FMTOC.qxd 8/30/10 5:33 PM Page xxiv FMTOC.qxd 8/30/10 5:33 PM Page i accessible, affordable, active learning «PRWLYDWHVVWXGHQWVZLWK FRQÀGHQFHERRVWLQJ IHHGEDFNDQGSURRIRI SURJUHVV «VXSSRUWVLQVWUXFWRUVZLWK UHOLDEOHUHVRXUFHVWKDW UHLQIRUFHFRXUVHJRDOV LQVLGHDQGRXWVLGHRI WKHFODVVURRP ,QFOXGHV ,QWHUDFWLYH 7H[WERRN  5HVRXUFHV :LOH\3/86/HDUQ0RUH ZZZZLOH\SOXVFRP FMTOC.qxd 8/30/10 5:33 PM Page ii $//7+(+(/35(6285&(6$1'3(5621$/6833257 <28$1'<285678'(1761((' ZZZZLOH\SOXVFRPUHVRXUFHV 0LQXWH7XWRULDOVDQGDOO RIWKHUHVRXUFHV\RX \RXU VWXGHQWVQHHGWRJHWVWDUWHG 3UHORDGHGUHDG\WRXVH DVVLJQPHQWVDQGSUHVHQWDWLRQV &UHDWHGE\VXEMHFWPDWWHUH[SHUWV 6WXGHQWVXSSRUWIURPDQ H[SHULHQFHGVWXGHQWXVHU )$4VRQOLQHFKDW DQGSKRQHVXSSRUW ZZZZLOH\SOXVFRPVXSSRUW &ROODERUDWHZLWK\RXUFROOHDJXHV ILQGDPHQWRUDWWHQGYLUWXDODQGOLYH HYHQWVDQGYLHZUHVRXUFHV ZZZ:KHUH)DFXOW\&RQQHFWFRP <RXU :LOH\3/86$FFRXQW0DQDJHU 3HUVRQDOWUDLQLQJDQG LPSOHPHQWDWLRQVXSSRUW FMTOC.qxd 8/30/10 5:33 PM Page iii 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 . FMTOC.qxd 8/30/10 5:33 PM Page iv To our wives . . . Maureen and Jan . . . and to our children Suvarna and Eric, Justin, and Julie VP & PUBLISHER ACQUISITIONS EDITOR PROJECT EDITOR SENIOR EDITORIAL ASSISTANT PRODUCTION MANAGER SENIOR PRODUCTION EDITOR CREATIVE DIRECTOR SENIOR DESIGNER SENIOR ILLUSTRATION EDITOR PHOTO RESEARCHER PRODUCTION MANAGEMENT SERVICES ASSOCIATE DIRECTOR OF MARKETING ASSISTANT MARKETING MANAGER EXECUTIVE MEDIA EDITOR MEDIA EDITOR COVER PHOTO George Hoffman Lacey Vitetta Jennifer Manias Emily McGee Dorothy Sinclair Janet Foxman Harry Nolan Madelyn Lesure Anna Melhorn Sheena Goldstein Furino Production Amy Scholz Diane Mars Allison Morris Greg Chaput Radius Images/Photolibrary This book was set in 10/12 Janson by Aptara Corp. and printed and bound by R.R. Donnelley/Jefferson City. The cover was printed by RR Donnelley, Inc. 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No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201) 748-6011, fax (201) 748-6008, website www.wiley.com/go/permissions. Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year. These copies are licensed and may not be sold or transferred to a third party. Upon completion of the review period, please return the evaluation copy to Wiley. Return instructions and a free of charge return shipping label are available at www.wiley.com/go/returnlabel. Outside of the United States, please contact your local representative. 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 FMTOC.qxd 8/30/10 5:33 PM Page v 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 FMTOC.qxd vi 8/30/10 5:33 PM Page 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. FMTOC.qxd 8/30/10 5:33 PM Page vii 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 ( b)(P Q). Since b 10 and Q 400 10P, 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 stat10 a b 3 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? 10 a b 0.33 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 D E 1 2 vii 8/30/10 5:33 PM viii Page 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 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. Price (dollars per bushel) FMTOC.qxd 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. FMTOC.qxd 8/30/10 9:05 PM Page ix 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 FMTOC.qxd x 8/30/10 5:33 PM Page 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. FMTOC.qxd 8/30/10 5:33 PM Page xi 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 FMTOC.qxd xii 8/30/10 5:33 PM Page 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. FMTOC.qxd 8/30/10 5:33 PM Page xiii 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 FMTOC.qxd 8/30/10 xiv 5:33 PM Page 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: Donald Bumpass, Sam Houston State University; Tina A. Carter, Florida State University; Carl Davidson, Michigan State University; Helen Knudsen, University of Pittsburgh; Robert P. McComb, Texas Tech University; Zuohong Pan, Western Connecticut State University; Irene Trela, University of Western Ontario; and several others who wish to remain anonymous. THIRD EDITION REVIEWERS: Javad Amid, Uppsala University, Sweden; Shahina Amin, University of Northern Iowa; Eihab Fathelrahman, Washington State University, Vancouver; Thomas Gresik, University of Notre Dame; Eric Jamelske, University of Wisconsin, Eau Claire; Dean Karlan, Yale University; Mark Killingsworth, Rutgers University, Qihong Liu, The University of Oklahoma; Robert P. McComb, Texas Technical University; Brian McManus, Washington University; Silve Parviainen, University of Illinois; Francisca G. C. Richter, Cleveland State University; Brain Simboli, Lehigh University; Charles N. Steele, Montana State University. SECOND EDITION REVIEWERS: James Burnell, College of Wooster; Whewon Cho, Tennessee Technological University; Paul Cowgill, Duke University; Ron Deiter, Iowa State University; Martine Duchatelet, Barry University; Otis W. Gilley, California State University, L.A.; Marvin A. Gordon, University of Illinois at Chicago; Gregory Green, Indiana State University; Umit Gurun, Michigan State University; Russell F. Hardy, University of New Mexico at Carlsbad; Dr. Naphtali Hoffman, Elmira University; Don Holley, Boise State University; Jiandong Ju, University of Oklahoma; Mary Kassis, State University of West Georgia; Sang H. Lee, Southeastern Louisiana University; Charles Mason, University of Wyoming; John J. Nader, Grand Valley State University; Charles M. North, Baylor University; Richard M. Peck, University of Illinois-Chicago; Brian Peterson, Manchester College; Thomas Pogue, University of Iowa; Malcolm Robinson, Thomas More College; Philip Rothman, East Carolina University; Christopher S. Ruebeck, Lafayette College; Barbara Schone, Georgetown University; Konstantinos Serfes, SUNY-Stony Brook; Mark Thoma, University of Oregon; Michele T. Villinski, DePauw University; Robert O. Weagley, Missouri State University. FIRST EDITION REVIEWERS: Anas Alhajji, Colorado School of Mines; Scott Atkinson, University of Georgia; Doris Bennett, Jacksonville State University; Arlo Biere, Kansas State University; Douglas Blair, Rutgers University; Michael Bognanno, Temple University; Stephen Bronars, University of Texas, Austin; Douglas Brown, Georgetown University; Kenneth Brown, University of Northern Iowa; Don Bumpass, Sam Houston State University; Colin Campbell, Ohio State University; Corey S. Capps, University of Illinois; Manual Carvajal, Florida International University; Kousik Chakrabarti, University of Michigan, Ann Arbor; Myong-Hun Chang, Cleveland State University; Ken Chapman, California State University; Yongmin Chen, University of ColoradoBoulder; Kui Kwon (Alice) Chong, University of North Carolina-Charlotte; Peter FMTOC.qxd 8/30/10 5:33 PM Page xv AC K N OW L E D G M E N T S Coughlin, University of Maryland; Steven Craig, University of Houston; Mike Curme, Miami University; Rudolph Daniels, Florida A & M University; James Dearden, Lehigh University; Stacey Deirgerconlin, Syracuse University; John Edwards, Tulane University; Matthew Eichner, Johns Hopkins University; Ronel Elul, Brown University; Maxim Engers, University of Virginia; Raymond Fisman, Columbia University; Eric Friedman, Rutgers University; Susan Gensemer, Syracuse University; Steven Marc Goldman, University of California, Berkeley; Barnali Gupta, Miami University; Claire Hammond, Wake Forest University; Shawkat Hammoudeh, Drexel University; Lyn Holmes, Temple University; Michael Jerison, SUNY-Albany; Jiandong Ju, University of Oklahoma; David Kamerschen, University of Georgia; 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; Guogiang Tian, Texas A & M; Theofanis Tsoulouhas, North Carolina State University; Geoffrey Turnbull, Louisiana State University; Mich Tvede, University of Pennsylvania; Mark Walbert, Illinois University; Mark Walker, University of Arizona; Larry Westphal, Swarthmore College; Kealoha Widdows, Wabash College; Chiounan Yeh, Alabama State University. xv FMTOC.qxd 8/30/10 5:33 PM Page xvi 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 FMTOC.qxd 8/30/10 5:33 PM Page xvii 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 FMTOC.qxd 8/30/10 5:33 PM Page xviii xviii CONTENTS CHAPTER 4 Consumer Choice 103 The Effects of a Change in Price or Income: An Algebraic Approach 160 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 Understanding Consumer Surplus from the Demand Curve 173 Understanding Consumer Surplus from the Optimal Choice Diagram: Compensating Variation and Equivalent Variation 175 5.4 Market Demand 182 Network Externalities 184 Market Demand with Network Externalities 5.5 The Choice of Labor and Leisure 132 187 Are Observed Choices Consistent with Utility Maximization? 133 As Wages Rise, Leisure First Decreases, Then Increases 187 The Backward-Bending Supply of Labor 189 APPENDIX 1 The Mathematics of Consumer Choice 143 5.6 Consumer Price Indices APPENDIX 2 The Time Value of Money 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 192 LEARNING-BY-DOING EXERCISES 144 CHAPTER 5 The Theory of Demand 184 152 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 FMTOC.qxd 8/30/10 5:33 PM Page xix 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 FMTOC.qxd xx 8/30/10 5:33 PM Page 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 FMTOC.qxd 8/30/10 5:33 PM Page xxi 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 FMTOC.qxd 8/30/10 5:33 PM Page xxii 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 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 636 606 606 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 FMTOC.qxd 8/30/10 5:33 PM Page xxiii 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 FMTOC.qxd 8/30/10 5:33 PM Page xxiv c01analyzingeconomicproblems.qxd 7/14/10 10:54 AM Page 1 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 c01analyzingeconomicproblems.qxd 7/14/10 10:55 AM Page 2 • 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 c01analyzingeconomicproblems.qxd 7/14/10 10:22 PM Page 3 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 c01analyzingeconomicproblems.qxd 4 7/14/10 CHAPTER 1 10:22 PM Page 4 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 c01analyzingeconomicproblems 6/14/10 1:38 PM Page 5 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. c01analyzingeconomicproblems 6 6/14/10 1:38 PM CHAPTER 1 Page 6 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.) c01analyzingeconomicproblems.qxd 7/14/10 11:05 AM Page 7 1 . 2 T H R E E K E Y A N A LY T I C A L TO O L S S 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 E 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  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  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.) c01analyzingeconomicproblems.qxd 8 7/14/10 CHAPTER 1 S 10:49 AM Page 8 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 E 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 )  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 )  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 c01analyzingeconomicproblems.qxd 7/14/10 10:49 AM Page 9 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). c01analyzingeconomicproblems.qxd 10 7/14/10 CHAPTER 1 11:05 AM Page 10 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 c01analyzingeconomicproblems 6/14/10 1:38 PM Page 11 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) 100,000, 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) 100,000 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 c01analyzingeconomicproblems 12 6/14/10 1:38 PM CHAPTER 1 Page 12 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 c01analyzingeconomicproblems.qxd 6/21/10 8:47 AM Page 13 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. 