《MATHEMATICS FOR ECONOMISTS》

PARTⅠIntroduction3

1Introduction3

1.1 MATHEMATICS IN ECONOMIC THEORY3

1.2MODELS OF CONSUMER CHOICE5

Two-Dimensional Model of Consumer Choice5

Multidimensional Model of Consumer Choice9

2One-Variable Calculus：Foundations10

2.1FUNCTIONS ON R110

Vocabulary of Functions10

Polynomials11

Graphs12

Increasing and Decreasing Functions12

Domain14

Interval Notation15

2.2LINEAR FUNCTIONS16

The Slope of a Line in the Plane16

The Equation of a Line19

Polynomials of Degree One Have Linear Graphs19

Interpreting the Slope of a Linear Function20

2.3 THE SLOPE OF NONLINEAR FUNCTIONS22

2.4COMPUTING DERIVATIVES25

Rules for Computing Derivatives27

2.5DIFFERENTIABILITY AND CONTINUITY29

A Nondifferentiable Function30

Continuous Functions31

Continuously Differentiable Functions32

2.6 HIGHER-ORDER DERIVATIVES33

2.7 APPROXIMATION BY DIFFERENTIALS34

3One-Variable Calculus：Applications39

3.1USING THE FIRST DERIVATIVE FOR GRAPHING39

Positive Derivative Implies Increasing Function39

Using First Derivatives to Sketch Graphs41

3.2 SECOND DERIVATIVES AND CONVEXITY43

3.3GRAPHING RATIONAL FUNCTIONS47

Hints for Graphing48

3.4TAILS AND HORIZONTAL ASYMPTOTES48

Tails of Polynomials48

Horizontal Asymptotes of Rational Functions49

3.5MAXIMA AND MINIMA51

Local Maxima and Minima on the Boundary and in the Interior51

Second Order Conditions53

Global Maxima and Minima55

Functions with Only One Critical Point55

Functions with Nowhere-Zero Second Derivatives56

Functions with No Global Max or Min56

Functions Whose Domains Are Closed Finite Intervals56

3.6APPLICATIONS TO ECONOMICS58

Production Functions58

Cost Functions59

Revenue and Profit Functions62

Demand Functions and Elasticity64

4One-Variable Calculus：Chain Rule70

4.1COMPOSITE FUNCTIONS AND THE CHAIN RULE70

Composite Functions70

Differentiating Composite Functions：The Chain Rule72

4.2INVERSE FUNCTIONS AND THEIR DERIVATIVES75

Deffnition and Examples of the Inverse of a Function75

The Derivative of the Inverse Function79

The Derivative of x m/n80

5Exponents and Logarithms82

5.1 EXPONENTIAL FUNCTIONS82

5.2 THE NUMBER e85

5.3LOGARITHMS88

Base 10 Logarithms88

Base e Logarithms90

5.4 PROPERTIES OF EXP AND LOG91

5.5 DERIVATIVES OF EXP AND LOG93

5.6APPLICATIONS97

Present Value97

Annuities98

Optimal Holding Time99

Logarithmic Derivative100

PARTⅡLinear Algebra107

6Introduction to Linear Algebra107

6.1 LINEAR SYSTEMS107

6.2EXAMPLES OF LINEAR MODELS108

Example 1：Tax Benefits of Charitable Contributions108

Example 2：Linear Models of Production110

Example 3：Markov Models of Employment113

Example 4：IS-LM Analysis115

Example 5：Investment and Arbitrage117

7Systems of Linear Equations122

7.1GAUSSIAN AND GAUSS-JORDAN ELIMINATION122

Substitution123

Elimination of Variables125

7.2 ELEMENTARY ROW OPERATIONS129

7.3 SYSTEMS WITH MANY OR NO SOLUTIONS134

7.4RANK—THE FUNDAMENTAL CRITERION142

Application to Portfolio Theory147

7.5 THE LINEAR IMPLICIT FUNCTION THEOREM150

8Matrix Algebra153

8.1MATRIX ALGEBRA153

Addition153

Subtraction154

Scalar Multiplication155

Matrix Multiplication155

Laws of Matrix Algebra156

Transpose157

Systems of Equations in Matrix Form158

8.2 SPECIAL KINDS OF MATRICES160

8.3 ELEMENTARY MATRICES162

8.4 ALGEBRA OF SQUARE MATRICES165

8.5INPUT- OUTPUT MATRICES174

