Table Of Content

AN EXPLANATION OF CONSTRAINED OPTIMIZATION FOR ECONOMISTS In a constrained optimization problem, the decisionmaker wants to select the “optimal” choice – the one most valuable to him or her – that meets all of the constraints imposed by the problem. Such problems are at the heart of modern economics, where the typical behavioral postulate is that a decisionmaker behaves “rationally”; that is, chooses optimally from a set of constrained choices. Most books on constrained optimization are technical and full of jargon that makes it hard for the inexperienced reader to gain an holis- tic understanding of the topic. Peter B. Morgan’s Explanation of Con- strained Optimization for Economists solves this problem by emphasiz- ingexplanations, bothwrittenandvisual, ofthemannerinwhichmany constrained optimization problems can be solved. Suitable as a text- book or a reference for advanced undergraduate and graduate students familiar with the basics of one-variable calculus and linear algebra, this book is an accessible, user-friendly guide to this key concept. PETER B. MORGAN is an associate professor in the Department of Economics at the University at Buffalo. This page intentionally left blank An Explanation of Constrained Optimization for Economists PETER B. MORGAN UNIVERSITY OF TORONTO PRESS Toronto Buffalo London (cid:2)c University of Toronto Press 2015 Toronto Buffalo London www.utpublishing.com Printed in the U.S.A. ISBN 978-1-4426-4278-2 (cloth) ISBN 978-1-4426-7777-4 (paper) Printed on acid-free paper. Library and Archives Canada Cataloguing in Publication Morgan, Peter B., 1949–, author An Explanation of Constrained Optimization for Economists / Peter B. Morgan Includes bibliographical references and index. ISBN 978-1-4426-4653-7 (bound). ISBN 978-1-4426-1446-8 (pbk.) 1. Constrained optimization. 2. Economics, Mathematical. I. Title. QA323.M67 2015 519.6 C2015-900428-4 University of Toronto acknowledges the financial assistance to its publishing program of the Canada Council for the Arts and the Ontario Arts Council, an agency of the Government of Ontario. an Ontario government agency un organisme du gouvernement de l’Ontario University of Toronto Press acknowledges the financial support of the Government of Canada through the Canada Book Fund for its publishing activities. To my grandmother, Phoebe Powley, for giving up so much to care for me. To my mother, June Powley, for fiercely emphasizing education. To my wife, Bea, with my love. ACKNOWLEDGMENTS Thanks are due to Professor Carl Simon for allowing my classes at the University at Buffalo to use a mathematical economics coursepack that he authored. The genesis of this book is in part a result of questions asked by students who sought to complete their understanding of Professor Simon’s fine coursepack. I sincerely thank Song Wei for pointing out various errors in an early draft of this manuscript. Thanks are also due to the many classes that have allowed me to practiceuponthemand, especially, fortheirinsistenceuponhavingclearexplanations provided to them. Special thanks are due to Nicole Hunter for her tireless checking of this manuscript and her discovery of many potential embarrassments. Several reviewers provided helpful suggestions and criticisms. I thank each for their time and efforts, and for their perspectives. Full responsibility for any remaining errors, omissions, ambiguities, eccentricities, and the like remain with the author. Contents List of Figures xii List of Tables xvii 1 Introduction 1 2 Basics 6 2.1 Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Real Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Some Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Direct Products of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Addition of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.7 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.8 Total Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.9 Continuous Differentiability . . . . . . . . . . . . . . . . . . . . . . . 44 2.10 Contour Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.11 Marginal Rates of Substitution . . . . . . . . . . . . . . . . . . . . . 47 2.12 Gradient Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.13 Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.14 Gradients, Rates of Substitution, and Inner Products . . . . . . . . . 55 2.15 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.16 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.17 Hessian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.18 Quadratic Approximations of Functions . . . . . . . . . . . . . . . . . 71 2.19 Saddle-Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.20 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.21 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 vii viii CONTENTS 2.22 How to Proceed from Here . . . . . . . . . . . . . . . . . . . . . . . . 83 2.23 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.24 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3 Basics of Topology 104 3.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.2 Topological Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.3 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.4 Metric Space Topologies . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.5 Bounded Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.6 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3.7 How Do Our Topological Ideas Get Used? . . . . . . . . . . . . . . . 116 3.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.9 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4 Sequences and Convergence 123 4.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.2 Bounded Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.5 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.8 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5 Continuity 145 5.1 Basic Ideas about Mappings . . . . . . . . . . . . . . . . . . . . . . . 146 5.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.3 Semi-Continuity and Continuity . . . . . . . . . . . . . . . . . . . . . 155 5.4 Hemi-Continuity and Continuity . . . . . . . . . . . . . . . . . . . . . 158 5.5 Upper-Hemi-Continuity by Itself . . . . . . . . . . . . . . . . . . . . . 169 5.6 Joint Continuity of a Function . . . . . . . . . . . . . . . . . . . . . . 175 5.7 Using Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.9 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 CONTENTS ix 6 Hyperplanes and Separating Sets 191 6.1 Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.2 Separations of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.3 Separating a Point from a Set . . . . . . . . . . . . . . . . . . . . . . 197 6.4 Separating a Set from a Set . . . . . . . . . . . . . . . . . . . . . . . 203 6.5 How Do We Use What We Have Learned? . . . . . . . . . . . . . . . 203 6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6.7 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 7 Cones 216 7.1 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 7.2 Convex Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 7.3 Farkas’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 7.4 Gordan’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 7.7 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 8 Constrained Optimization, Part I 238 8.1 The Constrained Optimization Problem . . . . . . . . . . . . . . . . . 238 8.2 The Typical Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.3 First-Order Necessary Conditions . . . . . . . . . . . . . . . . . . . . 242 8.4 What Is a Constraint Qualification? . . . . . . . . . . . . . . . . . . . 247 8.5 Fritz John’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 8.6 Constraint Qualifications and Farkas’s Lemma . . . . . . . . . . . . . 255 8.7 Particular Constraint Qualifications . . . . . . . . . . . . . . . . . . . 257 8.8 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 8.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 8.10 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 9 Lagrange Functions 283 9.1 What Is a Lagrange Function? . . . . . . . . . . . . . . . . . . . . . . 283 9.2 RevisitingtheKarush-Kuhn-TuckerCondition . . . . . . . . . . . . . . 286 9.3 Saddle-Points for Lagrange Functions . . . . . . . . . . . . . . . . . . 287 9.4 Saddle-Points and Optimality . . . . . . . . . . . . . . . . . . . . . . 291 9.5 Concave Programming Problems . . . . . . . . . . . . . . . . . . . . 293 9.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297