A Critique of Mathematical Economics

Mr. Reed of Beaver Falls, Pennsylvania, is a recent graduate of Grove City College, now studying for an advanced degree in American history with emphasis on economics.

In the public debate over current economic problems, a crucial issue involving the very foundations and procedures of economics has been obscured. What is the proper method for analyzing economic problems? Is it mathematical or theoretical? When Adam Smith first gave it a distinct place in social science, economics was a theoretical discipline. It was considered a deductive science using verbal logic, grounded on a few basic axioms. Today’s orthodoxy is quite different. It holds that economic knowledge can be discovered and extended by means of mathematical expressions.

The mathematical treatment of economic principles was first undertaken by the French economist, mathematician, and philosopher, A.A. Cournot, in 1838. Elaborate numerical formulations of the conditions of "general equilibrium" have been worked out, notably by Leon Walras in 1874 and Vilfredo Pareto in 1909. Alfred Marshall was very influential in popularizing mathematical economics but nevertheless relegated his own mathematics and even diagrams mainly to footnotes and appendices, and preferred to present his conclusions in the verbal or "literary" form of exposition.

The work of W. Stanley Jevons in the late 19th Century gave great impetus to the mathematical approach. In his preface to the second edition of The Theory of Political Economy, Jevons instructed other economists that "their science can only be satisfactorily treated on an explicitly mathematical basis."1 Jevons went even further in contending that "all economic writers must be mathematical so far as they are scientific at all, because they treat of economic quantities, and the relations of such quantities, and all quantities and relations of quantities come within the scope of mathematics."²

The Keynesian Revolution

With the Keynesian revolution of the 1930s, the transformation of economics was complete. Extensive use of graphs, equations and statistics replaced the verbal, deductive method. The majority of economists today are absorbed in the construction of economic "models," along the lines of classical mechanics, and use a myriad of complex numerical relationships. Only a small minority of unorthodox economists, consisting mostly of "free market" advocates, have raised serious objections to this methodology.

It is time the orthodox methodology be challenged vigorously. The mathematical approach is based on fallacious assumptions and intellectual arrogance. Though its impact has been enormous—that is not disputed here—it has done little more than make economics into a dull and confusing morass for the layman and a clever excuse for central planners to justify coercive government interference with the market.

John Maynard Keynes once wrote that "if we have all the facts before us, we shall have enough simultaneous equations to give us a determinate result."3 At first this description of the essence of mathematical economics appears most profound. But a second glance reveals that what Keynes is saying is, that if we already know the future we can predict it! If only a snake had arms it could do push-ups!

Ralph Waldo Emerson had a more sober view of what constitutes human knowledge. He said that "knowledge is knowing that we cannot know." Humans are mortals and cannot know for sure what tomorrow holds. All the equations, all the data and all the impressive algebraic concoctions in the world cannot remove the uncertainty of the future.

The Entrepreneurial Role

Non-mathematical economists have postulated for decades that the function of the entrepreneur is to anticipate changes in the marketplace. Once the entrepreneur has made a decision, he then exposes his wealth and income by arranging factors of production in such manner that he may satisfy future consumer demand. If he anticipates correctly, he will earn entrepreneurial profits; if his judgments are wrong he will incur losses. Any number of variable and unforeseen elements may arise to affect the outcome: changes in fashion and technology, government policy, labor union activities, competition, prices, and even the weather. None of these elements is entirely predictable; none can be accurately determined by past performance. Such is the nature of the entrepreneurial function and, indeed, of reality itself. Attempts to mathematically quantify these elements in advance or to attach numerical significance to the subjective judgments of the entrepreneurs themselves are pure folly. They are doomed to the failure which lies in gross simplicity and imprecision.

It is ironic that mathematical economics strives for the exactness of numbers and yet bogs down in static equations which necessarily cannot begin to account for all the relevant factors. Economist Henry Hazlitt tells us that if a mathematical equation is not precise, it is worse than worthless; it is a fraud:

It gives our results a merely spurious precision. It gives an illusion of knowledge in place of the candid confession of ignorance, vagueness, or uncertainty which is the beginning of wisdom.4

J.E. Cairnes, a contemporary of Jevons, concluded that economic truth could not be enhanced by mathematics. Answering Jevons, Cairnes declared that Unless it can be shown either that mental feelings admit of being expressed in precise quantitative forms, or, on the other hand, that economic phenomena do not depend on mental feelings, I am unable to see how this conclusion can be avoided.5

Historical Data

The "quantities" which mathematical economics claims to know can be little more than data of economic history. The equations which incorporate them are thus static and incapable of dynamic analysis. They are not directed to describing the ever-changing, volatile marketplace where entrepreneurs and speculators continually rearrange production in order to profit from price changes. In actuality, these equations attempt to depict "equilibrium," a static model—a purely hypothetical construction which can never be realized in a changing world. Moreover, these equations provide no information about the human action by means of which the hypothetical state of equilibrium has been reached.

A further weakness inherent in the mathematical approach is discussed by Henry Hazlitt in his great book, The Failure of the "New Economics." There Hazlitt shows that "a mathematical statement, to be scientifically useful, must, like a verbal statement, at least be verifiable, even when it is not verified."

If I say, for example, that John’s love of Alice varies in an exact and determinable relationship with Mary’s love of John, I ought to be able to prove that this is so. I do not prove my statement—in fact, I do not make it a whit more plausible or "scientific"—if I write, solemnly, let X equal Mary’s love of John, and Y equal John’s love of Alice, then Y = f(X)—and go on triumphantly from there.