Proof of Theorem 8.13178

8.6 PARTITIONED MATRICES （optional）180

8.7DECOMPOSING MATRICES （optional）183

Mathematical Induction185

Including Row Interchanges185

9Determinants：An Overview188

9.1THE DETERMINANT OF A MATRIX189

Defining the Determinant189

Computing the Determinant191

Main Property of the Determinant192

9.2 USES OF THE DETERMINANT194

9.3 IS-LM ANALYSIS VIA CRAMER’S RULE197

10Euclidean Spaces199

10.1 POINTS AND VECTORS IN EUCLIDEAN SPACE199

The Real Line199

The Plane199

Three Dimensions and More201

10.2 VECTORS202

10.3 THE ALGEBRA OF VECTORS205

Addition and Subtraction205

Scalar Multiplication207

10.4 LENGTH AND INNER PRODUCT IN Rn209

Length and Distance209

The Inner Product213

10.5LINES222

10.6PLANES226

Parametric Equations226

Nonparametric Equations228

Hyperplanes230

10.7ECONOMIC APPLICATIONS232

Budget Sets in Commodity Space232

Input Space233

Probability Simplex233

The Investment Model234

IS-LM Analysis234

11Linear Independence237

11.1LINEAR INDEPENDENCE237

Definition238

Checking Linear Independence241

11.2 SPANNING SETS244

11.3BASIS AND DIMENSION IN Rn247

Dimension249

11.4 EPILOGUE249

PARTⅢCalculus of Several Variables253

12Limits and Open Sets253

12.1SEQUENCES OF REAL NUMBERS253

Definition253

Limit of a Sequence254

Algebraic Properties of Limits256

12.2 SEQUENCES IN Rm260

12.3OPEN SETS264

Interior of a Set267

12.4CLOSED SETS267

Closure of a Set268

Boundary of a Set269

12.5 COMPACT SETS270

12.6 EPILOGUE272

13Functions of Several Variables273

13.1FUNCTIONS BETWEEN EUCLIDEAN SPACES273

Functions from Rn to R274

Functions from Rk to Rm275

13.2GEOMETRIC REPRESENTATION OF FUNCTIONS277

Graphs of Functions of Two Variables277

Level Curves280

Drawing Graphs from Level Sets281

Planar Level Sets in Economics282

Representing Functions from Rk to Rl for k ＞ 2283

Images of Functions from R1 to Rm285

13.3SPECIAL KINDS OF FUNCTIONS287

Linear Functions on Rk287

Quadratic Forms289

Matrix Representation of Quadratic Forms290

Polynomials291

13.4 CONTINUOUS FUNCTIONS293

13.5VOCABULARY OF FUNCTIONS295

Onto Functions and One-to-One Functions297

Inverse Functions297

Composition of Functions298

14Calculus of Several Variables300

14.1 DEFINITIONS AND EXAMPLES300

14.2ECONOMIC INTERPRETATION302

Marginal Products302

Elasticity304

14.3 GEOMETRIC INTERPRETATION305

14.4THE TOTAL DERIVATIVE307

Geometric Interpretation308

Linear Approximation310

Functions of More than Two Variables311

14.5THE CHAIN RULE313

Curves313

Tangent Vector to a Curve314

Differentiating along a Curve：The Chain Rule316

14.6DIRECTIONAL DERIVATIVES AND GRADIENTS319

Directional Derivatives319

The Gradient Vector320

14.7EXPLICIT FUNCTIONS FROM Rn TO Rm323

Approximation by Differentials324

The Chain Rule326

14.8HIGHER-ORDER DERIVATIVES328

Continuously Differentiable Functions328

Second Order Derivatives and Hessians329

Young’s Theorem330

Higher-Order Derivatives331

An Economic Application331

14.9 Epilogue333

15Implicit Functions and Their Derivatives334

15.1IMPLICIT FUNCTIONS334

Examples334

The Implicit Function Theorem for R2337

Several Exogenous Variables in an Implicit Function341

15.2LEVEL CURVES AND THEIR TANGENTS342

Geometric Interpretation of the Implicit Function Theorem342

Proof Sketch344

