Fundamental methods of mathematical economics / Alpha C. Chiang.

Format:

Book

Author/Creator:

Chiang, Alpha C., 1927-

Language:

English

Subjects (All):

Economics, Mathematical.

Physical Description:

xii, 788 pages : illustrations ; 25 cm

Edition:

Third edition.

Place of Publication:

New York : McGraw-Hill, [1984]

Summary:

The best-selling, best known text in Mathematical Economics course, Chiang teaches the basic mathematical methods indispensable for understanding current economic literature. the book's patient explanations are written in an informal, non-intimidating style. To underscore the relevance of mathematics to economics, the author allows the economist's analytical needs to motivate the study of related mathematical techniques; he then illustrates these techniques with appropriate economics models. Graphic illustrations often visually reinforce algebraic results. Many exercise problems serve as drills and help bolster student confidence. These major types of economic analysis are covered: statics, comparative statics, optimization problems, dynamics, and mathematical programming. These mathematical methods are introduced: matrix algebra, differential and integral calculus, differential equations, difference equations, and convex sets.

Contents:

1 The Nature of Mathematical Economics 3

1.1 Mathematical versus Nonmathematical Economics 3

1.2 Mathematical Economics versus Econometrics 5

2 Economic Models 7

2.1 Ingredients of a Mathematical Model 7

2.2 The Real-Number System 10

2.3 The Concept of Sets 11

2.4 Relations and Functions 17

2.5 Types of Function 23

2.6 Functions of Two or More Independent Variables 29

2.7 Levels of Generality 31

Part 2 Static (or Equilibrium) Analysis

3 Equilibrium Analysis in Economics 35

3.1 The Meaning of Equilibrium 35

3.2 Partial Market Equilibrium

A Linear Model 36

3.3 Partial Market Equilibrium

A Nonlinear Model 40

3.4 General Market Equilibrium 46

3.5 Equilibrium in National-Income Analysis 52

4 Linear Models and Matrix Algebra 54

4.1 Matrices and Vectors 55

4.2 Matrix Operations 58

4.3 Notes on Vector Operations 67

4.4 Commutative, Associative, and Distributive Laws 76

4.5 Identity Matrices and Null Matrices 79

4.6 Transposes and Inverses 82

5 Linear Models and Matrix Algebra (Continued) 88

5.1 Conditions for Nonsingularity of a Matrix 88

5.2 Test of Nonsingularity by Use of Determinant 92

5.3 Basic Properties of Determinants 98

5.4 Finding the Inverse Matrix 103

5.5 Cramer's Rule 107

5.6 Application to Market and National-Income Models 112

5.7 Leontief Input-Output Models 115

5.8 Limitations of Static Analysis 124

Part 3 Comparative-Static Analysis

6 Comparative Statics and the Concept of Derivative 127

6.1 The Nature of Comparative Statics 127

6.2 Rate of Change and the Derivative 128

6.3 The Derivative and the Slope of a Curve 131

6.4 The Concept of Limit 132

6.5 Digression on Inequalities and Absolute Values 141

6.6 Limit Theorems 145

6.7 Continuity and Differentiability of a Function 147

7 Rules of Differentiation and Their Use in Comparative Statics 155

7.1 Rules of Differentiation for a Function of One Variable 155

7.2 Rules of Differentiation Involving Two or More Functions of the Same Variable 159

7.3 Rules of Differentiation Involving Functions of Different Variables 169

7.4 Partial Differentiation 174

7.5 Applications to Comparative-Static Analysis 178

7.6 Note on Jacobian Determinants 184

8 Comparative-Static Analysis of General-Function Models 187

8.1 Differentials 188

8.2 Total Differentials 194

8.3 Rules of Differentials 196

8.4 Total Derivatives 198

8.5 Derivatives of Implicit Functions 204

8.6 Comparative Statics of General-Function Models 215

8.7 Limitations of Comparative Statics 226

Part 4 Optimization Problems

9 Optimization: A Special Variety of Equilibrium Analysis 231

9.1 Optimum Values and Extreme Values 232

9.2 Relative Maximum and Minimum: First-Derivative Test 233

9.3 Second and Higher Derivatives 239

9.4 Second-Derivative Test 245

9.5 Digression on Maclaurin and Taylor Series 254

9.6 Nth-Derivative Test for Relative Extremum of a Function of One Variable 263

10 Exponential and Logarithmic Functions 268

10.1 The Nature of Exponential Functions 269

10.2 Natural Exponential Functions and the Problem of Growth 274

