Introduction and Outline

1. What Will and What Will Not Be Treated

Mathematical economics currently includes, and perhaps is even dominated by, a number of branches with which we will have little to do. Thus, in order to define the subject of the present lectures, it is well to say something about these excluded branches. One topic that we shall not discuss to any great length is the subject that might be called efficiency economics in general, and is often called by the several names of its principal techniques — linear programming, operations research, perhaps also theory of games. In these subjects, the aim is to find the optimal adjustment, in one or another sense, to a given situation; they refer with greatest cogency and success to the profit-making possibilities of a single firm. As an omnibus reference to this area of thought let me cite Vajda's Linear Programming and the Theory of Games, and also von Neumann and Morgenstern's sparkling Theory of Games and Economic Behavior. Nor will we deal with econometrics, i.e. applied and theoretical economic statistics, except incidentally. Instead, we shall take economics as the cognitive study of a given object, the economy, and ask in the sense of natural science: what is this object like, how does it behave, and why? For this reason, we find the term analytical prefacing economics in our title. In spirit, our economics will be theoretical or speculative rather than directly empirical, and thus close in its basic approach to what has been called classical economics. In form, however, we will be more systematically mathematical. The branch of mathematics of which we will make greatest use will be the theory of matrices; let me here make reference to D. T. Finkbeiner's Introduction to Matrices and Linear Transformations, to Paul Halmos' Finite Dimensional Vector Spaces, Gantmacher's Theory of Matrices, and note the existence of numerous other introductory works on this subject. From time to time we will use a bit of calculus.

We will begin with a discussion of the theory of equilibrium prices — what has been traditionally called value theory — and go on to a discussion of business cycle theory, beginning with a model like that introduced by Lloyd Metzler, and developing the connection between this cycle theory and the equilibrium analysis that is more commonly called Keynesian. In the economic literature let me cite, in the first place, the famous General Theory of Keynes, which, as a pioneering work of science, is worth studying in spite of its numerous pedagogical and even theoretical mare's nests. A stimulating companion volume for the admirer of Keynes is Henry Hazlitt's The Failure of the New Economics. An Analysis of the Keynesian Fallacies. A superior mathematical exposition of the Keynesian theories is K. Kurihara's Introduction to Keynesian Dynamics; another, particularly fine, work of a similar sort is H. J. Brems's Output, Employment, Investment. Much of what we have to say will make reference to the " input-output " model of W. Leontief, on which there exists a vast literature. A good sample of this literature, full of references, is Activity Analysis of Production and Allocation, T. C. Koopmans, ed. Our attempts to compare speculative results with economic reality will be enormously facilitated by the extensive and painstaking work of the National Bureau of Economic Research, published in the form of a great many separate studies. A very fresh and stimulating empirical account of business cycles is the easily available Business Cycles and their Causes by W. C. Mitchell.

2. A Bouquet of Warnings

Mathematics may perhaps have a valuable role to play in economics — but its application brings several dangers. Mathematics necessarily works with exact models. In the course of investigating such a model, it is easy to forget that the mathematical exactness of one's reasoning has nothing to do with the exactness with which the model reflects economic reality. For this reason, a few dampening admonitions are in order. I quote the first and most severe from Ludwig von Mises' Human Action:

The problems of prices and costs have been treated also with mathematical methods. There have even been economists who held that the only appropriate method of dealing with economic problems is the mathematical method and who derided the logical economists as "literary" economists.

If this antagonism between the logical and the mathematical economists were merely a disagreement concerning the most adequate procedure to be applied in the study of economics, it would be superfluous to pay attention to it. The better method would prove its preeminence by bringing about better results. It may also be that different varieties of procedure are necessary for the solution of different problems and that for some of them one method is more useful than the other.

However, this is not a dispute about heuristic questions, but a controversy concerning the foundations of economics. The mathematical method must be rejected not only on account of its barrenness. It is an entirely vicious method, starting from false assumptions and leading to fallacious inferences. Its syllogisms are not only sterile; they divert the mind from the study of the real problems and distort the relations between the various phenomena.

The deliberations which result in the formulation of an equation are necessarily of a nonmathematical character. The formulation of the equation is the consummation of our knowledge; it does not directly enlarge our knowledge. Yet, in mechanics the equation can render very important practical services. As there exist constant relations between various mechanical elements and as these relations can be ascertained by experiments, it becomes possible to use equations for the solution of definite technological problems. Our modern industrial civilization is mainly an accomplishment of this utilization of the differential equations of physics. No such constant relations exist, however, between economic elements. The equations formulated by mathematical economics remain a useless piece of mental gynnastics and would remain so even if they were to express much more than they really do.

A corresponding sentiment is voiced by Keynes in his General Theory:

It is a great fault of symbolic pseudo-mathematical methods of formalizing a system of economic analysis, such as we shall set down in section VI of this chapter, that they expressly assume strict independence between the factors involved and lose all their cogency and authority if this hypothesis is disallowed; whereas, in ordinary discourse, where we are not blindly manipulating but know all the time what we are doing and what the words mean, we can keep "at the back of our heads" the necessary reserves and qualifications and the adjustments which we shall have to make later on, in a way in which we cannot keep complicated partial differentials "at the back" of several pages of algebra which assume that they all vanish. Too large a proportion of recent "mathematical" economics are mere concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentions and unhelpful symbols.

A more optimistic if still cautious opinion is stated by Professor F. H. Knight in the 1954 Britannica.

