meyer, J. Gomez-Ibanez, W. Tye, and C. Winston, eds. ( Washington, DC: Brookings
Institution Press, 1999).
697
External costs (or benefits) can be significant, as Mohring saw when studying the effects of rush hour
congestion in Minneapolis and St. Paul, Minnesota, using data on travel patterns in 1990. He found that
“the average peak-hour trip imposes costs on other travelers equal to roughly half of the cost directly
experienced by those taking the average trip.”
A public good benefits all consumers, even though individual consumers may not pay for the costs of its provision. Examples include national defense, public radio and television, and public parks. A public good has two features: (1) consumption of the good by one person does not reduce the amount that another can consume, and
(2) a consumer cannot be excluded from access to the good. For example, anyone can view a public television
station, and the reception of the signal by one person does not reduce the opportunity for others to receive it.
Why worry about externalities and public goods? As we will see in this chapter, with an externality or
a public good, the costs and benefits affecting some decision makers differ from those for society as a
whole, causing the market to undersupply public goods and creating situations where social costs differ
from social benefits. Thus, in a competitive market when there are externalities or public goods, the
invisible hand may not guide the market to an economically efficient allocation of resources.
CHAPTER PREVIEW
After reading and studying this chapter, you will be able to:
• Define externalities and public goods.
• Explain why externalities and public goods are a source of market failure.
• Distinguish between positive and negative externalities.
• Analyze how taxes or emissions standards could reduce the economic inefficiency that arises in a
competitive market with a negative externality.
• Analyze how a congestion toll can reduce the economic inefficiency due to negative externalities
from traffic congestion.
• Explain how a subsidy
could reduce the economic
inefficiency that arises in a
competitive market with a
positive externality.
• Describe the Coase
Theorem and discuss its
economic significance.
• Show how the efficient
quantity of a public good
is determined.
• Explain the free rider
problem.
698
699
17.1 INTRODUCTION
M
arkets with externalities and markets with public goods are two kinds of markets
that are unlikely to allocate resources efficiently. We first encountered externalities in
Chapter 5, where we studied network externalities. In general, the defining feature of an
externality is that the actions of one consumer or producer affect other consumers’ or
producers’ costs or benefits in a way not fully reflected by market prices (in our chapteropening example, for instance, the individual driver’s price for driving on the highway
doesn’t reflect the social cost of increased congestion). A public good, in general, has
two defining features: first, one person’s consumption of the good (e.g., driving x miles
on the highway) does not reduce the quantity that can be consumed by any other person (all other drivers can still drive as far as they want on the highway); and second, all
consumers have access to the good (any driver can drive on the highway).
Public goods include such services as national defense, public parks and highways,
and public radio and television. To see why public television, for example, is a public
good, note how it conforms to the definition above: when one viewer watches a public
television program, no other viewer is prevented from watching it (to put this another
way, the marginal cost of serving an additional viewer is zero); further, once the television program is broadcast, no viewer can be excluded from watching it.
In Chapter 10, we used partial equilibrium analysis to show that a competitive
market maximizes the sum of consumer and producer surplus. Since there are no
externalities or public goods in a perfectly competitive market, the private costs and
benefits that decision makers face are the same as the social costs and benefits. In this
case, the invisible hand guides the market to produce the efficient level of output, even
though each producer and consumer acts solely in his or her own self-interest. In
Chapter 16, we extended the analysis of competitive markets to a general equilibrium
setting and showed that the allocation of resources in a competitive equilibrium is
economically efficient (again assuming an absence of externalities and public goods).
When the market includes externalities or public goods, however, the market price
may not reflect the social value of the good, and the market may therefore not maximize total surplus—that is, the equilibrium may be economically inefficient. For this
reason, externalities and public goods are often identified as sources of market failure.
A P P L I C A T I O N
Since at least the 1970s, scientists have continued to
warn that many fish species are in danger of being
“overfished” due to increased human consumption.
Overfishing could ultimately lead to the irreparable
harm or even extinction of a species. For example, a
dramatic decline in Atlantic cod populations in the
early 1990s led the Canadian government to impose
an indefinite moratorium on cod fishing in the Grand
Banks, an area off the coast of Newfoundland with
2
INTRODUCTION
externality The effect
that an action of any decision
maker has on the well-being
of other consumers or producers, beyond the effects
transmitted by changes in
prices.
public good A good,
such as national defense,
that has two defining
features: first, one person’s
consumption does not
reduce the quantity that
can be consumed by any
other person; second, all
consumers have access to
the good.
17.1
How to Avoid “Collapse”
of a Fish Species
one of the richest fishing areas on the planet. In 2006
the Fisheries Service of the National Oceanic and
Atmospheric Administration estimated 20 percent of
U.S. fisheries to be overfished.2 At the same time a
study in Nature in 2006 estimated that 29 percent of
species studied had declined to 10 percent of their
original levels, what they term a “collapse” of a
species. The primary cause was overfishing, though
pollution and loss of habitat are also factors.
In 2008, a study in Science provided some hope
for the problem of overfishing.3 Scientists studied
more than 11,000 fisheries worldwide to try to find a
system that would avoid overfishing. They concluded
Cornelia Dean, “Study Sees ‘Global Collapse’ of Fish Species,” New York Times, November 3, 2006.
John Tierney, “How to Save Fish,” New York Times, September 18, 2008.
3
17.1
700
CHAPTER 17
EXTERNALITIES AND PUBLIC GOODS
that a system called “catch shares” holds promise. In
the catch shares system, a maximum allowable catch is
determined each year by the government with input
from fishery scientists. Specific fishermen own the
rights to a certain percentage of the annual quota,
and only those with such rights are allowed to catch that
type of fish. The quota rights can be bought and sold at
the current market price. If the fish population thrives,
the rights have more value. If the fish are overfished, the
rights go down in value. This gives incentives to the
fishermen to protect the species from overfishing. For
example, after a catch shares system was implemented
in Alaska, fishermen began using fewer hooks, resulting
in less harm to the fish population, since they no longer
had to “race to fish” in competition with each other. Of
course, limiting the maximum catch per year also helps
solve the overfishing problem.
In the Science study, researchers found that fisheries
using a catch shares system had only half the odds of a
species collapse. Moreover, the fish population became
stronger the longer the catch shares system had been
used. In some fisheries that use catch shares, the fishing
industry has actually lobbied to impose even stricter
17.2
EXTERNALITIES
limits than those suggested by biologists, in order to
further improve the economic value of the fishery.
Fishing grounds are an example of a common property resource, and the fishing done by one fisherman
imposes a negative externality on other fisherman. This
gives rise to a market failure. In this chapter, you will
learn how negative externalities can lead to market
failure, and you will study possible government interventions that can offset or eliminate the inefficiency
that the market failure gives rise to. You will find that
there may be solutions to externality problems that
largely play out in a private market. A catch shares
system is one such example.
Currently, about 1 percent of fisheries worldwide
use this system. Despite such promising results, the
catch shares system is still controversial. Some environmental groups oppose the system, though others have
become advocates given recent evidence on their effectiveness. If a catch shares system helps overcome inefficiencies due to a market failure, we would expect it to
catch on and become more widely used. It will be interesting to see if, over the next decade, this happens.
Externalities can arise in many ways, but, however they arise, their effects are always
the same: The actions of a consumer or producer may benefit or harm other consumers or producers.
Externalities are positive if they help other producers or consumers. We frequently observe positive externalities from consumption. For example, when a child
is vaccinated to prevent the spread of a contagious disease, that child receives a private benefit because the immunization protects her from contracting the disease.
Further, because she is less likely to transmit the disease, other children in the community benefit as well. The bandwagon effect we studied in Chapter 5 is a positive
externality because one consumer’s decision to buy a good improves the well-being
of other consumers.
There are also many examples of positive externalities from production. The development of a new technology like the laser or the transistor often benefits not only
the inventor, but also many other producers and consumers in the economy.
Externalities can also be negative if they impose costs on or reduce benefits for
other producers or consumers. For example, a negative externality from production
occurs if a manufacturer of an industrial good causes environmental damage by polluting the air or water. A negative externality from consumption occurs if there is a
snob effect, as we learned in Chapter 5.
Highway congestion, as discussed in the introduction to this chapter, is also an
example of a negative externality. You are no doubt also familiar with other examples
of congestion externalities, including those encountered on computer networks, in
telephone systems, and in air transportation.
701
17.2 EXTERNALITIES
Externalities can occur in a variety of market settings, including not only markets
with competition, but also those with monopoly and other imperfect markets discussed in earlier chapters. In this chapter we will focus on the effects of externalities
in otherwise competitive markets. As you read the chapter, you might think about
how you can apply the principles we introduce to study the effects of externalities in
markets that are not competitive.
A P P L I C A T I O N
17.2
Gone Surfing?
If you have ever surfed the Internet, you have no doubt
encountered an electronic experience similar to driving
on a freeway. Often you are moving quickly from one
Web page to another, while at other times you feel as
though you are in stop-and-go traffic, waiting for a reply
or slowly transmitting or downloading data. Everyone
who sends an e-mail or downloads a file shares bandwidth, that is, the capacity for carrying data over the network. Sometimes, the capacity is adequate to handle the
load without congestion. At other times, there is so much
traffic that the network becomes congested, and additional messages further slow the flow of traffic.
Often described as an information superhighway,
the Internet is a very large network connecting millions
of computers around the world. Some of the larger con-
nections serve as electronic pipelines, and the largest
pipelines are known collectively as the Internet backbone. The backbone is a collection of networks belonging to the major Internet service providers (ISPs) such as
UUNet (now a division of Verizon), AT&T, Sprint Nextel,
and Level 3. These networks connect with each other at
five points (Washington, D.C., New Jersey, Chicago, San
Francisco, and San Jose, California), allowing computers
to connect with each other in the United States and
with other computers in the rest of the world. There are
also many smaller electronic pipelines, made up of local
and regional ISPs, that often connect individual residential and business customers to the backbone.
When you connect to the Internet, you incur
private costs, including costs from network congestion
because your time is valuable. You may also pay
charges for each minute you are connected to the
North America response time (MS)
215
210
205
200
195
190
185
180
175
170
1:00 AM
3:05 AM
5:10 AM
7:15 AM
9:20 AM
11:25 AM
1:30 PM
3:35 PM
5:40 PM
7:45 PM
FIGURE 17.1 Congestion in the Internet
The speed with which traffic moves through the Internet varies during the day, depending on
the amount of congestion in the network. The graph shows the speed of data flow in North
America over February 19, 2010. The “response time” measures how long it takes for a set of
data to travel from point A to point B and back (round trip). The response time is measured in
milliseconds (thousands of a second). A response time of 200 ms means that it takes 2/10th of
one second for the data to complete a round trip.
Source: Internet Traffic Report (www.internettrafficreport.com/namerica.htm), February 19, 2010.
9:50 PM
11:55 PM
702
CHAPTER 17
EXTERNALITIES AND PUBLIC GOODS
network. If your benefits from connecting exceed
these private costs, you will stay online. If your private
costs are too high because of congestion, you may
decide to delay going online until another time.
Many users consult websites that provide current
information on the extent of congestion on the
Internet, much as they listen to traffic reports on
radio or television stations before deciding whether
to make a trip by auto. For example, the Internet
Traffic Report (http://www.internettrafficreport.com)
measures the round-trip travel time for messages sent
along major paths of the Internet. For a typical day,
February 19, 2010, Figure 17.1 shows that the Internet
in North America was relatively congested between
about 6:00 P.M. and 10:00 P.M. Mountain Standard
Time, and much less congested in the early hours of
the morning (e.g. between about 3 A.M. and 5 A.M.
This interesting site also reports response times in
Asia, Australia, Europe, and South America.
You may also impose external costs on other users
when you surf the Web because your own traffic
adds to congestion throughout the network. Like
the automobile commuter, while you think about the
private (internal) costs that you incur because of congestion, you probably do not think about the external
costs you impose on others as your own traffic adds to
congestion.
NEGATIVE EXTERNALITIES AND ECONOMIC EFFICIENCY
Why do firms produce too much in an otherwise competitive market when there are
negative externalities? Consider what happens when the production process for a
chemical product also generates toxic emissions that harm the environment. Let’s
assume that only one technology is available to produce the chemical. That technology
produces the chemical and the pollutant in fixed proportion: One unit of pollutant is
emitted along with each ton of the chemical produced. Each producer of the chemical is “small” in the market, so each producer acts as a price taker.
If the producers do not have to pay for the environmental damage their pollution
causes, each firm’s private cost will be less than the social cost of producing the chemical. The private cost will include the costs of capital, labor, raw materials, and energy
necessary to produce the chemical. However, the private cost will not include the cost of
the damage that the toxic waste does to the air or water around the plant. The social cost
includes both the private cost and the external cost of environmental damage.
Figure 17.2 illustrates the consequences of the externality in a competitive market. With a negative externality, the marginal social cost exceeds the marginal private
cost. The marginal private cost curve MPC measures the industry’s marginal cost of
producing the chemical. Because the technology produces the pollutant and the
chemical in a fixed proportion, the horizontal axis measures both the number of units
of the pollutant and the number of tons of chemical produced. The marginal external
cost of the pollutant is measured by MEC, which rises because the incremental damage to the environment increases as more pollution occurs. The marginal social cost
MSC exceeds the marginal private cost by the amount of the marginal external cost:
MSC MPC ⫹ MEC. That is, the marginal social cost curve is the vertical sum of the
marginal private cost curve and the marginal external cost curve.
If firms do not pay for the external costs, the market supply curve is the marginal
private cost curve for the industry (the horizontal sum of the individual firms’ marginal
private cost curves). The equilibrium price will be P1, and the market output will be Q1.
The first column of the table in Figure 17.2 shows the net economic benefits in
equilibrium with the negative externality. Consumer surplus is areas A ⫹ B ⫹ G ⫹
K—that is, the area below the market demand curve D and above the equilibrium
price P1. The private producer surplus is areas E ⫹ F ⫹ R ⫹ H ⫹ N (the area below
703
17.2 EXTERNALITIES
A
MSC
Price
MPC = Market supply
P*
P1
MPC at Q*
B
E
F
G K
H N
M
R
D
MEC
V
Z
Q1
Q*
Tons of the chemical per week
= units of pollutant per week
Externality
With a negative externality,
the marginal social cost MSC
exceeds the marginal private
cost MPC by the amount of the
marginal external cost MEC.