10.3 Logarithms 282

10.4 Logarithmic Functions 287

10.5 Derivatives of Exponential and Logarithmic Functions 292

10.6 Optimal Timing 298

10.7 Further Applications of Exponential and Logarithmic Derivatives 302

11 The Case of More than One Choice Variable 307

11.1 The Differential Version of Optimization Conditions 308

11.2 Extreme Values of a Function of Two Variables 310

11.3 Quadratic Forms

An Excursion 319

11.4 Objective Functions with More than Two Variables 332

11.5 Second-Order Conditions in Relation to Concavity and Convexity 337

11.6 Economic Applications 353

11.7 Comparative-Static Aspects of Optimization 364

12 Optimization with Equality Constraints 369

12.1 Effects of a Constraint 370

12.2 Finding the Stationary Values 372

12.3 Second-Order Conditions 379

12.4 Quasiconcavity and Quasiconvexity 387

12.5 Utility Maximization and Consumer Demand 400

12.6 Homogeneous Functions 410

12.7 Least-Cost Combination of Inputs 418

12.8 Some Concluding Remarks 431

Part 5 Dynamic Analysis

13 Economic Dynamics and Integral Calculus 435

13.1 Dynamics and Integration 436

13.2 Indefinite Integrals 437

13.3 Definite Integrals 447

13.4 Improper Integrals 454

13.5 Some Economic Applications of Integrals 458

13.6 Domar Growth Model 465

14 Continuous Time: First-Order Differential Equations 470

14.1 First-Order Linear Differential Equations with Constant Coefficient and Constant Term 470

14.2 Dynamics of Market Price 475

14.3 Variable Coefficient and Variable Term 480

14.4 Exact Differential Equations 483

14.5 Nonlinear Differential Equations of the First Order and First Degree 489

14.6 The Qualitative-Graphic Approach 493

14.7 Solow Growth Model 496

15 Higher-Order Differential Equations 502

15.1 Second-Order Linear Differential Equations with Constant Coefficients and Constant Term 503

15.2 Complex Numbers and Circular Functions 511

15.3 Analysis of the Complex-Root Case 523

15.4 A Market Model with Price Expectations 529

15.5 The Interaction of Inflation and Unemployment 535

15.6 Differential Equations with a Variable Term 541

15.7 Higher-Order Linear Differential Equations 544

16 Discrete Time: First-Order Difference Equations 549

16.1 Discrete Time, Differences, and Difference Equations 550

16.2 Solving a First-Order Difference Equation 551

16.3 The Dynamic Stability of Equilibrium 557

16.4 The Cobweb Model 561

16.5 A Market Model with Inventory 566

16.6 Nonlinear Difference Equations

The Qualitative-Graphic Approach 569

17 Higher-Order Difference Equations 576

17.1 Second-Order Linear Difference Equations with Constant Coefficients and Constant Term 577

17.2 Samuelson Multiplier-Acceleration Interaction Model 585

17.3 Inflation and Unemployment in Discrete Time 591

17.4 Generalizations to Variable-Term and Higher-Order Equations 596

18 Simultaneous Differential Equations and Difference Equations 605

18.1 The Genesis of Dynamic Systems 605

18.2 Solving Simultaneous Dynamic Equations 608

18.3 Dynamic Input-Output Models 616

18.4 The Inflation-Unemployment Model Once More 623

18.5 Two-Variable Phase Diagrams 628

18.6 Linearization of a Nonlinear Differential-Equation System 638

18.7 Limitations of Dynamic Analysis 646

Part 6 Mathematical Programming

19 Linear Programming 651

19.1 Simple Examples of Linear Programming 652

19.2 General Formulation of Linear Programs 661

19.3 Convex Sets and Linear Programming 665

19.4 Simplex Method: Finding the Extreme Points 671

19.5 Simplex Method: Finding the Optimal Extreme Point 676

19.6 Further Notes on the Simplex Method 682

20 Linear Programming (Continued) 688

20.1 Duality 688

20.2 Economic Interpretation of a Dual 696

20.3 Activity Analysis: Micro Level 700

20.4 Activity Analysis: Macro Level 709

21 Nonlinear Programming 716

21.1 The Nature of Nonlinear Programming 716

21.2 Kuhn-Tucker Conditions 722

21.3 The Constraint Qualification 731

21.4 Kuhn-Tucker Sufficiency Theorem: Concave Programming 738

21.5 Arrow-Enthoven Sufficiency Theorem: Quasiconcave Programming 744

21.6 Economic Applications 747

21.7 Limitations of Mathematical Programming 754

The Greek Alphabet 756

Mathematical Symbols 757

A Short Reading List 760

Answers to Selected Exercise Problems 763.

Notes:

Includes index.

Bibliography: pages 760-762.

ISBN:

0070108137 :

OCLC:

9970179