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Praxeological Economics and Mathematical Economics

04/28/2006Gene Callahan

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.

— Ludwig von Mises, Human Action

If not an economist, what am I? An outdated freak whose functional role in the general scheme of things has passed into history? Perhaps I should accept such an assessment, retire gracefully, and, with alcoholic breath, hoe my cabbages. Perhaps I could do so if the modern technicians had indeed produced "better" economic mousetraps. Instead of evidence of progress, however, I see a continuing erosion of the intellectual (and social) capital that was accumulated by "political economy" in its finest hours.

— James Buchanan, What Should Economists Do?

Today's economic mainstream — neoclassical economics — is thoroughly mathematical. The vast majority of papers in academic journals are dense with mathematical notation. Nonmathematical approaches to the subject are often viewed as unscientific and imprecise.

However, the success of the mathematical approach in breaking new economic ground has been minimal. Even Bryan Caplan, a critic of Austrian economics, admits, "[Mathematical approaches] have had fifty years of ever-increasing hegemony in economics. The empirical evidence on their contribution is decidedly negative."

That has led to a number of challenges to the mainstream. The part of neoclassical theory most frequently under attack has been the assumption of rational economic behavior. The neoclassical notion of rationality posits that human behavior should result in the same outcomes as a computer calculating how to employ certain "parameters" to achieve an "optimum" result. There is a great deal of fiddling around with what the parameters should be, and exactly how to categorize the optimum result. Many of the challenges to the neoclassical paradigm recommend modifying the current models with some new parameters, or adjusting what is considered optimal. Perhaps an "altruism" parameter would modify the degree of selfishness the models apparently suggest, or mixing some quantity of "social conformity" into the desired output might explain the whims of fashion better. But some of the criticism goes deeper: Mathematics, while not useless in economics, cannot convey the principles of human action.

Mises explained the fundamental gulf between praxeological economics and mathematics in Human Action:

Logic and mathematics deal with an ideal system of thought. The relations and implications of their system are coexistent and interdependent. We may say as well that they are synchronous or that they are out of time. A perfect mind could grasp them all in one thought. Man's inability to accomplish this makes thinking itself an action, proceeding step by step from the less satisfactory state of insufficient cognition to the more satisfactory state of better insight. But the temporal order in which knowledge is acquired must not be confused with the logical simultaneity of all parts of an aprioristic deductive system. Within such a system the notions of anteriority and consequence are metaphorical only. They do not refer to the system, but to our action in grasping it. The system itself implies neither the category of time nor that of causality. There is functional correspondence between elements, but there is neither cause nor effect.

What distinguishes epistemologically the praxeological system from the logical system is precisely that it implies the categories both of time and of causality.

Let us take a famous mathematical discovery, the Pythagorean theorem, as an example of what Mises is talking about. As is well known to geometry students everywhere, the theorem says that there is an immutable relationship between the three sides of a right triangle, where the sum of the squares of the legs is equal to the square of the hypotenuse (a2 + b2 = c2). None of the legs of a triangle causes any of the other legs to be a certain length. Neither Pythagoras's equation nor any of the infinite number of triangles that it describes have any temporal relationship to each other. We needn't go into the question of whether or not mathematical forms have an existence independent of the human mind. In either case, once we apprehend the Pythagorean relationship, then the universe of right triangles, along with the relationship of their sides and all other geometric facts about them, emerge as aspects of a completely timeless, ideal form. Although our limited minds must approach those aspects piecemeal, their existence is simultaneous with the very notion "right triangle," and none of those aspects are prior to or stand in a causal relationship to any other aspect of the ideal form.

Human action is different. Just as the idea of a right triangle implies the Pythagorean theorem, the idea of human action implies "before" and "after," "cause" and "effect." We cannot make sense of human plans unless we understand that there is a past that, for the human actor, provides the soil in which the seeds of action might be sown; there is a present during which the sowing might transpire; and there is a future in which the actor hopes to reap the fruit of any action. Similarly, we must see that the actor hopes that his action will be the cause of a desired effect, or he would not act.

Viewing the economy as if it were a mathematical form is coherent if it is seen, for instance, as the study of a limiting state — equilibrium — that the real economy may gravitate toward. But when used to explain human action it creates confusion, for it eliminates from view real human choices, the very phenomena that differentiate economics from other disciplines.

Let's look at an example. Steven Landsburg's microeconomics textbook, Price Theory, reminds students:

It is important to distinguish causes from effects. For an individual demander or supplier, the price is taken as a given and determines the quantity demanded or supplied. For the market as a whole, the demand and supply curves determine both price and quantity simultaneously.

Landsburg is telling students that they must not think of prices as being determined by the actions of individuals —  individuals simply take prices as a given. Instead, it is the abstract mathematical notions of supply and demand curves that "simultaneously" determine what occurs in the market.

"Austrian economics is the economics of people viewed as creative, intelligent agents."

