MATHEMATICS FOR ECONOMISTS Carl P Simon and Lawrence Blume W W NORTON & COMPANY NEW YORK LONDON l l l l Copyright 0 1994 by W. W. Norton & Company, Inc. ALL RIGHTS RESERVED PRINTED IN THE UNITED STATES OF AMERICA FIRST EDITION The text of this book is composed in Times Roman. with the display set in Optima. Composition by Integre Technical Publishing Company, Inc. Book design by Jack Meserole. Library of Congress Cataloging-in-Publication Data Blume, Lawrence. Mathematics for economists / Lawrence Blume and Carl Simon. P. cm. 1. Economics, Mathematical. I. Simon, Carl P., 1945- . II. Title. HB135.B59 1994 510’.24339-dc20 93-24962 ISBN 0-393-95733-O W. W. Norton & Company, Inc., 500 Fifth Avenue, New York, N.Y. 10110 W. W. Norton & Company Ltd., 10 Coptic Street, London WClA 1PU 7 8 9 0 Contents Preface xxi Introduction P A R T I 1 Introduction 3 1.1 MATHEMATICS IN ECONOMIC THEORY 3 1.2 MODELS OF CONSUMER CHOICE 5 Two-Dimensional Model of Consumer Choice 5 Multidimensional Model of Consumer Choice 9 2 One-Variable Calculus: Foundations 10 2.1 FUNCTIONS ON R’ 10 Vocabulary of Functions 10 Polynomials 11 Graphs 12 Increasing and Decreasing Functions 12 Domain 14 Interval Notation 15 2.2 LINEAR FUNCTIONS 16 The Slope of a Line in the Plane 16 The Equation of a Line 19 Polynomials of Degree One Have Linear Graphs 19 Interpreting the Slope of a Linear Function 20 2.3 THE SLOPE OF NONLINEAR FUNCTIONS 22 2.4 COMPUTING DERIVATIVES 25 Rules for Computing Derivatives 27 vi CONTENTS 2.5 DIFFERENTIABILITY AND CONTINUITY 29 A Nondifferentiable Function 30 Continuous Functions 31 Continuously Differentiable Functions 32 2.6 HIGHER-ORDER DERIVATIVES 33 2.7 APPROXIMATION BY DIFFERENTIALS 34 3 One-Variable Calculus: Applications 39 3.1 USING THE FIRST DERIVATIVE FOR GRAPHING 39 Positive Derivative Implies Increasing Function 39 Using First Derivatives to Sketch Graphs 41 3.2 SECOND DERIVATIVES AND CONVEXITY 43 3.3 GRAPHING RATIONAL FUNCTIONS 47 Hints for Graphing 48 3.4 TAILS AND HORIZONTAL ASYMPTOTES 48 Tails of Polynomials 48 Horizontal Asymptotes of Rational Functions 49 3.5 MAXIMA AND MINIMA 51 local Maxima and Minima on the Boundary and in the Interior 51 Second Order Conditions 53 Global Maxima and Minima 5.5 Functions with Only One Critical Point 55 Functions with Nowhere-Zero Second Derivatives 56 Functions with No Global Max or Min 56 Functions Whose Domains Are Closed Finite Intervals 56 3.6 APPLICATIONS TO ECONOMICS 58 Production Functions 58 Cost Functions 59 Revenue and Profit Functions 62 Demand Functions and Elasticity 64 4 One-Variable Calculus: Chain Rule 70 4.1 COMPOSITE FUNCTIONS AND THE CHAIN RULE 70 Composite Functions 70 Differentiating Composite Functions: The Chain Rule 72 4.2 INVERSE FUNCTIONS AND THEIR DERIVATIVES 75 Definition and Examples of the Inverse of a Function 7.5 The Derivative of the Inverse Function 79 The Derivative of x”“” 80 CONTENTS vii 5 Exponents and Logarithms 82 5.1 EXPONENTIAL FUNCTIONS 82 5.2 THE NUMBER e 85 5.3 LOGARITHMS 88 Base 10 Logarithms 88 Base e Logarithms 90 5.4 PROPERTIES OF EXP AND LOG 91 5.5 DERIVATIVES OF EXP AND LOG 93 5.6 APPLICATIONS 