Foundations Of Mathematical And Computational Economics Pdf

Unformatted text preview: 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 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign, Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media ( ) 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 economic 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 systems 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 differentiation, 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 1.1 Mathematics . . . . . . . . . . . . . . . 1.2 Philosophies of Mathematics . . . . . . 1.3 Women in Mathematics . . . . . . . . . 1.4 Computation . . . . . . . . . . . . . . . 1.5 Mathematics and Economics . . . . . . 1.6 Computation and Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 9 11 12 14 14 2 Basic Mathematical Concepts and Methods . . . . . . 2.1 Functions of Real Variables . . . . . . . . . . . . 2.1.1 Variety of Economic Relationships . . . 2.1.2 Exercises . . . . . . . . . . . . . . . . . 2.2 Series . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Summation Notation  . . . . . . . . . 2.2.2 Limit . . . . . . . . . . . . . . . . . . . 2.2.3 Convergent and Divergent Series . . . . 2.2.4 Arithmetic Progression . . . . . . . . . 2.2.5 Geometric Progression . . . . . . . . . . 2.2.6 Exercises . . . . . . . . . . . . . . . . . 2.3 Permutations, Factorial, Combinations, and the Binomial Expansion . . . . . . . . . . . . . . . . 2.3.1 Exercises . . . . . . . . . . . . . . . . . 2.4 Logarithm and Exponential Functions . . . . . . 2.4.1 Logarithm . . . . . . . . . . . . . . . . 2.4.2 Base of Natural Logarithm, e . . . . . . 2.4.3 Exercises . . . . . . . . . . . . . . . . . 2.5 Mathematical Proof . . . . . . . . . . . . . . . . 2.5.1 Deduction, Mathematical Induction, and Proof by Contradiction . . . . . . . . . . 2.5.2 Computer-Assisted Mathematical Proof . 2.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 17 22 23 23 24 25 27 29 31 34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 37 38 38 40 41 42 . . . . . . . . . . . . . . . . . . . . . 42 44 45 ix x Contents 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 50 51 51 56 3 Basic Concepts of Computation . . . . . . . . 3.1 Iterative Methods . . . . . . . . . . . . . 3.1.1 Naming Cells in Excel . . . . . . 3.2 Absolute and Relative Computation Errors 3.3 Efficiency of Computation . . . . . . . . 3.4 o and O . . . . . . . . . . . . . . . . . . 3.5 Solving an Equation . . . . . . . . . . . . 3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 57 60 61 62 63 66 68 4 Basic Concepts and Methods of Probability Theory and Statistics 4.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Random Variables and Probability Distributions . . . . . . . . 4.3 Marginal and Conditional Distributions . . . . . . . . . . . . 4.4 The Bayes Theorem . . . . . . . . . . . . . . . . . . . . . . . 4.5 The Law of Iterated Expectations . . . . . . . . . . . . . . . . 4.6 Continuous Random Variables . . . . . . . . . . . . . . . . . 4.7 Correlation and Regression . . . . . . . . . . . . . . . . . . . 4.8 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 69 72 74 79 81 82 85 88 90 2.7 Part II Trigonometry . . . . . . . . . . 2.6.1 Cycles and Frequencies 2.6.2 Exercises . . . . . . . . Complex Numbers . . . . . . . 2.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Algebra 5 Vectors . . . . . . . . . . . . . . . . . . . . . . . 5.1 Vectors and Vector Space . . . . . . . . . . 5.1.1 Vector Space . . . . . . . . . . . . 5.1.2 Norm of a Vector . . . . . . . . . . 5.1.3 Metric . . . . . . . . . . . . . . . 5.1.4 Angle Between Two Vectors and the Cauchy-Schwarz Theorem 5.1.5 Exercises . . . . . . . . . . . . . . 5.2 Orthogonal Vectors . . . . . . . . . . . . . 5.2.1 Gramm-Schmidt Algorithm . . . . 5.2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 96 100 102 104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 108 109 109 111 6 Matrices and Matrix Algebra . . . . . . . . . . 6.1 Basic Definitions and Operations . . . . . . 6.1.1 Systems of Linear Equations . . . 6.1.2 Computation with Matrices . . . . 6.1.3 Exercises . . . . . . . . . . . . . . 6.2 The Inverse of a Matrix . . . . . . . . . . . 6.2.1 A Number Called the Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 113 118 121 122 123 127 Contents xi 6.2.2 Rank and Trace of a Matrix . . . . . . . . . . . . . 6.2.3 Another Way to Find the Inverse of a Matrix . . . . 6.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . Solving Systems of Linear Equations Using Matrix Algebra . 6.3.1 Cramer's Rule . . . . . . . . . . . . . . . . . . . . 