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**Mathematical methods and models for economists.**
*(English)*
Zbl 0943.91001

Cambridge: Cambridge University Press. xii, 835 p. (2000).

This 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. The topics covered are separated into three parts: Preliminaries, Statics, Dynamics.

Part I includes basic concepts, metric and normed spaces, vector spaces and linear transformations, differential calculus.

Part II introduces linear models written as linear systems of equations, nonlinear models and comparative statics, and the implicit-function theorem. Concepts of the intermediate-value theorem and fixed-point theorems are establishing the existence of equilibrium in nonlinear models. Convex sets and separation theorems are crucial in optimization theory which is fundamental for modeling the behavior of economic agents as the outcome of constained optimization problems. Applications concern profit maximization by a competitive firm and implicit contracts. One chapter gives applications to microeconomics: Consumer preference relations, utility functions, Slutsky equations, Walrasian equilibrium, Nash equilibrium, models of imperfect competition, Cournot model.

Part III starts with Dynamical Systems I which considers a parameterized system of difference or differential equations. Dynamical Systems II treats the basic elements of the theory of difference and differential equations, the solution of linear systems, stability conditions, and autonomous nonlinear systems. In Dynamical Systems III some applications to economics are reviewed (IS-LM model, perfect-foresight models, neoclassical growth models). Finally, The dynamic optimization principle is explained and demonstrated for some applications (search models, optimal growth in discrete time, investment with installation costs, Cass-Koopmans model).

The textbook is highly recommended to graduate students of economics. Furthermore, it provides a useful mathematical reference for researchers in economics.

Part I includes basic concepts, metric and normed spaces, vector spaces and linear transformations, differential calculus.

Part II introduces linear models written as linear systems of equations, nonlinear models and comparative statics, and the implicit-function theorem. Concepts of the intermediate-value theorem and fixed-point theorems are establishing the existence of equilibrium in nonlinear models. Convex sets and separation theorems are crucial in optimization theory which is fundamental for modeling the behavior of economic agents as the outcome of constained optimization problems. Applications concern profit maximization by a competitive firm and implicit contracts. One chapter gives applications to microeconomics: Consumer preference relations, utility functions, Slutsky equations, Walrasian equilibrium, Nash equilibrium, models of imperfect competition, Cournot model.

Part III starts with Dynamical Systems I which considers a parameterized system of difference or differential equations. Dynamical Systems II treats the basic elements of the theory of difference and differential equations, the solution of linear systems, stability conditions, and autonomous nonlinear systems. In Dynamical Systems III some applications to economics are reviewed (IS-LM model, perfect-foresight models, neoclassical growth models). Finally, The dynamic optimization principle is explained and demonstrated for some applications (search models, optimal growth in discrete time, investment with installation costs, Cass-Koopmans model).

The textbook is highly recommended to graduate students of economics. Furthermore, it provides a useful mathematical reference for researchers in economics.

Reviewer: Roland Fahrion (Heidelberg)

### MSC:

91-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance |

00A06 | Mathematics for nonmathematicians (engineering, social sciences, etc.) |