Foundations of Mathematical and Computational Economics

Foundations of Mathematical and Computational

Economics

Second Edition

Kamran Dadkhah

Foundations of Mathematical

and Computational

Economics

Second Edition

123

Kamran Dadkhah

Northeastern University

Department of Economics

Boston

USA

[email protected]

First edition published by Thomson South-Western 2007, ISBN 978-0324235838

ISBN 978-3-642-13747-1 e-ISBN 978-3-642-13748-8

DOI 10.1007/978-3-642-13748-8

Springer Heidelberg Dordrecht London New York

© Springer-Verlag Berlin Heidelberg 2007, 2011

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To Karen and my daughter, Lara

Preface

Mathematics is both a language of its own and a way of thinking; applying

mathematics to economics reveals that mathematics is indeed inherent to eco￾nomic life. The objective of this book is to teach mathematical knowledge and

computational skills required for macro and microeconomic analysis, as well as

econometrics. In addition, I hope it conveys a deeper understanding and appreciation

of mathematics.

Examples in the following chapters are chosen from all areas of economics

and econometrics. Some have very practical applications, such as determining

monthly mortgage payments; others involve more abstract models, such as sys￾tems of dynamic equations. Some examples are familiar in the study of micro and

macroeconomics; others involve less well-known and more recent models, such as

real business cycle theory.

Increasingly, economists need to make complicated calculations. Systems of

dynamic equations are used to forecast different economic variables several years

into the future. Such systems are used to assess the effects of alternative policies,

such as different methods of financing Social Security over a few decades. Also,

many theories in microeconomics, industrial organization, and macroeconomics

require modeling the behavior and interactions of many decision makers. These

types of calculation require computational dexterity. Thus, this book provides an

introduction to numerical methods, computation, and programming with Excel and

Matlab. In addition, because of the increasing use of computer software such as

Maple and Mathematica, sections are included to introduce the student to differenti￾ation, integration, and solving difference and differential equations using Maple and

to the concept of computer-aided mathematical proof.

The second edition differs from the first in several respects. Parts of the book

are rearranged, some materials are deleted and some new topics and examples are

added. In the first edition most computational examples used Matlab and some

Excel. In the present edition, Excel and Matlab are given equal weights. These are

done in the hope of making the book more reader friendly. Similarly, more use

is made of the Maple program for solving non-numerical problems. Finally, many

errors had crept into the first edition, which are corrected in the present edition. I am

indebted to students in my math and stat classes for pointing out some of them.

vii

viii Preface

I would like to thank Barbara Fess of Springer-Verlag for her support in preparing

this new edition. I also would like to thank Saranya Baskar and her colleagues at

Integra for their excellent work in producing the book.

As always, my greatest appreciation is to Karen Challberg, who during the entire

project gave me support, encouragement, and love.

Contents

Part I Basic Concepts and Methods

1 Mathematics, Computation, and Economics ............ 3

1.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Philosophies of Mathematics .................. 9

1.3 Women in Mathematics . . . . . . . . . . . . . . . . . . . . . 11

1.4 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Mathematics and Economics . . . . . . . . . . . . . . . . . . 14

1.6 Computation and Economics . . . . . . . . . . . . . . . . . . 14

2 Basic Mathematical Concepts and Methods . . . . . . . . . . . . . 17

2.1 Functions of Real Variables . . . . . . . . . . . . . . . . . . . 17

2.1.1 Variety of Economic Relationships . . . . . . . . . . 22

2.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Summation Notation . . . . . . . . . . . . . . . . 24

2.2.2 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.3 Convergent and Divergent Series . . . . . . . . . . . 27

2.2.4 Arithmetic Progression . . . . . . . . . . . . . . . . 29

2.2.5 Geometric Progression . . . . . . . . . . . . . . . . . 31

2.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Permutations, Factorial, Combinations, and the

Binomial Expansion . . . . . . . . . . . . . . . . . . . . . . . 35

2.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4 Logarithm and Exponential Functions . . . . . . . . . . . . . 38

2.4.1 Logarithm . . . . . . . . . . . . . . . . . . . . . . . 38

2.4.2 Base of Natural Logarithm, e . . . . . . . . . . . . . 40

2.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 Mathematical Proof . . . . . . . . . . . . . . . . . . . . . . . 42