6.3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 133 135 137 139 142 7 Advanced Topics in Matrix Algebra . . . . . . . . . . . . . . . . 7.1 Quadratic Forms and Positive and Negative Definite Matrices . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . 7.2 Generalized Inverse of a Matrix . . . . . . . . . . . . . . . . 7.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . 7.3 Orthogonal Matrices . . . . . . . . . . . . . . . . . . . . . 7.3.1 Orthogonal Projection . . . . . . . . . . . . . . . . 7.3.2 Orthogonal Complement of a Matrix . . . . . . . . 7.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . 7.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . 7.4.1 Complex Eigenvalues . . . . . . . . . . . . . . . . 7.4.2 Repeated Eigenvalues . . . . . . . . . . . . . . . . 7.4.3 Eigenvalues and the Determinant and Trace of a Matrix . . . . . . . . . . . . . . . . . . . . . . . 7.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . 7.5 Factorization of Symmetric Matrices . . . . . . . . . . . . . 7.5.1 Some Interesting Properties of Symmetric Matrices 7.5.2 Factorization of Matrix with Real Distinct Roots . . 7.5.3 Factorization of a Positive Definite Matrix . . . . . 7.5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . 7.6 LU Factorization of a Square Matrix . . . . . . . . . . . . . 7.6.1 Cholesky Factorization . . . . . . . . . . . . . . . 7.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . 7.7 Kronecker Product and Vec Operator . . . . . . . . . . . . . 7.7.1 Vectorization of a Matrix . . . . . . . . . . . . . . 7.7.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 143 . . . . . . . . . . . 143 146 147 150 150 150 152 153 153 159 160 . . . . . . . . . . . . . 164 166 167 167 170 172 176 176 181 182 183 185 185 . . . . . . . . . 189 189 190 192 193 194 194 196 198 6.3 Part III Calculus 8 Differentiation: Functions of One Variable . . . 8.1 Marginal Analysis in Economics . . . . . . 8.1.1 Marginal Concepts and Derivatives 8.1.2 Comparative Static Analysis . . . . 8.1.3 Exercises . . . . . . . . . . . . . . 8.2 Limit and Continuity . . . . . . . . . . . . 8.2.1 Limit . . . . . . . . . . . . . . . . 8.2.2 Continuity . . . . . . . . . . . . . 8.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Contents 8.3 8.4 8.5 8.6 8.7 Derivatives . . . . . . . . . . . . . . . . . . . 8.3.1 Geometric Representation of Derivative 8.3.2 Differentiability . . . . . . . . . . . . 8.3.3 Rules of Differentiation . . . . . . . . 8.3.4 Properties of Derivatives . . . . . . . . 8.3.5 l'Hôpital's Rule . . . . . . . . . . . . 8.3.6 Exercises . . . . . . . . . . . . . . . . Monotonic Functions and the Inverse Rule . . . 8.4.1 Exercises . . . . . . . . . . . . . . . . Second- and Higher-Order Derivatives . . . . . 8.5.1 Exercises . . . . . . . . . . . . . . . . Differential . . . . . . . . . . . . . . . . . . . 8.6.1 Second- and Higher-Order Differentials 8.6.2 Exercises . . . . . . . . . . . . . . . . Computer and Numerical Differentiation . . . . 8.7.1 Computer Differentiation . . . . . . . 8.7.2 Numerical Differentiation . . . . . . . 8.7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Differentiation: Functions of Several Variables . . . . . . . . . . . 9.1 Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Second-Order Partial Derivatives . . . . . . . . . . . 9.1.2 Differentiation of Functions of Several Variables Using Computer . . . . . . . . . . . . . . . 9.1.3 The Gradient and Hessian . . . . . . . . . . . . . . . 9.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Differential and Total Derivative . . . . . . . . . . . . . . . . 9.2.1 Differential . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Total Derivative . . . . . . . . . . . . . . . . . . . . 9.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Homogeneous Functions and the Euler Theorem . . . . . . . . 9.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Implicit Function Theorem . . . . . . . . . . . . . . . . . . . 9.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Differentiating Systems of Equations . . . . . . . . . . . . . . 9.5.1 The Jacobian and Independence of Nonlinear Functions 9.5.2 Differentiating Several Functions . . . . . . . . . . . 9.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 10 The Taylor Series and Its Applications . . . . . . . . . 10.1 The Taylor Expansion . . . . . . . . . . . . . . . . 10.1.1 Exercises . . . . . . . . . . . . . . . . . . 10.2 The Remainder and the Precision of Approximation 10.2.1 Exercises . . . . . . . . . . . . . . . . . . 10.3 Finding the Roots of an Equation . . . . . . . . . . 