Linear Programming and Economic Analysis

Robert Dorfman, Paul A. Samuelson, Robert M. Solow

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486142111
• Topic: Mathématiques
• Subtopic: Programmation linéaire et non linéaire

Designed primarily for economists and those interested in management economics who are not necessarily accomplished mathematicians, this text offers a clear, concise exposition of the relationship of linear programming to standard economic analysis. The research and writing were supported by The RAND Corporation in the late 1950s.
Linear programming has been one of the most important postwar developments in economic theory, but until publication of the present volume, no text offered a comprehensive treatment of the many facets of the relationship of linear programming to traditional economic theory. This book was the first to provide a wide-ranging survey of such important aspects of the topic as the interrelations between the celebrated von Neumann theory of games and linear programming, and the relationship between game theory and the traditional economic theories of duopoly and bilateral monopoly.
Modern economists will especially appreciate the treatment of the connection between linear programming and modern welfare economics and the insights that linear programming gives into the determinateness of Walrasian equilibrium. The book also offers an excellent introduction to the important Leontief theory of input-output as well as extensive treatment of the problems of dynamic linear programming.
Successfully used for three decades in graduate economics courses, this book stresses practical problems and specifies important concrete applications.

1

Introduction

1-1. HISTORICAL SKETCH

At any time, an economy has at its disposal given quantities of various factors of production and a number of tasks to which those factors can be devoted. These factors of production can be allocated to the different tasks, generally, in a large number of different ways, and the results will vary. There is no more frequent problem in economic analysis than the inquiry into the characteristics of the “best” allocation in situations of this kind.

We have just outlined a rudimentary problem in welfare economics or in the theory of production. It is also a problem in linear economics, the word “linear” being introduced to call attention to the fact that the basic restrictions in the problem take the form of the simplest of all mathematical functions. In this case the restrictions state that the total amount of any factor devoted to all tasks must not exceed the total amount available; mathematically each restriction is a simple sum.

This illustration suggests that many familiar problems in economics fall within the scope of linear economics. Like Molière’s M. Jourdain and his prose, economists have been doing linear economics for more than forty years without being conscious of it. Why, then, a book on the subject at this date? Because until recently economists have passed over the linear aspects of their problems as obvious, trivial, and uninteresting. But in the last decade the stone which the builders rejected has become the headstone of the corner. New methods of analysis have been developed that depend heavily on the linear characteristics of economic problems and, indeed, accentuate them. The most flourishing of these methods are linear programming, input-output analysis, and game theory.

These three branches of linear economics originated separately and only gradually grew together. The first to be developed was game theory, the central theorem of which was announced by John von Neumann¹ in 1928. The main impact of game theory on economics was delayed, however, until the publication of Theory of Games and Economic Behavior² in 1944. Briefly stated, the theory of games rests on the notion that there is a close analogy between parlor games of skill, on the one hand, and conflict situations in economic, political, and military life, on the other. In any of these situations there are a number of participants with incompatible objectives, and the extent to which each participant attains his objective depends upon what all the participants do. The problem faced by each participant is to lay his plans so as to take account of the actions of his opponents, each of whom, of course, is laying his own plans so as to take account of the first participant’s actions. Thus each participant must surmise what each of his opponents will expect him to do and how these opponents will react to these expectations.

It was von Neumann’s remarkable achievement to demonstrate that something definite can be said about such a welter of cross-purposes and psychological interactions. He showed that under certain assumptions, which we shall have to examine, each participant can act so as to be guaranteed at least a certain minimum gain (or maximum loss). When each participant acts so as to secure his minimum guaranteed return, then he prevents his opponents from attaining any more than their minimum guar-anteeable gains. Thus the minimum gains become the actual gains, and the actions and returns for all participants are determinate.

The implications of this theory for economics are evident. It holds out the hope of banishing oligopolistic indeterminacy from economic situations in which von Neumann’s assumptions are satisfied. The military implications are also evident. And, it turns out, there are important implications for statistical theory as well. Since 1944 the development of these three fields of application of game theory has gone forward actively.

Input-output analysis was the second of the three branches of linear economics to appear. Leontief published the first clear statement of the method in 1936³ and a full exposition in 1941.⁴ Input-output analysis is based on the idea that a very considerable proportion of the effort of a modern economy is devoted to the production of intermediate goods, and the output of intermediate goods is closely linked to the output of final products. A change in the output of any final product (say automobiles) implies changes in the outputs of the intermediate goods (copper, glass, steel, etc., including automobiles) used in producing that final product and, indeed, in producing goods used in producing those intermediate goods, and so on.