Relationship to the Gradient345

Tangent to the Level Set Using Differentials347

Level Sets of Functions of Several Variables348

15.3SYSTEMS OF IMPLICIT FUNCTIONS350

Linear Systems351

Nonlinear Systems353

15.4 APPLICATION：COMPARATIVE STATICS360

15.5 THE INVERSE FUNCTION THEOREM （optional）364

15.6 APPLICATION：SIMPSON’S PARADOX368

PARTⅣOptimization375

16Quadratic Forms and Definite Matrices375

16.1 QUADRATIC FORMS375

16.2DEFINITENESS OF QUADRATIC FORMS376

Definite Symmetric Matrices379

Application：Second Order Conditions and Convexity379

Application：Conic Sections380

Principal Minors of a Matrix381

The Definiteness of Diagonal Matrices383

The Definiteness of 2 × 2 Matrices384

16.3LINEAR CONSTRAINTS AND BORDERED MATRICES386

Definiteness and Optimality386

One Constraint390

Other Approaches391

16.4 APPENDIX393

17Unconstrained Optimization396

17.1 DEFINITIONS396

17.2 FIRST ORDER CONDITIONS397

17.3SECOND ORDER CONDITIONS398

Sufficient Conditions398

Necessary Conditions401

17.4GLOBAL MAXIMA AND MINIMA402

Global Maxima of Concave Functions403

17.5ECONOMIC APPLICATIONS404

Profit-Maximizing Firm405

Discriminating Monopolist405

Least Squares Analysis407

18Constrained Optimization Ⅰ：First Order Conditions411

18.1 EXAMPLES412

18.2EQUALITY CONSTRAINTS413

Two Variables and One Equality Constraint413

Several Equality Constraints420

18.3INEQUALITY CONSTRAINTS424

One Inequality Constraint424

Several Inequality Constraints430

18.4 MIXED CONSTRAINTS434

18.5 CONSTRAINED MINIMIZATION PROBLEMS436

18.6 KUHN-TUCKER FORMULATION439

18.7EXAMPLES AND APPLICATIONS442

Application：A Sales-Maximizing Firm with Advertising442

Application：The Averch-Johnson Effect443

One More Worked Example445

19Constrained Optimization Ⅱ448

19.1THE MEANING OF THE MULTIPLIER448

One Equality Constraint449

Several Equality Constraints450

Inequality Constraints451

Interpreting the Multiplier452

19.2ENVELOPE THEOREMS453

Unconstrained Problems453

Constrained Problems455

19.3SECOND ORDER CONDITIONS457

Constrained Maximization Problems459

Minimization Problems463

Inequality Constraints466

Alternative Approaches to the Bordered Hessian Condition467

Necessary Second Order Conditions468

19.4 SMOOTH DEPENDENCE ON THE PARAMETERS469

19.5 CONSTRAINT QUALIFICATIONS472

19.6PROOFS OF FIRST ORDER CONDITIONS478

Proof of Theorems 18.1 and 18.2：Equality Constraints478

Proof of Theorems 18.3 and 18.4：Inequality Constraints480

2 0Homogeneous and Homothetic Functions483

20.1HOMOGENEOUS FUNCTIONS483

Definition and Examples483

Homogeneous Functions in Economics485

Properties of Homogeneous Functions487

A Calculus Criterion for Homogeneity491

Economic Applications of Euler’s Theorem492

20.2HOMOGENIZING A FUNCTION493

Economic Applications of Homogenization495

20.3 CARDINAL VERSUS ORDINAL UTILITY496

20.4HOMOTHETIC FUNCTIONS500

Motivation and Definition500

Characterizing Homothetic Functions501

21Concave and Quasiconcave Functions505

21.1CONCAVE AND CONVEX FUNCTIONS505

Calculus Criteria for Concavity509

21.2PROPERTIES OF CONCAVE FUNCTIONS517

Concave Functions in Economics521

21.3QUASICONCAVE AND QUASICONVEX FUNCTIONS522

Calculus Criteria525

21.4 PSEUDOCONCAVE FUNCTIONS527

21.5CONCAVE PROGRAMMING532

Unconstrained Problems532

Constrained Problems532

Saddle Point Approach534