10.3.1 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 200 201 204 207 214 215 216 219 220 221 221 223 224 224 224 225 226 227 227 230 232 232 234 235 235 237 240 240 243 244 248 248 248 250 255 257 257 266 267 270 270 270 Contents xiii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 273 276 276 279 11 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Indefinite Integral . . . . . . . . . . . . . . . . . 11.1.1 Rules of Integration . . . . . . . . . . . . . 11.1.2 Change of Variable . . . . . . . . . . . . . . 11.1.3 Integration by Parts . . . . . . . . . . . . . 11.1.4 Exercises . . . . . . . . . . . . . . . . . . . 11.2 The Definite Integral . . . . . . . . . . . . . . . . . 11.2.1 Properties of Definite Integrals . . . . . . . 11.2.2 Rules of Integration for the Definite Integral 11.2.3 Change of Variable . . . . . . . . . . . . . . 11.2.4 Integration by Parts . . . . . . . . . . . . . 11.2.5 Riemann-Stieltjes Integral . . . . . . . . . . 11.2.6 Exercises . . . . . . . . . . . . . . . . . . . 11.3 Computer and Numerical Integration . . . . . . . . . 11.3.1 Computer Integration . . . . . . . . . . . . 11.4 Numerical Integration . . . . . . . . . . . . . . . . . 11.4.1 The Trapezoid Method . . . . . . . . . . . . 11.4.2 The Lagrange Interpolation Formula . . . . 11.4.3 Newton-Cotes Method . . . . . . . . . . . . 11.4.4 Simpson's Method . . . . . . . . . . . . . . 11.4.5 Exercises . . . . . . . . . . . . . . . . . . . 11.5 Special Functions . . . . . . . . . . . . . . . . . . . 11.5.1 Exercises . . . . . . . . . . . . . . . . . . . 11.6 The Derivative of an Integral . . . . . . . . . . . . . 11.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 282 283 285 287 289 289 294 298 299 300 304 306 306 306 307 308 310 312 313 315 316 316 317 320 12 Static Optimization . . . . . . . . . . . . . . . . . . . . . . . 12.1 Maxima and Minima of Functions of One Variable . . . 12.1.1 Inflection Point . . . . . . . . . . . . . . . . . . 12.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . 12.2 Unconstrained Optima of Functions of Several Variables 12.2.1 Convex and Concave Functions . . . . . . . . . 12.2.2 Quasi-convex and Quasi-concave Functions . . 12.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . 12.3 Numerical Optimization . . . . . . . . . . . . . . . . . . 12.3.1 Steepest Descent . . . . . . . . . . . . . . . . . 12.3.2 Golden Section Method . . . . . . . . . . . . . 12.3.3 Newton Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 324 331 333 334 338 341 342 343 343 344 344 10.4 Part IV 10.3.2 The Bisection Method . . . . . . . . . . . 10.3.3 Newton's Method . . . . . . . . . . . . . 10.3.4 Exercises . . . . . . . . . . . . . . . . . . Taylor Expansion of Functions of Several Variables 10.4.1 Exercises . . . . . . . . . . . . . . . . . . Optimization xiv Contents 12.3.4 12.3.5 Matlab Functions . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . 345 346 13 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . 13.1 Optimization with Equality Constraints . . . . . . . . . . . 13.1.1 The Nature of Constrained Optima and the Significance of λ . . . . . . . . . . . . . . . 13.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . 13.2 Value Function . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . 13.3 Second-Order Conditions and Comparative Static . . . . . . 13.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . 13.4 Inequality Constraints and Karush-Kuhn-Tucker Conditions . 13.4.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 347 347 . . . . . . . . . 354 354 355 362 362 368 368 372 375 14 Dynamic Optimization . . . . . . . . . . . . . . . . 14.1 Dynamic Analysis in Economics . . . . . . . . 14.2 The Control Problem . . . . . . . . . . . . . . 14.2.1 The Functional and Its Derivative . . . 14.3 Calculus of Variations . . . . . . . . . . . . . . 14.3.1 The Euler Equation . . . . . . . . . . 14.3.2 Second-Order Conditions . . . . . . . 14.3.3 Generalizing the Calculus of Variations 14.3.4 Exercises . . . . . . . . . . . . . . . . 14.4 Dynamic Programming . . . . . . . . . . . . . 14.4.1 Exercises . . . . . . . . . . . . . . . . 14.5 The Maximum Principle . . . . . . . . . . . . 14.5.1 Necessary and Sufficient Conditions . 14.5.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 377 379 381 384 386 389 390 391 391 396 397 405 405 15 Differential Equations . . . . . . . . . . . . . . . . . . . . . . 15.1 Examples of Continuous Time Dynamic Economic Models 15.2 An Overview . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Initial Value Problem . . . . . . ...
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