2.5.1 Deduction, Mathematical Induction, and

Proof by Contradiction . . . . . . . . . . . . . . . . . 42

2.5.2 Computer-Assisted Mathematical Proof . . . . . . . . 44

2.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 45

ix

x Contents

2.6 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6.1 Cycles and Frequencies . . . . . . . . . . . . . . . . 50

2.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 51

2.7 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . 51

2.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Basic Concepts of Computation . . . . . . . . . . . . . . . . . . . 57

3.1 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.1 Naming Cells in Excel . . . . . . . . . . . . . . . . . 60

3.2 Absolute and Relative Computation Errors . . . . . . . . . . . 61

3.3 Efficiency of Computation . . . . . . . . . . . . . . . . . . . 62

3.4 o and O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.5 Solving an Equation . . . . . . . . . . . . . . . . . . . . . . . 66

3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Basic Concepts and Methods of Probability Theory and Statistics 69

4.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2 Random Variables and Probability Distributions . . . . . . . . 72

4.3 Marginal and Conditional Distributions . . . . . . . . . . . . 74

4.4 The Bayes Theorem . . . . . . . . . . . . . . . . . . . . . . . 79

4.5 The Law of Iterated Expectations . . . . . . . . . . . . . . . . 81

4.6 Continuous Random Variables . . . . . . . . . . . . . . . . . 82

4.7 Correlation and Regression . . . . . . . . . . . . . . . . . . . 85

4.8 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Part II Linear Algebra

5 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.1 Vectors and Vector Space . . . . . . . . . . . . . . . . . . . . 96

5.1.1 Vector Space . . . . . . . . . . . . . . . . . . . . . . 100

5.1.2 Norm of a Vector . . . . . . . . . . . . . . . . . . . . 102

5.1.3 Metric . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.1.4 Angle Between Two Vectors

and the Cauchy-Schwarz Theorem . . . . . . . . . . 105

5.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 108

5.2 Orthogonal Vectors . . . . . . . . . . . . . . . . . . . . . . . 109

5.2.1 Gramm-Schmidt Algorithm . . . . . . . . . . . . . . 109

5.2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 111

6 Matrices and Matrix Algebra . . . . . . . . . . . . . . . . . . . . 113

6.1 Basic Definitions and Operations . . . . . . . . . . . . . . . . 113

6.1.1 Systems of Linear Equations . . . . . . . . . . . . . 118

6.1.2 Computation with Matrices . . . . . . . . . . . . . . 121

6.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 122

6.2 The Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . 123

6.2.1 A Number Called the Determinant . . . . . . . . . . 127

Contents xi

6.2.2 Rank and Trace of a Matrix . . . . . . . . . . . . . . 132

6.2.3 Another Way to Find the Inverse of a Matrix . . . . . 133

6.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3 Solving Systems of Linear Equations Using Matrix Algebra . . 137