14 6/14/10 1:38 PM CHAPTER 1 FIGURE 1.3 Page 14 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) c01analyzingeconomicproblems 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). c01analyzingeconomicproblems 6/14/10 1:38 PM Page 15 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 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. 9 “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). c01analyzingeconomicproblems.qxd 16 7/14/10 CHAPTER 1 11:05 AM Page 16 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 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 E 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. c01analyzingeconomicproblems.qxd 7/14/10 10:55 AM Page 17 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 S2: supply of corn when rainfall is r2 Price of corn S1 S2 P1* P2* FIGURE 1.6 Comparative Statics: 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 c01analyzingeconomicproblems 18 6/14/10 1:38 PM CHAPTER 1 S Page 18 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 E 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 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  F4 and W  F 4. Therefore, L  F 4 and W  F 4. 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 1.3 Microeconomic analysis can be used to study both positive and normative questions. POSITIVE AND N O R M AT I V E A N A LYS I S 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 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. c01analyzingeconomicproblems 6/14/10 1:38 PM Page 19 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). c01analyzingeconomicproblems.qxd 20 7/14/10 CHAPTER 1 10:57 AM Page 20 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. c01analyzingeconomicproblems.qxd 7/14/10 11:02 AM Page 21 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. c01analyzingeconomicproblems 22 6/14/10 1:38 PM CHAPTER 1 Page 22 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. c01analyzingeconomicproblems 6/14/10 1:38 PM Page 23 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: c01analyzingeconomicproblems 6/14/10 24 1:38 PM CHAPTER 1 Emissions of CO2 by a Plant (metric tons per year) Annual Cost of Operating Plant 1 (millions) Page 24 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 $1 per kilogram and the equilibrium quantity is 9 million kilograms per year. T 30 50 65 70 Q* (millions of kilograms per year) 9 P* ($ 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. c01analyzingeconomicproblems 6/14/10 1:38 PM Page 25 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. c02demandandsupplyanalysis.qxd 2 6/14/10 1:39 PM Page 26 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 c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 27 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 c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 28 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 c02demandandsupplyanalysis.qxd 7/14/10 11:22 AM Page 29 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. C hapter 1 introduced equilibrium and comparative statics analysis. In this chapter, we 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: 2.1 D E M A N D, S U P P LY, A N D MARKET EQUILIBRIUM c02demandandsupplyanalysis.qxd 30 6/14/10 1:39 PM CHAPTER 2 Page 30 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 c02demandandsupplyanalysis.qxd 7/14/10 11:09 AM Page 31 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. law of demand The 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 D E Sketching a Demand Curve Suppose the demand for new automobiles in the United States is described by the equation Qd ⫽ 5.3 ⫺ 0.1P (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 Solution 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. c02demandandsupplyanalysis.qxd 1:39 PM CHAPTER 2 Page 32 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 6/14/10 $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. c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 33 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 D E 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. c02demandandsupplyanalysis.qxd 34 6/14/10 1:39 PM CHAPTER 2 Page 34 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 D 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 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, c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 35 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 1:39 PM CHAPTER 2 Page 36 D E M A N D A N D S U P P LY A N A LYS I S Price 36 6/14/10 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 c02demandandsupplyanalysis.qxd 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 c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 37 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 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 E 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 c02demandandsupplyanalysis.qxd 38 7/14/10 11:22 AM CHAPTER 2 Page 38 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. c02demandandsupplyanalysis.qxd 7/15/10 8:57 AM Page 39 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. 40 6/14/10 1:39 PM CHAPTER 2 Page 40 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) c02demandandsupplyanalysis.qxd 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. c02demandandsupplyanalysis.qxd 6/21/10 8:50 AM Page 41 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 c02demandandsupplyanalysis.qxd 42 6/14/10 1:39 PM CHAPTER 2 Page 42 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- 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) 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 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. c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 43 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. c02demandandsupplyanalysis.qxd 44 6/14/10 1:39 PM CHAPTER 2 Page 44 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 ΔP A D3 D2 ΔQ2 ΔQ1 D1 Q Quantity (thousands of units per year) c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 45 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 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 E 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. c02demandandsupplyanalysis.qxd 46 6/14/10 1:39 PM CHAPTER 2 Page 46 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 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 Q P= a – b b ,P b la et stic w ee 2b Q = a – bP 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 c02demandandsupplyanalysis.qxd 7/14/10 11:10 AM Page 47 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 ⫽ aP⫺b, 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 ⫽ aP⫺b where a and b are positive constants. The term ⫺b is the price elasticity of demand along this curve. S 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 E 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? ⑀Q, P ⫽ (⫺b)(PⲐQ) Since b ⫽ ⫺10 and Q ⫽ 400 ⫺ 10P, when P ⫽ 30, ⑀Q,P ⫽ ⫺10 a 30 b ⫽ ⫺3 400 ⫺ 10(30) (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? and when P ⫽ 10, Solution 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). (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 ⫽ ⫺10 a Similar Problems: 10 b ⫽ ⫺0.33 400 ⫺ 10(10) 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. c02demandandsupplyanalysis.qxd 48 6/14/10 1:39 PM CHAPTER 2 Page 48 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. c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 49 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 c02demandandsupplyanalysis.qxd 50 6/21/10 8:50 AM CHAPTER 2 Page 50 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. c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 51 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. c02demandandsupplyanalysis.qxd 52 7/14/10 11:23 AM CHAPTER 2 Page 52 D E M A N D A N D S U P P LY A N A LYS I S 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 c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 53 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- 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 15 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. 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. 17 c02demandandsupplyanalysis.qxd 54 6/14/10 1:39 PM CHAPTER 2 Page 54 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. c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 55 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 0 15 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. c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 56 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. D E M A N D A N D S U P P LY A N A LYS I S 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. 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. c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 57 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 c02demandandsupplyanalysis.qxd 58 6/14/10 1:39 PM CHAPTER 2 Page 58 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. c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 59 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 $10 per ton, and 33 million tons were sold. 59 60 6/14/10 1:39 PM CHAPTER 2 Page 60 D E M A N D A N D S U P P LY A N A LYS I S D2008 D2009 S2010 S2008 Price ($ per ton) c02demandandsupplyanalysis.qxd $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 c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 61 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 62 6/14/10 1:39 PM CHAPTER 2 D E M A N D A N D S U P P LY A N A LYS I S S1 Price (dollars per unit) Page 62 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) c02demandandsupplyanalysis.qxd 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. c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 63 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. c02demandandsupplyanalysis.qxd 64 6/14/10 1:39 PM CHAPTER 2 Page 64 D E M A N D A N D S U P P LY A N A LYS I S 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. c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 65 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 c02demandandsupplyanalysis.qxd 66 6/14/10 1:39 PM CHAPTER 2 Page 66 D E M A N D A N D S U P P LY A N A LYS I S 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) c02demandandsupplyanalysis.qxd 7/14/10 11:12 AM Page 67 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 c02demandandsupplyanalysis.qxd 68 7/14/10 11:14 AM CHAPTER 2 Page 68 D E M A N D A N D S U P P LY A N A LYS I S 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: P 0.10 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 c02demandandsupplyanalysis.qxd 6/14/10 1:39 PM Page 69 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. c02demandandsupplyanalysis.qxd 70 6/14/10 1:39 PM CHAPTER 2 Page 70 D E M A N D A N D S U P P LY A N A LYS I S 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 c02demandandsupplyanalysis.qxd 7/14/10 11:16 AM Page 71 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 c02demandandsupplyanalysis.qxd 72 7/14/10 11:21 AM CHAPTER 2 Page 72 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⫺(b⫹1) dP Forming the expression for the point elasticity of demand, we have ⑀Q, P ⫽ dQ P dP Q ⫽ ⫺baP⫺(b⫹1) ⫻ 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.) c03consumerpreferencesandtheconceptofutility.qxd 3 6/14/10 2:54 PM Page 73 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 c03consumerpreferencesandtheconceptofutility.qxd 7/14/10 12:06 PM Page 74 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 c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 75 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. c03consumerpreferencesandtheconceptofutility.qxd 76 CHAPTER 3 6/14/10 2:54 PM Page 76 CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY She prefers basket A to basket B (written A  B). She prefers basket B to basket A (written B  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  B and if B  E, then A  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. c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 77 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  B and B  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. c03consumerpreferencesandtheconceptofutility.qxd 78 CHAPTER 3 7/15/10 8:58 AM Page 78 CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY U( y), total utility of hamburgers (a) S 2.24 2.00 B U( y) = √ y C R A 1.00 0 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' C' 0.25 0.22 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. 