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PARTⅠ Introduction 3

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 R1 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

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 55

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

Deffnition and Examples of the Inverse of a Function 75

The Derivative of the Inverse Function 79

The Derivative of x m/n 80

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

PARTⅡ Linear Algebra 107

6 Introduction to Linear Algebra 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

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 Rn 209

Length and Distance 209

The Inner Product 213

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 Rn 247

Dimension 249

11.4 EPILOGUE 249

PARTⅢ Calculus of Several Variables 253

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

13 Functions of Several Variables 273

13.1 FUNCTIONS BETWEEN EUCLIDEAN SPACES 273

Functions from Rn to R 274

Functions from Rk to Rm 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 Rl for k ＞ 2 283

Images of Functions from R1 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

14.7 EXPLICIT FUNCTIONS FROM Rn TO Rm 323

Approximation by Differentials 324

The Chain Rule 326

14.8 HIGHER-ORDER DERIVATIVES 328

Continuously Differentiable Functions 328

Second Order Derivatives and Hessians 329

Young’s Theorem 330

Higher-Order Derivatives 331

An Economic Application 331

14.9 Epilogue 333

15 Implicit Functions and Their Derivatives 334

15.1 IMPLICIT FUNCTIONS 334

Examples 334

The Implicit Function Theorem for R2 337

Several Exogenous Variables in an Implicit Function 341

15.2 LEVEL CURVES AND THEIR TANGENTS 342

Geometric Interpretation of the Implicit Function Theorem 342

Proof Sketch 344

Relationship to the Gradient 345

Tangent to the Level Set Using Differentials 347

Level Sets of Functions of Several Variables 348

15.3 SYSTEMS OF IMPLICIT FUNCTIONS 350

Linear Systems 351

Nonlinear Systems 353

15.4 APPLICATION：COMPARATIVE STATICS 360

15.5 THE INVERSE FUNCTION THEOREM （optional） 364

15.6 APPLICATION：SIMPSON’S PARADOX 368

PARTⅣ Optimization 375

16 Quadratic Forms and Definite Matrices 375

16.1 QUADRATIC FORMS 375

16.2 DEFINITENESS OF QUADRATIC FORMS 376

Definite Symmetric Matrices 379

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 × 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