Any brief statement of principles is bound to make economic theories appear thinner and more remote from the concrete facts of economic life than they are. There is a place and a need for all degrees of generality. In recent decades this need has found increasing expression in the developing and spreading study of mathematical economics, in which exposition is made accurate and compact by the use of graphs and of algebraic formulae.

Only by the use of mathematics is it possible to bring together into a single comprehensible picture the variety, the complexity, and most of all the interdependence of the numerous factors which determine prices, costs, output and demand and the wages or hire of productive agents. ... The principal value of such elaborate and abstract systems lies in forcibly reminding the enquirer that a change in practically any economic variable has direct or indirect effects on innumerable other magnitudes, and so preventing him from fatally oversimplifying conceptions of economic cause and effect. ... The more theoretical parts of economics cannot be taken to be a complete and adequate account of the mechanism of modern economic life. They afford serviceable approximations to partial, but important aspects of the truth.

The most striking and possibly the most important characteristic of recent work in economics, as contrasted with the older, is its greater realism. It does not attempt to do without abstract conceptions, but it does attempt to take these from the world of affairs, or bring them into line with facts.

Hoping to approach Professor Knight's high goal, we may begin our investigations.

3. Introduction of a Model (Single Labor Sector)

By an economy we shall mean a complex of activities in which various commodities are produced and subsequently either consumed or utilized in the production of further commodities. If the economy absorbs commodities from outside itself, or if it supplies commodities to the outside, it is called open. On the other hand, if the economy is completely self-contained it is called closed. We wish to describe a model of an economy. Our model will in the first instance be open, in that labor must be supplied to the system by a "household sector" and products must be supplied to the household sector by the system. The model will then be "closed" by introducing labor as an additional commodity which is "used up" in the production of various other commodities, and for the production of which these various other commodities are required.

After formulating our model, we shall first indicate the manner in which it gives rise to a simple but interesting theory of prices; next give a brief discussion of the extent to which the model is a faithful reflection of the real economy; and subsequently pass to an extended mathematical analysis of the model, and to an investigation of the question of what additional and useful relationships among the various parameters of an economy can be elucidated by using the model.

We begin by establishing, in some definite but entirely arbitrary way, a certain standard physical unit for each commodity, as, e.g. 1 car, 1 ton of coal, 1 bushel of wheat, etc., hereinafter called one unit of the commodity. The process of production of any commodity requires appropriate amounts both of circulating and of fixed capital. Thus, for instance, to produce one ton of pig iron it is required, in the first place, that certain amounts of coal — say, half a ton, and certain amounts of iron ore — say, one and a half tons, be used up; but, in addition it is required that a blast furnace be tied up, for a certain period, say for half a day. The blast furnace is tied up but not used up, and hence reckons only as fixed, but not as circulating capital.

These two aspects of production will be described in our general model as follows. Let the economy involve a total of n commodities, i.e. let C1, ..., Cn be a total list of the commodities produced in an economy; cars, cigarettes, typewriters, etc. To produce one unit of any given commodity Ci, it is (technologically) required that various amounts pij of other commodities Cj be used up; in addition, it is (technologically) required that fij units of Cj be tied up (and thus not available for the manufacture of some other product) for a standard production period, say one day, even though these fij units of Cj are not necessarily used up. Note that if the standard unit of C1 is, say, a bushel, and that of C2 is, say, one ton, p12 has the dimensions tons per bushel, and p12 the dimensions ton-days per bushel.

pij is said to be the amount of Cj utilized in the production of one unit of Ci, while pij is said to be the amount of Cj consumed in the production of one unit of Ci. When a commodity is consumed it is also utilized and therefore we shall assume

(1.1) If pij > 0 then fij >0 i, j = 1, ..., n.

The model as we have thus far defined it is called the open Leontief model; the matrix pij is often called the input-output matrix, and analysis of such a model is often called input-output analysis. The matrix fij may be called the fixed capital matrix. It is clear upon a moment's reflection that this open model, as it has been defined, permits us to deduce, from a given "final demand" for a certain "bill of goods," what inputs are required; and thus, for instance, by considering the desired output of military goods in a wartime situation, to predict where "bottle-necks" are apt to develop. This sort of application has often been stressed; reference may be made to the work Activity Analysis of Production and Allocation cited above. Matrices pij for the American economy divided into fifty and into two hundred sectors have been computed by the Bureau of Labor Statistics. Our interest, however, will not be in this direct sort of "bottleneck analysis," but in the use of the input-output model as a framework for more abstract economic analysis. For this reason, we proceed at once to a description of a corresponding closed model.

To close our model, we must introduce labor as an input and as an output. Let pjo denote the amount of labor (measured say, in manhours) required for the production of Cj, and let poj be the amount of Cj which is "consumed in order to produce a man-hour of labor," i.e., the average real wages paid out per hour of labor. By the introduction of these matrix elements the model economy is rendered closed, i.e. the set of commodities produced is the same as the set of commodities utilized in production.

In our simple linear economic model (often called the closed Leontief model) we may readily set up a theory of prices. Let p0, p1, ..., pn be the prices of the various products produced; then p0, p1, ..., pn are also the prices of the commodities utilized and/or consumed in production.

We formulate the conditions that must be satisfied by the

The price of a commodity Ci is made up of the sum of two terms. The first is the sum of the values of all of the products consumed in the manufacture of Ci, i.e.