2 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(21y ) 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 ). c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 79 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 ¢U ¢y, the rate of change in total utility at that point as consumption rises or falls, which is what marginal utility measures (note that ¢U ¢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  ¢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. c03consumerpreferencesandtheconceptofutility.qxd CHAPTER 3 2:54 PM Page 80 CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY U( y), total utility of hamburgers 80 6/14/10 (a) S N B K L C U = ( y) A R 0 M 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 7 9 C' y, weekly consumption of hamburgers MUy 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 c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 81 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=8 U, level of utility 10 8 U=6 6 U=4 4 FIGURE 3.4 A B 2 C 12 10 y, 8 its 6 4 of clo 2 thi ng U=2 un 0 2 4 8 6 f food x, units o 10 12 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). c03consumerpreferencesandtheconceptofutility.qxd 82 CHAPTER 3 6/14/10 2:54 PM Page 82 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  1y(2 1x) and MUy  1x(2 1y). So, at basket A (with x  2 and y  8), MUx  18(2 12)  1 and MUy  12(2 18)  14. 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 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 E 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  1y(2 1x) and MUy  1x(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. c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 83 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 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 E 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 MUH  1 22H MUR  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. c03consumerpreferencesandtheconceptofutility.qxd 84 CHAPTER 3 6/14/10 2:54 PM Page 84 CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY 12 y, units of clothing 10 A 8 6 U=8 B 4 FIGURE 3.5 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). U=6 C 2 U=4 U=2 0 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. c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 85 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 c03consumerpreferencesandtheconceptofutility.qxd 86 CHAPTER 3 6/14/10 2:54 PM Page 86 CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY Preference directions y B A U0 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. 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  ¢ 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 c03consumerpreferencesandtheconceptofutility.qxd 7/14/10 2:09 PM Page 87 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 c03consumerpreferencesandtheconceptofutility.qxd 88 CHAPTER 3 6/14/10 2:54 PM Page 88 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 ¢y  ¢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. c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 89 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. c03consumerpreferencesandtheconceptofutility.qxd 90 CHAPTER 3 6/14/10 Page 90 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 2:54 PM 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 E 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 I 5 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  yx . On curve U1, the MRSx,y at basket G is 168  2; therefore, the slope of the indifference curve at G is 2. The MRSx,y at basket I is 432  18; therefore, the slope of the indifference curve at I is 18. 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. c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 91 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 MUy  yx, as we move along the indifference curve by increasing x and decreasing y, MRSx, y  yx 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 MUy  2Ax (2By)  Ax (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. c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 92 92 CHAPTER 3 3.3 A consumer’s willingness to substitute one good for another will depend on the com- SPECIAL PREFERENCES 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. CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY 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  MUBMUM  a a  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  MUPMUW  12. 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). c03consumerpreferencesandtheconceptofutility.qxd 7/14/10 12:07 PM Page 93 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. c03consumerpreferencesandtheconceptofutility.qxd 94 CHAPTER 3 6/14/10 2:54 PM Page 94 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 UAx  y z, where z measures the quantity of the third commodity, and A, , , and  are all positive constants. c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:55 PM Page 95 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 c03consumerpreferencesandtheconceptofutility.qxd 96 CHAPTER 3 6/14/10 2:55 PM Page 96 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) c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:55 PM Page 97 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 c03consumerpreferencesandtheconceptofutility.qxd 98 CHAPTER 3 7/14/10 2:11 PM Page 98 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. c03consumerpreferencesandtheconceptofutility.qxd 7/14/10 2:13 PM Page 99 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. c03consumerpreferencesandtheconceptofutility.qxd 100 CHAPTER 3 7/14/10 12:08 PM Page 100 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 ⫽ yⲐ(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 ⫽ 1Ⲑ(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 ⲐF. 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 c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:55 PM Page 101 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 2. 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 14 espresso and 3 4 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  1y(21x) and MUy  1x(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 x0.6 ) and MUy  0.6 (x0.4y0.4 ) . 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  1(2 1x) and MUy  1 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)  2 1x  y The marginal utilities are MUx  1 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 c03consumerpreferencesandtheconceptofutility.qxd 102 CHAPTER 3 7/14/10 2:17 PM Page 102 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 c04consumerchoice.qxd 7/14/10 4 3:15 PM Page 103 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 c04consumerchoice.qxd 7/14/10 3:16 PM Page 104 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 c04consumerchoice.qxd 7/14/10 3:16 PM Page 105 105 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 c04consumerchoice.qxd 6/18/10 5:31 PM Page 106 106 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. Px x  Py y  I (4.1) 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. c04consumerchoice.qxd 6/18/10 5:31 PM Page 107 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  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 Ⲑx  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 (yⲐx  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 c04consumerchoice.qxd 6/18/10 5:31 PM 108 Page 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 = Δy Δx Px =– Py =– 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. c04consumerchoice.qxd 6/18/10 5:31 PM Page 109 109 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 c04consumerchoice.qxd 6/18/10 110 5:31 PM Page 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 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 E 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. c04consumerchoice.qxd 6/18/10 5:31 PM Page 111 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  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  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  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 U2 5 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. c04consumerchoice.qxd 6/18/10 112 5:31 PM Page 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. c04consumerchoice.qxd 6/18/10 5:31 PM Page 113 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 YX —the ratio of total calories from food consumed at home to total calories of food consumed away from home—would equal 66 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. c04consumerchoice.qxd 114 6/18/10 5:31 PM Page 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 MUy  (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). c04consumerchoice.qxd 6/18/10 5:31 PM Page 115 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  1620  0.8 and that MUy Py  8/40  0.2. Eric’s marginal utility per dollar spent on food (MUx Px) is higher than his marginal utility per dollar spent on clothing (MUy Py). 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 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 E 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  MUy  Px Py, 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. c04consumerchoice.qxd 6/18/10 116 5:31 PM Page 116 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: max Utility  U(x, y) (4.5) (x, y) subject to: Px x  Py y  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: min expenditure  Px x  Py y (4.6) (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 FIGURE 4.5 20 BL1: spending = $640 per month 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 c04consumerchoice.qxd 6/18/10 5:31 PM Page 117 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 A 10 FIGURE 4.6 C 8 U2 = 200 5 U1 = 128 BL1 0 8 10 16 20 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. c04consumerchoice.qxd 6/18/10 118 5:31 PM Page 118 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 MUy Px Py, or (reversing the inequality) MUx MUy  Px Py. Then, by cross multiplying, MUx Px  MUy Py, 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 c04consumerchoice.qxd 6/18/10 5:31 PM Page 119 4.2 OPTIMAL CHOICE 119 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 D E 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)x = 12, 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 Px  10/1  10, while the marginal utility per dollar spent on y is MUy Py  10 2  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 Py)  12. 