Introduction to Mathematical Economics. by D. W. Bushaw; R. W. ClowerReview by: Victor E. SmithJournal of the American Statistical Association, Vol. 53, No. 281 (Mar., 1958), pp. 224-226Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2282593 .

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224 AMERICAN STATISTICAL ASSOCIATION JOURNAL, MARCH 1958

A final chapter deals with the aggregation problem, completely along the lines developed by Theil. Indeed, it is only this last chapter in which the book deals with problems of direct concern to the readers of this Journal. It would be unjust to find fault with a book of 768 pages in which there does not appear to be any excess verbos- ity because it has omitted certain topics. Nevertheless, it may be useful to note, without critical intent, the virtually complete absence of precisely those areas in which economics and statistics most closely abut. Even on aggregation, while the coverage of Theil's important work is excellent, there is no mention of the classical theory of index numbers or of its extension to functional index numbers (where use is made of an entire Engel curve) by Frisch, Wald, and others, although this topic is one of the most illuminating applications of consumers' demand theory to empirical analysis.

The relations of probability theory to economics are not touched on, except for a brief reference to the Bernoulli and von Neumann-Morgenstern-Ramsey utility theories. Economic behavior in risky situations is simply not discussed, though the theory of investment in both fixed capital (see Hart) and in inventories really has been greatly illuminated by an appreciation of the role played by the uncertainty of the future (actually, though theories about investment behavior play, of course, an important role in the macro-dynamic sections, there is no micro-economic discus- sion whatever). Finally, though the explanation of cyclical fluctuations as a stochastic process has been a recurrent, though perhaps not dominant theme, since the work of Yule, Slutzky, and Frisch, the macro-dynamic models are exclusively deterministic.

Allen's book will be an important reference for the mathematical economist, par- ticularly in the development of macro-dynamic models, and an indispensable adjunct in teaching. It has less significance for the statistician.

Introduction to Mathematical Economics. D. W. Bushaw and R. W. Clower. Homewood, Illinois: Richard D. Irwin, Inc., 1957. Pp. xii, 345. $7.00.

VICTOR E. SMITH, Michigan State University

ASIDE from R. G. D. Allen's new book, recent textbook material for a course in mathematical economics has been limited to books designed for the teaching of

mathematics rather than economics. Uuseful as these books have been, their orienta- tion and organization have been mathematical; bits and pieces of economics appear but in the form of illustrations of mathematical principles, rather than in their own right as the center of attraction. The student gains facility in mathematical manipula- tion and becomes acquainted with numerous functions useful as representations of economic phenomena, but he does not study economics as a coherent system of thought.

Probably introductory texts in mathematical economics would have been less scarce were it not true that an appropriate order of presentation for the economic material is liable to be heartily inappropriate for the mathematical material. Bushaw and Clower have met this problem partly by dealing with the economics of markets (both isolated and interconnected) before taking up the theory of the consumer and the firm and partly by consolidating the mathematical tools needed in Part II of the book ("The Mathematics"). Thus Part I is permitted to be a treatise on economics. Whenever a mathematical technique is used for the first time the reader is referred to the appropriate section of "The Mathematics." In general, the student who turns to the mathematics only when thus directed will find Part II so written that this order of presentation of the topics is feasible, although it is not the order a mathematician

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BOOK REEIEWS 225

would have chosen if mathematical convenience alone had determined the arrange- ment. The student who prefers a more mathematical arrangement may work through Part II from its beginning. He will find it a concise text covering a good part of the mathematics used by economists, but not such topics as matrix algebra and set theory.

The student of Part II is expected to have "a tolerable level of accomplishment in elementary algebra," plus a good deal of maturity. "The Mathematics" begins with functions, graphs, and equations, but proceeds fairly rapidly through chapters on the differential calculus, definite integrals and Taylor series, quadratic forms and de- terminants, difference and differential equations, and maxima and minima. It is a book to be worked through, not simply read through. The demonstrations are com- plete, but concise. The exercises induce thought as well as provide practice.

The student who possesses only college algebra and has only a modest aptitude for mathematics may find the pace too fast. The treatment of difference and differential equations, for instance, is more abstract and more compressed than in Baumol's Economic Dynamics (Macmillan, 1951). Nor will one find here such an abundance of algebraic functions appropriate for the use of the economist or as many opportuni- ties for practice in manipulation as one finds in R. G. D. Allen's Mathematical Analysis for Economists (Macmillan, 1939) or Tintner's Mathematics and Statistics for Econo- mists (Rinehart, 1953).