6.3.1 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . 139

6.3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 142

7 Advanced Topics in Matrix Algebra . . . . . . . . . . . . . . . . . 143

7.1 Quadratic Forms and Positive and Negative

Definite Matrices . . . . . . . . . . . . . . . . . . . . . . . . 143

7.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 146

7.2 Generalized Inverse of a Matrix . . . . . . . . . . . . . . . . . 147

7.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 150

7.3 Orthogonal Matrices . . . . . . . . . . . . . . . . . . . . . . 150

7.3.1 Orthogonal Projection . . . . . . . . . . . . . . . . . 150

7.3.2 Orthogonal Complement of a Matrix . . . . . . . . . 152

7.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 153

7.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 153

7.4.1 Complex Eigenvalues . . . . . . . . . . . . . . . . . 159

7.4.2 Repeated Eigenvalues . . . . . . . . . . . . . . . . . 160

7.4.3 Eigenvalues and the Determinant and Trace of

a Matrix . . . . . . . . . . . . . . . . . . . . . . . . 164

7.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 166

7.5 Factorization of Symmetric Matrices . . . . . . . . . . . . . . 167

7.5.1 Some Interesting Properties of Symmetric Matrices . 167

7.5.2 Factorization of Matrix with Real Distinct Roots . . . 170

7.5.3 Factorization of a Positive Definite Matrix . . . . . . 172

7.5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 176

7.6 LU Factorization of a Square Matrix . . . . . . . . . . . . . . 176

7.6.1 Cholesky Factorization . . . . . . . . . . . . . . . . 181

7.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 182

7.7 Kronecker Product and Vec Operator . . . . . . . . . . . . . . 183

7.7.1 Vectorization of a Matrix . . . . . . . . . . . . . . . 185

7.7.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 185

Part III Calculus

8 Differentiation: Functions of One Variable . . . . . . . . . . . . . 189

8.1 Marginal Analysis in Economics . . . . . . . . . . . . . . . . 189

8.1.1 Marginal Concepts and Derivatives . . . . . . . . . . 190

8.1.2 Comparative Static Analysis . . . . . . . . . . . . . . 192

8.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 193

8.2 Limit and Continuity . . . . . . . . . . . . . . . . . . . . . . 194

8.2.1 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 194

8.2.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . 196

8.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 198

xii Contents

8.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

8.3.1 Geometric Representation of Derivative . . . . . . . . 200

8.3.2 Differentiability . . . . . . . . . . . . . . . . . . . . 201

8.3.3 Rules of Differentiation . . . . . . . . . . . . . . . . 204

8.3.4 Properties of Derivatives . . . . . . . . . . . . . . . . 207

8.3.5 l’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . 214

8.3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 215

8.4 Monotonic Functions and the Inverse Rule . . . . . . . . . . . 216

8.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 219

8.5 Second- and Higher-Order Derivatives . . . . . . . . . . . . . 220

8.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 221

8.6 Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

8.6.1 Second- and Higher-Order Differentials . . . . . . . . 223

8.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 224

8.7 Computer and Numerical Differentiation . . . . . . . . . . . . 224

8.7.1 Computer Differentiation . . . . . . . . . . . . . . . 224

8.7.2 Numerical Differentiation . . . . . . . . . . . . . . . 225

8.7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 226

9 Differentiation: Functions of Several Variables . . . . . . . . . . . 227

9.1 Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . 227

9.1.1 Second-Order Partial Derivatives . . . . . . . . . . . 230

9.1.2 Differentiation of Functions of Several

Variables Using Computer . . . . . . . . . . . . . . . 232

9.1.3 The Gradient and Hessian . . . . . . . . . . . . . . . 232

9.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 234

9.2 Differential and Total Derivative . . . . . . . . . . . . . . . . 235

9.2.1 Differential . . . . . . . . . . . . . . . . . . . . . . . 235

9.2.2 Total Derivative . . . . . . . . . . . . . . . . . . . . 237

9.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 240

9.3 Homogeneous Functions and the Euler Theorem . . . . . . . . 240

9.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 243

9.4 Implicit Function Theorem . . . . . . . . . . . . . . . . . . . 244

9.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 248

9.5 Differentiating Systems of Equations . . . . . . . . . . . . . . 248

9.5.1 The Jacobian and Independence of Nonlinear Functions 248

9.5.2 Differentiating Several Functions . . . . . . . . . . . 250

9.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 255

10 The Taylor Series and Its Applications . . . . . . . . . . . . . . . 257

10.1 The Taylor Expansion . . . . . . . . . . . . . . . . . . . . . . 257

10.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 266

10.2 The Remainder and the Precision of Approximation . . . . . . 267

10.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 270

10.3 Finding the Roots of an Equation . . . . . . . . . . . . . . . . 270

10.3.1 Iterative Methods . . . . . . . . . . . . . . . . . . . 270

Contents xiii

10.3.2 The Bisection Method . . . . . . . . . . . . . . . . . 272

