Nonparametric Comparative Statics and Stability Nonparametric Comparative Statics and Stability DOUGLAS HALE GEORGE LADY JOHN MAYBEE JAMES QUIRK PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY Copyright © 1999 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved Library of Congress Cataloging-in-Publication Data Nonparametric comparative statics and stability / by Douglas Hale ... [et al.]. p. cm. Includes bibliographical references and index. ISBN 0-691-00690-3 (cloth : alk. paper) 1. Economics, Mathematical. 2. Macroeconomics—Mathematical models. 3. Equilibrium (Economics). I. Hale, Douglas, 1946- HB135.N664 1999 330'.01'51-dc21 98-55311 CIP This book has been composed in Times Roman with Helvetica Display The paper used in this publication meets the minimum requirements of ANSI/NISO Z39.48-1992 (R1997) ( Permanence of Paper) http: //pup.pnnceton.edu Printed in the United States of America 1 3 5 7 9 1 0 8 6 4 2 32101 032794628 To Yvonne M. M. Bishop Two True Scholars: David F. Lady and Dorothy E. Lady Carol Shirley Contents PREFACE ix 1. Nonparametric Analysis 3 1.1 Introduction 3 1.2 Quantitative Analysis 4 1.3 NonparametricAnalysis 6 1.4 An Example 8 1.5 Organization of the Book 11 2. Qualitative Comparative Statics and Stability 13 2.1 Introduction 13 2.2 Sign Solvability—Background 13 2.3 The Algorithmic Approach to Strong Sign Solvability 15 2.4 The Algebraic Approach to Sign Solvability 20 2.5 Sign Stability 27 2.6 Potential Stability 36 2.7 Conclusions 38 Appendix: Special Topics in Matrix Analysis 40 3. Information and Invertibility 49 3.1 Introduction 49 3.2 An Algebraic Analysis of Qualitative Invertibility 50 3.3 The Elimination Principle 56 3.4 Categorizing Information 60 3.5 The Invertibility of Matrices with Ranked Entries 63 3.6 The Invertibility of Matrices with Entries Specified within Given Intervals 81 3.7 Summary of the Algebraic Method 85 3.8 Stability 86 4. Applications in Qualitative Comparative Statics 92 4.1 Introduction 92 viii CONTENTS 4.2 Algorithmic Principles 93 4.3 A Qualitative Analysis of the Oil Market Simulation Model 98 4.4 A Qualitative Analysis of the Oil and Gas Supply Model 104 4.5 Klein's Model I 117 4.6 Summary 121 5. The Maximization Hypothesis 122 5.1 Introduction 122 5.2 Unconstrained Maximization 122 5.3 Application: Profit-Maximizing Choices of Inputs by a Competitive Firm 134 5.4 Constrained Maximization 136 5.5 Application: Minimizing the Cost of Production 141 Appendix: Advanced Topics in Matrix Analysis 143 6. The Correspondence Principle 150 6.1 Introduction 150 6.2 An Example 152 6.3 StabilityandSignSolvability 157 6.4 GM-Matrices 161 6.5 Scope of the Correspondence Principle 165 6.6 Interpretation 168 7. The Competitive Equilibrium: Comparative Statics 170 7.1 Introduction 170 7.2 A Competitive Economy 171 7.3 Some Historical Comments 174 7.4 Restatement of Sign Solvability 176 7.5 Sign Solvability under Walras's Law 180 7.6 Sign Solvability under Walras's Law and Homogeneity 196 8. The Competitive Equilibrium: Stability 206 8.1 Introduction 206 8.2 Stability under Walras's Law and Homogeneity 208 NOTES 223 BIBLIOGRAPHY 227 NAME INDEX 239 SUBJECT INDEX 241 Preface The fundamental issue addressed in this book concerns the degree to which subject matter such as economics can be considered a science, i.e., a source of refutable hypotheses. Such a concern is not limited to the realm of scholarly debate, but arrives with ever-increasing impor­ tance in an age when many governmental and commercial policies are derived from computer-based renditions of conceptual models from economics and elsewhere. Our particular experience with applied analysis includes the energy market forecasting systems developed by the Energy Information Ad­ ministration (EIA) of the U.S. Department of Energy over the decades since the 1973-1974 oil embargo. The issues at stake in the use of these systems include the degree of expected dependency upon foreign sources of energy supply; the effect on energy production and costs of adopting, or removing, the regulation of energy markets; and the consequences of undertaking alternative polices designed to ameliorate the environmen­ tal problems associated with the production and consumption of many energy products. The outcomes of policy issues such as these have substantial material significance for all of us; further, the design of many specific policies depends upon exactly how the data about the relationships at issue are used. Given this, the EIA has maintained an aggressive interest in reviewing and determining the operational quality of its policy analysis modeling. The particular circumstances that led to this book can be traced in part to the EIA model quality control program. In light of the impor­ tance of modeling to the energy policy debate, a symposium was held at the University of Colorado in 1980, with attendees invited from a broad range of disciplines (proceedings in Greenberg and Maybee [1981]). The papers and discussions at the symposium covered the technical issues that needed to be resolved, and promising approaches for resolving them, in order to understand the basis for the expected accuracy of fhe results from large computer-based forecasting systems. Many of the results reported here stem from investigations prompted by this sympo­ sium. Some of the examples presented in Chapter 4 were initially sponsored by the EIA as part of its continuing interest in model quality control.