If firms do not pay for the
external costs, the market supply
curve is the marginal private
cost of the industry MPC. The
equilibrium price will be P1,
and the market output will be
Q1. At the social optimum, firms
would be required to pay for
the external costs, leading to a
market price P* and quantity
Q*. The externality therefore
leads to overproduction in the
market by the amount (Q1 ⫺ Q*)
and to a deadweight loss equal
to area M.
Equilibrium
(price P1)
Consumer surplus
A
B
G
Private producer
surplus
E
F
R
Cost of externality
G
R
K
H
N
M
Net social benefits
(consumer surplus
private producer
surplus cost of
externality)
A
B
E
Deadweight loss
M
Difference
between Social
Optimum and
Equilibrium
Social
Optimum
(price P *)
FIGURE 17.2 Negative
K
H
A
N
B
B
R
E
H
R
F
M
A
F
G
H
B
B
G
E
Zero
the market price and above the market supply curve). The cost of the externality is
areas R ⫹ H ⫹ N ⫹ G ⫹ K ⫹ M (the area below the marginal social cost curve and
above the market supply curve), which is equal to areas Z ⫹ V. The net social benefits equal the sum of the consumer surplus and the private producer surplus, minus the
cost of the externality—that is, areas A ⫹ B ⫹ E ⫹ F ⫺ M.
G
G
K
N
M N K
(external cost
savings)
F
M
(increase in
net benefits
at social
optimum)
M
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Now let’s see why the competitive market fails to produce efficiently. In equilibrium
the marginal benefit of the last unit produced is P1, which is lower than the marginal
social cost of production for that unit. Thus, the net economic benefit from producing
that unit is negative.
The efficient amount of output in the market is Q* , the quantity at which the market
demand curve and the marginal social cost curve intersect. There the marginal benefit
of the last unit produced (P* ) just equals the marginal social cost. The production of
any units beyond Q* creates a deadweight loss because the marginal social cost curve
lies above the demand curve.
As shown in the second column of the table in Figure 17.2, if consumers pay the
price P* for the chemical, net economic benefits would increase. Consumer surplus
would fall to A (the area under the demand curve and above P*). Private producer surplus would be areas B ⫹ E ⫹ F ⫹ R ⫹ H ⫹ G (the area below the price P * and above
the market supply curve). The external cost is areas R ⫹ H ⫹ G (the area below the marginal social cost curve and above the market supply curve). The net social benefits equal
consumer surplus plus private producer surplus minus the external cost (⫺R ⫺ H ⫺
G)—that is, areas A ⫹ B ⫹ E ⫹ F.
The third column of the table in Figure 17.2 shows the differences between the
social optimum and the equilibrium in terms of consumer surplus, private producer surplus, and the cost of the externality. In terms of net social benefits, it also shows that the
market failure arising from the externality creates a deadweight loss equal to area M.
To summarize, the negative externality leads the market to overproduce by the
amount Q1 ⫺ Q*. It also reduces the net economic benefits by area M, the deadweight
loss arising from the externality.
Learning-By-Doing Exercise 17.1 will help you understand why generally it is not
socially optimal to prohibit industries from using technologies that produce negative
externalities.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 7 . 1
D
The Efficient Amount of Pollution
Problem
Evaluate the following argument: “Since pollution is a negative externality, it would
be socially optimal to declare illegal the use of any production process that creates pollution.”
Solution Refer to Figure 17.2. At the social optimum, net social benefits are areas A ⫹ B ⫹ E ⫹ F. While
it is true that there are costs from the externality (areas
R ⫹ H ⫹ G ), the net social benefits from producing the
chemical are nevertheless positive, even after taking the
external costs into account. If it were illegal to produce
the chemical because of the negative externality, society
would be deprived of the net benefits represented by
areas A ⫹ B ⫹ E ⫹ F. Thus, the optimal amount of
pollution is not zero.
If we were to outlaw all pollution, we would deprive
ourselves of many of the most important products and services in our lives, including gasoline and oil, electric power,
many processed foods, goods made from steel, iron, and
plastics, and most modern forms of transportation.
Similar Problems:
17.1, 17.3, 17.26
Emissions Standards
emissions standard A
governmental limit on the
amount of pollution that
may be emitted.
Figure 17.2 is useful in helping us understand why a market fails to produce efficiently
with the negative externality. But what can be done to eliminate or reduce economic
inefficiency? One possibility is for the government to intervene in the market by restricting the amount of the chemical that can be produced and, therefore, the amount
of pollution emitted as a by-product. A governmental limit on the amount of pollution
allowed is called an emissions standard.
17.2 EXTERNALITIES
705
In the United States, the Environmental Protection Agency (EPA) is the governmental agency primarily responsible for overseeing efforts to keep the air clean.
Under the 1990 Clean Air Act, the EPA specifies limits on the amount of pollutants
allowed in the air anywhere in the United States. The regulation of air quality is a
complex undertaking because there are so many kinds of air pollution, and the patterns of pollution change from year to year. The EPA concentrates on emissions that
might harm people, including smog, carbon monoxide, lead, particulate matter, sulfur
dioxide, and nitrogen dioxide. There are also many other airborne compounds, called
air toxins, that can be hazardous to people.
Under the Clean Air Act, federal and state governments can require large sources
of pollution, such as power plants or factories, to apply for a permit to release pollutants into the air. The permit specifies the types and quantities of pollutants that can
be emitted and the steps the source must take to monitor and control pollution. The
EPA can assess fines on sources that exceed allowed emissions. Approximately 35
states have implemented statewide permit programs for air pollution.
Unfortunately, it is not easy for the government to determine optimal emissions
standards. Consider again our example with the chemical manufacturers. To calculate the
optimal emissions in the entire market, the government would need to know the market
demand curve for the chemical, as well as the marginal private and social cost curves. If
the only way to reduce pollution is to cut back on the amount of the chemical produced,
the efficient emissions standard in Figure 17.2 would be Q* units of pollutant (the amount
of pollutant released into the air when Q* tons of the chemical are produced).
Even if the regulator could calculate the optimal size of the emissions in the entire market, it must decide how much pollution each firm will be allowed to release.
Some firms will be able to reduce (abate) emissions at lower costs than other firms.
The determination of the socially optimal pollution allowance for each firm will depend on the costs of abatement for each firm in the market.
To see why abatement costs matter, suppose the government wants to reduce
pollution in the market by one unit. Suppose, also, that it would cost Firm A $1,000
to reduce pollution by one unit, while Firm B could reduce pollution by the same
amount at a cost of only $100. It would cost society less to require Firm B to cut back
its pollution. To some extent, the government can simplify the task of determining the
efficient allocation of rights to pollute by allowing firms to trade emissions permits.
The government could initially allocate rights to pollute and then let firms trade the
rights in a competitive market. Firms with higher abatement costs will attach a higher
value to the right to emit a unit of pollution than will firms with lower abatement cost.
The latter firms may even have an incentive to trade some of their rights to pollute to
firms with high abatement costs. In equilibrium, the rights to pollute will be distributed so that the total costs of abatement are as low as possible.
Under the Clean Air Act it is possible to implement a system of tradeable emissions permits. For example, the program to clean up acid rain includes pollution
allowances that can be traded, bought, and sold.
Emissions Fees
The government may also reduce the economic inefficiency from a negative externality by imposing a tax on the firm’s output or on the amount of pollutant the firm emits.
An emissions fee is a tax imposed on pollution that is released into the environment.
Figure 17.3 illustrates the effect of an emissions fee for our example of chemical
manufacturing. Suppose the government collects a tax of $T on each ton of chemical produced. Because each firm emits one unit of pollutant for each ton of chemical produced,
we can also view the tax as an emissions fee of $T on each unit of pollutant.
emissions fee A tax
imposed on pollution that
is released into the
environment.
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Optimal emissions fee = P * – P s
Price
A
FIGURE 17.3 Optimal Emissions
Fee with a Negative Externality
An optimal emissions fee (or tax) will
lead to the economically efficient output Q* in a competitive market. With
an optimal fee, the price consumers
pay must cover not only the marginal
private cost of production, but also the
fee. The curve labeled “Market supply ⫹
Tax” shows what quantity producers
will offer for sale when the price
charged to consumers covers the marginal private cost plus the tax. At the
optimal tax, the demand curve intersects the “Market supply ⫹ Tax” curve
at the socially optimal quantity Q*.
Consumers pay P*, and producers
receive a price equal to P s. The government collects tax revenues equal to
areas B ⫹ G ⫹ E ⫹ H. There is no
deadweight loss with the optimal tax
because net benefits are as large as
possible (A ⫹ B ⫹ E ⫹ F ).
P*
P1
Ps
MSC
MPC + Tax = Market supply + Tax
MPC = Market supply
B
E
F
G K
H N
R
D
MEC
V
Z
Q*
Q1
Tons of the chemical per week
= units of pollutant per week
Equilibrium (with tax)
Consumer surplus
A
Private producer surplus
F⫹R
⫺Cost of externality
⫺R ⫺ H ⫺ G
Government receipts from emissions tax
B⫹G⫹E⫹H
Net social benefits (consumer surplus ⫹
private producer surplus ⫺ cost of externality)
A⫹B⫹E⫹F
One way to understand the effect of the tax is to draw a new curve that adds the
amount of the tax vertically to the market supply curve, just as we did in Chapter 10
when we studied the effects of an excise tax in a competitive market. The curve labeled
“Market supply ⫹ Tax” in Figure 17.3 tells us how much producers will offer for sale
when the price charged to consumers covers the marginal private cost of production
plus the tax. The equilibrium with the tax is determined at the intersection of the
demand curve and the “Market supply ⫹ Tax” curve.
We have chosen the tax to maximize total surplus in Figure 17.3. The marketclearing quantity is Q*, the same level of output we identified as economically efficient
in Figure 17.2. At Q* the marginal social benefit is P*, the price consumers pay for each
ton of the chemical. Producers receive P s, which just covers their marginal private cost
of production. The government collects a tax of P* ⫺ P s per ton of the chemical sold
(equivalently viewed as an emissions fee of P* ⫺ P s per unit of pollutant). As the graph
shows, the tax just equals the marginal external cost of the pollution emitted when the
industry produces the last ton of the chemical. Thus, the marginal social benefit (P*)
equals the marginal private cost (P s ) plus the marginal external cost.
17.2 EXTERNALITIES
707
The table in Figure 17.3 gives us another way to see that the tax in the graph is
economically efficient. Consumers pay the price P* for the chemical, resulting in a consumer surplus equal to area A, the area under the demand curve and above P*. Private
producer surplus is areas F ⫹ R, the area below the price producers receive P s and above
the marginal private cost curve. The external cost is areas R ⫹ H ⫹ G, which is the
same as area Z. The government receives tax revenues equal to areas B ⫹ G ⫹ E ⫹ H.
The net social benefits equal consumer surplus, plus private producer surplus, plus the
tax receipts, minus the external cost (⫺R ⫺ H ⫺ G)—that is, areas A ⫹ B ⫹ E ⫹ F.
This is the same net benefit that we showed to be socially optimal in Figure 17.2. 4
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 7 . 2
D
Emissions Fee
Consider a variation of the chemical manufacturing example. Suppose the inverse demand curve
for the chemical (which is also the marginal benefit
curve) is P d ⫽ 24 ⫺ Q, where Q is the quantity consumed
(in millions of tons per year) when the price consumers
pay (in dollars per ton) is P d.
The inverse supply curve (also the marginal private
cost curve) is MPC ⫽ 2 ⫹ Q, where MPC is the marginal
private cost when the industry produces Q.
The industry emits one unit of pollutant for each
ton of chemical it produces. As long as there are fewer
than 2 million units of pollutant emitted each year, the
external cost is zero. But when the pollution exceeds
2 million units, the marginal external cost is positive.
The marginal external cost curve is
MEC ⫽ e
0, when Q ⱕ 2
⫺2 ⫹ Q, when Q 7 2
where MEC is marginal external cost in dollars per unit
of pollutant when Q units of pollutant are released.
Also suppose the government wants to use an emissions fee of $T per unit of emissions to induce the market
to produce the economically efficient amount of the
chemical.
Problem
(a) Construct a graph and a table comparing the equilibria
with and without the emissions fee:
• Graph the demand, supply (with no emissions fee),
marginal external cost, and marginal social cost
4
curves. Label two points on the graph: the point
that represents the equilibrium price and quantity
when there is no correction for the externality (i.e.,
no emissions fee) and the point that represents the
amount of the chemical the market should supply at
the social optimum. Indicate the actual price and
quantity at each point.
• Graph the supply curve after the imposition of an
emissions fee that induces the production of an
economically efficient amount of the chemical.
Indicate the price consumers will pay and the price
producers will receive.
• In the table, indicate the amount of the emissions
fee (dollars per unit) that will lead to the economically efficient production of the chemical. Fill
in the table with the following information for
the equilibria with and without the fee (indicate
both the areas on the graph and the actual dollar
amounts): consumer surplus, private producer
receipts from the fee, net social benefits, and
deadweight loss.
(b) Explain why the following sum is the same with and
without the fee: consumer surplus ⫹ private producer
surplus ⫺ external cost ⫹ government receipts from the
fee ⫹ deadweight loss.
Solution
(a) See Figure 17.4. The demand (marginal benefit)
curve is D. The supply (marginal private cost) curve is
MPC. The marginal external cost curve is MEC (it has a
As we indicated in Chapter 10, one must be careful when using a partial equilibrium analysis like the one
in Figure 17.3. A change in the amount of the good consumed in one market may affect market prices,
and therefore welfare, elsewhere. Further, there may be additional welfare effects when the government
distributes the revenues from the emissions fee somewhere else in the economy. The welfare analysis in
Figure 17.3 does not capture these effects.