97 Present Value 97 Annuities 98 Optimal Holding Time 99 Logarithmic Derivative 100 Linear Algebra P A R T I I 6 Introduction to ALlgienberaar 107 6.1 LINEAR SYSTEMS 107 6.2 EXAMPLES OF LINEAR MODELS 108 Example 1: Tax Benefits of Charitable Contributions 108 Example 2: Linear Models of Production 110 Example 3: Markov Models of Employment 113 Example 4: IS-LM Analysis 115 Example 5: Investment and Arbitrage 117 7 Systems of Linear Equations 122 7.1 GAUSSIAN AND GAUSS-JORDAN ELIMINATION 122 Substitution 123 Elimination of Variables 125 7.2 ELEMENTARY ROW OPERATIONS 129 7.3 SYSTEMS WITH MANY OR NO SOLUTIONS 134 7.4 RANK-THE FUNDAMENTAL CRITERION 142 Application to Portfolio Theory 147 7.5 THE LINEAR IMPLICIT FUNCTION THEOREM 150 . . . VIII CONTENTS 8 Matrix Algebra 153 8.1 MATRIX ALGEBRA 153 Addition 153 Subtraction 154 Scalar Multiplication 155 Matrix Multiplication 155 Laws of Matrix Algebra 156 Transpose 157 Systems of Equations in Matrix Form 158 8.2 SPECIAL KINDS OF MATRICES 160 8.3 ELEMENTARY MATRICES 162 8.4 ALGEBRA OF SQUARE MATRICES 165 8.5 INPUT-OUTPUT MATRICES 174 Proof of Theorem 8.13 178 8.6 PARTITIONED MATRICES (optional) 180 8.7 DECOMPOSING MATRICES (optional) 183 Mathematical Induction 185 Including Row Interchanges 185 9 Determinants: An Overview 188 9.1 THE DETERMINANT OF A MATRIX 189 Defining the Determinant 189 Computing the Determinant 191 Main Property of the Determinant 192 9.2 USES OF THE DETERMINANT 194 9.3 IS-LM ANALYSIS VIA CRAMER’S RULE 197 10 Euclidean Spaces 199 10.1 POINTS AND VECTORS IN EUCLIDEAN SPACE 199 The Real Line 199 The Plane 199 Three Dimensions and More 201 10.2 VECTORS 202 10.3 THE ALGEBRA OF VECTORS 205 Addition and Subtraction 205 Scalar Multiplication 207 10.4 LENGTH AND INNER PRODUCT IN R” 209 Length and Distance 209 The Inner Product 213 CONTENTS ix 10.5 LINES 222 10.6 PLANES 226 Parametric Equations 226 Nonparametric Equations 228 Hyperplanes 230 10.7 ECONOMIC APPLICATIONS 232 Budget Sets in Commodity Space 232 Input Space 233 Probability Simplex 233 The Investment Model 234 IS-LM Analysis 234 11 Linear Independence 237 11.1 LINEAR INDEPENDENCE 237 Definition 238 Checking Linear Independence 241 11.2 SPANNING SETS 244 11.3 BASIS AND DIMENSION IN R” 247 Dimension 249 11.4 EPILOGUE 249 P A R T I I I Calculus of Several Variables 12 Limits and Open Sets 253 12.1 SEQUENCES OF REAL NUMBERS 253 Definition 253 Limit of a Sequence 254 Algebraic Properties of Limits 256 12.2 SEQUENCES IN Rm 260 12.3 OPEN SETS 264 Interior of a Set 267 12.4 CLOSED SETS 267 Closure of a Set 268 Boundary of a Set 269 12.5 COMPACT SETS 270 12.6 EPILOGUE 272 X CONTENTS 13 Functions of Several Variables 273 13.1 FUNCTIONS BETWEEN EUCLIDEAN SPACES 273 Functions from R” to R 274 Functions from Rk to R” 275 13.2 GEOMETRIC REPRESENTATION OF FUNCTIONS 277 Graphs of Functions of Two Variables 277 Level Curves 280 Drawing Graphs from Level Sets 281 Planar Level Sets in Economics 282 Representing Functions from Rk to R’ for k > 2 283 Images of Functions from R’ to Rm 285 13.3 SPECIAL KINDS OF FUNCTIONS 287 Linear Functions on Rk 287 Quadratic Forms 289 Matrix