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 MUy  ( y  10)x] 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 = – Px 1 =– 2 Py BL MUx MUy = –1 FIGURE 4.8 R 0 2 4 6 8 x, units of food 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 12. 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. c04consumerchoice.qxd 120 6/18/10 5:31 PM Page 120 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 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 E 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 V, units of vanilla ice cream A Budget line BL P C = –3 slope = – 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 c04consumerchoice.qxd 6/18/10 5:31 PM Page 121 121 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 Ph 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 Py  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 c04consumerchoice.qxd 122 6/18/10 5:31 PM Page 122 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 A =– B U1 hA hB Py = –Ph U2 J G I Ph (I + S) Ph h, units of housing Ph 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 c04consumerchoice.qxd 6/18/10 5:31 PM Page 123 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 c04consumerchoice.qxd 6/18/10 124 5:31 PM Page 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 R U3 U2 A U4 U1 Slope of budget lines KJ, EG, and MN =– 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 c04consumerchoice.qxd 6/18/10 5:31 PM Page 125 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). c04consumerchoice.qxd 6/18/10 126 5:31 PM Page 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. c04consumerchoice.qxd 6/18/10 5:31 PM Page 127 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. c04consumerchoice.qxd 128 6/18/10 5:31 PM Page 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 C2ⲐC1  (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. c04consumerchoice.qxd 6/18/10 5:31 PM Page 129 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 6/18/10 130 FIGURE 4.16 5:31 PM Page 130 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 c04consumerchoice.qxd Preference directions $44,450 E Slope of budget line EA = –1.015 $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 Slope of budget line AG = –1.15 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. c04consumerchoice.qxd 6/18/10 5:31 PM Page 131 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 c04consumerchoice.qxd 6/18/10 132 5:31 PM Page 132 CHAPTER 4 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? c04consumerchoice.qxd 6/18/10 5:31 PM Page 133 4.4 REVEALED PREFERENCE 133 F Units of clothing Preference directions B C 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? c04consumerchoice.qxd 134 6/18/10 5:31 PM Page 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 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  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 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 E 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. c04consumerchoice.qxd 6/18/10 5:31 PM Page 135 4.4 REVEALED PREFERENCE 135 12 10 FIGURE 4.19 8 B C y 6 4 A 2 D BL1 0 2 4 5 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 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 c04consumerchoice.qxd 136 6/18/10 5:31 PM Page 136 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 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 E 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). c04consumerchoice.qxd 6/18/10 5:31 PM Page 137 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- A B BL1 BL2 x (a) Case 1 A Similar Problems: 4.21, 4.23, 4.24, 4.28 y A y B 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 BL1 BL2 x (b) Case 2 137 BL1 BL2 x (c) Case 3 BL1 BL2 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. c04consumerchoice.qxd 7/14/10 138 3:18 PM Page 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 c04consumerchoice.qxd 6/18/10 5:31 PM Page 139 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? c04consumerchoice.qxd 140 6/18/10 5:31 PM Page 140 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 c04consumerchoice.qxd 6/18/10 5:31 PM Page 141 141 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. c04consumerchoice.qxd 6/18/10 142 5:31 PM Page 142 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 c04consumerchoice.qxd 6/18/10 5:31 PM Page 143 143 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 BL2 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) (A4.1) (x, y) 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) C BL1 BL3 X c04consumerchoice.qxd 144 7/14/10 3:19 PM Page 144 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 c04consumerchoice.qxd 6/18/10 5:31 PM Page 145 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 c04consumerchoice.qxd 146 6/18/10 5:31 PM Page 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 C2 CT  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 $1,200,000 $1,500,000    $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 c04consumerchoice.qxd 6/18/10 5:31 PM Page 147 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 c04consumerchoice.qxd 148 6/18/10 5:31 PM Page 148 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 1r 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) 1r 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. c04consumerchoice.qxd 6/18/10 5:31 PM Page 149 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) 1r 7 U¿(I2) 1r 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 c05Thetheoryofdemand.qxd 5 7/23/10 8:51 AM Page 150 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 c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 151 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. 151 c05Thetheoryofdemand.qxd 7/23/10 152 8:51 AM CHAPTER 5 Page 152 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 Where do demand curves come from? In Chapter 4, we showed how to determine OPTIMAL CHOICE AND DEMAND 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 153 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 c05Thetheoryofdemand.qxd 7/23/10 154 price consumption curve The set of utilitymaximizing baskets as the price of one good varies (holding constant income and the prices of other goods). 8:51 AM CHAPTER 5 Page 154 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.” 6 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 155 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. c05Thetheoryofdemand.qxd 7/23/10 156 8:51 AM CHAPTER 5 A P P L I C A T I O N Page 156 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.” c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 157 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 158 158 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) c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 159 5.1 OPTIMAL CHOICE AND DEMAND S 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 E 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. c05Thetheoryofdemand.qxd 160 7/23/10 8:51 AM CHAPTER 5 Page 160 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 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 E 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 c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 161 161 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 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 E 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 c05Thetheoryofdemand.qxd 7/23/10 162 8:51 AM CHAPTER 5 Solution Page 162 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 163 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, 163 c05Thetheoryofdemand.qxd 8:51 AM CHAPTER 5 Page 164 T H E T H E O RY O F D E M A N D Step 1: Find the initial basket A. Px Slope of BL1 = – 1 Py A y, clothing 164 7/23/10 BL1 U1 xA x, food (a) 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 x, food (b) 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 B Slope of BLd = – BL2 U1 BL1 xA (c) xB xC x, food U2 Px 2 Py c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 165 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). 165 c05Thetheoryofdemand.qxd 166 7/23/10 8:51 AM CHAPTER 5 Page 166 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 U2 Px 1 Py Px 2 Py Px 2 Py 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 = – FIGURE 5.8 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. y, clothing Slope of BL2 = – C Slope of BLd = – A U2 Px Px Px xA xC xB x, food 2 Py U1 BLd 2 Py B BL1 1 Py BL2 c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 167 5.2 CHANGE IN THE PRICE OF A GOOD: SUBSTITUTION EFFECT AND INCOME Slope of BL1 = – y, clothing C Slope of BL2 = – U2 Slope of BLd = – A Px Px 2 Py Px 2 Py FIGURE 5.9 U1 BL1 xB 1 Py B xC xA 167 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. c05Thetheoryofdemand.qxd 168 7/23/10 8:51 AM CHAPTER 5 A P P L I C A T I O N Page 168 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 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 E 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 169 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. c05Thetheoryofdemand.qxd 170 7/23/10 8:51 AM CHAPTER 5 A P P L I C A T I O N Page 170 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 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 E 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 171 5.2 CHANGE IN THE PRICE OF A GOOD: SUBSTITUTION EFFECT AND INCOME 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 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. Similar Problems: 5.9, 5.21, 5.33 B y, other goods C A U1 FIGURE 5.11 U2 BL2 xC xB S 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 E 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? c05Thetheoryofdemand.qxd 172 7/26/10 9:39 PM CHAPTER 5 Page 172 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 2 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  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. BL2 B 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 173 173 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. 8:51 AM CHAPTER 5 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. Page 174 T H E T H E O RY O F D E M A N D $25 time (dollars per hour) 174 7/23/10 P, price consumer is willing to pay for court c05Thetheoryofdemand.qxd 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 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 E 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 c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 175 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. c05Thetheoryofdemand.qxd 176 7/23/10 8:51 AM CHAPTER 5 Page 176 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. 