10.3.3 Newton’s Method . . . . . . . . . . . . . . . . . . . 273

10.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 276

10.4 Taylor Expansion of Functions of Several Variables . . . . . . 276

10.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 279

11 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

11.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . 282

11.1.1 Rules of Integration . . . . . . . . . . . . . . . . . . 283

11.1.2 Change of Variable . . . . . . . . . . . . . . . . . . . 285

11.1.3 Integration by Parts . . . . . . . . . . . . . . . . . . 287

11.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 289

11.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . 289

11.2.1 Properties of Definite Integrals . . . . . . . . . . . . 294

11.2.2 Rules of Integration for the Definite Integral . . . . . 298

11.2.3 Change of Variable . . . . . . . . . . . . . . . . . . . 299

11.2.4 Integration by Parts . . . . . . . . . . . . . . . . . . 300

11.2.5 Riemann-Stieltjes Integral . . . . . . . . . . . . . . . 304

11.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 306

11.3 Computer and Numerical Integration . . . . . . . . . . . . . . 306

11.3.1 Computer Integration . . . . . . . . . . . . . . . . . 306

11.4 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . 307

11.4.1 The Trapezoid Method . . . . . . . . . . . . . . . . . 308

11.4.2 The Lagrange Interpolation Formula . . . . . . . . . 310

11.4.3 Newton-Cotes Method . . . . . . . . . . . . . . . . . 312

11.4.4 Simpson’s Method . . . . . . . . . . . . . . . . . . . 313

11.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 315

11.5 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . 316

11.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 316

11.6 The Derivative of an Integral . . . . . . . . . . . . . . . . . . 317

11.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 320

Part IV Optimization

12 Static Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 323

12.1 Maxima and Minima of Functions of One Variable . . . . . . 324

12.1.1 Inflection Point . . . . . . . . . . . . . . . . . . . . . 331

12.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 333

12.2 Unconstrained Optima of Functions of Several Variables . . . 334

12.2.1 Convex and Concave Functions . . . . . . . . . . . . 338

12.2.2 Quasi-convex and Quasi-concave Functions . . . . . 341

12.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 342

12.3 Numerical Optimization . . . . . . . . . . . . . . . . . . . . . 343

12.3.1 Steepest Descent . . . . . . . . . . . . . . . . . . . . 343

12.3.2 Golden Section Method . . . . . . . . . . . . . . . . 344

12.3.3 Newton Method . . . . . . . . . . . . . . . . . . . . 344

xiv Contents

12.3.4 Matlab Functions . . . . . . . . . . . . . . . . . . . 345

12.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 346

13 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . 347

13.1 Optimization with Equality Constraints . . . . . . . . . . . . 347

13.1.1 The Nature of Constrained Optima

and the Significance of λ . . . . . . . . . . . . . . . . 354

13.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 354

13.2 Value Function . . . . . . . . . . . . . . . . . . . . . . . . . 355

13.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 362

13.3 Second-Order Conditions and Comparative Static . . . . . . . 362

13.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 368

13.4 Inequality Constraints and Karush-Kuhn-Tucker Conditions . . 368

13.4.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . 372

13.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 375

14 Dynamic Optimization . . . . . . . . . . . . . . . . . . . . . . . . 377

14.1 Dynamic Analysis in Economics . . . . . . . . . . . . . . . . 377

14.2 The Control Problem . . . . . . . . . . . . . . . . . . . . . . 379

14.2.1 The Functional and Its Derivative . . . . . . . . . . . 381

14.3 Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . 384

14.3.1 The Euler Equation . . . . . . . . . . . . . . . . . . 386

14.3.2 Second-Order Conditions . . . . . . . . . . . . . . . 389

14.3.3 Generalizing the Calculus of Variations . . . . . . . . 390

14.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 391

14.4 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . 391

14.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 396

14.5 The Maximum Principle . . . . . . . . . . . . . . . . . . . . 397

14.5.1 Necessary and Sufficient Conditions . . . . . . . . . 405

14.5.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 405

Part V Differential and Difference Equations

15 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 409

15.1 Examples of Continuous Time Dynamic Economic Models . . 409

15.2 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 412

15.2.1 Initial Value Problem . . . . . . . . . . . . . . . . . 414

15.2.2 Existence and Uniqueness of Solutions . . . . . . . . 416

15.2.3 Equilibrium and Stability . . . . . . . . . . . . . . . 417

15.3 First-Order Linear Differential Equations . . . . . . . . . . . . 418

15.3.1 Variable Coefficient Equations . . . . . . . . . . . . 420

15.3.2 Particular Integral, the Method

of Undetermined Coefficients . . . . . . . . . . . . . 421

15.3.3 Separable Equations . . . . . . . . . . . . . . . . . . 424

15.3.4 Exact Differential Equations . . . . . . . . . . . . . . 426

15.3.5 Integrating Factor . . . . . . . . . . . . . . . . . . . 430

15.3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 433