Fundamenta Methods of Mathematical Economics - GBV [PDF]

Fundamenta Methods of Mathematical Economics Fourth Edition

Alpha C. Chiang Professor Emeritus University of Connecticut

Kevin Wainwright British Columbia Institute of Technology and Simon Fraser University

Me Graw Hill Boston Burr Ridge, IL Dubuque, IA Madison, Wl New York San Francisco St. Louis Bangkok Bogota Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto

Contents PART ONE

PART TWO

INTRODUCTION 1

STATIC (OR EQUILIBRIUM) ANALYSIS 29

Chapter 1 The Nature of Mathematical Economics 2

Chapter 3 Equilibrium Analysis in Economics 30

1.1 Mathematical versus Nonmathematical

3.1 3.2

1.2

Economics 2 Mathematical Economics versus Econometrics 4

Chapter 2 Economic Models 5 Ingredients of a Mathematical Model 5 Variables, Constants, and Parameters 5 Equations and Identities 6 2.2 The Real-Number System 7 2.3 The Concept of Sets 8 Set Notation 9 Relationships between Sets 9 Operations on Sets 11 Laws of Set Operations 12 Exercise 2.3 14 2.4 Relations and Functions 15 Ordered Pairs 15 Relations and Functions 16 Exercise 2.4 19 2.5 Types of Function 20 > Constant Functions 20 Polynomial Functions 20 Rational Functions 21 Nonalgebraic Functions 23 A Digression on Exponents 23 Exercise 2.5 24 2.6 Functions of Two or More Independent Variables 25 2.7 Levels of Generality 27

3.3

2.1

3.4

3.5

The Meaning of Equilibrium 30 Partial Market Equilibrium—A Linear Model 31 Constructing the Model 31 Solution by Elimination of Variables 33 Exercise 3.2 34 Partial Market Equilibrium—A Nonlinear Model 35 Quadratic Equation versus Quadratic Function 35 The Quadratic Formula 36 Another Graphical Solution 37 Higher-Degree Polynomial Equations 38 Exercise 3.3 40 General Market Equilibrium 40 Two-Commodity Market Model 41 Numerical Example 42 n-Commodity Case 43 ' Solution of a General-Equation System 44 Exercise 3.4 45 Equilibrium in National-Income Analysis 46 Exercise 3.5 47

Chapter 4 Linear Models and Matrix Algebra

48

4.1 Matrices and Vectors 49

4.2

Matrices as Arrays 49 Vectors as Special Matrices 50 Exercise 4.1 51 Matrix Operations 51 Addition and Subtraction of Matrices 51 Scalar Multiplication 52

xii

4.3

4.4

4.5

4.6

4.7

Contents

Multiplication of Matrices 53 The Question of Division 56 The S Notation 56 Exercise 4.2 58 Notes on Vector Operations 59 Multiplication of Vectors 59 Geometric Interpretation of Vector Operations 60 Linear Dependence 62 Vector Space 63 Exercise 4.3 65 Commutative, Associative, and Distributive Laws 67 Matrix Addition 67 Matrix Multiplication 68 Exercise 4.4 69 Identity Matrices and Null Matrices 70 Identity Matrices 70 Null Matrices 71 Idiosyncrasies of Matrix Algebra 72 Exercise 4.5 72 Transposes and Inverses 73 Properties ofTransposes 74 Inverses and Their Properties 75 Inverse Matrix and Solution of Linear-Equation System 77 Exercise 4.6 78 Finite Markov Chains 78 Special Case: Absorbing Mfirkov Chains 81 Exercise 4.7 81

5.3

5.4

5.5

5.6

5.7

Chapter 5 Linear Models and Matrix Algebra (Continued) 82 5.1

5.2

Conditions for Nonsingularity of a Matrix 82 Necessary versus Sufficient Conditions 82 Conditions for Nonsingularity 84 Rank of a Matrix 85 Exercise 5.1 87 Test of Nonsingularity by Use of Determinant 88 Determinants and Nonsingularity 88 Evaluating a Third-Order Determinant 89 Evaluating an nth-Order Determinant by Laplace Expansion 91 Exercise 5.2 93