2 K L E A C U2 B BL1 O U1 BL2 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 c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 177 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 2 y, clothing J 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. c05Thetheoryofdemand.qxd 178 7/26/10 11:57 PM CHAPTER 5 S Page 178 T H E T H E O RY O F D E M A N D L E A R N I N G - B Y- D O I N G E X E R C I S E 5 . 8 D E 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) ⫽ 2 1x ⫹ 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 179 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. c05Thetheoryofdemand.qxd 180 7/23/10 8:51 AM CHAPTER 5 S Page 180 T H E T H E O RY O F D E M A N D L E A R N I N G - B Y- D O I N G E X E R C I S E 5 . 9 D E 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 c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 181 5.3 CHANGE IN THE PRICE OF A GOOD: THE CONCEPT OF CONSUMER SURPLUS 181 P, price of food (dollars per unit) $10 Demand for food, x = 8 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. c05Thetheoryofdemand.qxd 182 7/23/10 8:51 AM CHAPTER 5 Page 182 T H E T H E O RY O F D E M A N D 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 c05Thetheoryofdemand.qxd 7/23/10 1:36 PM Page 183 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 c05Thetheoryofdemand.qxd 7/23/10 184 8:51 AM CHAPTER 5 Page 184 T H E T H E O RY O F D E M A N D 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 185 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 c05Thetheoryofdemand.qxd 7/23/10 186 8:51 AM CHAPTER 5 Page 186 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). c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 187 187 5.5 THE CHOICE OF LABOR AND LEISURE Annual membership price Demand $1,500 A $1,200 $1,900 B 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 c05Thetheoryofdemand.qxd 188 7/23/10 8:51 AM CHAPTER 5 Page 188 T H E T H E O RY O F D E M A N D 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 $600 w = 25 $480 w = 20 $360 w = 15 U4 U3 w = 10 $240 U2 $120 w=5 B U5 U1 H I G 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 189 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 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 FIGURE 5.25 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 190 7/23/10 8:51 AM CHAPTER 5 Page 190 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 c05Thetheoryofdemand.qxd 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 191 5.5 THE CHOICE OF LABOR AND LEISURE S 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 E 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. 192 5.6 CONSUMER PRICE INDICES 7/23/10 8:51 AM CHAPTER 5 Page 192 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 c05Thetheoryofdemand.qxd 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 193 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 c05Thetheoryofdemand.qxd 194 7/23/10 8:51 AM CHAPTER 5 A P P L I C A T I O N Page 194 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 c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 195 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? c05Thetheoryofdemand.qxd 7/23/10 196 8:51 AM CHAPTER 5 Page 196 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. c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 197 197 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 q1 E 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 c05Thetheoryofdemand.qxd 7/23/10 198 8:51 AM CHAPTER 5 Page 198 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 c05Thetheoryofdemand.qxd 7/23/10 8:51 AM Page 199 199 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 c06Inputsandproductionfunctions.qxd 6 6/28/10 1:29 PM Page 200 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 c06Inputsandproductionfunctions.qxd 6/28/10 1:29 PM Page 201 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 c06Inputsandproductionfunctions.qxd 202 6/28/10 CHAPTER 6 1:29 PM Page 202 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 c06Inputsandproductionfunctions.qxd 6/28/10 1:29 PM Page 203 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  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. c06Inputsandproductionfunctions.qxd 204 7/28/10 CHAPTER 6 A P P L I C A T I O N 3:28 PM Page 204 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. c06Inputsandproductionfunctions.qxd 6/28/10 1:29 PM Page 205 205 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. c06Inputsandproductionfunctions.qxd 6/28/10 1:29 PM Page 206 206 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. c06Inputsandproductionfunctions.qxd 6/28/10 1:29 PM Page 207 207 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  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. c06Inputsandproductionfunctions.qxd 208 6/28/10 CHAPTER 6 1:29 PM Page 208 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. c06Inputsandproductionfunctions.qxd 6/28/10 1:29 PM Page 209 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. c06Inputsandproductionfunctions.qxd 210 6/28/10 CHAPTER 6 1:29 PM Page 210 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 The single-input production function is useful for developing key concepts, such as PRODUCTION FUNCTIONS WITH MORE THAN ONE INPUT 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). c06Inputsandproductionfunctions.qxd 6/28/10 1:29 PM Page 211 211 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 semiconductor chips per day) = height of hill at any point thousands K of machine30 hours per day North B 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. c06Inputsandproductionfunctions.qxd 212 6/28/10 CHAPTER 6 1:29 PM Page 212 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, c06Inputsandproductionfunctions.qxd 6/28/10 1:29 PM Page 213 213 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 C 24 E 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 c06Inputsandproductionfunctions.qxd 214 6/28/10 CHAPTER 6 1:29 PM Page 214 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 7750 7500 7500 7000 00 65 62 50 60 00 (a) Glisan Glacier 7750 000 82 50 8 Pulpit Rock Coe Glacier 8500 7750 50 50 00 The Chimney Newton Clark 00 Glacier 0 Coalman Glacier 11 Newton Clark Glacier 8000 Zigzag Glacier FIGURE 6.7 Mississippi 82 (b) 50 87 50 8000 7500 7250 Steel Cliff 9250 1050 1075 0 0 0 775 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 c06Inputsandproductionfunctions.qxd 6/28/10 1:29 PM Page 215 K, thousands of machine-hours per day 6.3 PRODUCTION FUNCTIONS WITH MORE THAN ONE INPUT B 18 Q3 > Q2 D 6 0 215 Q2 > 25 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. c06Inputsandproductionfunctions.qxd 216 6/28/10 CHAPTER 6 S 1:29 PM Page 216 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 E 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. c06Inputsandproductionfunctions.qxd 6/28/10 1:29 PM Page 217 217 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. c06Inputsandproductionfunctions.qxd 218 6/28/10 CHAPTER 6 1:29 PM Page 218 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 ) c06Inputsandproductionfunctions.qxd 6/28/10 1:29 PM Page 219 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 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 E 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. 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 Solution 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. c06Inputsandproductionfunctions.qxd 220 6/28/10 CHAPTER 6 A P P L I C A T I O N 1:29 PM Page 220 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). c06Inputsandproductionfunctions.qxd 6/28/10 1:29 PM Page 221 221 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 0 100 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 Q=1 million c06Inputsandproductionfunctions.qxd 222 6/28/10 CHAPTER 6 1:29 PM Page 222 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 c06Inputsandproductionfunctions.qxd 6/28/10 1:30 PM Page 223 223 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%Ⲑ75%  1). S 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 E Calculating the Elasticity of Substitution from a Production Function Problem Consider a production function whose equation is given by the formula Q  1K L, which has corresponding marginal products, MPL  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 c06Inputsandproductionfunctions.qxd 224 6/28/10 CHAPTER 6 1:30 PM Page 224 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. c06Inputsandproductionfunctions.qxd 6/28/10 1:30 PM Page 225 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. c06Inputsandproductionfunctions.qxd 6/28/10 1:30 PM Page 226 226 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. c06Inputsandproductionfunctions.qxd 6/28/10 1:30 PM Page 227 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. c06Inputsandproductionfunctions.qxd 228 6/28/10 CHAPTER 6 A P P L I C A T I O N 1:30 PM Page 228 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 s1 s  bK ] s1 s s s1 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. c06Inputsandproductionfunctions.qxd 7/28/10 3:29 PM Page 229 229 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 c06Inputsandproductionfunctions.qxd 230 6/28/10 CHAPTER 6 1:30 PM Page 230 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 c06Inputsandproductionfunctions.qxd 6/28/10 1:30 PM Page 231 231 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: • 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. increasing returns to scale A proportionate 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 Q=2 1 0 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. c06Inputsandproductionfunctions.qxd 232 6/28/10 CHAPTER 6 S 1:30 PM Page 232 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 E 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 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 16 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). c06Inputsandproductionfunctions.qxd 6/28/10 1:30 PM Page 233 233 K, units of capital per year 6.6 TECHNOLOGICAL PROGRESS E 30 D 20 10 0 Q = 300 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 c06Inputsandproductionfunctions.qxd 234 6/28/10 CHAPTER 6 1:30 PM Page 234 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). c06Inputsandproductionfunctions.qxd 7/28/10 3:29 PM Page 235 235 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 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 E 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 ⱖ 1 and K ⱖ 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 c06Inputsandproductionfunctions.qxd 236 6/28/10 CHAPTER 6 1:30 PM Page 236 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). c06Inputsandproductionfunctions.qxd 6/28/10 1:30 PM Page 237 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). c06Inputsandproductionfunctions.qxd 238 6/28/10 CHAPTER 6 1:30 PM Page 238 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 c06Inputsandproductionfunctions.qxd 6/30/10 12:58 PM Page 239 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 c06Inputsandproductionfunctions.qxd 240 6/28/10 CHAPTER 6 1:30 PM Page 240 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 1 0 0 1 2 2 4 3 6 4 8 2 4 6 8 10 2 4 6 8 10 12 3 6 8 10 12 14 4 8 10 12 14 16 c06Inputsandproductionfunctions.qxd 6/28/10 1:30 PM Page 241 241 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  L 2K. 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  1 2K 2 2L MPK  1 2L 2 2K Q  2L  2K MPL  1 1 2 2L MPK  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 MRTSL,K Diminishing Marginal Product of Labor? Diminishing Marginal Product of Capital? Diminishing Marginal Rate of Technical Substitution? 