708
CHAPTER 17
$24
EXTERNALITIES AND PUBLIC GOODS
MSC = MPC + MEC
A
L
22
MPC + T, = Supply with emissions fee
(when T = 6)
Price (dollars per ton)
16
13
MPC = Supply
M
B
MEC
J
H
E
10
9
6
N
W
D = Demand (marginal benefit)
K
X
V
4
2 F
0
I
G
2
R
8
U
11
24
Quantity (millions of tons of chemical per year; Millions of units of pollution per year)
No Emissions
Fee
Emissions Fee
of $6 per Unit
Consumer surplus
AJH
$60.5 million
ABM
$32 million
Private producer surplus
FJH
$60.5 million
FEN
$32 million
⫺Cost of externality
⫺VLH (⫽ ⫺GIU)
⫺$40.5 million
⫺VNM (⫽ ⫺GKR)
⫺$18 million
Government receipts from emissions fee
zero
ENMB
$48 million
Net social benefits (consumer surplus ⫹
private producer surplus ⫺ cost of externality ⫹
government receipts)
AMVF ⫺ MLH
$80.5 million
AMVF
$94 million
FIGURE 17.4 Emissions Fee
The economically efficient output is 8 million tons, determined by the intersection of the
demand and MSC curves at point M. An emissions fee of $6 per unit of pollutant leads to the
efficient level of output. With no emissions fee, the price of the chemical is $13 per ton, and
11 million tons are sold each year. The negative externality leads to an inefficiently high level
of pollution and a deadweight loss of $13.5 million per year.
709
17.2 EXTERNALITIES
kink in it, at point G, because MEC 0 when Q ⱕ 2).
The marginal social cost curve is MSC (the vertical sum
of MPC and MEC, with a kink at point V corresponding
to the kink in MEC ).
The equilibrium with no emissions fee is at point H,
where the demand and supply curves intersect. When
supply equals demand, 24 ⫺ Q ⫽ 2 ⫹ Q, or Q ⫽ 11;
since P d ⫽ 24 ⫺ Q, when Q ⫽ 11, P d ⫽ 24 ⫺ Q ⫽ 13—
that is, at this equilibrium, consumers pay a price of $13
per ton and producers supply 11 million tons per year.
The socially optimal amount of production is at
point M, where the demand and marginal social cost
curves intersect. When demand equals marginal social
cost, 24 ⫺ Q ⫽ (2 ⫹ Q) ⫹ (⫺2 ⫹ Q) (marginal social cost
is the sum of marginal private cost and marginal external
cost), or Q ⫽ 8; when Q ⫽ 8, P d ⫽ 24 ⫺ Q ⫽ 16—that is,
at the social optimum, consumers pay a price of $16 per
ton and producers supply 8 million tons per year.
After the imposition of an emissions fee that induces the production of an economically efficient
amount of the chemical, the supply curve will pass
through point M (at the socially optimal level of production, Q ⫽ 8) and will be the sum of the marginal private
cost and the fee—that is, the curve MPC ⫹ T. When
Q ⫽ 8, MPC ⫽ 2 ⫹ Q ⫽ 10. Thus, at this equilibrium,
consumers pay $16 per ton and producers receive $10 per
ton, so the emissions fee T ⫽ $16 ⫺ $10 ⫽ $6 per unit of
emissions.
For each equilibrium the table shows the consumer
surplus, private producer surplus, cost of the externality,
government receipts from the emissions fee (when a fee
is imposed), and the net social benefits.
(b) As the figures in the table show, consumer surplus ⫹
private producer surplus ⫺ external cost ⫹ government
receipts ⫹ deadweight loss ⫽ $94 million, both with and
without the emissions fee. This figure represents the
potential net benefit in the market, which is the same
whether or not there is a fee. When there is no fee, the
market performs inefficiently because of the negative
externality, and there is a deadweight loss. (Only $80.5
million of the $94 million potential net benefit is captured as net social benefit.) When there is a fee, the
market performs efficiently, and the entire potential net
benefit is captured. (There is no deadweight loss.)
Similar Problems:
17.4, 17.10, 17.11, 17.12
Common Property
Emissions fees and standards are measures that can help correct economic inefficiency
arising when a technology produces an undesired by-product along with some good or
service that society values. Negative externalities can also occur in markets that do not
involve a by-product, as we have already seen in the use of roadways or the Internet.
These are examples of common property, that is, resources that anyone can access.
With common property we often observe congestion, a negative externality leading to overuse of a facility. Figure 17.5 illustrates how congestion generates economic
inefficiency. The horizontal axis shows the volume of traffic on a highway, measured
in vehicles per hour. The vertical axis shows the price of driving (i.e., gas and oil, wear
and tear on the car, and the cost of the driver’s time spent on this activity). When the
traffic volume is below Q1, there is no congestion. Thus, the marginal external cost is
zero for traffic volumes below Q1. This means that the marginal private cost and the
marginal social cost are the same at these low volumes.
When the traffic volume exceeds Q1, congestion arises. Each new vehicle entering the system adds to the transit time for all vehicles. That is why the marginal
external cost rises as traffic volume grows.
Now let’s consider the effects of congestion at two different times of the day. In the
peak period (rush hour), the demand for use of the highway is high. Absent any government intervention, the equilibrium traffic level would be Q5, determined by the intersection of the peak demand curve and the marginal private cost curve, at point A. At that
point, the marginal benefit for the last vehicle is $5. The marginal private cost is also $5.
However, the marginal social cost imposed by the last vehicle is $8 (point G). Thus, the
marginal external cost is the amount by which the last vehicle increases the costs for other
vehicles, that is, $3, the length of the segment AG (also the length of the segment TU ).
common property A
resource, such as a public
park, a highway, or the
Internet, that anyone can
access.
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Price of driving
MSC = MPC + MEC
G
$8.00
B
5.75
Optimal peak
toll = $1.75
Optimal offpeak toll
$0.50
MPC (supply)
A
MEC
5.00
4.00
2.50
2.00
0
M
N
Q1
E
I
T
L
Q2 Q3
Q4
U
Q5
Peak demand
(marginal benefit)
Off-peak demand
(marginal benefit)
Traffic volume (vehicles per hour)
FIGURE 17.5 Congestion Pricing
There is no congestion as long as the level of traffic is lower than Q1. With higher levels of
traffic, the negative congestion externality grows. An optimal toll will lead to a traffic volume
where marginal benefit equals marginal social cost. In the peak period, equilibrium with no toll
is at point A, where deadweight loss is equal to area ABG. A toll of $1.75 (the length of line
segment BE ) moves the equilibrium to the economically efficient point B. The efficient off-peak
toll would be $0.50, the length of the line segment MN. In the off-peak period, equilibrium with
no toll is at point L, where deadweight loss is equal to area LMN. In this case, a toll of $0.50 (the
length of line segment MN ) moves the equilibrium to the economically efficient point M.
The socially optimal level of traffic is Q4, determined by the intersection of the
peak demand curve and the marginal social cost curve, at point B. At that point, the
marginal benefit and the marginal social cost for the last vehicle are both $5.75.
The marginal private cost is $4.00 (point E ). The highway authority could correct for
the externality by imposing a toll of $1.75 during the rush hour, bringing the traffic
volume to Q4.
In an off-peak period, the demand for highway use is lower. Without a toll, the
equilibrium traffic level would be Q3, at the intersection of the off-peak demand curve
and the marginal private cost curve (point L), where marginal benefit for the last
vehicle is $2.00. The socially optimal traffic level would be Q2, at the intersection of
the off-peak demand curve and the marginal social cost curve (point M ), where marginal benefit for the last vehicle is $2.50. Thus, the efficient off-peak toll would be
$0.50, the length of the line segment MN.
The congestion toll, like an emissions fee, is a tax that can be used to correct for
negative externalities. Today, the automated collection devices on most toll roads are
not capable of collecting tolls that vary during the day. However, as Application 17.2
shows, with new technology the widespread use of variable tolls is not far away.
Besides congestion, there are other examples of negative externalities with common property. For example, most lakes and rivers, and many hunting grounds, are
common property. When one person catches fish, a negative externality is imposed on
17.2 EXTERNALITIES
A P P L I C A T I O N
711
17.3
Congestion Pricing in California
The state of California has had regular fiscal crises in recent years. The recession of 2008–2010 has created the
worst in the state’s history as tax revenues plummeted.
In 2009, California furloughed state workers, froze
spending on many projects, and even issued IOUs to
contractors and some citizens who were due tax
refunds. By early 2010 California had the lowest credit
rating of any state. Because of its poor rating, the state
would have to pay high interest rates in order to raise
funds by issuing general obligation bonds. In fact, in
January 2010 the state halted the sale of all state
bonds.
An alternative source of funds that the state is considering is revenue-backed bonds. These are bonds that
are secured by a dedicated source of funding, such as
revenue from toll roads. California was the site of an
innovative toll road, Route 91, which in 1995 became
the first to be privately financed, and also the first to
use congestion pricing, with tolls that vary during the
day to keep traffic freely moving.
Traffic congestion has long been a problem in
Southern California. Route 91 connects the major employment centers of Orange and Los Angeles counties
with the rapidly growing residential areas in Riverside
and San Bernardino counties. In 1995 a 10-mile, 4-lane
toll road was located within the median of the existing
8-lane freeway. In order to use the tollway, motorists
must obtain a transponder (electronic device) and prepay money into an account. The transponder functions
much like a credit card, containing information on the
amount of money that motorists have in their account.
Each time a motorist uses the toll road, antennas
situated above the highway communicate with the
transponder and deduct the toll from that account.
There are no toll booths. The rate varies with time of
day. The rate on the busiest hour, 4:00 P.M. to 5:00 P.M.
eastbound on Thursdays, is $9.90, the highest toll for
any road in the country.
Under a franchise granted by the California
Department of Transportation (Caltrans), the $130 million construction cost for the project was financed
by a private entity, the California Private Transportation
Company (CPTC). Upon completion of construction, the
CPTC transferred ownership of the tollway to Caltrans
and leased the facility back from the agency for 35
years. CPTC collected tolls and paid state agencies to
provide law enforcement and road maintenance.
However, this deal proved controversial (Caltrans had
agreed to not widen the freeway alongside the toll
road, so as to not increase competition for it), so in
2003 the Orange County Transportation Authority purchased it from the CPTC for $207.5 million.
others who would like to fish. The negative externality can become significant when
rivalry among commercial fishing enterprises leads to a serious depletion in the stock
of fish, jeopardizing fishing harvests in future years. Governments can limit the depletion by imposing taxes or by limiting the quantity of fish that may be caught.
Negative externalities also arise in the petroleum industry, where there are a number
of owners of the mineral rights in large reservoirs of oil or natural gas. When one producer extracts a barrel of oil from a reservoir, it depletes the stock of oil available to other
producers. The amount of oil that can be successfully recovered from an oil reservoir depends on the way the oil is extracted. If individual producers vigorously compete to extract
oil as quickly as they can, they may damage the reservoir, reducing the total amount that
producers can ultimately recover. To enhance total recovery, and to minimize the effects
of the negative externality, producers often coordinate production. Frequently, this involves “unitizing” a field, with production operations carried out through a joint venture.
POSITIVE EXTERNALITIES AND
ECONOMIC EFFICIENCY
Positive externalities surround us in everyday life. Examples include education, health
care, research and development, public transit, and the bandwagon effect we studied
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EXTERNALITIES AND PUBLIC GOODS
17.4
London’s Congestion Charge5
On February 17, 2003, the city of London put microeconomic theory into practice when it initiated a £8
charge (about $12) aimed at reducing traffic congestion in the center of the city. Between 7:00 A.M. and
6:30 P.M., Monday through Friday, motorists traveling
within a 21-square-kilometer area of London known
as the charging zone were required to pay the fee.
The charging zone encompassed much of downtown
London, including the City, which contains the financial
district, and the West End, London’s main commercial
and entertainment hub.
With a system of streets that had hardly changed
since medieval times, central London has long struggled
with the problem of traffic congestion. Seventeenthcentury author Samuel Pepys wrote about being tangled up in traffic jams with horses and buggies.6
Modern estimates of the cost of traffic congestion in
London were on the order of $300 million per year.
Because London offered realistic alternatives to driving
(most notably, extensive bus and subway services), a
central theme in the debate over traffic congestion had
been how to entice people out of their cars and onto
public mass transportation.
The first congestion pricing scheme was implemented in Singapore in 1975. Drivers entering the
downtown area were charged a fee to reduce traffic.
This system was later extended to segments of several
freeways. In 1998 the system was fully automated with
transponders in cars and automatic charging of fees,
so that traffic can flow uninterrupted. Similar systems
have been implemented in Edinburgh, Stockholm, and
Milan, among other cities.
When the London system was introduced, skepticism about whether the plan would work was wide-
spread, with newspapers talking about “Carmageddon,”
and the Labour government of Tony Blair disassociating
itself from the mayor who had pushed for the system.
A number of prominent business groups vocally
opposed congestion pricing, arguing that it would
severely hurt retailers located in the charging zone.
Groups representing motorists and some labor organizations also opposed congestion pricing. Others worried that public transportation would be inadequate
to handle the flood of commuters who would turn to
it to avoid the fee.
The day-to-day operation of the congestion pricing
scheme was outsourced to a private company. Drivers
can pay the congestion charge at machines located
throughout the zone, as well as at selected retail locations and via the Internet. There are 174 entry and exit
points around the charging zone. When a vehicle drives
into the zone, its picture is taken by one of 203 video
cameras located at entry and exit points and within the
charging zone. These cameras (initially developed for
antiterrorism efforts) record license plates and match
them to lists of individuals who have paid the charge in
advance. Owners of vehicles that have not paid the fee
are fined from £60 to £180 (about $90 to $270).
Contrary to the fears, the scheme works well. Well
over 100,000 vehicles enter the zone and pay the charge
each day. Revenues from congestion charges have
topped £200,000, well above administrative expenses.
There has been a noticeable impact on traffic volume
and speed. The number of vehicles entering the zone
decreased by about 23 percent, while the average speed
increased by about 21 percent. One indication of the
success of London’s plan was that several parts of the
city outside the zone lobbied to be included within it.
In 2007 the charging zone was extended westward,
roughly doubling in size.
in Chapter 5. With a positive externality, the marginal social benefit from the good or
service exceeds the marginal private benefit. Other people around a consumer also
benefit when the consumer furthers her education or keeps herself in good health.
Similarly, when one firm succeeds in developing a new product or technology with a
program of research and development, the benefits often spill over to other firms and,
ultimately, to consumers.
5
Georgia Santos and Blake Shaffer, “Preliminary Results of the London Congestion Charging Schemes,”
Public Works Management & Policy (2004).
6
Randy Kennedy, “The Day the Traffic Disappeared,” New York Times, April 20, 2003.
17.2 EXTERNALITIES
Just as firms overproduce when there are negative externalities, so do firms underproduce when there are positive externalities. And just as the overproduction is the result
of consumers’ not taking external costs into account, so is the underproduction a result of
consumers’ not taking external benefits into account. That is, when you decide whether
to buy a good, you consider the benefits you will receive (the marginal private benefit),
but you do not consider the benefits your consumption will have for others. Figure 17.6
shows why this underproduction arises in a competitive market with a positive externality.
