This book is written for those students of economics intent on learning the basic mathematical methods that have become indispensable for a proper understanding of the current economic literature. Unfortunately, studying mathematics is, for many, something akin to taking bitter-tasting medicine—absolutely necessary, but extremely unpleasant. Such an attitude, referred to as “math anxiety,” has its roots—we believe—largely in the inauspicious manner in which mathematics is often presented to students. In the belief that conciseness means elegance, explanations offered are frequently too brief for clarity, thus puzzling students and giving them an undeserved sense of intellectual inadequacy. An overly formal style of presentation, when not accompanied by any intuitive illustrations or demonstrations of “relevance,” can impair motivation. An uneven progression in the level of material can make certain mathematical topics appear more difficult than they actually are. Finally, exercise problems that are excessively sophisticated may tend to shatter student’s confidence, rather than stimulate thinking as intended.

With that in mind, we have made a serious effort to minimize anxiety-causing features. To the extent possible, patient rather than cryptic explanations are offered. The style is deliberately informal and “reader-friendly.” As a matter of routine, we try to anticipate and answer questions that are likely to arise in the students’ minds as they read. To underscore the relevance of mathematics to economics, we let the analytical needs of economists motivate the study of the related mathematical techniques and then illustrate the latter with appropriate economic models immediately afterward. Also, the mathematical tool kit is built up on a carefully graduated schedule, with the elementary tools serving as stepping stones to the more advanced tools discussed later. Wherever appropriate, graphic illustrations give visual reinforcement to the algebraic results. And we have designed the exercise problems as drills to help solidify grasp and bolster confidence, rather than exacting challenges that might unwittingly frustrate and intimidate the novice.

In this book, the following major types of economic analysis are covered: statics (equilibrium analysis), comparative statics, optimization problems (as a special type of statics), dynamics, and optimal control theory. To tackle these, the following mathematical methods are introduced in due course: matrix algebra, differential and integral calculus, differential equations, and difference equations. Because of the substantial number of illustrative economic models—both macro and micro—appearing here, this book should be useful also to those who are already mathematically-trained but still in need of a guide to usher them from the realm of mathematics to the land of economics. For the same reason, the book should not only serve as a text for a course on mathematical methods but also as supplementary reading in such courses as microeconomic theory, macroeconomic theory, and economic growth and development.

We have attempted to retain the principal objectives and style of the previous editions. However, the present edition contains several significant changes. The material on mathematical programming is now presented earlier in a new Chap. 13 entitled “Further Topics in Optimization”. This chapter has two major themes: optimization with inequality constraints, and the envelope theorem. Under the first theme, the Kuhn-Tucker conditions are chi09109_fm.qxd 9/16/04 23:13 Page vii developed in much the same manner as in the previous edition. However, the topic has been enhanced with several new economic applications, including peak-load pricing and consumer rationing. The second theme is related to the development of the envelope theorem, the maximum-value function, and the notion of duality. By applying the envelope theorem to various economic models, we derive important results such as Roy’s identity, Shephard’s lemma, and Hotelling’s lemma.

The second major addition to this edition is a chapter on optimal control theory (Chap. 20). The purpose of this chapter is to introduce the reader to the basics of optimal control and demonstrate how it may be applied in economics, including examples from natural resource economics and macroeconomic growth theory. The material in this chapter is drawn in great part from the discussion of optimal control theory in Elements of Dynamic Optimization by Alpha C. Chiang (McGraw-Hill 1992, now published by Waveland Press, Inc.) which presents a thorough treatment of both optimal control and its precursor, calculus of variations. The chapter on optimal control is meant only to get one “started” in this field; readers looking for a richer understanding of the topic should proceed to the aforementioned book.

Aside from the two new chapters, there are several significant additions and refinements to this edition. In Chap. 3 we have expanded the discussion of solving higher-order polynomial equations by factoring (Sec. 3.3). In Chap. 4, a new section on Markov chains has been added (Sec. 4.7). And, in Chap. 5, we have introduced the checking of the rank of a matrix via an echelon matrix (Sec. 5.1), and the Hawkins-Simon condition in connection with the Leontief input-output model (Sec. 5.7). With respect to economic applications, many new examples have been added and some of the existing applications have been enhanced. A linear version of the IS-LM model has been included in Sec. 5.6, and a more general form of the model in Sec. 8.6 has been expanded to encompass both a closed and open economy, thereby demonstrating a much richer application of comparative statics to general-function models. Other additions include: a discussion of expected utility and risk preferences (Sec. 9.3), a profit-maximization model that incorporates the Cobb-Douglas production function (Sec. 11.6), and a two-period intertemporal choice problem (Sec. 12.3). Finally, the exercise problems have been revised and augmented, giving students a greater opportunity to hone their skills.