WELFARE ECONOMICS AND SOCIAL CHOICE THEORY WELFARE ECONOMICS AND SOCIAL CHOICE THEORY Allan M. Feldman Brown University Kluwer • Nijhoff Publishing Boston/The Hague/ Dordrecht/ Lancastel Distributors for North America: Kluwer Academic Publishers 101 Philip Drive Assinippi Park Norwell, Massachusetts 02061 Distributors outside North America: Kluwer Academic Publishers Group Distribution Centre P.O. Box 322 3300 AH Dordrecht, The Netherlands Library of Congress Cataloging in Publication Data Feldman, Allan, 1943- Welfare economics and social choice theory. Bibliography: p. Includes index. 1. Welfare economics. 2. Social choice. I. Title. HB99.3.F45 330.15'5 79-25744 ISBN-13: 978-0-89838-034-7 e-ISBN-\3: 978-1-4615-8141-3 DOl: 10.1007/978-1-4615-8141-3 Copyright © 1980 by Martinus Nijhoff Publishing. Seventh pri nt i ng 1989 Softcover reprint of the hardcover I st edition 1989 No part of this book may be reproduced in any form by print, photoprint, microfilm or any other means, without written permission from the publisher. CONTENTS Preface ix Introduction 1 The Concerns of Welfare Economics 1 The Concerns of Social Choice Theory 4 Practical Concerns of Welfare Economics and Social Choice Theory 7 1 Preferences and Utility 9 Fundamental Assumptions 9 Best Alternatives and Utility Functions 11 The Formal Model of Preferences 12 Introduction to Social Preferences 16 Exercises 20 Appendix 21 Selected References 22 2 Barter Exchange 23 Introduction 23 Allocations 24 v vi CONTENTS The Edgeworth Box Diagram 26 Pareto Optimal Allocations and the Core 28 Algebraic Examples 30 Final Notes on the Core: The Number of Coalitions 36 Exercises 36 Selected References 37 3 Market Exchange and Optimality 39 Introduction 39 The Two-Person, Two-Goods Model 40 Competitive Equilibrium in an Exchange Economy: Formal Preliminaries 45 The First Fundamental Theorem of Welfare Economics 47 The Second Fundamental Theorem of Welfare Economics 51 An Algebraic Example 58 Exercises 61 Selected References 63 4 Production and Optimality 65 Introduction 65 Optimal Production Plans 68 Competitive Equilibrium Production Plans 71 The First Fundamental Theorem of Welfare Economics, Production Version 74 The Second Fundamental Theorem of Welfare Economics, Production Version 76 Extending the Production Model, and Combining Production and Exchange 79 An Algebraic Example in a Simple Production Model 82 Exercises 85 Appendix 86 Selected References 87 5 Externalities 89 Introduction 89 Externalities in an Exchange Economy: An Example 92 Pigouvian Taxes and Subsidies: The Exchange Example Continued 96 Pigouvian Taxes and Subsidies: A Production Example 100 Exercises 103 Selected References 105 CONTENTS vii 6 Public Goods 106 Introduction 106 The Public Goods Model 107 The Samuelson Public Good Optimality Condition 109 Private Financing of the Public Good and the Free Rider Problem 112 The Wicksell-Lindahl Tax Scheme 114 Fixed Tax Shares and Majority Voting 119 The Demand-Revealing Tax Scheme 122 The Groves-Ledyard Tax Scheme 129 Selected References 134 7 Compensation Criteria 138 Introduction 138 Notational Preliminaries 140 The Pareto Criterion 140 The Kaldor Criterion 142 The Scitovsky Criterion 144 The Samuelson Criterion 145 Exercises 147 Selected References 148 8 Fairness and the Rawls Criterion 150 Introduction 150 Fairness 151 The Rawls Criterion 155 Exercises 158 Selected References 159 9 Majority Voting 161 Introduction 161 The Majority Voting Criterion 162 Majority Voting and Single-Peakedness 164 Majority Voting and Single-Peakedness: The Multidimensional Case 171 Exercises 175 Selected References 176 10 Arrow's Impossibility Theorem 178 Introduction 178 The Model 179 viii CONTENTS Requirements on the Collective Choice Rule 180 Applying the Requirements 183 Arrow's Impossibility Theorem 186 Relaxing the Universality Requirement 190 Reaction to Arrow's Impossibility Theorem 191 Exercises 194 Selected References 195 11 Strategic Behavior 196 Introduction 196 Examples of Strategic Behavior 198 The Gibbard-Satterthwaite Impossibility Theorem 202 Significance of the Gibbard-Satterthwaite Theorem 209 Exercises 210 Appendix 212 Selected Reference:; 214 12 Epilog 216 Subject Index 223 Author Index 229 PREFACE This book covers the main topics of welfare economics - general equilib rium models of exchange and production, Pareto optimality, externalities and public goods - and some of the major topic of social choice the ory - compensation criteria, fairness, voting, Arrow's Theorem, and stra tegic behavior. The underlying question is this: "Is a particular economic or voting mechanism good or bad for society?" Welfare economics is mainly about whether the market mechanism is good or bad; social choice is largely about whether voting mechanisms can improve upon the results of the market. The book grew out of my undergraduate welfare economics course at Brown University, and it is intended for the undergraduate student who has some prior familiarity with microeconomics. However the book is also use ful for graduate students and professionals, economists and non-econo mists, who want an overview of welfare and social choice results unbur dened by detail and mathematical complexity. Welfare economics and social choice both suffer, I think, from exces sively technical treatments in professional journals and monographs. One purpose of this book is to present these fields in a way that reduces mathe matical technicalities. Knowledge of calculus and linear algebra is not a pre- ix x PREFACE requisite for reading this. However, the results are presented rigorously; there are theorems and proofs (occasionally relegated to appendices), and the reader should be familiar enough with logic to know when A implies B and when it doesn't. What mathematical notation is used is defined and ex plained as it is introduced. Each analytical chapter contains examples to il lustrate the concepts and the theorems, and most of the chapters provide exercises for the reader. At the ends of most chapters are lists of selected references. These refer ences sections are not meant to be scholarly bibliographies; rather, they should be used by the reader who wants to dig a little deeper into a subject, or who wants to find out a little about the source of a possibly ancient idea. I would like to thank all my past students of welfare economics and so cial choice for helping me distill this material. I would like to thank the edi tors of Economic Inquiry and The American Economic Review, in whose journals parts of Chapters 8, 10, and 11 were previously published. My wife Barbara patiently read much of the manuscript while it was being prepared, and Mrs. Marion Wathey heroically typed it. Providence, Rhode Island A.F. INTRODUCTION THE CONCERNS OF WELFARE ECONOMICS Welfare economics is the normative branch of economics: it is concerned with what is good and what is bad, rather than what is. This distinguishes it from the positive branches of economics, such as labor economics, health economics, economic history, the theory of financial markets, development and international trade, monetary and macroeconomics. Each of the posi tive branches is largely devoted to explaining why things are the way they are: e.g., why doctors are paid more than nurses, why some countries ex port agricultural commodities and some export machinery, and why busi ness fluctuations occur. Many positive economists make policy prescrip tions; some say we ought to have no minimum wage, or we ought to have a higher minimum wage, or we ought to have lower tariffs, or we ought to have a 3 percent annual growth in the money supply. These prescriptions are all based partly on positive economics (the prescribers have information that indicates the likely consequences of actions that might be taken) and partly on normative economics (the prescribers have some ideas about which consequences are good and which are bad).