Mathematical Methods and Models for EconomistsThis book is intended as a textbook for a first-year Ph. D. course in mathematics for economists and as a reference for graduate students in economics. It provides a self-contained, rigorous treatment of most of the concepts and techniques required to follow the standard first-year theory sequence in micro and macroeconomics. The topics covered include an introduction to analysis in metric spaces, differential calculus, comparative statics, convexity, static optimization, dynamical systems and dynamic optimization. The book includes a large number of applications to standard economic models and over two hundred fully worked-out problems. |
Contents
Review of Basic Concepts | 6 |
Compactness and the ExtremeValue Theorem | 8 |
Relations | 17 |
Functions | 27 |
Complex Numbers | 36 |
558 | 61 |
2838 | 79 |
Vector Spaces and Linear Transformations | 117 |
Problems and Applications 316 | 319 |
Some Applications to Microeconomics | 325 |
Consumer Theory | 339 |
Walrasian General Equilibrium in a Pure Exchange Economy 354 | 356 |
Games in Normal Form and Nash Equilibrium | 375 |
Bibliography | 385 |
Basic Concepts and Scalar Systems | 391 |
Autonomous Systems | 401 |
Eigenvalues and Eigenvectors | 146 |
Polynomial Equations | 152 |
Differential Calculus | 160 |
Differentiability | 179 |
Homogeneous Functions | 189 |
Static Models and Comparative Statics | 195 |
Existence of Equilibrium | 218 |
Problems | 224 |
Concave Functions | 245 |
Quasiconcave Functions | 261 |
Quadratic Forms | 268 |
Static Optimization | 274 |
The Lagrange Problem | 282 |
The KuhnTucker Problem | 291 |
Concave Programming without Differentiability | 297 |
Autonomous Differential Equations | 411 |
Solution of Nonautonomous Linear Equations | 428 |
Higher Dimensions | 449 |
Problems 489 | 465 |
Neoclassical Growth Models 518 | 481 |
An Introduction to Dynamic Optimization 549 | 494 |
Optimal Control 566 | 502 |
Bibliography 580 | 510 |
Optimal Growth in Discrete Time 598 | 521 |
The CassKoopmans Model and Some Applications 622 | 530 |
Problems 643 | 541 |
Bibliography 653 | 547 |
Notes 654 | |
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Common terms and phrases
agent arbitrary assume assumption Chapter closed compact set concave concave function consider constraint consumption contains continuous function contradiction convergent subsequence convex combination convex set correspondence d₂ defined definition Df(x differentiable economic eigenvalues eigenvectors element equation equilibrium equivalent exists feasible Figure finite first-order conditions fixed point function f given Hamel basis hemicontinuity Hence implies income increase inequality interval inverse Lagrange problem Let f limit linear transformation mapping matrix maximal metric space Moreover natural numbers nonempty norm normed vector space Notice obtain open set output parameters partial derivatives phase line preordering Problem Proof prove quasiconcave real numbers result s.th scalar sequence solve steady strictly subset sufficient supremum Theorem tion triangle inequality u₁ unique utility function value function variables vector space x₁ y₁ zero