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? c06Inputsandproductionfunctions.qxd 242 6/30/10 CHAPTER 6 1:56 PM Page 242 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 ⫽ 1 2K 2 2L MPK ⫽ 1 2L 2 2K Q ⫽ 2L ⫹ 2K MPL ⫽ 1 1 2 2L MPK ⫽ 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? c06Inputsandproductionfunctions.qxd 7/28/10 3:30 PM Page 243 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): MPL ⫽ MPK ⫽ 0f 0L 0f 0K ⫽ aALa⫺1K b ⫽ bALaK b⫺1 Now, recall that, in general, MRTSL,K ⫽ MPL MPK c06Inputsandproductionfunctions.qxd 244 6/28/10 CHAPTER 6 1:30 PM Page 244 INPUTS AND PRODUCTION FUNCTIONS Thus, for this Cobb–Douglas production function, MRTSL,K   aALa1K b bALaK b1 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 a 1 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. c07costsandcostminimization.qxd 7 6/28/10 1:42 PM Page 245 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 c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 246 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 c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 247 247 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. c07costsandcostminimization.qxd 248 6/28/10 CHAPTER 7 1:42 PM Page 248 C O S T S A N D C O S T M I N I M I Z AT I O N 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. c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 249 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. c07costsandcostminimization.qxd 250 6/28/10 CHAPTER 7 A P P L I C A T I O N 1:42 PM Page 250 C O S T S A N D C O S T M I N I M I Z AT I O N 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 c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 251 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. c07costsandcostminimization.qxd 252 6/28/10 CHAPTER 7 A P P L I C A T I O N 1:42 PM Page 252 C O S T S A N D C O S T M I N I M I Z AT I O N 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 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 E 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. c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 253 253 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. c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 254 254 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. c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 255 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 TC0 TC1 TC2 w w w L, labor services per year FIGURE 7.1 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. c07costsandcostminimization.qxd 256 6/28/10 CHAPTER 7 1:42 PM Page 256 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 c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 257 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. c07costsandcostminimization.qxd 258 6/28/10 CHAPTER 7 1:42 PM Page 258 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 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 E 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 MPK MPL 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 c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 259 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 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 E 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 c07costsandcostminimization.qxd CHAPTER 7 1:14 PM Page 260 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 7/25/10 200-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 c07costsandcostminimization.qxd 7/25/10 1:15 PM Page 261 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. c07costsandcostminimization.qxd 262 6/28/10 1:42 PM CHAPTER 7 Page 262 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). c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 263 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). c07costsandcostminimization.qxd 264 6/28/10 CHAPTER 7 1:42 PM Page 264 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 K1 B A 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. c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 265 265 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. CHAPTER 7 1:42 PM Page 266 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 266 6/28/10 B C Q = 200 A Q = 100 L, labor services per year w, dollars per unit labor c07costsandcostminimization.qxd $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. c07costsandcostminimization.qxd 7/25/10 1:16 PM Page 267 267 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 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 E 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? 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 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 r a KbK B w or K⫽ Q w 50 A r 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 6/28/10 CHAPTER 7 1:42 PM Page 268 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 Low elasticity of substitution implies . . . 15 10 5 0 B Q = 100 isoquant 10 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 15 Wage rate, $ per unit labor (a) A High elasticity of substitution implies . . . K, capital services per year c07costsandcostminimization.qxd (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. c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 269 269 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 c07costsandcostminimization.qxd 270 6/28/10 CHAPTER 7 1:42 PM Page 270 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 c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 271 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. c07costsandcostminimization.qxd 272 6/30/10 1:24 PM CHAPTER 7 A P P L I C A T I O N Page 272 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. c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 273 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 K B D C 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 c07costsandcostminimization.qxd 274 6/28/10 CHAPTER 7 1:42 PM Page 274 C O S T S A N D C O S T M I N I M I Z AT I O N 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 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 E 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 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. 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 c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 275 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 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 E 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 c07costsandcostminimization.qxd 276 6/28/10 1:42 PM CHAPTER 7 Page 276 C O S T S A N D C O S T M I N I M I Z AT I O N 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. c07costsandcostminimization.qxd 6/30/10 1:00 PM Page 277 277 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 c07costsandcostminimization.qxd 278 6/28/10 CHAPTER 7 1:42 PM Page 278 C O S T S A N D C O S T M I N I M I Z AT I O N 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 c07costsandcostminimization.qxd 6/28/10 1:42 PM Page 279 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 c07costsandcostminimization.qxd 280 6/28/10 CHAPTER 7 1:43 PM Page 280 C O S T S A N D C O S T M I N I M I Z AT I O N 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 2 1 1 capital, labor, and materials are MPK ⫽ 13K⫺3L3M 3, 1 2 1 1 1 2 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 c07costsandcostminimization.qxd 6/28/10 1:43 PM Page 281 281 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 c07costsandcostminimization.qxd 282 6/28/10 CHAPTER 7 1:43 PM Page 282 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)K r(1 ⫺ d)2K 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. c07costsandcostminimization.qxd 6/28/10 1:43 PM Page 283 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¶ ⫽01w⫽l 0L 0L (A7.3) 0f (L, K ) 0¶ ⫽01r⫽l 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 c07costsandcostminimization.qxd 284 6/28/10 CHAPTER 7 1:43 PM Page 284 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 w⫽a Q 2 br 50L • Step 2. Substitute the solution for w into the capital demand function K ⫽ (QⲐ50) 1(wⲐr): Q 1 Q ( 50L )2r 2 b K⫽ a r 50 which simplifies to K ⫽ 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. c08costscurves.qxd 7/2/10 4:40 PM 8 Page 285 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 c08costscurves.qxd 7/2/10 4:41 PM Page 286 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 c08costscurves.qxd 7/2/10 4:41 PM Page 287 287 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 B TC2 = wL2 + rK2 A TC1 = wL1 + rK1 0 (b) 1 million Q, TVs per year 2 million TC(Q) 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. c08costscurves.qxd 7/2/10 4:41 PM 288 Page 288 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 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 E 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 c08costscurves.qxd 7/2/10 4:41 PM Page 289 8 . 1 L O N G - RU N C O S T C U RV E S 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  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. c08costscurves.qxd 7/2/10 290 4:41 PM Page 290 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 B $50 million A 0 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. c08costscurves.qxd 7/2/10 4:41 PM Page 291 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 c08costscurves.qxd 7/2/10 4:41 PM 292 Page 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. c08costscurves.qxd 7/2/10 4:41 PM Page 293 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). c08costscurves.qxd 294 7/2/10 4:41 PM Page 294 CHAPTER 8 S D 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 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? 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. 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 Solution 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). c08costscurves.qxd 7/2/10 4:41 PM Page 295 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. c08costscurves.qxd 7/2/10 4:41 PM 296 Page 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 Q, thousands of full-time students 40 50 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 c08costscurves.qxd 7/2/10 4:41 PM Page 297 297 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. c08costscurves.qxd 7/2/10 4:41 PM 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. Page 298 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 c08costscurves.qxd 7/2/10 4:41 PM Page 299 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. c08costscurves.qxd 300 7/2/10 4:41 PM Page 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 Q=L 2 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 √ Q= √ L Q=L L = Q √ TC = w Q √ AC = w/ Q L = Q2 TC = wQ2 AC = wQ L =Q TC = wQ AC = w Decreasing Increasing Constant Economies of scale Diseconomies of scale Neither Increasing Decreasing Constant c08costscurves.qxd 7/2/10 4:41 PM Page 301 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 MC < AC MC > AC MC = AC Decreases Increases Constant Economies of scale Diseconomies of scale Neither 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. c08costscurves.qxd 7/2/10 4:41 PM 302 Page 302 CHAPTER 8 C O S T C U RV E S 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 Electronic computers Computer storage devices Computer terminals Computer peripheral equipment TC,Q 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. c08costscurves.qxd 7/2/10 4:41 PM Page 303 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 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 E 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. 