The great advantage of Part I, "The Economics," is that the economic problem re- ceives the principal emphasis. It is not brought in merely to illustrate the mathemat- ics. The text begins with the analysis of an isolated market, requiring only simple algebra. Then stock and flow markets are distinguished; investment demand is dis- cussed; and a model for an isolated market having both stock and flow demands is presented, the algebra meanwhile becoming slightly more complicated. Next come models of multiple markets and the explicit recognition of money. The calculus first appears in connection with a discussion of these models from the viewpoint of com- parative statics.

The distinctive emphasis of the book is on dynamic models, in the sense of models in which the values of the variables are functions of time. The major problem of the book is the problem of stability, in isolated markets and in multiple markets. Thus differential and difference equations appear in Chap. III, with an analysis of con- vergent and explosive systems, and derivation of the stability conditions. By the time one has finished Chap. IV, on the dynamics of multiple markets, he has gained a rather thorough acquaintance with the problem.

Consumer behavior and the theory of the firm do not appear until Chaps. V and VI. The theory of consumer choice is essentially that of Hicks' Value and Capital (Oxford, 1939), but with the indifference map given an axiomatic basis. The authors develop the comparative statics propositions of Hicks' mathematical appendix rather carefully before going on to present their own stock-flow model and a dynamic model of consumer choice.

The theory of the firm, in addition to conventional input and output equilibrium analyses, presents a generalized flow model, a stock-flow model, and a simple dynamic model for output adjustment. The final chapter of "The Economics" includes two sections which deal with price determination in a market which is not purely com- petitive. This is a stimulating formulation, quite different from the usual analyses of imperfect competition.

In one instance the spirit of innovation that characterized this book has extended to terminology. "Macroeconomics" in this volume does not refer to aggregative eco- nomics, but to the economics of a market or markets. The authors' only venture into

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226 AMERICAN STATISTICAL ASSOCIATION JOURNAL, MARCH 1958

aggregative economics, an eight-page appendix on the Keynesian system, attached to Chap. II, does not strike me as either helpful or successful.

The book has very few typographical errors. It is apparent that this is not merely an intermediate theory text restated in

mathematical terms. The volume raises questions which are significant for the economist; it deserves to be read for this alone. At a time when dynamic models, despite their importance, are largely neglected in theory texts, the emphasis of this book will help redress the balance. Unfortunately, many students who would have been fascinated by dynamics in the form of growth and cyclical models will be in- clined to consider the same dynamics applied to the fundamental question of market stability as too academic to deserve their attention.

Since the book was directed toward an audience both better grounded in economics and more mature than the usual advanced undergraduate or beginning graduate student, it may prove a bit too taxing when used as a text for such students, if they are not already quite well trained in mathematics. Yet I feel that it will be so used, both because of its intrinsic interest and because it has the great advantage of making the necessary mathematics available to the serious student, along with the economics, and at the time when needed.

Linear Programming: An Explanation of the Simplex Algorithm. Dakota Ulrich Greenwald. New York: The Ronald Press Company. Pp. vii, 75. $3.00.

DAVID VALINSKY, City College of New York D ESPITE the vast volume of writing that has been done in the general field of linear programming during the past decade, few of the contributions to the

literature have been full-scale books and, until the present volume appeared, none has been primarily directed to the interested but nonmathematically oriented student.

Patently, it is impossible to encompass an adequate discussion of the whole com- plex field of linear programming within the confines of a small volume, and, the book's title to the contrary, the author has not attempted to do so. His purpose is properly described in the volume's sub-title, which aptly describes its content.

Linear programming, which is a planning tool of management, is concerned with finding the optimum pattern of the variables and restrictions entering into industrial or military programming activities. The present volume deals with the "simplex tableau," an algorithm which has proved to be the most useful of many techniques applicable to the solution of linear programming problems. It is particularly well adapted to small-scale problems, but, under some circumstances may be expanded to deal with problems of considerable size.

Essentially, this book has been designed for senior and graduate engineering or management students and makes no demands for advanced mathematical back- ground. It comprises an exposition of the simplex algorithm as applied to linear pro- gramming problems susceptible of solution by hand computation.

The plan of the book is simple and straightforward. It limits itself to an exposition and demonstration of 19 problems involving application of the subject to work sched- uling, product mix under given selling prices and production limitations, minimiza- tion and nonoptimum programs, minimization of distribution costs, etc. Eight of these problem areas exemplify the use of the simplex tableau for their solution and are used for demonstration of method. The remaining 11 are presented for solution by the reader, and serve to familiarize him, by practice, with the use of the algorithm in practical problem situations.

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