Representation of Quadratic Forms 290 Polynomials 291 13.4 CONTINUOUS FUNCTIONS 293 13.5 VOCABULARY OF FUNCTIONS 295 Onto Functions and One-to-One Functions 297 Inverse Functions 297 Composition of Functions 298 14 Calculus of Several Variables 300 14.1 DEFINITIONS AND EXAMPLES 300 14.2 ECONOMIC INTERPRETATION 302 Marginal Products 302 Elasticity 304 14.3 GEOMETRIC INTERPRETATION 305 14.4 THE TOTAL DERIVATIVE 307 Geometric Interpretation 308 Linear Approximation 310 Functions of More than Two Variables 311 14.5 THE CHAIN RULE 313 Curves 313 Tangent Vector to a Curve 314 Differentiating along a Curve: The Chain Rule 316 14.6 DIRECTIONAL DERIVATIVES AND GRADIENTS 319 Directional Derivatives 319 The Gradient Vector 320 xii CONTENTS Application: Second Order Conditions and Convexity 379 Application: Conic Sections 380 Principal Minors of a Matrix 381 The Definiteness of Diagonal Matrices 383 The Definiteness of 2 X 2 Matrices 384 16.3 LINEAR CONSTRAINTS AND BORDERED MATRICES 386 Definiteness and Optimality 386 One Constraint 390 Other Approaches 391 16.4 APPENDIX 393 ? 7 Unconstrained Optimization 396 17.1 DEFINITIONS 396 17.2 FIRST ORDER CONDITIONS 397 17.3 SECOND ORDER CONDITIONS 398 Sufficient Conditions 398 Necessary Conditions 401 17.4 GLOBAL MAXIMA AND MINIMA 402 Global Maxima of Concave Functions 403 17.5 ECONOMIC APPLICATIONS 404 Profit-Maximizing Firm 405 Discriminating Monopolist 405 Least Squares Analysis 407 18 Constrained Optimization I: First Order Conditions 411 18.1 EXAMPLES 412 18.2 EQUALITY CONSTRAINTS 413 Two Variables and One Equality Constraint 413 Several Equality Constraints 420 18.3 INEQUALITY CONSTRAINTS 424 One Inequality Constraint 424 Several Inequality Constraints 430 18.4 MIXED CONSTRAINTS 434 18.5 CONSTRAINED MINIMIZATION PROBLEMS 436 18.6 KUHN-TUCKER FORMULATION 439 C O N T E N T SXII I‘** 18.7 EXAMPLES AND APPLICATIONS 442 Application: A Sales-Maximizing Firm with Advertising 442 Application: The Averch-Johnson Effect 443 One More Worked Example 445 19 Constrained Optimization II 448 19.1 THE MEANING OF THE MULTIPLIER 448 One Equality Constraint 449 Several Equality Constraints 450 Inequality Constraints 451 Interpreting the Multiplier 452 19.2 ENVELOPE THEOREMS 453 Unconstrained Problems 453 Constrained Problems 455 19.3 SECOND ORDER CONDITIONS 457 Constrained Maximization Problems 459 Minimization Problems 463 Inequality Constraints 466 Alternative Approaches to the Bordered Hessian Condition 467 Necessary Second Order Conditions 468 19.4 SMOOTH DEPENDENCE ON THE PARAMETERS 469 19.5 CONSTRAINT QUALIFICATIONS 472 19.6 PROOFS OF FIRST ORDER CONDITIONS 478 Proof of Theorems 18.1 and 18.2: Equality Constraints 478 Proof of Theorems 18.3 and 18.4: Inequality Constraints 480 20 Homogeneous and Homothetic Functions 483 20.1 HOMOGENEOUS FUNCTIONS 483 Definition and Examples 483 Homogeneous Functions in Economics 485 Properties of Homogeneous Functions 487 A Calculus Criterion for Homogeneity 491 Economic Applications of Euler’s Theorem 492 20.2 HOMOGENIZING A FUNCTION 493 Economic Applications of Homogenization 495 20.3 CARDINAL VERSUS ORDINAL UTILITY 496