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Preface PART 1: MATRIX ALGEBRA AND LINEAR ECONOMIC MODELS 1. Matrix Algebra 1.1 Basic Concepts 2 1.2 Determinants 9 1.3 The Inverse of a Matrix 17 1.4 Linear Dependence of Vectors and the Rank of a Matrix 25 *1.5 Kronecker Products and Vecs of Matrices 33 2. Simultaneous Linear Equations 2.1 Definitions 37 2.2 Homogeneous Case 39 2.3 Nonhomogeneous Case 44 2.4 Special Case m=n 48 3. Linear Economic Models 3.1 Introduction and Definitions 54 3.2 Examples of Linear Economic Models 56 3.3 The Use of Matrix Algebra in Statistics and Econometrics 63 4. Quadratic Forms and Positive Definite Matrices 4.1 Introduction 69 4.2 Eigen Values of a Symmetric Matrix 70 4.3 Eigen Values of Special Matrices 73 4.4 Eigen Vectors of a Symmetric Matrix 75 4.5 Matrix whose Columns are the Eigen Vectors of a Symmetric Matrix 80 4.6 Diagonalization of Quadratic Forms 83 4.7 Eigen Values and 85 4.8 An Alternative Approach using Determinants 86 PART 2: FUNCTIONS OF MANY VARIABLES AND OPTIMIZATION 5. Functions of Many Variables 5.1 Functions in General 93 5.2 Partial Differentiation 95 5.3 Special Sorts of Functions 102 5.4 Comparative Statics and Nonlinear Economic Models 116 5.5 Differentials and Taylor¿s Approximation 123 6. Optimization 6.1 Unconstrained Optimization 131 6.2 Local Optima and Global Optima 142 6.3 Constrained Optimization 146 6.4 Constrained Local Optima versus Constrained Global Optima 153 *6.5 An Introduction to Matrix Calculus 156 7. Comparative Static Analysis in Optimization Problems 7.1 Introduction 166 7.2 Unconstrained Optimization 166 7.3 Constrained Optimization 170 7.4 Slutsky¿s Equation 174 7.5 Applications of the Envelope Theorems in Economics 178 PART 3: DYNAMIC ANALYSIS 8. Integration 8.1 Introduction 201 8.2 Definite Integrals 202 8.3 Integration as Anti Differentiation 207 8.4 Indefinite Integrals 212 8.5 Further Considerations 215 8.6 Economic Applications 218 9. Continuous Time: Differential Equations 9.1 Definitions 225 9.2 Linear Differential Equations 226 9.3 First Order Linear Differential Equations with Constant Coefficients 227 9.4 Economic Dynamics Using First Order Differential Equations 233 9.5 Second Order Linear Differential Equations with Constant Coefficents 243 9.6 Economic Application: A Dynamic Supply and Demand Model 250 9.7 Higher Order Linear Differential Equations 252 9.8 Descriptive Analysis of Nonlinear Differential Equations 254 10. Discrete Time: Difference Equations 10.1 Introduction and Definitions 259 10.2 First Order Linear Difference Equations with Constant Coefficients 261 10.3 Second Order Linear Difference Equations with Constant Coefficients 262 10.4 Investigating the Nature of the Roots of a Quadratic Equation 269 10.5 Economic Applications 272 10.6 Higher Order Linear Difference Equations 280 11. Dynamic Optimization *11.1 Introduction 283 *11.2 Dynamic Optimization versus Static Optimization 283 *11.3 The Basic Optimal Control Problem and Pontryagin¿s Maximum Principle 285 *11.4 Extensions to the Basic Problem 293 *11.5 Economic Application: Ramsey/Solow Model 305 Answers to exercises Further Reading 315 Index

Library of Congress Subject Headings for this publication:

Economics, Mathematical.

Economics -- Mathematical models.