5.8

Basic Properties of Determinants 94 Determinantal Criterion for Nonsingularity 96 Rank of a Matrix Redefined 97 Exercise 5.3 98 Finding the Inverse Matrix 99 Expansion of a Determinant by Alien Cofactors 99 Matrix Inversion 100 -" Exercise 5.4 102 Cramer's Rule 103 Derivation of the Rule 103 Note on Homogeneous-Equation Systems 105 Solution Outcomes for a Linear-Equation System 106 Exercise 5.5 107 Application to Market and National-Income Models 107 Market Model 107 National-Income Model 108 IS-LM Model: Closed Economy 109 Matrix Algebra versus Elimination of Variables 111 Exercise 5.6 111 Leontief Input-Output Models 112 Structure of an Input-Output Model 112 The Open Model 113 A Numerical Example 115 The Existence of Nonnegative Solutions 116 Economic Meaning of the Hawkins-Simon Condition 118 The Closed Model 119 Exercise 5.7 120 Limitations of Static Analysis 120

PART THREE COMPARATIVE-STATIC ANALYSIS 123 Chapter 6 Comparative Statics and the Concept of Derivative 124 6.1 6.2

The Nature of Comparative Statics 124 Rate of Change and the Derivative 125 The Difference Quotient 125

Contents

The Derivative 126 Exercise 6.2 127 6.3 The Derivative and the Slope of a Curve 128 6.4 The Concept of Limit 129 Left-Side Limit and Right-Side Limit 129 Graphical Illustrations 130 Evaluation of a Limit 131 Formal View of the Limit Concept 133 Exercise 6.4 135 6.5 Digression on Inequalities and Absolute s Values 136 Rules of Inequalities 136 Absolute Values and Inequalities 137 Solution of an Inequality 138 Exercise 6.5 139 6.6 Limit Theorems 139 Theorems Involving a Single Function 139 Theorems Involving Two Functions 140 Limit of a Polynomial Function 141 . Exercise 6.6 141 6.7 Continuity and Differentiability of a Function 141 Continuity of a Function 141 Polynomial and Rational Functions 142 Differentiability of a Function 143 Exercise 6.7 146

7.3

Chapter 7

8.1

7.4

7.5

7.6

Chapter 8 Comparative-Static Analysis of General-Function Models 178

Rules of Differentiation and Their Use in Comparative Statics 148 7.1 Rules of Differentiation for a Function of One Variable 148 Constant-Function Rule 148 Power-Function Rule 149 Power-Function Rule Generalized 151 Exercise 7.1 152 7.2 Rules of Differentiation Involving Two or More Functions of the Same Variable 152 Sum-Difference Rule 152 Product Rule 155 Finding Marginal-Revenue Function from Average-Revenue Function 156 Quotient Rule 158 Relationship Between Marginal-Cost and Average- Cost Functions 159 Exercise 7.2 160

Rules of Differentiation Involving Functions of Different Variables 161 Chain Rule 161 Inverse-Function Rule 163 Exercise 7.3 165 Partial Differentiation 165 Partial Derivatives 165 Techniques of Partial Differentiation 166 Geometric Interpretation of Partial Derivatives 167 Gradient Vector 168 Exercise 7.4 169 Applications to Comparative-Static Analysis 170 Market Model 170 National-Income Model 172 Input-Output Model 173 Exercise 7.5 .175 Note on Jacobian Determinants 175 Exercise 7.6 177

8.2 8.3 8.4

8.5

Differentials 179 Differentials and Derivatives 179 Differentials and Point Elasticity 181 Exercise 8.1 184 Total Differentials 184 Exercise 8.2 186 Rules of Differentials 187 Exercise 8.3 189 Total Derivatives 189 Finding the Total Derivative 189 A Variation on the Theme 191 Another Variation on the Theme 192 Some General Remarks 193 Exercise 8.4 193 Derivatives of Implicit Functions 194 Implicit Functions 194 Derivatives of Implicit Functions 196 Extension to the Simultaneous-Equation Case 199 Exercise 8.5 204