21.6APPENDIX537

Proof of the Sufficiency Test of Theorem 21.14537

Proof of Theorem 21.15538

Proof of Theorem 21.17540

Proof of Theorem 21.20541

22Economic Applications544

22.1UTILITY AND DEMAND544

Utility Maximization544

The Demand Function547

The Indirect Utility Function551

The Expenditure and Compensated Demand Functions552

The Slutsky Equation555

22.2ECONOMIC APPLICATION：PROFIT AND COST557

The Profit-Maximizing Firm557

The Cost Function560

22.3PARETO OPTIMA565

Necessary Conditions for a Pareto Optimum566

Sufficient Conditions for a Pareto Optimum567

22.4 THE FUNDAMENTAL WELFARE THEOREMS569

Competitive Equilibrium572

Fundamental Theorems of Welfare Economics573

PARTⅤ Eigenvalues and Dynamics579

23Eigenvalues and Eigenvectors579

23.1 DEFINITIONS AND EXAMPLES579

23.2SOLVING LINEAR DIFFERENCE EQUATIONS585

One-Dimensional Equations585

Two-Dimensional Systems：An Example586

Conic Sections587

The Leslie Population Model588

Abstract Two-Dimensional Systems590

K-Dimensional Systems591

An Alternative Approach：The Powers of a Matrix594

Stability of Equilibria596

23.3PROPERTIES OF EIGENVALUES597

Trace as Sum of the Eigenvalues599

23.4REPEATED EIGENVALUES601

2 × 2 Nondiagonalizable Matrices601

3 × 3 Nondiagonalizable Matrices604

Solving Nondiagonalizable Difference Equations606

23.5COMPLEX EIGENVALUES AND EIGENVECTORS609

Diagonalizing Matrices with Complex Eigenvalues609

Linear Difference Equations with Complex Eigenvalues611

Higher Dimensions614

23.6 MARKOV PROCESSES615

23.7 SYMMETRIC MATRICES620

23.8 DEFINITENESS OF QUADRATIC FORMS626

23.9APPENDIX629

Proof of Theorem 23.5629

Proof of Theorem 23.9630

24Ordinary Differential Equations：Scalar Equations633

24.1 DEFINITION AND EXAMPLES633

24.2EXPLICIT SOLUTIONS639

Linear First Order Equations639

Separable Equations641

24.3LINEAR SECOND ORDER EQUATIONS647

Introduction647

Real and Unequal Roots of the Characteristic Equation648

Real and Equal Roots of the Characteristic Equation650

Complex Roots of the Characteristic Equation651

The Motion of a Spring653

Nonhomogeneous Second Order Equations654

24.4EXISTENCE OF SOLUTIONS657

The Fundamental Existence and Uniqueness Theorem657

Direction Fields659

24.5PHASE PORTRAITS AND EQUILIBRIA ON R1666

Drawing Phase Portraits666

Stability of Equilibria on the Line668

24.6APPENDIX：APPLICATIONS670

Indirect Money Metric Utility Functions671

Converse of Euler’s Theorem672

25Ordinary Differential Equations：Systems of Equations674

25.1PLANAR SYSTEMS：AN INTRODUCTION674

Coupled Systems of Differential Equations674

Vocabulary676

Existence and Uniqueness677

25.2LINEAR SYSTEMS VIA EIGENVALUES678

Distinct Real Eigenvalues678

Complex Eigenvalues680

Multiple Real Eigenvalues681

25.3 SOLVING LINEAR SYSTEMS BY SUBSTITUTION682

25.4STEADY STATES AND THEIR STABILITY684

Stability of Linear Systems via Eigenvalues686

Stability of Nonlinear Systems687

25.5PHASE PORTRAITS OF PLANAR SYSTEMS689

Vector Fields689

Phase Portraits：Linear Systems692

Phase Portraits：Nonlinear Systems694

25.6FIRST INTEGRALS703

The Predator-Prey System705

Conservative Mechanical Systems707

25.7 LIAPUNOV FUNCTIONS711

25.8 APPENDIX：LINEARIZATION715

PARTⅥ Advanced Linear Algebra719

26Determinants：The Details719

26.1 DEFINITIONS OF THE DETERMINANT719

26.2 PROPERTIES OF THE DETERMINANT726

