Mathematical Economics

November 26, 1956

In his fascinating four-volume anthology, The World of Mathematics (reviewed in Newsweek of Oct. 15), James R. Newman has chosen, as his two outstanding examples of the use of mathematics in economics, excerpts from the works of Cournot and Jevons. He has chosen wisely. These are the two outstanding pioneer works upon whose fundamental postulates practically all current mathematical economics is based. If these postulates are unsound, the whole towering structure of mathematical economics is built on sand.

Mathematical economics today enjoys immense academic prestige. Open at random almost any issue of, say, The American Economic Review, and your eye will probably fall on an unintelligible maze of algebraic symbols, differential equations, and diagrams and graphs full of crisscrossing curves. Open almost any “advanced” or even “elementary” college textbook on economics today and you will find the same thing, interspersed with contemptuous references to mere “literary” economists—i.e., economists who manage to get along without an ostentatious arsenal of x’s and y’s, ordinates and abscissa, big Deltas and little d’s.

FUNCTIONS AND CURVES

Then read Cournot and Jevons, and you will find that our contemporary mathematical economists tacitly accept the same underlying postulates, and have since added practically nothing but elaborations. Cournot, more than a century age (in 1838), was already writing such sentences as: “Let us admit therefore that the sales or the annual demand D is, for each article, a particular function F(p) of the price p of such article.” And he was also explaining how “a curve can be made to represent the function in question.”

Cournot’s work seemed so strange and abstruse that it languished in neglect for 40 years. It was not until 1871, when William Stanley Jevons once more applied mathematical methods in his Theory of Political Economy, that mathematical economics began to be taken up in earnest. Today it is a virulent epidemic.

It is precisely because Jevons was bolder (and rasher) in his claims than Cournot that it is easier to detect his unproved or invalid assumptions. The subject is far too basic and intricate to argue in this space, and I am compelled to fall back on citations from authorities. In 1875, the eminent British economist J.E. Cairnes, commenting on Jevons’s work, declared: “So far as I can see, economic truths are not discoverable through the instrumentality of mathematics. If this view be unsound, there is at hand an easy means of refutation—the production of an economic truth, not before known, which has been thus arrived at.”

‘A FIGURE OF SPEECH’

Cairnes’s challenge has remained unanswered. But as he was not himself distinguished as a mathematician, his objections have been disregarded. Yet Paul Painlevé, a mathematician of the first rank, in his introduction in 1909 to the French translation of Jevons’s work, denied the usefulness or legitimacy of mathematical methods in economics except in a quite “humble” and “occasional” role. He pointed out that “the assimilation of the laws of economic equilibrium to the principles of static mechanics” was no more than “a figure of speech.”

The most uncompromising and compact refutation of the postulates of mathematical economics has been made by Ludwig von Mises: “The subject matter of economics is not potatoes, shirts, or razor blades,” he points out, “but human action, which is directed by value judgments. A judgment of value does not measure, it grades. It does not say A equals B. It says: I prefer A to B. Only out of such choices does action come into being. Production and exchange do not result from equality of value, but from value difference. Therefore, in the field of human action, there is no unit of measurement and no measuring. Prices are not measured in money; they are expressed in money.”

Most mathematical economics, in short, rests on invalid assumptions. Businessmen make use of statistics, of course; but they pay no attention to the hypothetical equations and curves of the mathematical economists, who are playing with a false analogy and a wonderful toy that diverts their attention from real problems.