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8.6

Contents

Comparative Statics of General-Function Models 205 Market Model 205 Simultaneous-Equation Approach 207 Use of Total Derivatives 209 National-Income Model (IS-LM) 210 Extending the Model: An Open Economy 213 Summary of the Procedure 216 Exercise 8.6 217

8.7

Limitations of Comparative Statics

218

PART FOUR OPTIMIZATION PROBLEMS 219 Chapter 9 Optimization: A Special Variety of Equilibrium Analysis 220 9.1 9.2

Optimum Values and Extreme Values 221 Relative Maximum and Minimum: First-Derivative Test 222 Relative versus Absolute Extremum First-Derivative Test 223 Exercise 9.2 226

9.3

9.4

9.5

Second and Higher Derivatives

222

227

Derivative of a Derivative 227 Interpretation of the Second Derivative 229 An Application 231 Attitudes toward Risk 231 Exercise 9.3 233 Second-Derivative Test 233 Necessary versus Sufficient Conditions 234 Conditions for Profit Maximization 235 Coefficients of a Cubic Total-Cost Function 238 Upward-Sloping Marginal-Revenue Curve 240 Exercise 9.4 241

Maclaurin and Taylor Series

242

Maclaurin Series of a Polynomial Function 242 Taylor Series of a Polynomial Function 244 Expansion of an Arbitrary Function 245 Lagrange Form of the Remainder 248 Exercise 9.5 250

9.6

Mh-Derivative Test for Relative Extremum of a Function of One Variable 250 Taylor Expansion and Relative Extremum 250 Some Specific Cases 251 Nth-Derivative Test 253 Exercise 9.6 254

Chapter 10 Exponential and Logarithmic Functions 255 10.1 The Nature of Exponential Functions 256 Simple Exponential Function 256 Graphical Form 256 Generalized Exponential Function 25 7 A Preferred Base 259 Exercise 10.1 260 10.2 Natural Exponential Functions and the Problem of Growth 260 The Number e 260 An Economic Interpretation ofe 262 Interest Compounding and the Function Ae" 262 Instantaneous Rate of Growth 263 Continuous versus Discrete Growth 265 Discounting and Negative Growth 266 Exercise 10.2 267

10.3 Logarithms

267

The Meaning of Logarithm 267 Common Log and Natural Log 268 Rules of Logarithms 269 An Application 271 Exercise 10.3 272 10.4 Logarithmic Functions 272 Log Functions and Exponential Functions 272 The Graphical Form 273 Base Conversion 274 Exercise 10.4 276 10.5 Derivatives of Exponential and Logarithmic Functions 277 Log-Function Rule 277 Exponential-Function Rule 278 The Rules Generalized 278 The Case of Base b 280 Higher Derivatives 280 An Application 281 Exercise 10.5 282

Contents

10.6 Optimal Timing 282 A Problem of Wine Storage 282 Maximization Conditions 283 A Problem of Timber Cutting 285 Exercise 10.6 286 10.7 Further Applications of Exponential and Logarithmic Derivatives 286 Finding the Rate of Growth 286 Rate of Growth of a Combination of Functions 287 Finding the Point Elasticity 288 Exercise 10.7 290

Chapter 11 The Case of More than One Choice Variable 291 11.1 The Differential Version of Optimization Conditions 291 First-Order Condition 291 Second-Order Condition 292 Differential Conditions versus Derivative Conditions 293 11.2 Extreme Values of a Function of Two Variables 293 First-Order Condition 294 Second-Order Partial Derivatives 295 Second-Order Total Differential 297 Second-Order Condition 298 Exercise 11.2 300 11.3 Quadratic Forms—An Excursion 301 Second-Order Total Differential as a Quadratic Form 301 Positive and Negative Definiteness 302 Determinantal Test for Sign Definiteness 302 Three-Variable Quadratic Forms 305 n-Variable Quadratic Forms 307 Characteristic-Root Test for Sign Definiteness 307 Exercise 11.3 312 11.4 Objective Functions with More than Two Variables 313 First-Order Condition for Extremum 313 Second-Order Condition 313 n-Variable Case 316 Exercise 11.4 317