26.3USING DETERMINANTS735

The Adjoint Matrix736

26.4ECONOMIC APPLICATIONS739

Supply and Demand739

26.5APPENDIX743

Proof of Theorem 26.1743

Proof of Theorem 26.9746

Other Approaches to the Determinant747

27Subspaces Attached to a Matrix750

27.1VECTOR SPACES AND SUBSPACES750

Rn as a Vector Space750

Subspaces of Rn751

27.2 BASIS AND DIMENSION OF A PROPER SUBSPACE755

27.3 ROW SPACE757

27.4COLUMN SPACE760

Dimension of the Column Space of A760

The Role of the Column Space763

27.5NULLSPACE765

Affine Subspaces765

Fundamental Theorem of Linear Algebra767

Conclusion770

27.6 ABSTRACT VECTOR SPACES771

27.7APPENDIX774

Proof of Theorem 27.5774

Proof of Theorem 27.10775

28Applications of Linear Independence779

28.1GEOMETRY OF SYSTEMS OF EQUATIONS779

Two Equations in Two Unknowns779

Two Equations in Three Unknowns780

Three Equations in Three Unknowns782

28.2 PORTFOLIO ANALYSIS783

28.3VOTING PARADOXES784

Three Alternatives785

Four Alternatives788

Consequences of the Existence of Cycles789

Other Voting Paradoxes790

Rankings of the Quality of Firms790

28.4ACTIVITY ANALYSIS：FEASIBILITY791

Activity Analysis791

Simple Linear Models and Productive Matrices793

28.5ACTIVITY ANALYSIS：EFFICIENCY796

Leontief Models796

PARTⅦ Advanced Analysis803

29Limits and Compact Sets803

29.1 CAUCHY SEQUENCES803

29.2 COMPACT SETS807

29.3 CONNECTED SETS809

29.4ALTERNATIVE NORMS811

Three Norms on Rn811

Equivalent Norms813

Norms on Function Spaces815

29.5APPENDIX816

Finite Covering Property816

Heine-Borel Theorem817

Summary820

30Calculus of Several Variables Ⅱ822

30.1WEIERSTRASS’S AND MEAN VALUE THEOREMS822

Existence of Global Maxima on Compact Sets822

Rolle’s Theorem and the Mean Value Theorem824

30.2TAYLOR POLYNOMIALS ON R1827

Functions of One Variable827

30.3 TAYLOR POLYNOMIALS IN Rn832

30.4SECOND ORDER OPTIMIZATION CONDITIONS836

Second Order Sufficient Conditions for Optimization836

Indefinite Hessian839

Second Order Necessary Conditions for Optimization840

30.5 CONSTRAINED OPTIMIZATION841

PARTⅧAppendices847

A1Sets，Numbers，and Proofs847

A1.1SETS847

Vocabulary of Sets847

Operations with Sets847

A1.2NUMBERS848

Vocabulary848

Properties of Addition and Multiplication849

Least Upper Bound Property850

A1.3PROOFS851

Direct Proofs851

Converse and Contrapositive853

Indirect Proofs854

Mathematical Induction855

A2Trigonometric Functions859

A2.1 DEFINITIONS OF THE TRIG FUNCTIONS859

A2.2 GRAPHING TRIG FUNCTIONS863

A2.3 THE PYTHAGOREAN THEOREM865

A2.4 EVALUATING TRIGONOMETRIC FUNCTIONS866

A2.5 MULTIANGLE FORMULAS868

A2.6 FUNCTIONS OF REAL NUMBERS868

A2.7 CALCULUS WITH TRIG FUNCTIONS870

A2.8 TAYLOR SERIES872

A2.9 PROOF OF THEOREM A2.3873

A3Complex Numbers876

A3.1BACKGROUND876

Denitions877

Arithmetic Operations877

A3.2 SOLUTIONS OF POLYNOMIAL EQUATIONS878

A3.3 GEOMETRIC REPRESENTATION879

A3.4 COMPLEX NUMBERS AS EXPONENTS882

A3.5 DIFFERENCE EQUATIONS884

A4Integral Calculus887

A4.1ANTIDERIVATIVES887

Integration by Parts888

A4.2 THE FUNDAMENTAL THEOREM OF CALCULUS889

A4.3APPLICATIONS890

Area under a Graph890

Consumer Surplus891

Present Value of a Flow892

A5Introduction to Probability894

A5.1 PROBABILITY OF AN EVENT894

A5.2 EXPECTATION AND VARIANCE895

A5.3 CONTINUOUS RANDOM VARIABLES896

A6 Selected Answers899

Index921