In Figure 17.6, the market demand curve MPB is the horizontal sum of the marginal private benefit curves of all the individuals in the market. The market supply
curve MC is also the industry marginal cost curve. If there is no correction for the externality, the market will be in equilibrium at the intersection of the demand curve and
the supply curve, where the price is P1 and the market output is Q1. In equilibrium,
private consumer surplus is the area below the MPB curve and above P1 (areas B ⫹ E ⫹
F ). Producer surplus is the area below P1 and above the MC curve (areas G ⫹ R).
Because of the positive externality, there is also an external benefit in the market,
as indicated by the marginal external benefit curve MEB. The marginal social benefit
MSB exceeds the marginal private benefit by the amount of the marginal external
benefit—that is, MSB ⫽ MPB ⫹ MEB. Again at the equilibrium without any correction for the externality (where market output is Q1), the size of the external benefit is
the area below the MSB curve and above the MPB curve (areas A ⫹ H ⫹ J ), which is
equal to the area under the MEB curve (areas U ⫹ V). Thus, at this equilibrium, the
net social benefit is the sum of the private consumer surplus, the producer surplus, and
the benefit from the externality (areas A ⫹ B ⫹ E ⫹ F ⫹ G ⫹ H ⫹ J ⫹ R).
Why does the competitive market fail to produce an economically efficient amount
of output? In equilibrium the marginal cost of the last unit produced is P1, which is lower
than the marginal social benefit for that unit. Thus, the net social benefit from producing another unit is positive. The economically efficient market output is Q*, where the
marginal social benefit equals the marginal cost for the last unit produced. Net benefits
would increase if the market expanded production to Q*. The failure to produce these
additional units introduces a deadweight loss equal to areas M ⫹ N.
How might public policy correct for the economic inefficiency resulting from underproduction with a positive externality? One possible way would be to subsidize
production of the good. (Recall from Chapter 10 that a subsidy is like a negative tax.
We learned there how a subsidy on each unit supplied stimulates production.)
How large must the subsidy be to lead the market to produce the efficient output
Q*? As shown in Figure 17.6, to supply the last unit, producers will need to receive
the price P s. However, consumers are willing to pay only P* for that unit. Thus, there
is a gap of P s ⫺ P * between the price producers require and the one consumers will
pay. Therefore, if the government provides a subsidy equal to P s ⫺ P *, it will induce
producers to provide that unit and consumers to purchase it.
The table in Figure 17.6 compares the equilibrium with no subsidy to the equilibrium at the social optimum (the equilibrium induced by the government subsidy). With
the subsidy, private consumer surplus increases by areas G ⫹ K ⫹ L, producer surplus increases by areas F ⫹ J ⫹ M, the external benefit increases by areas M ⫹ N ⫹ T, and the
cost to the government is equal to areas F ⫹ G ⫹ J ⫹ K ⫹ L ⫹ M ⫹ T. Thus, with the
subsidy, the net social benefit increases by areas M ⫹ N, and there is no deadweight loss.7
7
Once again, we observe that one must use caution when using a partial equilibrium analysis like the one
in Figure 17.6. If the government subsidizes one market, it must collect the funds for the subsidy (perhaps
introducing a deadweight loss) somewhere else in the economy. The welfare analysis in Figure 17.6 does not
capture these effects.
713
714
CHAPTER 17
EXTERNALITIES AND PUBLIC GOODS
MSB = MPB + MEB
Supply (MC)
Price
A
B
MSB at Q1
Supply – subsidy
per unit
H
E
Ps
F
P1
P*
G
N
J M
K
Optimal subsidy
per unit = P s – P*
T
L
R
MPB = market demand
U
MEB
W
V
Q1
Q*
Market quantity
Equilibrium
(no subsidy)
Social Optimum
(equilibrium
with subsidy)
Difference in Benefits
between Social
Optimum and Equilibrium
with No Subsidy
Private consumer
surplus
B⫹E⫹F
B⫹E⫹F⫹G⫹K⫹L
G⫹K⫹L
Producer surplus
G⫹R
F⫹G⫹R⫹J⫹M
F⫹J⫹M
Benefit from externality
A⫹H⫹J
A⫹H⫹J⫹M⫹N⫹T
M⫹N⫹T
⫺Government cost
from subsidy
zero
⫺F ⫺ G ⫺ J ⫺ K ⫺
L⫺M⫺T
⫺F ⫺ G ⫺ J ⫺ K ⫺
L⫺M⫺T
Net social benefits
(private consumer
surplus ⫹ producer surplus ⫹
benefit from externality ⫺
government cost)
A⫹B⫹E⫹F⫹
G⫹H⫹J⫹R
A⫹B⫹E⫹F⫹G⫹
H⫹J⫹M⫹N⫹R
M⫹N
FIGURE 17.6 Optimal Subsidy with a Positive Externality
With a positive externality, the marginal social benefit MSB equals the marginal private benefit MPB
plus the marginal external benefit MEB. In a competitive market with no correction for the externality, the equilibrium is determined by the intersection of the demand curve (i.e., the marginal
private benefit curve MPB) and the supply curve. The equilibrium price is P1 and the quantity is Q1.
The socially optimal output is Q*, determined by the intersection of the supply curve and
the marginal social benefit curve. The externality leads the market to underproduce by the
amount (Q* ⫺ Q1). The social optimum can be reached with a government subsidy. The optimal
subsidy per unit is the difference between the price received by producers P s and the price paid
by consumers P* at the efficient quantity Q*. The optimal subsidy eliminates the deadweight
loss (area M ⫹ N) that would arise without the subsidy.
17.2 EXTERNALITIES
A P P L I C A T I O N
17.5
Knowledge Spillovers
and Innovation
Economists have long recognized that an important
positive externality is knowledge spillovers. Exchange
of new ideas, and learning from the creativity of
others, often inspire innovations by others. For example, one pharmaceutical firm may develop a new
blockbuster drug using a specific type of organic
molecule. Other firms may be inspired by this development to focus research and development on similar
molecules, which may lead to additional new drugs.
Because of such knowledge spillovers, governments
often subsidize investments in research and development, especially through universities.
Like most externalities, knowledge spillovers
arise because of imperfect property rights. If a firm
could obtain legal protection for all of the economic
applications of a new idea (via patent, copyright, or
trademark protection), it would certainly do so. The
firm would then be able to profit from all of these
applications, possibly by selling or renting the rights
to some of those applications to others. In principal
this would lead to greater innovation. However, as
we have discussed earlier in the text, there is a tradeoff, since such protections would also create monopoly profits for the firm. In addition, it is not obvious
that a single firm would be able to profitably exploit
all of the possible applications of its new ideas.
Creativity often arises from combining ideas and information from different people or firms, applying
one idea in an unexpected new setting. For these
reasons, economists generally argue that knowledge
spillovers have strong positive effects on innovation
and economic growth.
Economists who study innovation describe two
relevant sources of knowledge spillovers. The first is
MAR Spillovers, named after the economists who first
analyzed them, Alfred Marshall (in 1890), Kenneth
Arrow, and Paul Romer. MAR spillovers occur when
there is a concentration of firms in the same industry
8
715
located in the same geographic area. Silicon Valley
is a prime example of MAR spillovers, with thousands of high-technology companies located in a
small area. When there is a concentration of firms
using related technologies, employees from different firms are more likely to interact with each other
professionally or socially, or switch employers. These
interactions increase the likelihood that new ideas
will proliferate across firms, generating additional
innovations.
The second form of knowledge spillovers is Jacobs
Spillovers, named after Jane Jacobs. She argued that
knowledge spillovers may be created by having a concentration of firms from diverse industries (in contrast
to MAR spillovers), because innovations in one industry may be applicable in other industries too. Indeed,
Jacobs spillovers are based on the idea that much creativity comes from interdisciplinary interactions. This
same idea is why universities are now building interdisciplinary research labs, hoping for new innovations
across departments (e.g., application of information
technology to medicine).
A recent study by Gerard Carlino summarizes
prior empirical research on these questions and provides some new evidence.8 Evidence suggests that
spillovers from concentration of firms in the same
area are indeed important to innovation. For example, in the 1990s 92 percent of all patents went to
residents of metropolitan areas, even though metropolitan areas comprise only 75 percent of the U.S.
population. For example, San Jose, California had
17.6 patents per 10,000 citizens, compared to 2.5
nationally. Table 17.1 shows the top and bottom 10
U.S. cities, ranked by number of patents per 10,000
citizens in the 1990s. Economists have also studied
which existing patents are cited by a new patent
application. Cited patents are 5–10 times more likely
to originate in the same metropolitan area as the
new patent, providing strong evidence that personal
interactions between employees across firms create
knowledge spillovers.
Gerald Carlino, “Knowledge Spillovers: Cities’ Role in the New Economy,” Business Review, Federal
Reserve Bank of Philadelphia, Quarter 4, 2001.
716
CHAPTER 17
TABLE 17.1
EXTERNALITIES AND PUBLIC GOODS
Patents per 10,000 Citizens, U.S. Metropolitan Areas
Top 10
Bottom 10
Metropolitan Area
Patents per
10,000 Citizens
San Jose, CA
Boise City, ID
Rochester, NY
Boulder, CO
Trenton, NJ
Burlington, VT
Rochester, MN
Poughkeepsie, NY
Ann Arbor, MI
Austin, TX
17.6
14.1
13.0
11.2
10.5
9.0
9.0
8.8
8.3
8.0
Metropolitan Area
Rockford, IL
Cincinnati, OH
Hartford, CT
Monmouth-Ocean, NJ
Akron, OH
Allentown, PA
Greeley, CO
Seattle, WA
Kalamazoo, MI
Sheyboygan, WI
Patents per
10,000 Citizens
4.0
3.9
3.8
3.8
3.8
3.8
3.8
3.8
3.8
3.8
Source: Carlino (2001).
P R O P E R T Y R I G H T S A N D T H E C OA S E T H E O R E M
property right The
exclusive control over the
use of an asset or resource.
So far we have examined how the government might correct for externalities using
taxes (emissions fees and tolls) and regulating quantity (emissions standards). As an
alternative, the government can assign a property right, that is, the exclusive control
over the use of an asset or resource, without interference by others.
Why are property rights important in dealing with externalities? Let’s return to
our example of a chemical manufacturing process that emits pollution as a by-product.
When we described the negative externality, we observed that manufacturers did not
have to compensate anyone when they released pollutants into the air. That is why
the firms based their production decisions on private marginal costs that did not
include the harm that pollution brought to the environment. The costs of pollution
were external to the manufacturers.
In that example we also assumed that no one in the surrounding community had
a legal right to clean air. If the community owned a property right to clean air, it could
have required firms to compensate it for the right to pollute. If a firm were to continue producing the chemical, its marginal private cost would then include the cost of
pollution. In other words, the costs of pollution would be internal to the firm instead
of external.
In 1960 Ronald Coase developed a fundamental theorem demonstrating how
the problem of externalities could be addressed by assigning property rights.9 He
illustrated the idea with an example involving two farms. Farm A raises cattle, and
the cattle occasionally stray onto the land of a neighboring farm, Farm B, which
raises crops. Farm A’s cattle impose a negative externality by damaging the crops
on Farm B.
9
Ronald H. Coase, “The Problem of Social Cost,” Journal of Law and Economics 3 (1960): 1–44.
17.2 EXTERNALITIES
Coase addressed the following issues: Should the cattle be allowed to roam on the
property of Farm B? Can the owner of Farm B require the owner of Farm A to construct a fence to restrain the cattle? If so, who should pay for the fence? Does it
matter whether the property rights are assigned to the owners of Farm A or Farm B?
The Coase Theorem states that, regardless of how property rights are assigned
with an externality, the allocation of resources will be efficient when the parties can
costlessly bargain with each other. If the owner of A has the right to let his cattle roam
on B’s land, B’s owner will pay A’s owner to build a fence when the damage to B’s crops
exceeds the cost of the fence. If the cost of the fence exceeds the damage to the crops,
it will not be in the interest of owner B to pay for the fence, and the cattle will roam.
In other words, when it is socially efficient to construct the fence, the fence will be
built to eliminate the externality.
Suppose, instead, that the property rights are assigned to owner B, so that A has
to compensate B for any damage. Owner A would build a fence if the damage to B’s
crops exceeds the cost of the fence. However, if the cost of the fence is greater than
the damage to the crops, then owner A will compensate owner B for the damage, and,
once again, the cattle will roam.
The example nicely demonstrates the remarkable point of the Coase Theorem.
Regardless of whether the property rights are assigned to the owner of Farm A or to
the owner of Farm B, the outcome is the same and it is socially efficient. The fence
will be built when the fence costs less than the damage to the crops, and it will not be
built when the fence costs more than the damage.
S
E
717
Coase Theorem The
theorem which states that
regardless of how property
rights are assigned with an
externality, the allocation of
resources will be efficient
when the parties can costlessly bargain with each
other.
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 7 . 3
D
The Coase Theorem
Problem
(a) In the case of the roaming cattle just described, suppose it is costless for the parties to bargain. Verify the
Coase Theorem when the cost of the fence is $2,000 and
the cost of the damage is $1,000.
(b) Verify the Coase Theorem if the fence costs $2,000
and the damage cost is $4,000.
Solution
(a) Suppose the property rights are assigned to A.
Owner B can either pay for a fence costing $2,000, or
live with the damage of $1,000. B therefore does not find
it worthwhile to pay for a fence, and the cattle will roam.
Owner B receives no compensation for the damage of
$1,000.
Suppose the property rights are assigned to B.
Owner A can either spend $2,000 to build a fence to prevent damage or build no fence and pay $1,000 to owner
B to compensate for damage. Owner A does not find it
worthwhile to pay for a fence, and the cattle will roam.
The damage to B is $1,000, but A will compensate B.
With either property rights assignment, the outcome is the same: the cattle will roam. It is economically
efficient to build no fence because the fence costs more
than the damage from roaming cattle.
(b) Suppose the property rights are assigned to A.
Owner B now finds it worthwhile to pay for a fence, and
the cattle will not roam.
Suppose the property rights are assigned to B.
Owner A now finds it worthwhile to pay for a fence, and
the cattle will not roam.
Once again, with either assignment of the property
right, the outcome is the same: the cattle will not roam.