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11.5 Second-Order Conditions in Relation to Concavity and Convexity 318 Checking Concavity and Convexity 320 Differentiable Functions 324 Convex Functions versus Convex Sets 327 Exercise 11.5 330 11.6 Economic Applications 331 Problem of a Multiproduct Firm 331 Price Discrimination 333 - _ Input Decisions of a Firm 336 Exercise 11.6 341 11.7 Comparative-Static Aspects of Optimization 342 Reduced-Form Solutions 342 General-Function Models 343 Exercise 11.7 345

Chapter 12 Optimization with Equality Constraints 347 12.1 Effects of a Constraint 347 12.2 Finding the Stationary Values 349 Lagrange-Multiplier Method 350 Total-Differential Approach 352 An Interpretation of the Lagrange Multiplier 353 n- Variable and Multiconstraint Cases 354 Exercise 12.2 355 12.3 Second-Order Conditions 356 Second-Order Total Differential 356 Second-Order Conditions 357 The Bordered Hessian 358 n-Variable Case 361 Multiconstraint Case 362 Exercise 12.3 363 12.4 Quasiconcavity and Quasiconvexity 364 Geometric Characterization 364 Algebraic Definition 365 Differentiable Functions 368 A Further Look at the Bordered Hessian 3 71 A bsolute versus Relative Extrema 3 72 Exercise 12.4 374 12.5 Utility Maximization and Consumer Demand 374 First-Order Condition 375 Second-Order Condition 376

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Contents

Comparative-Static Analysis 378 Proportionate Changes in Prices and Income 381 Exercise 12.5 382 12.6 Homogeneous Functions 383 Linear Homogeneity 383 Cobb-Douglas Production Function 386 Extensions of the Results 388 Exercise 12.6 389 12.7 Least-Cost Combination of Inputs 390 First-Order Condition 390 Second-Order Condition 392 The Expansion Path 392 Homothetic Functions 394 Elasticity of Substitution 396 CES Production Function 397 Cobb-Douglas Function as a Special Case of the CES Function 399 Exercise 12.7 401

Chapter 13 Further Topics in Optimization

The Arrow-Enthoven Sufficiency Theorem: Quasiconcave Programming 425 A Constraint-Qualification Test 426 Exercise 13.4 427 13.5 Maximum-Value Functions and the Envelope Theorem 428 The Envelope Theorem for Unconstrained Optimization 428 The Profit Function 429 __. Reciprocity Conditions 430 The Envelope Theorem for Constrained Optimization 432 Interpretation of the Lagrange Multiplier 434

13.6 Duality and the Envelope Theorem 435 The Primal Problem 435 The Dual Problem 436 Duality 436 Roy's Identity 43 7 Shephard's Lemma 438 Exercise 13.6 441 x

13.7 Some Concluding Remarks 442 402

13.1 Nonlinear Programming and Kuhn-Tucker Conditions 402 Step I: Effect of Nonnegativity Restrictions 403 Step 2: Effect of Inequality Constraints 404 Interpretation of the Kuhn-Tucker Conditions 408 The n-Variable, m-Constraint Case 409 Exercise 13.1 411 13.2 The Constraint Qualification 412 Irregularities at Boundary Points 412 The Constraint Qualification 415 Linear Constraints 416 Exercise 13.2 418 13.3 Economic Applications 418 War-Time Rationing 418 Peak-Load Pricing 420 Exercise 13.3 423 13.4 Sufficiency Theorems in Nonlinear Programming 424 The Kuhn-Tucker Sufficiency Theorem: Concave Programming 424