17 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

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

2 0 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

20.4 HOMOTHETIC FUNCTIONS 500

Motivation and Definition 500

Characterizing Homothetic Functions 501

21 Concave and Quasiconcave Functions 505

21.1 CONCAVE AND CONVEX FUNCTIONS 505

Calculus Criteria for Concavity 509

21.2 PROPERTIES OF CONCAVE FUNCTIONS 517

Concave Functions in Economics 521

21.3 QUASICONCAVE AND QUASICONVEX FUNCTIONS 522

Calculus Criteria 525

21.4 PSEUDOCONCAVE FUNCTIONS 527

21.5 CONCAVE PROGRAMMING 532

Unconstrained Problems 532

Constrained Problems 532

Saddle Point Approach 534

21.6 APPENDIX 537

Proof of the Sufficiency Test of Theorem 21.14 537

Proof of Theorem 21.15 538

Proof of Theorem 21.17 540

Proof of Theorem 21.20 541

22 Economic Applications 544

22.1 UTILITY AND DEMAND 544

Utility Maximization 544

The Demand Function 547

The Indirect Utility Function 551

The Expenditure and Compensated Demand Functions 552

The Slutsky Equation 555

22.2 ECONOMIC APPLICATION：PROFIT AND COST 557

The Profit-Maximizing Firm 557

The Cost Function 560

22.3 PARETO OPTIMA 565

Necessary Conditions for a Pareto Optimum 566

Sufficient Conditions for a Pareto Optimum 567

22.4 THE FUNDAMENTAL WELFARE THEOREMS 569

Competitive Equilibrium 572

Fundamental Theorems of Welfare Economics 573

PART Ⅴ Eigenvalues and Dynamics 579

23 Eigenvalues and Eigenvectors 579

23.1 DEFINITIONS AND EXAMPLES 579

23.2 SOLVING LINEAR DIFFERENCE EQUATIONS 585

One-Dimensional Equations 585

Two-Dimensional Systems：An Example 586

Conic Sections 587

The Leslie Population Model 588

Abstract Two-Dimensional Systems 590

K-Dimensional Systems 591

An Alternative Approach：The Powers of a Matrix 594

Stability of Equilibria 596

23.3 PROPERTIES OF EIGENVALUES 597

Trace as Sum of the Eigenvalues 599

23.4 REPEATED EIGENVALUES 601

2 × 2 Nondiagonalizable Matrices 601

3 × 3 Nondiagonalizable Matrices 604

Solving Nondiagonalizable Difference Equations 606

23.5 COMPLEX EIGENVALUES AND EIGENVECTORS 609

Diagonalizing Matrices with Complex Eigenvalues 609

Linear Difference Equations with Complex Eigenvalues 611

Higher Dimensions 614

23.6 MARKOV PROCESSES 615

23.7 SYMMETRIC MATRICES 620

23.8 DEFINITENESS OF QUADRATIC FORMS 626

23.9 APPENDIX 629

Proof of Theorem 23.5 629

Proof of Theorem 23.9 630

24 Ordinary Differential Equations：Scalar Equations 633

24.1 DEFINITION AND EXAMPLES 633

24.2 EXPLICIT SOLUTIONS 639

Linear First Order Equations 639

Separable Equations 641

24.3 LINEAR SECOND ORDER EQUATIONS 647

Introduction 647

Real and Unequal Roots of the Characteristic Equation 648

Real and Equal Roots of the Characteristic Equation 650

Complex Roots of the Characteristic Equation 651

The Motion of a Spring 653

Nonhomogeneous Second Order Equations 654

24.4 EXISTENCE OF SOLUTIONS 657

The Fundamental Existence and Uniqueness Theorem 657

Direction Fields 659

24.5 PHASE PORTRAITS AND EQUILIBRIA ON R1 666

Drawing Phase Portraits 666

Stability of Equilibria on the Line 668

24.6 APPENDIX：APPLICATIONS 670

Indirect Money Metric Utility Functions 671

Converse of Euler’s Theorem 672

25 Ordinary Differential Equations：Systems of Equations 674

25.1 PLANAR SYSTEMS：AN INTRODUCTION 674

Coupled Systems of Differential Equations 674

Vocabulary 676

Existence and Uniqueness 677

25.2 LINEAR SYSTEMS VIA EIGENVALUES 678

Distinct Real Eigenvalues 678

Complex Eigenvalues 680

Multiple Real Eigenvalues 681

25.3 SOLVING LINEAR SYSTEMS BY SUBSTITUTION 682

25.4 STEADY STATES AND THEIR STABILITY 684

Stability of Linear Systems via Eigenvalues 686

Stability of Nonlinear Systems 687

25.5 PHASE PORTRAITS OF PLANAR SYSTEMS 689

Vector Fields 689

Phase Portraits：Linear Systems 692