It is economically efficient to pay for the fence because
the fence costs less than the damage that would have occurred from roaming cattle.
Similar Problems:
17.20
17.14, 17.15, 17.16, 17.19,
718
CHAPTER 17
EXTERNALITIES AND PUBLIC GOODS
While the Coase Theorem claims that the allocation of resources will be economically efficient, regardless of the assignment of property rights, the distribution of
resources very much depends on who holds the property rights. In Learning-By-Doing
Exercise 17.3, suppose the cost of the fence is $2,000 and the cost of the damage is
$1,000. No one pays for a fence. Thus, the owner of the property rights is $1,000 better
off than he or she would be without the property rights.
If the cost of the damage is $4,000, someone will pay for a fence. If A owns the
property rights, B pays for the fence. However, if B owns the rights, A pays for it.
Thus, the owner of the property rights is $2,000 better off than he or she would be
without the property rights.
In this example, the “bargaining” between the parties is extremely simple once the
property rights are defined. If any money is transferred between the parties, the
amount of the transfer is the lesser of two amounts: the cost of the fence or the cost
of the damage to the crops.
Coase did not explore richer opportunities for bargaining in his work. However, his
ideas can be applied to more complex settings where bargaining is possible. Suppose the
cost of crop damage is $4,000 if the cattle stray to Farm B, but now let’s add another
fencing option. The cost of fencing owner A’s property is $2,000; alternatively, at a cost
of $3,000 owner B could build a fence around his property to keep the cattle out.
What happens when we assign the property rights to owner B? Owner A has three
options: (1) fence in Farm A at a cost of $2000, (2) offer owner B $3,000 to fence in
Farm B, or (3) let the cattle roam and pay owner B $4,000 to cover crop damage. To
minimize his cost, owner A will fence in Farm A.
Suppose the property rights belong to owner A. Owner B has three options:
(1) fence in Farm B at a cost of $3,000, (2) offer owner A a payment (to be discussed
below) to fence in Farm A, or (3) do nothing and incur $4,000 worth of crop damage.
Under the second option, there is now room for bargaining. Owner B would be willing to offer owner A up to $3,000 if A will fence in his property. (Owner B would offer
no more than $3,000 to A because B can fence in Farm B at that cost.) At the same
time, owner A will accept no less than $2,000 to fence in his property. There is an opportunity for both parties to be better off if they agree that B will pay A some amount
between $2,000 and $3,000 to fence in Farm A. For example, the two parties may
agree to split the difference, with owner A receiving a payment of $2,500 to build a
fence around his farm.
As before, the outcome is the same, regardless of who owns the property right:
Farm A will be fenced. Further, the outcome is socially efficient because the cost to
fence in Farm A is less than the cost to fence in Farm B and less than the damage
caused to the crop farmer if the cattle roam.
To summarize, the Coase Theorem shows that, as long as bargaining is costless,
assigning property rights for an externality leads to an efficient outcome, regardless of
who owns the rights. However, this powerful proposition depends crucially on the assumption that bargaining is costless. If the bargaining process itself is costly, then the
parties might not find it worthwhile to negotiate. Consider our earlier example of the
manufacturers who pollute the air as they produce a chemical. If pollution harms
thousands of people, it may not be easy for the victims of the negative externality to
organize themselves to bargain about compensation. Similarly, if there are many firms
in the industry, it may also be costly for them to organize.
There are other potential difficulties with bargaining. If the parties do not know the
costs and benefits of reducing the externality, or if they have different perceptions about
these costs and benefits, then bargaining may not lead to an efficient outcome. Finally,
719
17.3 PUBLIC GOODS
both parties must be willing to enter into agreements that are mutually beneficial. If one
of the parties simply refuses to bargain or refuses to give the other party an acceptable
compensation, it may not be possible to achieve an efficient resource allocation.
We have now learned why a competitive market fails to produce the socially opti- 17.3
mal output when there are externalities. For goods with positive externalities, consumers make purchasing decisions based on the marginal private benefits, which are
lower than marginal social benefits. Thus, the market produces a lower quantity than
the social optimum. Private benefits may be so low that a good is simply not provided
at all, even though production of the good would lead to positive net social benefits.
In this section we examine another kind of good that will be undersupplied by the
market, public goods. Public goods benefit all consumers even though individual consumers do not pay for the provision of the good. Public goods have two characteristics: They are nonrival goods and nonexclusive goods.
With a nonrival good, consumption by one person does not reduce the quantity
that can be consumed by others. An example of a nonrival good is public broadcasting.
When one viewer tunes in, the number of others who can watch or listen is not diminished. National defense is also a nonrival good. When one person in a community receives protection, the amount of protection available to other consumers is not reduced.
The marginal cost of providing output to another consumer of a nonrival good is zero.
By contrast, most goods we encounter in everyday life are rival goods. With a
given level of production of a rival good, the consumption of the good by one person
reduces the amount available to others. For example, when you buy a pair of jeans, a
soccer ball, or a computer, you have foreclosed the possibility that anyone else can buy
that particular item.
A nonexclusive good is a good that, once produced, is accessible to all consumers; no one can be excluded from consuming the good after it is produced. Once
a nonexclusive good is produced, a consumer can benefit from the good even if he
does not pay for it. Examples of nonexclusive goods are abundant, including national
defense, public parks, television and radio signals, and artwork in public places. By
contrast, an exclusive good is one to which consumers may be denied access.
Many goods are both exclusive and rival. Examples include computers, paintings,
items of clothing, and automobiles. Suppose a manufacturer makes 1,000 automobiles. When a consumer buys one of them, only 999 are left for others to purchase
(i.e., the good is rival). In addition, the manufacturer can deny consumers access to the
automobile—to enjoy the benefits of an automobile, the consumer must pay for it
(i.e., the good is exclusive).
Some goods are nonexclusive but rival. Anyone may reserve a picnic table at a
public park, but when one person reserves the table on a given day, it is not available
to others at that time. Hunting in public game areas is nonexclusive because everyone
has access to the game; however, hunters reduce the stock of game left for others when
they bag their quarry.
Finally, a good can be nonrival but exclusive. A pay-TV channel is exclusive
because producers can scramble the channel to control access. But the channel is also
nonrival. When someone purchases the right to view the channel, this action does not
reduce the opportunity for other viewers to do the same.
As we have observed, public goods, such as national defense and public broadcasting,
are both nonrival and nonexclusive. To avoid confusion as we study public goods, it is important to keep in mind that many goods that are publicly provided are not public goods,
PUBLIC
GOODS
nonrival good When
consumption of a good by
one person does not reduce
the quantity that can be
consumed by others.
rival goods When consumption of a good by one
person reduces the quantity
that can be consumed by
others.
nonexclusive good A
good that, once produced, is
accessible to all consumers;
no one can be excluded
from consuming such a
good after it is produced.
exclusive good A
good to which consumers
may be denied access.
720
CHAPTER 17
EXTERNALITIES AND PUBLIC GOODS
being either rival or exclusive or even both. For example, because a public university has
a limited capacity, education there can be a rival good. When one student enrolls, another
prospective student might be displaced. Further, education at a public university can be
an exclusive good because the university can deny admission to an applicant and because
the university can exclude any student who does not pay the required tuition.
E F F I C I E N T P R OV I S I O N O F A P U B L I C G O O D
How much of a public good should be provided to maximize net social benefits? As
with other goods, a public good should be provided as long as the marginal benefit of
an additional unit is at least as great as the marginal cost of that unit. The marginal
cost of a public good is the opportunity cost of using economic resources to produce
that good rather than other goods. Because public goods are nonrival, many consumers may enjoy the benefits of an additional unit. The marginal benefit is thus the
sum of the benefits of all the people who value the additional unit.
Figure 17.7 illustrates the efficient level of production for a public good. For simplicity, let’s assume that there are only two consumers in the market. D1 is the demand
curve for the public good by the first consumer, and D2 is the demand curve for the
second consumer. The height of a consumer’s demand curve at any quantity shows
the marginal benefit of an additional unit of the good to that consumer. For example,
the first consumer has a marginal benefit of $30 per year for the 70th unit. The second
consumer has a marginal benefit of $130 for the same unit.
MC = 400
$400
FIGURE 17.7
300 E
MSB
Price (dollars per unit)
Efficient Provision of a
Public Good
The marginal social benefit of a public good is
the vertical sum of the
demand curves for the
consumers in the market.
The marginal social benefit curve is EGH. When
the marginal cost of the
public good is $240, the
economically efficient
level of production is
30 units, the output at
which the marginal cost
and marginal social benefit curves intersect.
If the marginal cost is
$50, the efficient level of
production is 150 units;
if the marginal cost is
$400, it is inefficient to
provide the good at all.
240
MC = 240
F
200
170
160
130
D2
100
G
70
50
K
D1
30
0
10
30
70
MC = 50
H
100
150
Quantity of the public good (units)
200
17.3 PUBLIC GOODS
721
Because the public good is nonexclusive, both consumers have access to the good.
Thus, the marginal social benefit of the 70th unit is just the vertical sum of the marginal benefits for the two consumers: $130 ⫹ $30 ⫽ $160. In Figure 17.7, the marginal
social benefit curve is the kinked curve EGH. Between G and H (that is, when Q ⬎ 100)
the marginal social benefit curve coincides with D2 because the first consumer is not
willing to pay anything for these units. (Beyond point H—that is, when Q ⬎ 200—the
marginal social benefit curve coincides with the horizontal access because neither consumer is willing to pay anything for those units.)
We can now determine the economically efficient level of production for the public good. Suppose that the marginal cost of the public good is $240. The economically
efficient quantity is the quantity at which marginal social benefit equals marginal cost,
or 30 units. It would not be efficient to produce more than 30 units because the marginal cost would exceed the marginal social benefit for each additional unit produced.
For example, as we have already shown, the marginal social benefit of the 70th unit is
$160. However, this is less than the marginal cost, $240. Therefore, it would not be
socially efficient to provide the 70th unit of the public good.
Similarly, it would not be efficient to produce less than 30 units of the good. Over
this range of production, the marginal social benefit exceeds the marginal cost. Thus,
it would be economically efficient to expand production until the marginal social benefit just equals the marginal cost.
At the efficient level of output of 30 units, the marginal benefit for the first consumer
is $70, and the marginal benefit for the second consumer is $170. Thus, the marginal
social benefit of the 30th unit is $240, which just equals the marginal cost of that unit.
This example shows that it may be socially optimal to provide the good even if no
consumer alone is willing to pay enough to cover the marginal cost. Because the good
is nonrival, marginal social benefit is the sum of the willingness to pay by all consumers, not simply the willingness to pay by any individual alone.
Learning-By-Doing Exercise 17.4 will help you better understand how to find the
optimal amount of a public good, both graphically and algebraically. It will also help
you understand how to sum demand curves vertically.
S
E
L E A R N I N G - B Y- D O I N G E X E R C I S E 1 7 . 4
D
Optimal Provision of a Public Good
In Figure 17.7, demand curve D1 is P1 ⫽
100 ⫺ Q, and demand curve D2 is P2 ⫽ 200 ⫺ Q. (We
have written these in inverse form, with price on the left
and quantity on the right, for reasons explained below.)
Problem
(a) Suppose the marginal cost of the public good is
$240. Determine the efficient level of production of the
public good algebraically.
(b) Suppose the marginal cost of the public good is $50.
Determine the efficient level of production of the public
good both graphically and algebraically.
(c) Suppose the marginal cost of the public good is
$400. Determine the efficient level of production of the
public good both graphically and algebraically.
Solution
(a) The marginal social benefit curve MSB with a public
good is the vertical sum of the individual consumer
demand curves. When we sum vertically, we add prices
(i.e., willingness to pay); thus, MSB ⫽ P1 ⫹ P2 ⫽ (100 ⫺
Q) ⫹ (200 ⫺ Q) ⫽ 300 ⫺ 2Q. At the efficient level of
production, MSB ⫽ MC, or 300 ⫺ 2Q ⫽ 240, or Q ⫽ 30
units. (As noted above, we need to use the inverse form
of the demand curves in order to add prices.)
(b) If the marginal cost is $50, we find the efficient level
of production graphically by finding the intersection of
the MSB and MC curves. As shown in Figure 17.7, this
occurs at point K, where Q ⫽ 150 units. To find this optimum algebraically, we must recall that P1 ⫽ 0 when
Q ⬎ 100. In this case, then, MSB ⫽ P1 ⫹ P2 ⫽ 0 ⫹ P2 ⫽
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EXTERNALITIES AND PUBLIC GOODS
P2 200 ⫺ Q. When MSB ⫽ MC, 200 ⫺ Q ⫽ 50, or
Q ⫽ 150.
(c) If the marginal cost is $400, the marginal cost curve
lies above the entire marginal social benefit curve, as
shown in Figure 17.7. Therefore, it is not efficient to
produce any of the public good. Algebraically, if MSB ⫽
MC, then 300 ⫺ 2Q ⫽ 400, or Q ⫽ ⫺50. This tells us
that the MSB and MC curves do not intersect when Q ⬎ 0
(i.e., there is no positive efficient level of production of
the public good).
Here is a hint that you may find useful in adding
demand curves. First, you need to know whether you
should add the demand curves vertically or horizontally.
As we have shown in this chapter, if you need to find the
optimal level of a public good, you need to add demands
vertically. To add the demand curves vertically, write the
individual demand curves as inverse demands and then
add them up, as we have just done.
By contrast, in Chapter 5 we showed that, to construct
an ordinary market demand curve from individual demand
curves, you must add the demand curves horizontally
because you want to know the total quantity demanded at
any price. The goods we considered in Chapter 5 were
rival goods. That is why we did not add consumers’ willingness to pay to determine the value of an extra unit of
the good. To add the demand curves horizontally, write the
individual demand curves in their normal form, with Q on
the left-hand side and P on the right-hand side. To review
how to add demand curves horizontally, you might refer to
the discussion following Table 5.1.
Similar Problems:
17.21, 17.22, 17.23, 17.24, 17.25
THE FREE-RIDER PROBLEM
free rider A consumer
or producer who does not
pay for a nonexclusive
good, anticipating that
others will pay.
There are often thousands, or even millions, of consumers of public goods such as a dam,
a public park, or public broadcasting. To finance an efficient level of output for a public
good, consumers must jointly agree that everyone contributes an amount equal to his
own willingness to pay. However, since the provision of a public good is nonexclusive,
everyone benefits once the public good is provided. Consequently, individuals have no
incentive to pay as much as the good is really worth to them. A consumer can behave as
a free rider, paying nothing for a good while anticipating that others will contribute.