304 7/2/10 4:41 PM Page 304 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 c08costscurves.qxd C O S T C U RV E S Expansion path C K2 A K1 B 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 c08costscurves.qxd 7/2/10 4:41 PM Page 305 305 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. c08costscurves.qxd 306 7/2/10 4:41 PM Page 306 CHAPTER 8 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. C $110 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. SAC3(Q), when K = K3 SAC2(Q), when K = K2 D $60 $50 A $35 B 1 million 2 million Q, TVs per year 3 million AC(Q) c08costscurves.qxd 7/2/10 4:41 PM Page 307 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 A SAC2(Q) 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. c08costscurves.qxd 7/2/10 4:41 PM 308 Page 308 CHAPTER 8 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 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 E 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 c08costscurves.qxd 7/2/10 4:41 PM Page 309 309 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. c08costscurves.qxd 7/2/10 310 4:41 PM Page 310 CHAPTER 8 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 c08costscurves.qxd 7/2/10 4:41 PM Page 311 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 c08costscurves.qxd 312 7/2/10 4:41 PM Page 312 CHAPTER 8 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.” c08costscurves.qxd 7/2/10 4:41 PM Page 313 313 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. c08costscurves.qxd 7/2/10 4:41 PM 314 Page 314 CHAPTER 8 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. c08costscurves.qxd 7/2/10 4:41 PM Page 315 315 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. c08costscurves.qxd 7/2/10 316 4:41 PM Page 316 CHAPTER 8 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) c08costscurves.qxd 7/2/10 4:41 PM Page 317 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. c08costscurves.qxd 318 7/2/10 4:41 PM Page 318 CHAPTER 8 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? c08costscurves.qxd 7/2/10 4:41 PM Page 319 319 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 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. AFC 1 AC MC 50 3 30 AC MC 30 5 80 AVC 16 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 18 15 4 72 5 6 AVC 10 3 10 330 TFC 10 2 4 6 AVC 100 2 18 TVC 5 170 TVC 1 4 95 TC TC 3 20 4 Q Q 2 160 3 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 c08costscurves.qxd 320 7/2/10 4:41 PM Page 320 CHAPTER 8 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. c08costscurves.qxd 7/2/10 4:41 PM Page 321 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 ]L2 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  K1L. 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 c08costscurves.qxd 322 7/2/10 4:41 PM Page 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 Q 6 8 10 c08costscurves.qxd 7/2/10 4:41 PM Page 323 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)  100w2 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 323 c08costscurves.qxd 7/2/10 324 4:41 PM Page 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. c08costscurves.qxd 7/2/10 4:41 PM Page 325 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) 0 for (w, r) (w0, r0) (A8.4) 325 c08costscurves.qxd 326 7/2/10 4:41 PM Page 326 CHAPTER 8 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. c09perfectlycompetitivemarkets.qxd 9 7/23/10 11:40 AM Page 327 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 c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 328 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=rosesecuador&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 c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 329 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 c09perfectlycompetitivemarkets.qxd 330 7/23/10 CHAPTER 9 11:40 AM Page 330 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 c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 331 331 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. c09perfectlycompetitivemarkets.qxd 332 7/23/10 CHAPTER 9 11:40 AM Page 332 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 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. 9.2 We begin our analysis of perfect competition by studying decision making by a PROFIT MAXIMIZATION BY A PRICETA K I N G F I R M 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). c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 333 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. c09perfectlycompetitivemarkets.qxd 334 7/23/10 CHAPTER 9 A P P L I C A T I O N 11:40 AM Page 334 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 9.2 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. c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 335 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. c09perfectlycompetitivemarkets.qxd CHAPTER 9 Total revenue, total cost, and total profit (thousands of dollars per month) 336 7/23/10 11:40 AM Page 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 MR = P $1 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. c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 337 337 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. c09perfectlycompetitivemarkets.qxd 338 7/23/10 CHAPTER 9 A P P L I C A T I O N 11:40 AM Page 338 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. c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 339 339 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). c09perfectlycompetitivemarkets.qxd 340 7/23/10 CHAPTER 9 11:40 AM Page 340 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 $0.35 B Price (dollars per rose) 0.30 C SAC SMC AVC A 0.25 SAC40 0.18 AVC25 AVC40 PS = minimum AVC 0.10 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. c09perfectlycompetitivemarkets.qxd 7/27/10 8:03 PM Page 341 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 D 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 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? 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: (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. s(P) ⫽ • 0, when P 6 20 ⫺10 ⫹ Similar Problems: 1 P, when P ⱖ 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 c09perfectlycompetitivemarkets.qxd 342 7/23/10 CHAPTER 9 11:40 AM Page 342 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) SMC ANSC35 SAC ANSC AVC 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*). c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 343 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 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 E 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. c09perfectlycompetitivemarkets.qxd 344 7/27/10 CHAPTER 9 8:04 PM Page 344 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) 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) ⫽ • 0, when P 6 36 1 ⫺10 ⫹ P, when P ⱖ 36 2 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. c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 345 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. c09perfectlycompetitivemarkets.qxd 346 7/23/10 CHAPTER 9 11:40 AM Page 346 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 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 c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 347 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. c09perfectlycompetitivemarkets.qxd 348 7/23/10 CHAPTER 9 11:40 AM Page 348 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 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 c09perfectlycompetitivemarkets.qxd 7/27/10 8:05 PM Page 349 349 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 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 E 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. c09perfectlycompetitivemarkets.qxd 350 7/23/10 CHAPTER 9 11:40 AM Page 350 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) 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. c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 351 351 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. Price (dollars per rose) $0.55 0.22 May 1991–1993 August and November 1991–1993 0 SS January/ February 1991–1993 DJ F DAN 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 c09perfectlycompetitivemarkets.qxd 352 7/23/10 CHAPTER 9 11:40 AM Page 352 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 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 c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 353 353 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 MR = P Price (dollars per rose) $0.40 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. c09perfectlycompetitivemarkets.qxd CHAPTER 9 11:40 AM Page 354 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 7/23/10 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. c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 355 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*) Q*  10,000,000 50,000  200). c09perfectlycompetitivemarkets.qxd 356 7/27/10 CHAPTER 9 S 8:06 PM Page 356 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 E 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 c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 357 357 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 Quantity (thousands of units per year) 0 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 c09perfectlycompetitivemarkets.qxd 358 7/23/10 CHAPTER 9 11:40 AM Page 358 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 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. c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 359 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. c09perfectlycompetitivemarkets.qxd 360 7/23/10 CHAPTER 9 11:40 AM Page 360 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 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) c09perfectlycompetitivemarkets.qxd 8/18/10 2:27 AM Page 361 361 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 c09perfectlycompetitivemarkets.qxd 362 7/23/10 CHAPTER 9 11:40 AM Page 362 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 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. c09perfectlycompetitivemarkets.qxd 8/18/10 2:28 AM Page 363 363 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 c09perfectlycompetitivemarkets.qxd 364 7/23/10 CHAPTER 9 A P P L I C A T I O N 11:40 AM Page 364 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 9.8 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. 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 22 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. 23 24 See, for example, “Biofuels Boom,” CQ Researcher 16, no. 34 (September 29, 2006). “Ethanol’s Boom Stalling as Glut Depresses Prices,” New York Times (September 30, 2007). 