Phase Portraits：Nonlinear Systems 694

25.6 FIRST INTEGRALS 703

The Predator-Prey System 705

Conservative Mechanical Systems 707

25.7 LIAPUNOV FUNCTIONS 711

25.8 APPENDIX：LINEARIZATION 715

PART Ⅵ Advanced Linear Algebra 719

26 Determinants：The Details 719

26.1 DEFINITIONS OF THE DETERMINANT 719

26.2 PROPERTIES OF THE DETERMINANT 726

26.3 USING DETERMINANTS 735

The Adjoint Matrix 736

26.4 ECONOMIC APPLICATIONS 739

Supply and Demand 739

26.5 APPENDIX 743

Proof of Theorem 26.1 743

Proof of Theorem 26.9 746

Other Approaches to the Determinant 747

27 Subspaces Attached to a Matrix 750

27.1 VECTOR SPACES AND SUBSPACES 750

Rn as a Vector Space 750

Subspaces of Rn 751

27.2 BASIS AND DIMENSION OF A PROPER SUBSPACE 755

27.3 ROW SPACE 757

27.4 COLUMN SPACE 760

Dimension of the Column Space of A 760

The Role of the Column Space 763

27.5 NULLSPACE 765

Affine Subspaces 765

Fundamental Theorem of Linear Algebra 767

Conclusion 770

27.6 ABSTRACT VECTOR SPACES 771

27.7 APPENDIX 774

Proof of Theorem 27.5 774

Proof of Theorem 27.10 775

28 Applications of Linear Independence 779

28.1 GEOMETRY OF SYSTEMS OF EQUATIONS 779

Two Equations in Two Unknowns 779

Two Equations in Three Unknowns 780

Three Equations in Three Unknowns 782

28.2 PORTFOLIO ANALYSIS 783

28.3 VOTING PARADOXES 784

Three Alternatives 785

Four Alternatives 788

Consequences of the Existence of Cycles 789

Other Voting Paradoxes 790

Rankings of the Quality of Firms 790

28.4 ACTIVITY ANALYSIS：FEASIBILITY 791

Activity Analysis 791

Simple Linear Models and Productive Matrices 793

28.5 ACTIVITY ANALYSIS：EFFICIENCY 796

Leontief Models 796

PART Ⅶ Advanced Analysis 803

29 Limits and Compact Sets 803

29.1 CAUCHY SEQUENCES 803

29.2 COMPACT SETS 807

29.3 CONNECTED SETS 809

29.4 ALTERNATIVE NORMS 811

Three Norms on Rn 811

Equivalent Norms 813

Norms on Function Spaces 815

29.5 APPENDIX 816

Finite Covering Property 816

Heine-Borel Theorem 817

Summary 820

30 Calculus of Several Variables Ⅱ 822

30.1 WEIERSTRASS’S AND MEAN VALUE THEOREMS 822

Existence of Global Maxima on Compact Sets 822

Rolle’s Theorem and the Mean Value Theorem 824

30.2 TAYLOR POLYNOMIALS ON R1 827

Functions of One Variable 827

30.3 TAYLOR POLYNOMIALS IN Rn 832

30.4 SECOND ORDER OPTIMIZATION CONDITIONS 836

Second Order Sufficient Conditions for Optimization 836

Indefinite Hessian 839

Second Order Necessary Conditions for Optimization 840

30.5 CONSTRAINED OPTIMIZATION 841

PARTⅧ Appendices 847

A1 Sets，Numbers，and Proofs 847

A1.1 SETS 847

Vocabulary of Sets 847

Operations with Sets 847

A1.2 NUMBERS 848

Vocabulary 848

Properties of Addition and Multiplication 849

Least Upper Bound Property 850

A1.3 PROOFS 851

Direct Proofs 851

Converse and Contrapositive 853

Indirect Proofs 854

Mathematical Induction 855

A2 Trigonometric Functions 859

A2.1 DEFINITIONS OF THE TRIG FUNCTIONS 859

A2.2 GRAPHING TRIG FUNCTIONS 863

A2.3 THE PYTHAGOREAN THEOREM 865

A2.4 EVALUATING TRIGONOMETRIC FUNCTIONS 866

A2.5 MULTIANGLE FORMULAS 868

A2.6 FUNCTIONS OF REAL NUMBERS 868

A2.7 CALCULUS WITH TRIG FUNCTIONS 870

A2.8 TAYLOR SERIES 872

A2.9 PROOF OF THEOREM A2.3 873

A3 Complex Numbers 876

A3.1 BACKGROUND 876

Denitions 877

Arithmetic Operations 877

A3.2 SOLUTIONS OF POLYNOMIAL EQUATIONS 878

A3.3 GEOMETRIC REPRESENTATION 879

A3.4 COMPLEX NUMBERS AS EXPONENTS 882

A3.5 DIFFERENCE EQUATIONS 884

A4 Integral Calculus 887

A4.1 ANTIDERIVATIVES 887

Integration by Parts 888

A4.2 THE FUNDAMENTAL THEOREM OF CALCULUS 889

A4.3 APPLICATIONS 890

Area under a Graph 890

Consumer Surplus 891

Present Value of a Flow 892

A5 Introduction to Probability 894

A5.1 PROBABILITY OF AN EVENT 894

A5.2 EXPECTATION AND VARIANCE 895

A5.3 CONTINUOUS RANDOM VARIABLES 896

A6 Selected Answers 899

Index 921