PART FIVE DYNAMIC ANALYSIS 443 Chapter 14 Economic Dynamics and Integral Calculus 444 14.1 Dynamics and Integration 444 14.2 Indefinite Integrals 446 The Nature of Integrals 446 Basic Rules of Integration 447 Rules of Operation 448 Rules Involving Substitution 451 Exercise 14.2 453 14.3 Definite Integrals 454 Meaning of Definite Integrals 454 A Definite Integral as an Area under a Curve 455 Some Properties of Definite Integrals 458 Another Look at the Indefinite Integral 460 Exercise 14.3 460

Contents

14.4 Improper Integrals 461 Infinite Limits of Integration 461 Infinite Integrand 463 Exercise 14.4 464 14.5 Some Economic Applications of Integrals 464 From a Marginal Function to a Total Function 464 Investment and Capital Formation 465 Present Value of a Cash Flow 468 Present Value of a Perpetual Flow 470 Exercise 14.5 470 14.6 Domar Growth Model 471 The Framework 471 Finding the Solution 472 The Razor's Edge 473 Exercise 14.6 474

Chapter 15 Continuous Time: First-Order Differential Equations 475 15.1 First-Order Linear Differential Equations with Constant Coefficient and Constant Term 475 The Homogeneous Case 476 The Nonhomogeneous Case 476 Verification of the Solution 478 Exercise 15.1 479 15.2 Dynamics of Market Price 479 The Framework 480 The Time Path 480 The Dynamic Stability of Equilibrium 481 An Alternative Use of the Model 482 Exercise 15.2 483 15.3 Variable Coefficient and Variable Term 483 The Homogeneous Case 484 The Nonhomogeneous Case 485 Exercise 15.3 486 15.4 Exact Differential Equations 486 Exact Differential Equations 486 Method of Solution 487 Integrating Factor 489 Solution of First-Order Linear Differential Equations 490 Exercise 15.4 491

xvii

15.5 Nonlinear Differential Equations of the First Order and First Degree 492 Exact Differential Equations 492 Separable Variables 492 Equations Reducible to the Linear Form 493 Exercise 15.5 495 15.6 The Qualitative-Graphic Approach 495 The Phase Diagram 495 Types of Time Path 496 Exercise 15.6 498 15.7 Solow Growth Model 498 The Framework 498 A Qualitative-Graphic Analysis 500 A Quantitative Illustration 501 Exercise 15.7 502

Chapter 16 Higher-Order Differential Equations 503 16.1 Second-Order Linear Differential Equations with Constant Coefficients and Constant Term 504 The Particular Integral 504 The Complementary Function 505 The Dynamic Stability of Equilibrium 510 Exercise 16.1 511 16.2 Complex Numbers and Circular Functions 511 Imaginary and Complex Numbers 511 Complex Roots 512 Circular Functions 513 Properties of the Sine and Cosine Functions 515 Euler Relations 517 Alternative Representations of Complex Numbers 519 Exercise 16.2 521 1 6 . 3 Analysis of the Complex-Root Case 522 The Complementary Function 522 An Example of Solution 524 - The Time Path 525 The Dynamic Stability of Equilibrium 527 Exercise 16.3 527

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Contents

16.4 A Market Model with Price Expectations 527 Price Trend and Price Expectations 52 7 A Simplified Model 528 The Time Path of Price 529 Exercise 16.4 532 16.5 The Interaction of Inflation and Unemployment 532 The Phillips Relation 532 The Expectations-Augmented Phillips Relation 533 The Feedback from Inflation to Unemployment 534 The Time Path ofn 534 Exercise 16.5 537 16.6 Differential Equations with a Variable Term 538 Method of Undetermined Coefficients 538 A Modification 539 Exercise 16.6 540 16.7 Higher-Order Linear Differential Equations 540 Finding the Solution 540 Convergence and the Routh Theorem 542 Exercise 16.7 543

