Mathematical Methods and Models for Economists [ペーパーバック]

'Mathematical Methods' is the best math econ text you can buy. It does a far better job of explaining math modeling than Takayama or Simone and Blume. It reads better than Chiang. Its' broad coverage of techniques should be enough to satisfy most any instructor.

It starts off by running through some important basics- set theory, Venn diagrams, proofs. It then works up to calculus and optimization. It could use some more game theory, and should ditch the section on ISLM. The main strength of this book is that, unlike other math econ texts, one can read and understand it without prior knowledge of advanced mathmatics.

Of course, nobody really needs to learn all, or even most of, the math in this book. To get credentails as an economist, students must jump through many a mathematical hoop. This book helps students through this better than any of the alternatives. It has a reasonable paperback price too. Do not expect to have much fun reading 'Mathematical Methods'. Just bear in mind that there are far worse books to use in studying math econ.

Mathematical economics has been around for about 175 years, although as a discipline it has only been recognized for about five decades. Professional economists have had various levels of confidence in its validity and applicability, and mathematical economists have been criticized for the esoteric nature of the mathematics they deploy and some have been ostracized from academic departments for this very reason. This book emphasizes the mathematical tools, these being primarily the theory of optimization and dynamical systems, but the author does find time to discuss applications. Some of these could be classified as "classical" applications, but some are very contemporary in their scope and intersect the work done in financial engineering.

Part 1 of the book introduces the reader to the necessary background in real analysis, topology, differential calculus, and linear algebra. All of this mathematics is straightforward and can be found in many books.

In chapter 5, the author considers static economic models, which are described by collections of parametrized systems of equations. The equations are dependent on parameters describing the environment and `endogenous' variables. The goal is to find the values of the endogenous variables at equilibrium, and to find out if the equilibrium solutions are unique. In addition, it is interest to find out how the solution set changes when the parameters are changed. This is what the author calls `comparative statics'. Linear models are considered first, their analysis being amenable to the techniques of linear and multilinear algebra. The comparative statics for linear models is straightforward, with the shift in equilibrium as a parameter is change readily calculated. The comparative statics of nonlinear models involves the use of the implicit function theorem, and the author derives a formula for doing comparative statics in differentiable models. The discussion here, involving concepts such as transversality, critical points, regular values, and genericity, should be viewed as a warm-up to a more advanced treatment using differential topology.

The author studies static optimization in chapter 7, with the postulate of rationality assumed throughout. This allows the study of the behavior of economic agents to be reduced to a constrained optimization problem. The techniques of nonlinear programming are used to find solutions to the constrained optimization problem. Throughout this chapter one sees discussion of the ubiquitous `agent' who is embedded in a collection of possible environments, and is able to carry out a certain collection of actions.

The author finally gets to economic applications in chapter 8, wherein the author studies the behavior of a single agent under a set of restrictions imposed on it by its environment. This rather simplistic study is then generalized to the case of many interacting agents who are taken to be rational. The concept of `equilibrium', so entrenched in economic theory and economic modeling, makes its appearance here. In a condition of equilibrium, no agent has an incentive to change its behavior, and the actions of each individual are mutually compatible. Some of the usual concepts of equilibrium are discussed in the chapter, such as Walrasian equilibrium in exchange economies, and Nash equilibrium in game theory. The (subjective) preferences of consumers are modeled by binary relations and differentiable utility functions. The differentiability allows the techniques of chapter 7 to be used. The author asks the reader to work through some examples of `imperfect' competition at the end of the chapter.

After a straightforward review of dynamical systems in chapters 9 and 10, the author discusses applications of dynamical systems in chapter 11. He begins with a discussion of a dynamic IS-LM model, using assumptions on the evolution of the money supply, the formation of expectations, and price dynamics. This model consists of two first-order ordinary differential equations, and the author studies its fixed-point structure via a standard phase-space analysis. This analysis allows the author to study the effect of a change in parameters, such as change in the rate of money creation, i.e. the effects of a certain monetary policy. Also discussed are `perfect-foresight models', which address the difficult issue of boundary conditions in economic models based on dynamical systems. Two of these models are discussed, one is a stock price model based on the no-arbitrage principle from finance, and the other is a model of exchange-rate determination. The stock price model is the most interesting discussion in the book. It requires one to specify how expectations are formed, and, depending on how this is done, some very unexpected results occur. For example, if the agents have adaptive expectations, the author shows that the forecast error is predictable, and that agents who understand the structure of the model will have an incentive to deviate from the predicted behavior. This behavior on the part of the agents will invalidate the theory since the agents will have an incentive to compute the trajectory of prices, contrary to the assumption of the model. The author concludes that this is in direct conflict with the assumption that individuals are rational and maximize utility, i.e. that in a world without uncertainty, adaptive expectations are inconsistent with the assumption of rationality. The author avoids this problem by assuming that `perfect foresight' holds for the agents, i.e. the agents form expectations that are consistent with the structure of the model. He shows that the assumption of perfect foresight eliminates the inconsistency that was found in the adaptive expectations model. In the perfect foresight model, every agent uses the correct model to predict prices, and no agent has any incentive to act differently. The author then uses this model to study the response of share prices to a change in the tax rate on dividends. The rest of the chapter discusses neoclassical growth models and the software language Mathematica is introduced as a tool for solving nonlinear differential equations.

I did not read the last two chapters of the book, which cover dynamic optimization and its applications, and so I will omit their review.

For the first time (at least as far as I know) there is a fully self-contained book on all major mathematical tools needed for intermediary to advanced topics in economics (it even has a small section on logic and methods of proof!). Unfortunatly, there is no treatment whatsoever of integration and measure theory, which is regretable.