A P P L I C A T I O N
17.6
Free Riding on the Public Airwaves10
Public television and public radio are examples of
public goods. They are nonrival and nonexclusive.
With millions of viewers, it is not surprising that there
are many free riders in public broadcasting.
PBS (Public Broadcasting System), a private, nonprofit
media enterprise, provides much of the programming
for the (approximately) 365 public television stations in
the United States. Each week public television serves
nearly 65 million viewers. But most viewers are free riders.
Fewer than 5 million individuals and families contribute
to public television each year, with donations, pledges,
and membership fees that compromise approximately
25 percent of PBS’s total revenues. PBS receives another
18 percent from businesses, about 8 percent from foun-
10
dations, and 10 percent from miscellaneous sources.
Roughly 40 percent of all of its funding comes from
federal, state, and local governments.
The story is much the same for public radio. NPR
(National Public Radio) is a private, nonprofit company with approximately 800 member radio stations
and about 26 million listeners per week. However,
only 32 percent of its funding comes from subscribers.
About 21 percent comes from businesses and 10 percent from foundations; 16 percent of NPR’s funding
comes from the government at various levels. Because
of the free-rider problem, funds to support public broadcasting must come from a variety of other sources. For
decades governmental subsidies have remained important for the financial viability of the industry.
“Public Broadcasting Revenue, Fiscal Year 2008,” Corporation for Public Broadcasting (2009).
REVIEW QUESTIONS
723
The free-rider problem makes it difficult for a private market to provide public
goods efficiently. It is generally easier to organize effective efforts to collect voluntary
funding when the number of people involved in paying for a project is small because
each person recognizes that his or her contribution is important. However, when the
number of consumers of a public good becomes large, it is more likely that many consumers will act as free riders. Public intervention may be necessary to ensure the provision of a socially beneficial public good. The government therefore often produces
a public good itself or subsidizes the enterprises that produce the good.
CHAPTER SUMMARY
• An externality arises when the actions of any decision
maker, either a consumer or a producer, affect the benefits of other consumers or production costs of other
firms in the market in ways other than through changes
in prices. An externality that reduces the well-being of
others is a negative externality. An externality that brings
benefit to others is a positive externality.
• With a positive externality (like education or immunization to prevent the spread of contagious diseases), the
private marginal benefit is less than the social marginal
benefit. Consequently, a competitive market produces
less of the positive externality than is socially optimal.
The government may attempt to improve efficiency by
stimulating output with a production subsidy.
• Externalities cause market failure in competitive markets. With an externality, the invisible hand does not lead
an otherwise competitive market to produce an economically efficient level of the good.
• Inefficiencies arising from externalities may be eliminated if property rights to externalities are clearly assigned
and parties can bargain. The Coase Theorem shows that
when parties can costlessly bargain, the outcome of the
bargain will be economically efficient, regardless of which
party holds the property rights. However, it may be difficult to achieve an efficient outcome with bargaining if
there are many parties involved, or if bargaining is a costly
process. Although the assignment of the property rights
does not affect economic efficiency, it will affect the distribution of income. (LBD Exercise 17.3)
• With a negative production externality (like pollution), the private marginal cost to a producer is less
than the social marginal cost. With a negative consumption externality (like secondhand smoke from
cigarettes), a consumer does not pay for the cost of his
own actions imposed on other people. Consequently, a
competitive market produces more of the good than is
socially optimal. The government may attempt to improve economic efficiency by reducing the amount of
the good by imposing a quota (such as an emissions
standard) or a tax (such as an emissions fee). (LBD
Exercises 17.1, 17.2)
• A public good is a good that is nonrival and nonexclusive. The marginal social benefit curve for a public
good is the vertical sum of the individual demand curves
for that good. A public good is provided efficiently when
its marginal social benefit equals its marginal cost.
• Negative externalities can also arise in markets that
involve a common property (a resource anyone can access). With common property, the negative externality
of congestion often occurs. In such cases, government
can impose a tax on use of the common property in
order to achieve economic efficiency.
• A public good is likely to be underproduced because
consumers often act as free riders, benefiting from the
good but not paying for it. To ensure the provision of a
socially beneficial public good, the government often
produces the good itself or subsidizes enterprises that
produce the good. (LBD Exercise 17.4)
REVIEW QUESTIONS
1. What is the difference between a positive externality
and a negative externality? Describe an example of each.
2. Why does an otherwise competitive market with a
negative externality produce more output than would be
economically efficient?
3. Why does an otherwise competitive market with a
positive externality produce less output than would be
economically efficient?
4. When do externalities require government intervention, and when is such intervention unlikely to be necessary?
724
CHAPTER 17
EXTERNALITIES AND PUBLIC GOODS
5. How might an emissions fee lead to an efficient level
of output in a market with a negative externality?
8. How does a nonrival good differ from a nonexclusive
good?
6. How might an emissions standard lead to an efficient
level of output in a market with a negative externality?
9. What is a public good? How can one determine the
optimal level of provision of a public good?
7. What is the Coase Theorem, and when is it likely
to be helpful in leading a market with externalities to
provide the socially efficient level of output?
10. Why does the free-rider problem make it difficult or
impossible for markets to provide public goods efficiently?
PROBLEMS
17.1. Why is it not generally socially efficient to set an
emissions standard allowing zero pollution?
17.2. Education is often described as a good with positive externalities. Explain how education might generate
positive external benefits. Also suggest a possible action
the government might take to induce the market for
education to perform more efficiently.
17.3. a) Explain why cigarette smoking is often described
as a good with negative externalities.
b) Why might a tax on cigarettes induce the market for
cigarettes to perform more efficiently?
c) How would you evaluate a proposal to ban cigarette
smoking? Would a ban on smoking necessarily be economically efficient?
17.4. Consider Learning-By-Doing Exercise 17.2,
with a socially efficient emissions fee. Suppose a technological improvement shifts the marginal private cost
curve down by $1. If the government calculates the optimal fee given the new marginal private cost curve, what
will happen to the following?
a) The size of the optimal tax
b) The price consumers pay
c) The price producers receive
17.5. Consider the congestion pricing problem illustrated in Figure 17.5.
a) What is the size of the deadweight loss from the negative externalities if there is no toll imposed during the
peak period?
Equilibrium Price
and Quantity ⴝ
Consumer surplus
Private producer surplus
ⴚCost of externality
Net social benefits
Deadweight loss
b) Why is the optimal toll during the peak period not
$3, the difference between the marginal social cost
and the marginal private cost when the traffic volume
is Q5?
c) How much revenue will the toll authority collect per
hour if it charges the economically efficient toll during
the peak period?
17.6. The accompanying graph (on next page) shows
the demand curve for gasoline and the supply curve for
gasoline. The use of gasoline creates negative externalities, including CO2, which is an important source of
global warming. Using the graph and the table below,
identify:
• The equilibrium price and quantity of gasoline
• The producer and consumer surplus at the market
equilibrium
• The cost of the externality at the free-market
equilibrium
• The net social benefits arising at the free-market
equilibrium
• The socially optimal price of gasoline
• The consumer and producer surplus at the social
optimum
• The cost of the externality at the social optimum
• The net social benefits arising at the social
optimum
• The deadweight loss due to the externality
Social Optimum
Price and Quantity ⴝ
Difference Between Social
Optimum and Equilibrium
PROBLEMS
MSC
Price of gasoline per week
S
A
P1
M
G K
H N
B
P2
E
F
MEC
R
D
Z
V
Q1 Q2
Gallons of gasoline per week
(Graph for Problem 17.6)
17.7. The graph below shows conditions in a perfectly
competitive market in which there is some sort of externality. In this market, a consumer purchases at most one
unit of the good. There are many such consumers, and
725
they have different maximum willingnesses to pay.
Assume that the graph is drawn to scale.
a) What type of externality is present in this market: positive or negative?
b) What is the maximum level of social surplus that is potentially attainable in this market?
c) What is the deadweight loss that arises in a competitive equilibrium in this market?
d) Suppose a subsidy is given to producers: What
is the magnitude of the subsidy per unit that would
enable this market to attain the socially efficient outcome?
For the remaining questions, please indicate whether
the following government interventions would increase
social efficiency relative to the competitive equilibrium
outcome with no government intervention, decrease
social efficiency, or keep it unchanged:
e) A subsidy per unit equal to 0F given to consumers who
purchase the good.
f ) The government replaces private sellers and offers
the good at a price of zero. (Assume that government has
no inherent cost advantage or disadvantage relative to
private producers. Assume, too, the government’s cost of
production is financed by levying taxes.)
$ per unit
Supply curve = MPC
A
Demand curve = MPB
U
V
G
N
B
K
C
D
H
R
E
F
0
MSC
L
I
J
M
S
T
g) The government imposes a price ceiling that sets a
maximum price for the good equal to 0D.
W
Quantity
h) The government imposes a tax equal to NR on consumers who do not purchase the good.
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17.8. A competitive refining industry produces one
unit of waste for each unit of refined product. The industry disposes of the waste by releasing it into the atmosphere. The inverse demand curve for the refined
product (which is also the marginal benefit curve) is
P d 24 ⫺ Q, where Q is the quantity consumed when
the price consumers pay is P d. The inverse supply curve
(also the marginal private cost curve) for refining is
MPC ⫽ 2 ⫹ Q, where MPC is the marginal private cost
when the industry produces Q units. The marginal
external cost curve is MEC ⫽ 0.5Q, where MEC is the
marginal external cost when the industry releases Q units
of waste.
a) What are the equilibrium price and quantity for the
refined product when there is no correction for the
externality?
b) How much of the chemical should the market supply
at the social optimum?
c) How large is the deadweight loss from the externality?
d) Suppose the government imposes an emissions fee of
$T per unit of emissions. How large should the emissions fee be if the market is to produce the economically
efficient amount of the refined product?
17.9. Consider a manufactured good whose production process generates pollution. The annual demand
for the good is given by Qd ⫽ 100 ⫺ 3P. The annual
market supply is given by Qs ⫽ P. In both equations, P
is the price in dollars per unit. For every unit of output
produced, the industry emits one unit of pollution.
The marginal damage from each unit of pollution is
given by 2Q.
a) Find the equilibrium price and quantity in a market
with no government intervention.
b) At the equilibrium you computed, calculate: (i) consumer surplus; (ii) producer surplus; (iii) total dollars of
pollution damage. What are the overall social benefits in
the market?
c) Find the socially optimal quantity of the good. What is
the socially optimal market price?
d) At the social optimum you computed, calculate: (i) consumer surplus; (ii) producer surplus; and (iii) total dollars
of pollution damage. What are the overall social benefits
in the market?
e) Suppose an emissions fee is imposed on producers.
What emissions fee would induce the socially optimal
quantity of the good?
17.10. The demand for widgets is given by P ⫽ 60 ⫺
Q. Widgets are competitively supplied according to the
inverse supply curve (and marginal private cost) MPC ⫽
c. However, the production of widgets releases a toxic
gas into the atmosphere, creating a marginal external
cost of MEC ⫽ Q.
a) Suppose the government is considering imposing a tax
of $T per unit. Find the level of the tax, T, that ensures
the socially optimal amount of widgets will be produced
in a competitive equilibrium.
b) Suppose a breakthrough in widget technology lowers
the marginal private cost, c, by $1. How will this effect
the optimal tax you found in part (a)?
17.11. The market demand for gadgets is given by
P d ⫽ 120 ⫺ Q, where Q is the quantity consumers demand
when the price they consumers pay is P d. Gadgets are
competitively supplied according to the inverse supply
curve (and marginal private cost) MPC ⫽ 2Q, where Q is
the amount suppliers will produce when they receive a
price equal to MPC. The production of gadgets releases
a toxic effluent into the water supply, creating a marginal
external cost of MEC ⫽ Q. The government wants to
impose a sales tax on gadgets to correct for the externality.
When producers receive a price equal to MPC, the
amount consumers must pay is (1 ⫹ t)MPC, where t is
the sales tax rate. Find the level of the tax rate that ensures
the socially optimal amount of gadgets will be produced
in a competitive equilibrium.
17.12. Amityville has a competitive chocolate industry
with the (inverse) supply curve P s ⫽ 440 ⫹ Q. While the
market demand for chocolate is P d ⫽ 1200 ⫺ Q, there are
external benefits that the citizens of Amityville derive from
having a chocolate odor wafting through town. The marginal external benefit schedule is MEB ⫽ 6 ⫺ 0.05Q.
a) Without government intervention, what would be the
equilibrium amount of chocolate produced? What is the
socially optimal amount of chocolate production?
b) If the government of Amityville used a subsidy of $S
per unit to encourage the optimal amount of chocolate
production, what level should that subsidy be?
17.13. The only road connecting two populated islands is currently a freeway. During rush hour, there is
congestion because of the heavy traffic. The marginal external cost from congestion rises as the amount of traffic
on the road increases. At the current equilibrium, the
marginal external cost from congestion is $5 per vehicle.
Would a toll charge of $5 per vehicle lead to an economically efficient amount of traffic? If not, would you expect
the economically efficient toll to be larger than, or less
than $5?
17.14. A firm can produce steel with or without a filter
on its smokestack. If it produces without a filter, the external costs on the community are $500,000 per year. If
it produces with a filter, there are no external costs on the
community, and the firm will incur an annual fixed cost
of $300,000 for the filter.
a) Use the Coase Theorem to explain how costless bargaining will lead to a socially efficient outcome, regardless
727
PROBLEMS
of whether the property rights are owned by the community or the producer.
b) How would your answer to part (a) change if the extra
yearly fixed cost of the filter were $600,000?
17.15. Two farms are located next to each other. During
storms, sewage from Farm 1 flows into a stream located
on Farm 2. Farm 2 relies on this stream as a source of
drinking water for its livestock, and when the stream is
polluted with sewage, the livestock become sick and die.
The annual damage to Farm 2 from this form of pollution
is $100,000 per year. It is possible that Farm 1 can prevent
the runoff of sewage by installing storm drains. The cost
of the storm drains is $200,000.
a) Provide an argument that the Coase Theorem holds in
this situation.
b) Suppose that the damage to Farm 2 is $500,000 per
year, not $100,000 per year (with the cost of storm drains
remaining fixed at $200,000). Provide an argument that
the Coase Theorem holds in this case.