In its original version, input-output analysis dealt with an entirely closed economic system—one in which all goods were intermediate goods, consumables being regarded as the intermediate goods needed in the production of personal services. Equilibrium in such a system exists when the outputs of the various products are in balance in the sense that just enough of each is produced to meet the input requirements of all the others. The specification of this balance and its pricing implications was Leontief’s first objective.

Beginning with World War II, interest has shifted to a different view of Leontief’s model. In this view final demand is regarded as being exogenously determined, and input-output analysis is used to find levels of activity in the various sectors of the economy consistent with the specified pattern of final demand. For example, Cornfield, Evans, and Hoffenberg have computed employment levels in the various sectors and, hence, total employment consequent upon a presumed pattern of final demand,⁵ and Leontief has estimated the extent to which fluctuations in foreign trade influenced activity in various domestic sectors.⁶ The input-output model, obviously, lends itself well to mobilization planning and planning for economic development.⁷

The last of the three branches of linear economics to originate was linear programming. Linear programming was developed by George B. Dantzig in 1947 as a technique for planning the diversified activities of the U.S. Air Force.⁸ The problem solved by Dantzig has important similarities to the one studied by Leontief. In any operating period the Air Force has certain goals to achieve, and its various activities of procurement, recruitment, maintenance, training, etc., are intended to serve those goals. The relationship between goals and activities in an Air Force plan is analogous to the relationship between final products and industrial-sector outputs in Leontief’s model; in each case there is an end-means connection. The novelty in Dantzig’s problem arises from the fact that in Leontief’s scheme there is only a single set of sector output levels that is consistent with a specified pattern of final products, while in Air Force planning, or in planning for any similar organization, there are generally found to be several different plans that fulfill the goals. Thus a criterion is needed for deciding which of these satisfactory plans is best, and a procedure is needed for actually finding the best plan.

This problem is an instance of the kind of optimizing that has long been familiar to economics. Traditionally it is solved by setting up a production function and determining that arrangement of production which yields the desired outputs at lowest cost or which conforms to some other criterion of superiority. This approach cannot be applied to the Air Force, or to any other organization made up of numerous components, because it is impossible to write down a global production function relating the final products to the original inputs.⁹ Instead it is necessary to consider a number (perhaps large) of interconnected partial production functions, one for each type of activity in the organization. The technique of linear programming is designed to handle this type of problem.

The solution of the linear-programming problem for the Air Force stimulated two lines of development. First was the application of the technique to managerial planning in other contexts. A group at the Carnegie Institute of Technology took the lead in this direction.¹⁰ Second, a number of economists, with T. C. Koopmans perhaps in the forefront, began exploring the implications of the new approach for economic theory generally.¹¹ The present volume belongs to this general direction of effort. We shall regard linear programming as a flexible and powerful tool of economic analysis and hope that the applications to be presented below will justify our position.

These are the three major branches of linear economics. The relationship between input-output analysis and linear programming is evident. Input-output analysis may be thought of as a special case of linear programming in which there is no scope for choice once the desired pattern of final outputs has been determined.

The connection of these two with game theory is more obscure. Indeed, after the sketches we have given of the problems handled by the three techniques, it may seem surprising that there is any relationship, and, as a matter of history, the connection was not perceived for some time after the three individual problems and their solutions were well known. The connection resides in the fact that the mathematical structures of linear programming and of game theory are practically identical. Is this a pure coincidence?¹² Probably it does not pay to search for an economic interpretation. It may make the connection seem less mysterious if we put it this way: Both game theory and linear programming are applications of the same branch of mathematics—the analysis of linear inequalities—a branch which has many other applications as well, both in and out of economics. The connection is analogous with the connection between the growth of investments at compound interest and Malthusian population theory.

1-2. OUTLINE OF THE BOOK

Linear programming is the core of linear economics, and we take it up first. Chapter 2 sets forth the basic concepts and assumptions of linear programming and illustrates them by two examples, one from home economics and one from the theory of international trade. The truism that the problem of allocation and the problem of valuation are inseparable applies as well to linear programming as to other modes of economic analysis. The valuation aspect of linear programming is explored in Chap. 3.

Chapters 2 and 3 together take up the leading ideas of linear programming; Chap. 4 goes on to the mathematical properties of linear-programming problems and practical methods of solution. This latter chapter is somewhat technical and may be omitted since it adds no new economic concepts. Readers who are interested in actual solutions will find it indispensable, however.

