Mathematical Methods and Models for Economists

Rating:
Summary: A good overview
Review: 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.

Rating:
Summary: Second Best Optimal
Review: '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.

Rating:
Summary: Excellent book
Review: Excellent book for mathematicians who are working in a field of economics, economists, who are solving applied problems. All math algorithms for economics included. Highly recommended for not only students

Rating:
Summary: Excellent self-contained book
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.

Rating:
Summary: The Worst of Three Math Econ Books
Review: I bought this book for my Ph.D.-level mathematical economics course, and although I'd gotten an A+ both in Calc I and the honors section of Calc II the year before, I could make neither head nor tails out of the book. Sure, de la Fuente might work alright if you already think fluently in mathematics, but even if you do, you'll probably find Simon and Blume's book superior. If you don't think fluently in math, I'd suggest Alpha C. Chiang's math econ text instead. If you're just trying to learn mathematical economics, don't even open de le Fuente or you might find yourself unnecessarily abandoning the study of economics.

Rating:
Summary: Great value, for the right user
Review: I used this book extensively during my first year Ph.D. Econ. It has almost all the basics of the math I used. I agree that it is not a novel-style, Math. Maturity is required and as any First Edition has some typos (but to discover them -by constructing counterexamples- is a great way to show yourself you are understading the concepts and questioning all the assumptions). It has a good presentation of the Berge's (Theorem of the Maximum) and Static Optimization. Its section on dynamic optimization is mostly under continuum time, that I find not too popular nowadays. It is a great reading for the summer before Grad.School but never hesitate to consult lower-level books also -e.g. Simon and Blume. After this, I would read either Debreu's Theory of Value Math. Chapters or Takayama's.

Rating:
Summary: Overkill
Review: If you have not studied mathematics extensively, at least an undergraduate major, DO NOT BUY THIS BOOK. It is dense, abstract, has very few concrete examples, and has quite a few errors to boot. De La Fuente is a mathematician in full glory, and the student of economics is often left wondering when he will get to the relevence to economics.

To be fair the book does cover most topics at a higher level of sophistication than most other mathematical economics textbooks. It does a nice job covering calculus and the implicit function theorem, if you have seen these topics before. However, unless you have had about 8 semesters of college level mathematics of more, you will not be able to read and comprehend this book on your own without a class.

Most of my colleagues at UC Berekeley have been unhappy with this book. If you must use this book for a course, try buying another book as supplement. I recommend Mathematics for Economists by Simon and Blume.

Rating:
Summary: Too many typos, complete lack of illustrative examples
Review: If you want an exhaustive compendium of definitions, theorems, and their proofs for mathematical economomics, then this book may be extremely useful to you. However, if you're a first-year graduate student in economics who needs some help learning the required mathematics, you'll want a text that has example problems, and this doesn't.

And my professor (at UC-Berkeley) kept pointing out typos or errors in the text. Perhaps they'll be corrected in future editions, but for now, be careful!

Rating:
Summary: hard book
Review: yes, those people from UC,USA are right. this book is really hard for a student not specialized in mathematics. I do think mathematics for economics written by c. simon is an excellent book. But for those who want to deepen his knowledge, you should read these two books together.