Mathematical Ecnomics - Texas A&M University

Lecture Notes

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Mathematical Ecnomics

Guoqiang TIAN

Department of Economics

Texas A&M University

College Station, Texas 77843

(gtian@tamu.edu)

This version: August 2018

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The lecture notes are only for the purpose of my teaching and convenience of my students in class,

please not put them online or pass to any others.

Contents

1 The Nature of Mathematical Economics

1

1.1

Economics and Mathematical Economics . . . . . . . . . . . . . . . . . . .

1

1.2

Advantages of Mathematical Approach . . . . . . . . . . . . . . . . . . . .

2

2 Economic Models

4

2.1

Ingredients of a Mathematical Model . . . . . . . . . . . . . . . . . . . . .

4

2.2

The Real-Number System . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.3

The Concept of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.4

Relations and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.5

Types of Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.6

Functions of Two or More Independent Variables . . . . . . . . . . . . . . 10

2.7

Levels of Generality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Equilibrium Analysis in Economics

12

3.1

The Meaning of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2

Partial Market Equilibrium - A Linear Model . . . . . . . . . . . . . . . . 12

3.3

Partial Market Equilibrium - A Nonlinear Model . . . . . . . . . . . . . . . 14

3.4

General Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5

Equilibrium in National-Income Analysis . . . . . . . . . . . . . . . . . . . 19

4 Linear Models and Matrix Algebra

20

4.1

Matrix and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2

Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3

Linear Dependance of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.4

Commutative, Associative, and Distributive Laws . . . . . . . . . . . . . . 26

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4.5

Identity Matrices and Null Matrices . . . . . . . . . . . . . . . . . . . . . . 27

4.6

Transposes and Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Linear Models and Matrix Algebra (Continued)

32

5.1

Conditions for Nonsingularity of a Matrix . . . . . . . . . . . . . . . . . . 32

5.2

Test of Nonsingularity by Use of Determinant . . . . . . . . . . . . . . . . 34

5.3

Basic Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . 38

5.4

Finding the Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.5

Cramer¡¯s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.6

Application to Market and National-Income Models . . . . . . . . . . . . . 49

5.7

Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.8

Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.9

Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 Comparative Statics and the Concept of Derivative

62

6.1

The Nature of Comparative Statics . . . . . . . . . . . . . . . . . . . . . . 62

6.2

Rate of Change and the Derivative . . . . . . . . . . . . . . . . . . . . . . 63

6.3

The Derivative and the Slope of a Curve . . . . . . . . . . . . . . . . . . . 64

6.4

The Concept of Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.5

Inequality and Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . 68

6.6

Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.7

Continuity and Di?erentiability of a Function . . . . . . . . . . . . . . . . 70

7 Rules of Di?erentiation and Their Use in Comparative Statics

73

7.1

Rules of Di?erentiation for a Function of One Variable . . . . . . . . . . . 73

7.2

Rules of Di?erentiation Involving Two or More Functions of the Same Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.3

Rules of Di?erentiation Involving Functions of Di?erent Variables . . . . . 79

7.4

Integration (The Case of One Variable) . . . . . . . . . . . . . . . . . . . . 82

7.5

Partial Di?erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.6

Applications to Comparative-Static Analysis . . . . . . . . . . . . . . . . . 87

7.7

Note on Jacobian Determinants . . . . . . . . . . . . . . . . . . . . . . . . 89

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8 Comparative-static Analysis of General-Functions

92

8.1

Di?erentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8.2

Total Di?erentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

8.3

Rule of Di?erentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8.4

Total Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8.5

Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.6

Comparative Statics of General-Function Models . . . . . . . . . . . . . . . 104

8.7

Matrix Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

9 Derivatives of Exponential and Logarithmic Functions

107

9.1

The Nature of Exponential Functions . . . . . . . . . . . . . . . . . . . . . 107

9.2

Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

9.3

Derivatives of Exponential and Logarithmic Functions . . . . . . . . . . . . 109

10 Optimization: Maxima and Minima of a Function of One Variable

111

10.1 Optimal Values and Extreme Values . . . . . . . . . . . . . . . . . . . . . 111

10.2 General Result on Maximum and Minimum . . . . . . . . . . . . . . . . . 112

10.3 First-Derivative Test for Relative Maximum and Minimum . . . . . . . . . 113

10.4 Second and Higher Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 115

10.5 Second-Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

10.6 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

10.7 Nth-Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

11 Optimization: Maxima and Minima of a Function of Two or More Variables

122

11.1 The Di?erential Version of Optimization Condition . . . . . . . . . . . . . 122

11.2 Extreme Values of a Function of Two Variables . . . . . . . . . . . . . . . 123

11.3 Objective Functions with More than Two Variables . . . . . . . . . . . . . 128

11.4 Second-Order Conditions in Relation to Concavity and Convexity . . . . . 130

11.5 Economic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

12 Optimization with Equality Constraints

136

12.1 E?ects of a Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

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12.2 Finding the Stationary Values . . . . . . . . . . . . . . . . . . . . . . . . . 137

12.3 Second-Order Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

12.4 General Setup of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . 143

12.5 Quasiconcavity and Quasiconvexity . . . . . . . . . . . . . . . . . . . . . . 145

12.6 Utility Maximization and Consumer Demand . . . . . . . . . . . . . . . . . 149

13 Optimization with Inequality Constraints

152

13.1 Non-Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

13.2 Kuhn-Tucker Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

13.3 Economic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

14 Linear Programming

164

14.1 The Setup of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

14.2 The Simplex Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

14.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

15 Continuous Dynamics: Di?erential Equations

177

15.1 Di?erential Equations of the First Order . . . . . . . . . . . . . . . . . . . 177

15.2 Linear Di?erential Equations of a Higher Order with Constant Coe?cients 180

15.3 Systems of the First Order Linear Di?erential Equations . . . . . . . . . . 184

15.4 Economic Application: General Equilibrium . . . . . . . . . . . . . . . . . 190

15.5 Simultaneous Di?erential Equations. Types of Equilibria . . . . . . . . . . 194

16 Discrete Dynamics: Di?erence Equations

197

16.1 First-order Linear Di?erence Equations . . . . . . . . . . . . . . . . . . . . 199

16.2 Second-Order Linear Di?erence Equations . . . . . . . . . . . . . . . . . . 201

16.3 The General Case of Order n . . . . . . . . . . . . . . . . . . . . . . . . . 202

16.4 Economic Application:A dynamic model of economic growth . . . . . . . . 203

17 Introduction to Dynamic Optimization

205

17.1 The First-Order Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 205

17.2 Present-Value and Current-Value Hamiltonians

. . . . . . . . . . . . . . . 207

17.3 Dynamic Problems with Inequality Constraints . . . . . . . . . . . . . . . 208

17.4 Economics Application:The Ramsey Model . . . . . . . . . . . . . . . . . . 208

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