17.16. Suppose a factory located next to a river discharges pollution that causes $2 million worth of environmental damage to the residents downstream. The factory
could completely eliminate the pollution by treating the
water on location at a cost of $1.6 million. Alternatively,
the residents could construct a water purification plant
just upstream of their town, at a cost of $0.8 million,
which would not completely eliminate the environmental
damage to them but reduce it to $0.5 million. Under current law, the factory must compensate the town for any
environmental damage the factory causes. Bargaining
between the factory owner and the town is costless. What
would the Coase Theorem imply about the outcome of
bargaining between the town and the factory owner?
17.17. The demand for energy-efficient appliances
is given by P 100/Q, while the inverse supply (and
marginal private cost) curve is MPC Q. By reducing
demand on the electricity network, energy-efficient appliances generate an external marginal benefit according
to MEB eQ.
a) What is the equilibrium amount of energy-efficient
appliances traded in the private market?
b) If the socially efficient number of energy-efficient
appliances is Q 20, what is the value of e?
c) If the government subsidized production of energyefficient appliances by $S per unit, what level of the
subsidy would induce the socially efficient level of production?
17.18. The demand for air-polluting backhoes in Peoria
is P D 48 ⫺ Q. The air pollution creates a marginal
external cost according to MEC ⫽ 2 ⫹ Q. Supply of
backhoes is given by P S ⫽ 10 ⫹ cQ. If the socially efficient
level of backhoes is Q* ⫽ 12, find the tax that induces the
socially efficient level of backhoes in equilibrium and the
value of c.
17.19. The town of Steeleville has three steel factories,
each of which produces air pollution. There are 10 citizens of Steeleville, each of whose marginal benefits from
reducing air pollution is represented by the curve p(Q) ⫽
5 ⫺ Q/10, where Q is the number of units of pollutants
removed from the air. The reduction of pollution is a
public good. For each of the three sources of air pollution,
the following table lists the current amount of pollution
being produced along with the constant marginal cost of
reducing it.
Source
Factory A
Factory B
Factory C
Units of Pollution
MC of Pollution
Currently Being Produced
Reduction
20
40
60
$10
$20
$30
a) On a graph, illustrate marginal benefits (“demand”)
and the marginal costs (“supply”) of reducing pollution.
What is the efficient amount of pollution reduction?
Which factories should be the ones to reduce pollution,
and what would the total costs of pollution reduction be?
In a private market, would any units of this public good
be provided?
b) The Steeleville City Council is currently considering
the following policies for reducing pollution:
i. Requiring each factory to reduce pollution by 10 units
ii. Requiring each factory to produce only 30 units of
pollution
iii. Requiring each factory to reduce pollution by onefourth
Calculate the total costs of pollution reduction associated
with each policy. Compare the total costs and amount of
pollution reduction to the efficient amount you found in
part (a). Do any of these policies create a deadweight
loss?
c) Another policy option would create pollution permits,
to be allocated and, if desired, traded among the firms. If
each factory is allocated tradeable permits allowing it to
produce 30 units of pollution, which factories, if any,
would trade them? (Assume zero transactions costs.) If
they do trade, at what prices would the permits be
traded?
d) How does your answer in part (c) relate to that in
part (a)? Explain how the Coase Theorem factors into
this relationship.
17.20. A chemical producer dumps toxic waste into a
river. The waste reduces the population of fish, reducing
profits for the local fishing industry by $100,000 per year.
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CHAPTER 17
EXTERNALITIES AND PUBLIC GOODS
The firm could eliminate the waste at a cost of $60,000
per year. The local fishing industry consists of many
small firms.
a) Using the Coase Theorem, explain how costless bargaining will lead to a socially efficient outcome, regardless of whether the property rights are owned by the
chemical firm or the fishing industry.
b) Why might bargaining not be costless?
c) How would your answer to part (a) change if the waste
reduces the profits for the fishing industry by $40,000?
(Assume, as before, that the firm could eliminate the
waste at a cost of $60,000 per year.)
17.21. Consider an economy with two individuals.
Individual 1 has (inverse) demand curve for a public good
given by P1 60 ⫺ 2Q1, while individual 2 has (inverse)
demand curve for the public good given by P2 ⫽ 90 ⫺
5Q2. The prices are measured in $ per unit. Suppose the
marginal cost of producing the public good is $10 per
unit. What is the efficient level of the public good?
17.22. There are three consumers of a public good.
The demands for the consumers are as follows:
Consumer 1: P1 ⫽ 60 ⫺ Q
Consumer 2: P2 ⫽ 100 ⫺ Q
Consumer 3: P3 ⫽ 140 ⫺ Q
where Q measures the number of units of the good and
P is the price in dollars.
The marginal cost of the public good is $180. What
is the economically efficient level of production of the
good? Illustrate your answer on a clearly labeled graph.
17.23. Suppose that the good described in Problem 17.22
is not provided at all because of the free rider problem.
What is the size of the deadweight loss arising from this
market failure?
17.24. In Problem 17.22, how would your answer
change if the marginal cost of the public good is $60?
What if the marginal cost is $350?
17.25. A small town in Florida is considering hiring an
orchestra to play in the park during the year. The music
from the orchestra is nonrival and nonexclusive. A careful study of the town’s music tastes reveals two types of
individuals: music lovers and intense music lovers. If
forced to pay for an outdoor concert, the demand curve
for music lovers would be Q1 ⫽ 100 ⫺ (1/20)P1, where
Q1 is the number of concerts that would be attended and
P1 is the price per (hypothetical) ticket (in dollars) to the
concert. The demand curve for intense music lovers
would be Q2 ⫽ 200 ⫺ (1/10)P2. Assuming the marginal
cost of a concert is $2800, what is the efficient number of
concerts to offer each year?
17.26. Some observers have argued that the Internet is
overused in times of network congestion.
a) Do you think the Internet serves as common property? Are people ever denied access to the Internet?
b) Draw a graph illustrating why the amount of traffic is
higher than the efficient level during a period of peak demand when there is congestion. Let your graph reflect
the following characteristics of the Internet:
i. At low traffic levels, there is no congestion, with
marginal private cost equal to marginal external cost.
ii. However, at higher usage levels, marginal external
costs are positive, and the marginal external cost increases as traffic grows.
c) On your graph explain how a tax might be used to
improve economic efficiency in the use of the Internet
during a period of congestion.
d) As an alternative to a tax, one could simply deny access
to additional users once the economically efficient volume
of traffic is on the Internet. Why might an optimal tax be
more efficient than denying access?
17.27. There are two types of citizens in Pulmonia.
The first type has an inelastic demand for public broadcasting at Q ⫽ 8 hours per day; however, they are willing
to pay only up to $30 per hour for each hour up to Q ⫽ 8.
The second type demands public broadcasting according
to P ⫽ 60 ⫺ 3Q.
a) Suppose the marginal cost of public broadcasting is
MC ⫽ 15. What is the economically efficient level of
public broadcasting? Hint: it will help if you draw a careful sketch of the demand curve of each type of citizen.
b) Repeat part (a) for MC ⫽ 45.
MATHEMATICAL
APPENDIX
A.1 F U N C T I O N A L R E L AT I O N S H I P S
A.2 W H AT I S A “ M A R G I N ” ?
A.3 D E R I VAT I V E S
A.4 H OW TO F I N D A D E R I VAT I V E
A.5 M A X I M I Z AT I O N A N D M I N I M I Z AT I O N P R O B L E M S
A.6 M U LT I VA R I A B L E F U N C T I O N S
A.7 C O N S T R A I N E D O P T I M I Z AT I O N
A.8 L AG R A N G E M U LT I P L I E R S
This appendix provides an overview of some of the mathematical concepts that you will find useful as you study microeconomics. In addition to introducing and summarizing
the concepts, we will illustrate them by referring to selected
examples from the textbook.
A . 1 F U N C T I O N A L R E L AT I O N S H I P S
Economic analysis often requires that we understand how
to relate economic variables to one another. There are
three primary ways of expressing the relationships among
variables: graphs, tables, and algebraic functions. For example, Figure A.1 contains information about the demand for
paint in a market. The table at the bottom of the figure indicates how much paint consumers would purchase at various prices. For example, if the price of paint is $10 per liter,
consumers in the market will buy 3 million liters per year.
This information is also shown in the graph at point T. By
convention, economists draw demand curves with price on
the vertical axis and quantity on the horizontal axis. Since
quantity is measured along the horizontal axis (in millions
of liters), point T has the coordinates (3, 10). Similarly, at a
price of $8 per liter, consumers would buy 4 million liters
[indicated on the graph at point U, with coordinates (4, 8)].
Other points from data in the table are plotted at points S,
V, and W. As the figure shows, tables and graphs can be
very helpful in showing the relationships among variables.
We also often find it useful to express economic relationships with equations. We can express the relationship
between price and quantity using functional notation:
Q f (P )
(A.1)
where the function f tells us Q, the quantity of paint consumed (measured in millions of liters) when the price is P
(measured in $ per liter). A specific function that describes
the data in Figure A.1 is
Q 8 0.5P
(A.2)
Equation (A.2) is therefore the demand function that contains all of the points shown in Figure A.1. We have written
equations (A.1) and (A.2) with Q on the left-hand side and
P on the right-hand side. This is the natural way to write a
729
730
M AT H E M AT I C A L A P P E N D I X
$16
R
Demand for paint:
two algebraic forms
P = 16 – 2Q
Q = 8 – 0.5P
S
P, Price (dollars per liter)
12
∆P = –2
T
10
Slope of graph:
∆P
= –2
∆Q
∆Q = +1
U
8
V
6
W
4
FIGURE A.1 Functional
Relationships: Example with
Demand Curve
The graph and table show the
relationship between the
quantity of paint purchased in
a market (Q) and the price of
paint (P). For example, the
first row of the table indicates
that when the price is $12 per
liter, 2 million liters would be
purchased each year. This corresponds to point S. The functional relationship between
quantity and price can be represented algebraically in two
ways. If we write price as a
function of quantity, the form
of the demand curve is P
16 2Q. Equivalently, we may
write quantity as a function of
price, with Q 8 0.5P.
D
Z
0
2
3
4
5
6
8
Q, Quantity (millions of liters per year)
Point on
Graph
Price of Paint
($ per liter)
Millions of Liters
Purchased per Year
S
12
2
T
10
3
U
8
4
V
6
5
W
4
6
demand function if we want to ask the following question:
“How does the number of units sold depend on the price?”
The variable on the left-hand side (Q) is the dependent
variable, and the variable on the right-hand side (P ) is the
independent variable.
Let’s use equation (A.2) to find out how much consumers will buy when the price is $8 per liter. When P 8,
then Q 8 0.5(8) 4. Thus, consumers will buy 4 million liters per year. To emphasize that Q is a function of P,
equation (A.2) might also be written as Q(P ) 8 0.5P.
We might also use a demand function to answer a different question: “What price will induce consumers to demand
any specified quantity?” Now we are asking how the price
depends on the quantity we wish to sell. In other words, how
does P depend on Q? We can let P take the role of the dependent variable and Q the independent variable. To see how
P depends on Q, we can “invert” equation (A.2) by solving it
for P in terms of Q. When we do so, we find that the inverse
demand function can be expressed as equation (A.3):
P 16 2Q
(A.3)
All of the combinations of price and quantity in the
table in Figure A.1 also satisfy this equation. Let’s use
equation (A.3) to find out what price will make consumers
demand 4 million liters per year. When we substitute Q
4 into the equation, we find that P 16 2(4) 8. Thus,
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A . 1 F U N C T I O N A L R E L AT I O N S H I P S
if we want consumers to demand 4 million liters per year,
we should set the price at $8 per liter. To emphasize that P
is a function of Q, we might also write equation (A.3) as
P(Q) 16 2Q.
When we draw a demand curve with P on the vertical
axis and Q on the horizontal axis, the slope of the graph is
just the “rise over the run,” that is, the change in price (the
vertical distance) divided by the change in quantity (the
horizontal distance) as we move along the curve. For example, as we move from point S to point T, the change in price
is ⌬P 2, and the change in quantity is ⌬Q ⫽ ⫹1. Thus,
the slope is ⌬P/⌬Q 2. Since the demand curve in
the example is a straight line, the slope is a constant everywhere on the curve. The vertical intercept of the demand
curve occurs at point R, at a price of $16 per liter. This
means that no paint would be sold at that price or any
higher price.1 If the price of paint were zero, then people
would demand 8 million liters. This is the horizontal intercept in the graph, at point Z.
1
You may recall from a course in algebra that the equation of a
straight line is y mx b, where y is plotted on the vertical axis
and x is measured on the horizontal axis. With such a graph m is
the slope of the graph and b is the vertical intercept. In Figure A.1
the “y” variable is P because it is plotted on the vertical axis and the
“x” variable (the one on the horizontal axis) is Q. Thus, instead
of having the equation y 2x 16, with the example we have
P 2Q 16. The slope is 2 and the vertical intercept is 16.
TABLE A.1
For practice drawing supply and demand curves from
an equation, you might review Learning-By-Doing Exercises 2.1 and 2.2.
L E A R N I N G - B Y- D O I N G E X E R C I S E A . 1
Graphing Total Cost
This example will help you see how to draw a graph and
construct a table for a total cost function. Suppose that the
function representing the relationship between the total
costs of production (C ) and the quantity produced (Q) is as
follows:
C(Q) Q3 10Q2 40Q
Problem In a table, show the total cost of producing each
of the amounts of output: Q 0, Q 1, Q 2, Q 3,
Q 4, Q 5, Q 6, Q 7. Draw the total cost function
on a graph with total cost on the vertical axis and quantity
on the horizontal axis.
Solution The first two columns of Table A.1 show the
total cost for each level of output. For example, to produce
three units, we evaluate C(Q) when Q 3. We find that
C(3) (3)3 10(3)2 40(3) 57. (Do not worry about the
other columns in the table. We will refer to them later.)
The total cost curve is plotted in panel (a) in Figure A.2.
[Do not worry about panel (b). We will refer to it later.]