25 c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 365 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 40 80 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). c09perfectlycompetitivemarkets.qxd 366 7/23/10 CHAPTER 9 11:40 AM Page 366 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 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). c09perfectlycompetitivemarkets.qxd 7/23/10 11:40 AM Page 367 367 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. c09perfectlycompetitivemarkets.qxd 368 7/23/10 CHAPTER 9 11:41 AM Page 368 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 c09perfectlycompetitivemarkets.qxd 7/23/10 11:41 AM Page 369 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* 700,000  70,000 700,000  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 c09perfectlycompetitivemarkets.qxd 370 7/23/10 11:41 AM Page 370 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 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. c09perfectlycompetitivemarkets.qxd 7/23/10 11:41 AM Page 371 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. c09perfectlycompetitivemarkets.qxd 372 7/23/10 CHAPTER 9 11:41 AM Page 372 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. c09perfectlycompetitivemarkets.qxd 7/23/10 11:41 AM Page 373 9.5 ECONOMIC RENT AND PRODUCER SURPLUS MC P2 G Price (dollars per unit) P1 ANSC 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 c09perfectlycompetitivemarkets.qxd 374 7/23/10 CHAPTER 9 11:41 AM Page 374 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 Quantity (units per year) (a) Typical firm 200 Quantity (thousands of units per year) (b) Market of 1000 firms 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. c09perfectlycompetitivemarkets.qxd 7/27/10 8:09 PM Page 375 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 375 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 D E 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. c09perfectlycompetitivemarkets.qxd CHAPTER 9 11:41 AM Page 376 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 7/23/10 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. c09perfectlycompetitivemarkets.qxd 7/23/10 11:41 AM Page 377 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. c09perfectlycompetitivemarkets.qxd 378 7/23/10 CHAPTER 9 11:41 AM Page 378 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 c09perfectlycompetitivemarkets.qxd 7/23/10 11:41 AM Page 379 379 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. c09perfectlycompetitivemarkets.qxd 380 7/23/10 CHAPTER 9 11:41 AM Page 380 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 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; c09perfectlycompetitivemarkets.qxd 7/23/10 11:41 AM Page 381 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 10 Each firm of type B operates with the supply curve: Supply QB  2P, for P 0. Demand QMarket  108  10P. a) Suppose the market demand is 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 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. c09perfectlycompetitivemarkets.qxd 382 7/23/10 CHAPTER 9 11:41 AM Page 382 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 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 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 $3.00 per pound 100,000 pounds per month 1,000 pounds per month 100 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 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 $3.00 per pound 100,000 pounds per month 1,000 pounds per month 100 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 c09perfectlycompetitivemarkets.qxd 7/23/10 11:41 AM Page 383 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  12 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? c09perfectlycompetitivemarkets.qxd 384 7/23/10 CHAPTER 9 11:41 AM Page 384 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 w01P 0L 0L MPL (A9.1) 0f 0p r P r01P 0K 0K MPK (A9.2) c09perfectlycompetitivemarkets.qxd 7/23/10 11:41 AM Page 385 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 c10competitive markets applications.qxd 7/15/10 10 4:58 PM Page 386 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 c10competitive markets applications.qxd 7/15/10 4:58 PM Page 387 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 c10competitive markets applications.qxd 7/15/10 4:58 PM Page 388 388 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 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. 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 c10competitive markets applications.qxd 7/15/10 4:58 PM Page 389 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. c10competitive markets applications.qxd 390 7/15/10 CHAPTER 10 4:58 PM Page 390 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. c10competitive markets applications.qxd 7/15/10 4:58 PM Page 391 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 ). c10competitive markets applications.qxd 392 7/15/10 CHAPTER 10 4:58 PM Page 392 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 $8 C 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. c10competitive markets applications.qxd 7/15/10 4:58 PM Page 393 393 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 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 E 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. c10competitive markets applications.qxd 394 7/15/10 CHAPTER 10 4:58 PM Page 394 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 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) c10competitive markets applications.qxd 7/15/10 4:58 PM Page 395 395 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 c10competitive markets applications.qxd 396 7/15/10 CHAPTER 10 A P P L I C A T I O N 4:58 PM 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? 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? FIGURE 10.5 Page 396 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) c10competitive markets applications.qxd 7/15/10 4:58 PM Page 397 397 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. c10competitive markets applications.qxd 398 7/15/10 CHAPTER 10 4:58 PM Page 398 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 • 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 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 –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). c10competitive markets applications.qxd 7/15/10 4:58 PM Page 399 399 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 c10competitive markets applications.qxd 400 7/15/10 CHAPTER 10 S D E 4:58 PM Page 400 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 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). c10competitive markets applications.qxd 7/15/10 4:58 PM Page 401 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). 401 c10competitive markets applications.qxd 402 7/15/10 CHAPTER 10 4:58 PM Page 402 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 Rental price (dollars per month) Y A B E P * = $1,600 PR = $1,000 S U C F G W A B V X D Z S U 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 c10competitive markets applications.qxd 7/15/10 4:58 PM Page 403 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 c10competitive markets applications.qxd 404 7/15/10 CHAPTER 10 A P P L I C A T I O N 4:58 PM Page 404 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 10.3 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. c10competitive markets applications.qxd 7/15/10 4:58 PM Page 405 405 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. c10competitive markets applications.qxd 406 7/15/10 CHAPTER 10 4:58 PM Page 406 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 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. c10competitive markets applications.qxd 7/15/10 4:58 PM Page 407 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 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 . 3 D E 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 c10competitive markets applications.qxd 408 7/15/10 CHAPTER 10 Price (dollars per unit) $20 4:58 PM 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 Page 408 Supply V A X Price $6 ceiling S $2 Z R W 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. c10competitive markets applications.qxd 7/15/10 4:58 PM Page 409 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 c10competitive markets applications.qxd 410 7/15/10 w, wage rate (dollars per hour) CHAPTER 10 4:58 PM Page 410 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 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. c10competitive markets applications.qxd 7/15/10 4:58 PM Page 411 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. 411 c10competitive markets applications.qxd 412 7/15/10 CHAPTER 10 4:58 PM Page 412 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 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. 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 . 4 D E 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. c10competitive markets applications.qxd 7/15/10 4:58 PM Page 413 413 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 c10competitive markets applications.qxd 414 7/15/10 CHAPTER 10 4:58 PM Page 414 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 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. c10competitive markets applications.qxd 7/15/10 4:58 PM Page 415 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 F Deadweight loss $8 C E $6 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 c10competitive markets applications.qxd 416 7/15/10 CHAPTER 10 A P P L I C A T I O N 4:58 PM 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 Page 416 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). c10competitive markets applications.qxd 7/15/10 4:58 PM Page 417 417 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 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 E 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 c10competitive markets applications.qxd 418 7/15/10 CHAPTER 10 4:58 PM Page 418 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 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. c10competitive markets applications.qxd 7/15/10 4:58 PM Page 419 419 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 c10competitive markets applications.qxd 420 7/15/10 CHAPTER 10 4:58 PM Page 420 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 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) c10competitive markets applications.qxd 7/15/10 4:58 PM Page 421 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. c10competitive markets applications.qxd 422 10.5 IMPORT Q U OTA S A N D TA R I F F S 7/15/10 CHAPTER 10 4:58 PM Page 422 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 C onsumers in a country will want to import a good when the world price of the good 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). c10competitive markets applications.qxd 7/15/10 4:58 PM Page 423 423 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) Free Trade (with no quota) With 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. c10competitive markets applications.qxd 424 7/15/10 CHAPTER 10 A P P L I C A T I O N 4:58 PM 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 10.7 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 Page 424 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. c10competitive markets applications.qxd 7/15/10 4:58 PM Page 425 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 425 c10competitive markets applications.qxd 426 7/15/10 CHAPTER 10 4:58 PM Page 426 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 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. c10competitive markets applications.qxd 7/15/10 4:58 PM Page 427 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, wit