Chapter 17 Discrete Time: First-Order Difference Equations 544 17.1 Discrete Time, Differences, and Difference Equations 544 17.2 Solving a First-Order Difference Equation 546 ^ Iterative Method 546 General Method 548 Exercise 17.2 551 17.3 The Dynamic Stability of Equilibrium 551 The Significance of b 551 The Role of A 553 Convergence to Equilibrium 554 Exercise 17.3 554 17.4 The Cobweb Model 555 The Model 555 The Cobwebs 556 Exercise 17.4 558

17.5 A Market Model with Inventory 559 The Model 559 The Time Path 560 Graphical Summary of the Results 561 Exercise 17.5 562 17.6 Nonlinear Difference Equations— The Qualitative-Graphic Approach 562 Phase Diagram 562 Types of Time Path 564 . ..-• A Market with a Price Ceiling 565 Exercise 17.6 567

Chapter 18 Higher-Order Difference Equations

568

18.1 Second-Order Linear Difference Equations with Constant Coefficients and Constant Term 569 Particular Solution 569 Complementary Function 570 The Convergence of the Time Path 573 Exercise 18.1 575 18.2 Samuelson Multiplier-Acceleration Interaction Model 576 The Framework 576 The Solution 577 Convergence versus Divergence 578 A Graphical Summary 580 Exercise 18.2 581 18.3 Inflation and Unemployment in Discrete Time 581 The Model 581 The Difference Equation in p 582 The Time Path ofp 583 The Analysis ofU 584 The Long-Run Phillips Relation 585 Exercise 18.3 585 18.4 Generalizations to Variable-Term and Higher-Order Equations 586 Variable Term in the Form of cm' 586 Variable Term in the Form ct" 587 Higher-Order Linear Difference Equations 588 Convergence and the Schur Theorem 589 Exercise 18.4 591

Contents xix

Chapter 19 Simultaneous Differential Equations and Difference Equations 592 19.1 The Genesis of Dynamic Systems 592 Interacting Patterns of Change 592 The Transformation of a High-Order Dynamic Equation 593 19.2 Solving Simultaneous Dynamic Equations 594 Simultaneous Difference Equations 594 Matrix Notation 596 Simultaneous Differential Equations 599 Further Comments on the Characteristic Equation 601 Exercise 19.2 602 19.3 Dynamic Input-Output Models 603 Time Lag in Production 603 Excess Demand and Output Adjustment 605 Capital Formation 607 Exercise 19.3 608 19.4 The Inflation-Unemployment Model Once More 609 Simultaneous Differential Equations 610 Solution Paths 610 Simultaneous Difference Equations 612 Solution Paths 613 Exercise 19.4 614 19.5 Two-Variable Phase Diagrams 614 The Phase Space 615 The Demarcation Curves 615 Streamlines 617 Types of Equilibrium 618 Inflation and Monetary Rule a la Obst 620 Exercise 19.5 623 19.6 Linearization of a Nonlinear DifferentialEquation System 623 Taylor Expansion and Linearization 624

The Reduced Linearization 625 Local Stability Analysis 625 Exercise 19.6 629

Chapter 20 Optimal Control Theory

631

20.1 The Nature of Optimal Control 631 Illustration: A Simple Macroeconomic Model 632 Pontryagin s Maximum Principle 633 20.2 Alternative Terminal Conditions 639 Fixed Terminal Point 639 Horizontal Terminal Line 639 Truncated Vertical Terminal Line 639 Truncated Horizontal Terminal Line 640 Exercise 20.2 643 20.3 Autonomous Problems 644 20.4 Economic Applications 645 Lifetime Utility Maximization 645 Exhaustible Resource 647 Exercise 20.4 649 20.5 Infinite Time Horizon 649 Neoclassical Optimal Growth Model 649 The Current-Value Hamiltonian 651 Constructing a Phase Diagram 652 Analyzing the Phase Diagram 653 20.6 Limitations of Dynamic Analysis 654

The Greek Alphabet 655 Mathematical Symbols 656 A Short Reading List 659 Answers to Selected Exercises 662 Index 677