Relating Total, Average, and Marginal Cost with a Table*
(1)
Quantity
Produced
(units)
Q
(2)
Total
Cost
($)
C
0
0
(3)
“Arc”
Marginal Cost
($/unit)
C(Q) ⴚ C(Q ⴚ 1)
(4)
“Point”
Marginal Cost
($/unit)
dC/dQ
(5)
Average
Cost
($/unit)
C/Q
40
C(1) C(0) 31
1
31
23
31
12
24
7
19
8
16
15
15
28
16
47
19
C(2) C(1) 17
2
48
C(3) C(2) 9
3
57
C(4) C(3) 7
4
64
C(5) C(4) 11
5
75
C(6) C(5) 21
6
96
C(7) C(6) 37
7
133
(A.4)
*The table shows the values of total cost, marginal cost, and average cost curves when the cost
function is C(Q) ⴝ Q3 ⴚ 10Q2 40Q.
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M AT H E M AT I C A L A P P E N D I X
$140
Total cost (C)
120
∆Q = 1
C, cost (dollars)
100
80
60
B
Tangent line
at point B.
Slope = $28/unit
∆C = 21
Tangent line
at point E.
Slope = $12/unit
48
40
A
Tangent line
at point A.
Slope = $15/unit
E
20
0
1
2
3
4
5
6
7
Q, quantity (units)
(a)
$50
FIGURE A.2 Relating Total,
Average, and Marginal Cost
Graphically
Panel (a) shows the total cost of
producing any specified amount
of output. The units on the vertical axis of the top graph are
monetary (dollars). The bottom
graph shows the marginal and
average cost curves corresponding
to the total cost curve in the top
graph. The units on the vertical
axis of the bottom graph are
dollars per unit. In panel (b), the
value of the marginal cost at each
quantity is the same as the slope
of the total cost in panel (a).
C, cost (dollars per unit)
40
Marginal cost (MC)
Average cost (AC)
30
MC = 28
AC = 24
AC = 15
MC = 15
20
AC = 16
10
0
MC = 12
1
2
3
4
5
6
7
Q, quantity (units)
(b)
A . 2 W H AT I S A “ M A R G I N ” ?
Decision makers are often interested in the marginal value of
a dependent variable. The marginal value measures the change
in a dependent variable associated with a one-unit change in an
independent variable. The marginal cost therefore measures
the rate of change of cost, that is, ⌬C/⌬Q. A decision maker
may be interested in the marginal cost because it tells her how
much more it will cost to produce one more unit.
Consider once again Table A.1, which shows the
total cost based on equation (A.4). The dependent variable is total cost, and the independent variable is the
quantity produced. The table shows two ways of measuring the marginal cost. Column three illustrates the first
way by showing how the total cost changes when one
more unit is produced. The column is labeled “Arc”
Marginal Cost because it measures the change in total
cost over an arc, or region, over which the quantity
A . 2 W H AT I S A “ M A R G I N ” ?
increases by one unit. For example, when the quantity
increases from Q 2 to Q 3, total cost increases from
C(2) 48 to C(3) 57. Thus, the marginal cost over this
region of the cost curve is C(3) C(2) 9. Similarly,
the marginal cost over the arc from Q 5 to Q 6 is
C(6) C(5) 21.
We can also represent the marginal cost on a graph.
Consider Figure A.2(a). The vertical axis measures total
cost, and the horizontal axis indicates the quantity produced. We can show that the arc marginal cost approximates the slope of the total cost curve over a region of
interest. For example, let’s determine the marginal cost when
we increase quantity from Q 5 (at point A) to Q 6 (at
point B). We can construct a straight-line segment connecting points A and B. The slope of this segment is the change
in cost (the “rise”), which is 21, divided by the change in the
quantity (the “run”), which is 1. Thus, the slope of the segment connecting points A and B is the arc measure of the
marginal cost. Note that over the region the slope of the
total cost function changes. The arc marginal cost provides
us with an approximate value of the slope of the graph over
the region of interest.
Instead of approximating the marginal cost by measuring it over an arc, we could measure the marginal cost at any
specified point (i.e., at a particular quantity). For example,
at point A, the slope of the total cost curve is the slope of a
line tangent to the total cost curve at A. The slope of this
tangent line measures the rate of change of total cost at
point A. Thus, the slope of the line tangent to the total cost curve
at point A measures the marginal cost at point A. Similarly, the
slope of the line tangent to the total cost curve at point B
measures the marginal cost there.
How can we determine the value of the marginal cost
at a point? One way to do this would be to construct a
carefully drawn graph, and then measure the slope of the
line tangent to the graph at the point of interest. For
example, the slope of the total cost curve at point B (when
Q 6) is $28 per unit. Thus, the marginal cost when Q 6
is $28 per unit. Similarly, the marginal cost when Q 2 is
$12 per unit because that is the slope of the line tangent
to the total cost curve at point E. Column 4 in Table A.1
shows the exact “point” value of the marginal cost at each
quantity.
As we will show later, instead of drawing and carefully
measuring the slope of the graph, we can also use calculus
to find the marginal cost at a point. (See Learning-ByDoing Exercise A.5.)
Relating Average and Marginal Values
The average value is the total value of the dependent variable divided by the value of the independent variable.
Table A.1 also shows the average cost, that is, total cost
divided by output, C/Q. The average cost is calculated in
column 5.
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We can also show the average cost curve on a graph.
Consider the top graph in Figure A.2. We can show that the
average cost at any quantity is the slope of a segment connecting the origin with the total cost curve. For example,
let’s determine the average cost when the quantity is Q 2
(at point E ). We can construct a line segment 0E connecting the origin to point E. The slope of this segment is the
total cost (the rise), which is 48, divided by the quantity (the
run), which is 2. Thus, the slope of the segment is the average cost, 24.
The value of the average cost is generally different
from the value of the marginal cost. For example, the average cost at Q 2 (again, the slope of the segment connecting the origin to point E ) is 24, while the marginal cost (the
slope of the line tangent to the total cost curve) is 12. We
have plotted the values of the marginal and average cost on
Figure A.2(b).
We need one graph to plot the value of the total cost
and another to show the values of the average and marginal
cost curves. The units of total cost are monetary, for example, dollars. Thus, the units along the vertical axis in the top
graph are measured in dollars. However, the units of marginal cost, ⌬C/⌬Q, and average cost, C/Q, are dollars per
unit. The dimensions of total cost differ from the dimensions of average and marginal cost.
It is important to understand the relationship between
marginal and average values. Since the marginal value represents the rate of change in the total value, the following
statements must be true:
• The average value must increase if the marginal value
is greater than the average value.
• The average value must decrease if the marginal value
is less than the average value.
• The average value will be constant if the marginal value
equals the average value.
These relationships hold for the marginal and average
values of any measure. For example, suppose the average
height of the students in your class is 180 centimeters. Now
a new student, Mr. Margin, whose height is 190 centimeters, enters the class. What happens to the average height in
the class? Since Mr. Margin’s height exceeds the average
height, the average height must increase.
Similarly, if Mr. Margin’s height is 160 centimeters,
the average height in the class must decrease. Finally, if
Mr. Margin’s height is exactly 180 centimeters, the average
height in the class will remain unchanged.
This basic arithmetic insight helps us to understand
the relationship between average and marginal product
(see Figures 6.3 and 6.4), average and marginal cost (see
Figures 8.7, 8.8, 8.9, and 8.10), average and marginal
revenues for a monopolist (see Figures 11.2 and 11.4), and
average and marginal expenditures for a monopsonist (see
Figure 11.18).
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M AT H E M AT I C A L A P P E N D I X
L E A R N I N G - B Y- D O I N G E X E R C I S E A . 2
A . 3 D E R I VAT I V E S
Relating Average and Marginal Cost
In Figure A.2, we showed that one way to find the marginal
cost is to plot the total cost curve and carefully measure the
slope at each quantity. This is a tedious process, and it is not
always easy to draw a precise tangent line and measure its
slope accurately. Instead, we can use the powerful techniques of differential calculus to find the marginal cost or
other marginal values we might want to know about.
Let’s suppose that y is the dependent variable and x the
independent variable in a function:
This example will reinforce your understanding of the relationship between marginal and average values. Consider
the average and marginal cost curves in Figure A.2(b).
Problem Use the relationship between marginal and average cost to explain why the average cost curve is rising,
falling, or constant at each of the following quantities:
(a) Q 2
(b) Q 5
y f (x)
(c) Q 6
Consider Figure A.3, which depicts the value of the
dependent variable on the vertical axis and the value of the
independent variable on the horizontal axis.
As we have already discussed, if y measures the total
value, then the slope of the graph at any point measures the
marginal value. (For example, if y measures total cost and x
the quantity, then the slope of the cost function is the marginal cost at any quantity.) We can use a concept called a
derivative to help us find the slope of a function at any
point, such as point A in the figure.
We illustrate how a derivative works using Figure A.3.
Let’s begin with an algebraic approximation of the slope of
the graph. The function y f (x) is curved, so we know that
its slope will change as we move along the curve. We might
approximate the slope of the graph at E by selecting two
points on the curve, E and F. Let’s draw a segment connecting these two points and call the segment EF. The slope
of the segment is just the rise (⌬y ⫽ y3 ⫺ y1) over the run
Solution
(a) When Q 2, the marginal cost curve lies below the average
cost curve. Thus the average cost curve must be falling ( have
a negative slope).
(b) When Q 5, the marginal cost curve is equal to the
average cost curve (they intersect). Thus the average cost
curve must be neither increasing nor decreasing (have a
slope of zero) at that level of output. In this case, we see that
this means we are at the minimum point on the average cost
curve. ( We will discuss minimum and maximum points of
functions below.)
(c) When Q 6, the marginal cost curve lies above the average
cost curve. Thus the average cost curve must be rising (have a
positive slope).
y (dependent variable)
y
FIGURE A.3 The Meaning of a
Derivative
When x x1, the derivative of y with respect to x (i.e., dy/dx) is the slope of the
line tangent to point E.
F
3
B
y2
Tangent line
at point E
y1
0
E
x1
x2
x3
x (independent variable)
Relationship
between y and x
y = f (x)
A . 4 H OW TO F I N D A D E R I VAT I V E
(⌬x ⫽ x3 ⫺ x1). Thus, the slope of EF is ¢ y/¢x
( y3 y1)/(x3 x1). The graph indicates that the slope of
EF will not exactly measure the slope of the tangent line at
E, but it does give us an approximation of the slope. As the
graph is drawn, the slope of EF will be less than the slope of
the line tangent to the function at point E.
We can get a better approximation to the slope at E if
we choose another point on the graph closer to E, such as
point B. Let’s draw a segment connecting these two points
and call the segment EB. The slope of the segment EB is
¢y/¢x ( y2 y1)/(x2 x1). Once again, the graph tells us
that the slope of EB will not exactly measure the slope of
the tangent line at E (it still underestimates the slope at E ),
but it does give us a better approximation of the slope at E.
If we choose a point very close to E, the approximate
calculation of the slope will approach the actual slope at
point E. When the two points become very close to each
other, ⌬x approaches zero. The value of the approximation
as ⌬x approaches zero is the derivative, written dyⲐdx. We
express the idea of the derivative mathematically as follows:
dy
¢y
lim
d x ¢x S0 ¢ x
(A.5)
where the expression “lim¢xS0” tells us to evaluate the slope
¢y/ ¢ x “in the limit” as ¢ x approaches zero. The value of
the derivative dy/dx at point E is the slope of the graph at
that point.
A . 4 H O W TO F I N D A D E R I VAT I V E
y
In this section we will show you how to find a derivative for
a few of the functional forms commonly encountered in
economic models. You can refer to any standard calculus
4
0
A
x1
735
book to learn more about derivatives, including derivatives
of other types of functions not included here.
Derivative of a Constant
If the dependent variable y is a constant, its derivative with
respect to x is zero. In other words, suppose y k, where k
is a constant. Then d y/dx 0.
Consider, for example, the function y 4. Figure A.4
graphs this function. We can find the slope of this function
in two ways. First, because the graph is flat, we know that
the value of y does not vary as x changes. Thus, by inspection we observe that the slope of the graph is zero.
The second way to find the slope is to take the derivative.
Since the derivative of a constant is zero, then dy/dx 0.
Since the derivative is always zero, the slope of the graph of
the function y 4 is always zero.
Derivative of a Power Function
A power function has the form:
y axb
(A.6)
where a and b are constants. For such a function the
derivative is
dy
bax b1
dx
(A.7)
Let’s consider an example. Suppose y 4x. The left
graph of Figure A.5 shows this function. Since the function
is a straight line, it has a constant slope. We can find the
slope in two ways. First, take any two points on the graph,
such as A and B. We find that the slope ⌬y/⌬x ⫽ (16 ⫺ 8)/
(4 ⫺ 2) 4.
y=4
x
FIGURE A.4 Derivative of a Constant
The graph shows the function y 4. Since the
value of y does not vary as x changes, the graph is
a horizontal line. The slope of the graph is always
0. The derivative (dy/dx) 0 confirms the fact that
the slope of the function is always 0.
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M AT H E M AT I C A L A P P E N D I X
y = 4x
B
y
16
8
dy
dx
A
4
0
2
x
4
(a)
x
(b)
FIGURE A.5
Derivative of y 4x
Panel (a) shows the function y 4x. The slope of this graph is 4. Using the rule for the derivative
of a power function, we find that the derivative, (dy/dx) 4, and plot the derivative in panel (b).
The fact that the derivative is always 4 means that that slope of the function in panel (a) is
always 4.
The second way to find the slope is to take the derivative. We recognize that y 4x is a power function like the
one in equation (A.6), with a 4 and b 1. As equation (A.5)
shows, the derivative is dy/dx bax b1 4x0 4. Since the
derivative dy/dx is always 4, the slope of the graph of the
function y 4x is always 4.
L E A R N I N G - B Y- D O I N G E X E R C I S E A . 3
tells us that the slope of the function y 3x 2 [at point A in
panel(a)] is 6.
(b) When x 0, the value of the derivative is d y/d x
6(0) 0. Thus, the slope of the function y 3x 2 at point
B is 0.
(c) When x 2, the value of the derivative is d y/d x
6(2) 12. Therefore, the slope of the function y 3x 2 at
point C is 12.
Derivative of a Power Function
Consider the function y 3x 2, shown in Figure A.6(a).
Problem Find the slope of this function when
(a) x 1
(b) x 0
(c) x 2
Solution
(a) We recognize that y 3x2 is a power function like the
one in equation (A.6), with a 3 and b 2. As equation
(A.7) shows, the derivative is d y/d x bax b1 6x. [The
graph of the derivative is shown in Figure A.6(b).] Thus, the
slope of the function y 3x 2 will be 6x. When x 1,
the value of the derivative is d y/d x 6(1) 6. This
To summarize one of the uses of derivati