Yet this is the kind of assertion constantly being made by mathematical economists, and especially by Keynes.6

Mathematical economists are fond of drawing hypothetical "demand curves" and deriving from them hypothetical "functional relationships" between demand and price. But Hazlitt reveals the truism that "out of a merely hypothetical equation or set of equations they can never pull anything better than a merely hypothetical conclusion."7 Whether a hypothetical demand curve corresponds to any real "demand curve" is utterly impossible to determine. Supply and demand curves, then, are nothing more than "analogies, metaphors, and visual aids to thought," which should never be confused with realities.

At a Point in Time

A graph showing "the intersection of supply and demand" at a certain point may be suitable as a visual aid but not as a tool for extending economic knowledge. At best, it hypothetically illustrates a tiny segment of the market at one fleeting instant in time and has nothing to do with the ongoing changes in the market. If we have a snapshot of a dog jumping over a fence just as he clears the top rail, we do not assume that dogs live on top of fences. And the snapshot does not tell us the path the dog took to get to the fence or what he will be doing at 8:00 tomorrow morning.

A good modern example of the failure of assigning mathematical meaning to human experience in order to guide us in the future is the famous "Phillips Curve." Once a cornerstone in the thinking of government planners, the Phillips Curve is now a discredited statistical device employed by only the most diehard and naive mathematical economists. The "Curve" was supposed to show a "trade-off" relationship between inflation and unemployment: when inflation goes up, unemployment goes down, and vice versa. Along came the 1973-75 recession, with both prices and unemployment soaring, to send the mathematical economists back to their drawing boards.

The old "Quantity Theory of Money" was an early attempt to assert a fixed, constant relationship between the quantity of money and the prices of goods. The quantity theorists held that a change in the money supply would result in a proportional change in the prices of goods. Later economists showed that no such predictable relationship exists. Prices rise or fall neither in tandem with one another nor with the change in the money supply. How a change in the supply of money will affect prices is determined by how acting, motivated human beings react. In the real world of individual, subjective valuations, there are no constant relationships and consequently no measurement is possible.

But Man Reacts

Mathematics is made for the physical sciences, not the science of human action. It is very useful in physics, for example, because there one deals with a certain regularity of motion by unmotivated particles of matter. These particles, economist Murray Rothbard tells us, "move according to certain precisely observable, exact, quantitative laws."8 For instance, a brick dropped from atop a ten-story building will most assuredly fall to the ground below. Why? Because it has no choice in the matter and no means of making a choice. A regular, immutable force governs the unmotivated brick: the law of gravity. Science can proceed, via equations and "laboratory" experimentation, to determine the brick’s speed of descent with a precision so accurate that it can be duplicated with any number of bricks of the same weight dropped from the same height under the same conditions. Mathematical equations, then, are appropriate where there are constant quantitative relations among unmotivated variables.

In economics, the situation is entirely different. Here we are dealing with human ideas and human motivations leading to human action. Professor Rothbard writes:

Furthermore, since the data of human action are always changing, there are no precise, quantitative relationships in human history. In physics, the quantitative relationships, or laws, are constant; they are considered to be valid for any point in human history, past, present, or future. In the field of human action, there are no such quantitative constants.9

Politics as a branch of human action has fortunately escaped the mathematical jargon which plagues economics. Perhaps this is due to the resulting absurdity that would be readily apparent. Imagine someone manufacturing a set of equations to predict the outcome of an election! About the most mathematics could say would be that X=f(Y) [X is a function of Y] with X equaling the outcome of the election and Y equaling the voters.

Not Subject to Equation

No equation could indicate the path by which a man was elected or rejected, no equation could describe what the voters thought of a candidate at any particular time, no equation could account for shifts in public opinion and no equation for a current election could tell us anything about how any future election may turn out. These are all features of individual human action and are not capable of quantification. Only the verbal, logical analysis of the political scientist is applicable. Why is it any different in the equally volatile, equally subjective field of economics?

No wonder economics is labeled "the dismal science," one hundred years after Thomas Carlyle coined the phrase. Under the spell of mathematics, it has been reduced to cold, hard statistics. Acting man somehow has been left out of the picture, replaced by lifeless graphs and equations. Economics really deserves better. It is the science of human action and thus is a living thing. To live is to change, to be imperfect. To die is to become a statistic, to cease changing, to be perfectly dead.

Doubly Unwelcome

The mathematical approach is thus frowned upon by believers in the market. The typical citizen—the non-economist—is likewise repelled by it but for a different reason: the confusing array of complicated equations is simply beyond his understanding. When economics is treated in this manner, the typical citizen prefers to "leave it to the experts" with their slide rules, computers, and crystal balls. These "experts" invariably are proponents of expanding government power by manipulating the private economy. Mathematical economics is well-suited to the central planner because its static, impersonal and collective analysis "says" to acting individuals: "Slow down! I want to get a handle on you."

So it is that mathematics as methodology is incapable of making substantive contributions to economic knowledge. That task, today and always, belongs to deductive logic and verbal exposition. Economists must come to realize that their science is based on human action; that "aggregates" do not act, only individuals do. When that happens, we may be well on our way to solving many of our economic difficulties.


1W. Stanley Jevons, The Theory of Political Economy (New York: Sentry Press, 1965), p. xiv.

2Ibid., p. xxi.

3J.M. Keynes, The General Theory of Employment, Interest and Money (New York: Harcourt, Brace & World, Inc., 1936), p. 299.

4Henry Hazlitt, The Failure of the "New Economics" (Princeton, New Jersey: D. Van Nostrand Company, Inc., 1959), p. 99.

5J.E. Cairnes, The Character and Logical Method of Political Economy (2nd ed.; London : Macmillan & Co., 1888), pp. iv-v.

6Hazlitt, p. 101.

7 Ibid., p. 100.

8Murray Rothbard, Man, Economy and State (Los Angeles: Nash Publishing, 1970), p. 277.

9Ibid., pp. 277-8.


More by Lawrence W. Reed