Chapter 5 presents a particularly simple and important application of linear programming. It deals with this problem: Suppose that a homogeneous commodity is produced at a number of places and consumed at a number of places, and suppose also that the total demand at each point of consumption and total supply at each point of production are known. How much should each consuming point purchase from each producing point so that all demands are satisfied and total costs of transportation are kept as small as possible? This “transportation” or “assignment” problem is interesting not only for its own sake but because it has useful generalizations.

In Chap. 6 the linear-programming approach is applied to the theory of the competitive firm. The conclusions are consistent with those of the marginalist theory of production. But, as we noted earlier, the marginalist theory invokes the concept of a global production function comprehending all the activities of the firm, while in a multiproduct or multistage firm it may be more convenient to work with a number of partial production functions. Chapter 7 covers the imputation of values to the resources used by a competitive firm.

Chapters 6 and 7 were restricted to competitive firms because of one of the linearity assumptions. In a competitive firm, gross revenue is a linear function of the physical volume of sales, namely, the sum over all the kinds of commodity sold by the firm of price times quantity sold. In a firm not in perfect competition the relationship between revenue and physical sales volume is more complicated; it is, in fact, nonlinear. Chapter 8 discusses the analysis of such firms and the problem of relaxing some of the linearity assumptions in linear programming.

Input-output analysis is taken up next. The basic input-output system is set forth, illustrated, and discussed in Chap. 9. Chapter 10 is a more technical discussion of the system and may be omitted by readers who wish to avoid the more mathematical aspects of the subject. It deals with more difficult questions of interpretation than does Chap. 9, including an examination of Leontief’s strongest assumption—that there is a unique combination of factor and material inputs for the product of each economic sector.

Chapters 11 and 12 extend the input-output model dynamically, i.e., to a sequence of time periods, and link it up with the theory of capital. In this pair of chapters, again, the earlier chapter is primarily conceptual and the later is devoted to the more difficult and technical problems. Here, almost uniquely in this volume, our presentation takes issue with previously published results. We have mentioned above that in Leontief’s static system there is only one set of levels of sector outputs that will produce a specified pattern of final products. There is therefore no room for choice once the pattern of final output has been determined. Leontief has extended his system dynamically in a way that preserves this fully determined character. Our position is that the possibility of holding intermediate and final products in inventory makes choices inevitable, so that Leontief’s analysis ignores an important aspect of economic dynamics. But we cannot pursue the issues here; the reader will have to wait until Chap. 11. These chapters also arrive at some new criteria for economic efficiency in a dynamic context and some new conclusions concerning the operation of competitive markets in a dynamic context.

Rather surprisingly, linear programming has turned out to be the most powerful method available for resolving the problems of general equilibrium left unsolved by Walras and his immediate successors. Under what conditions will there exist an equilibrium position for an economy in which all prices and all outputs are nonnegative? Under what conditions is this equilibrium unique? The techniques at Walras’ disposal did not permit him to reach satisfactory answers to these questions. Solutions by means of linear programming are given in Chap. 13. Linear programming has also proved to be an easy and powerful method for deriving the basic theorems of welfare economics and is used for this purpose in Chap. 14.

The final two chapters deal with game theory. Chapter 15 deals with the basic concepts of game theory as applied to economic problems and discusses some methods of practical solution of game situations. Chapter 16 explores thoroughly the mathematical connections between game theory and linear programming.

The crucial dependence of game theory on the measurability of utility warrants some discussion, particularly in view of the old issue of the relevance of the measurability of utility for economics. Appendix A is devoted to this issue.

The reader will shortly become aware that linear economics makes liberal use of the results of matrix algebra. The text is nearly, but not completely, free of matrices. Nevertheless, to help readers who wish to gain some insight into matrix methods we have added Appendix B on matrix algebra, which, it is hoped, despite being called an appendix, will not be a useless appendage.

2

Basic Concepts of Linear Programming

2-1. INTRODUCTION

Since at least the time of Adam Smith and Cournot, economic theory has been concerned with maximum and minimum problems. Modern “neoclassical marginalism” represents the culmination of this interest.

In comparatively recent times mathematicians concerned with the complex problems of internal planning in the U.S. Air Force and other large organizations have developed a set of theories and procedures closely related to the maximization problems of economic theory. Since these procedures deal explicitly with the problem of planning the activities of large organizations, they are known as “linear programming.” The ma...