We can agree with Landsburg that it is important to distinguish causes from effects. At the same time, we must contend that, from the point of view of a science of human action, he has gotten them backward. Prices and quantities only change as the result of human action. Where in the world can a new price come from if not a human bidding or asking above or below the market price? It is the striving of individuals to better their circumstances, in the face of an uncertain future, that drives the market process.

Landsburg is forced into his odd posture because of his desire to capture human action with mathematical equations. Those equations cannot take into account creative human decisions based on the categories of cause, effect, before, and after. What they describe is a world of timeless correlations from which causation is absent. Human intentions play no part in the model, as the model assumes all humans can only accept it as a given. Faced with the prospect of acknowledging the limits of his model, Landsburg opts for eliminating human action from the economy.

The fact that supply and demand curves can give us a rough picture of market behavior is an effect of human action, and certainly not the cause of it. No one acts with the goal of bringing supply and demand into balance. People act in the market in order to profit, in the broadest sense of the word: they exchange because they feel they will be better off after the exchange than they were beforehand. That their search for profit tends to bring supply and demand into balance is a byproduct of their actual goals. As Hayek says in Individualism and Economic Order, "the modern theory of competitive equilibrium assumes the situation to exist which a true explanation ought to account for as the effect of the competitive process."

In order to make economic theory amenable to a mathematical treatment, neoclassical economists remove the very subject matter of praxeological economics, human action, from their theories. Mises says,

The mathematical economists disregard the whole theoretical elucidation of the market process and evasively amuse themselves with an auxiliary notion [i.e., equilibrium] employed in its context and devoid of any sense when used outside of this context. (Human Action)

The study of the correlations provided by mathematical descriptions of events is central to physics and chemistry because in those fields we can determine constants of correlation that allow us to make predictions. We feel confident that electrons will not suddenly decide that they aren't quite so attracted to protons, and that oxygen will not come to the conclusion that it would really prefer to bond with three hydrogen molecules rather than two.

Such constants are absent in human action. The subject matter of economics, in the Austrian view, is the logic of economic events, not the correlations existing between them. The study of those correlations is the subject matter of economic history and will never reveal fundamental laws of economics, because of the absence of constants. If we determine that last year a 10¢ rise in the price of bread resulted in a 2% reduction in demand — in neoclassical terms, we measure the elasticity of demand for bread — that does not tell us what will happen this year if there is another 10¢ increase.

Mathematical equations can be useful for modeling the result of people following through on previously made plans. Once a batter in a baseball game decides to swing at a pitch, we can use an equation that, based on the initial force the batter chose to apply to the bat, predicts the bat's progress. This equation will be of little use, however, in predicting whether the batter will change his mind and check his swing.

Similarly, the relative price of the two stocks in a proposed corporate merger may move in line with the predictions of a mathematical model for some time. But should market participants gain knowledge of something that alters their perception of the merger, the relative price of the stocks may differ greatly from the model's prediction. If rumors emerge indicating that the merger might fall through, the relative price of the seller might plunge. Arbitrage traders must employ their historical understanding in an attempt to grasp how other market participants will react to that news. Once that reevaluation is completed, a new risk factor for deal failure can be fed into the model and it may again function reasonably well. However, the model, cannot capture the change of perception, which is the beginning of the creation of a new plan. And it is precisely the implications of that planning that is the subject of Austrian economics. That is the moment of human choice, as the plan must aim for one goal while setting aside others, and choose some means to achieve that goal while rejecting others. Mathematical economics models the phases of markets when plans are not being created or revised, in other words, when the events that are of interest to Austrian economists, human choices, are absent.

None of the above should be taken to mean that the mathematical approach to economics is useless, only that it cannot capture the essence of human action. The British philosopher Michael Oakeshott says that we can theorize about a particular phenomenon as either a mechanical system, characterized by measurably constant responses to identical conditions, or as an intelligent activity, seen as intelligent precisely because it is not seen as the outcome of a mechanical process. In commenting on the two different approaches to the social sciences, Oakeshott says:

[In the formulation of a mechanical] "science of society" … a society is understood as a process, or structure, or an ecology; that is, it is an unintelligent "going-on," like a genetic process, a chemical structure, or a mechanical system. The components of this system are not agents performing actions; they are birth-rates, age groups, income brackets, intelligence quotients, life-styles, evolving "states of societies," environmental pressures, average mental ages, distributions in space and time, "numbers of graduates," patterns of child-bearing or of expenditure, systems of education, statistics concerning disease, poverty, unemployment, etc. And the enterprise is to make these identities more intelligible in terms of theorems displaying their functional interdependencies or causal relationships…. It is not an impossible undertaking. But it has little to do with human [action] and nothing at all to do with the performances of assignable agents. Whatever an environmental pressure, a behaviour-style, or the distribution of gas-cookers may be said to be correlated with or to cause (a rise in the suicide rate? a fall in the use of detergents?) these are not terms in which the choice of an agent to do or say this rather than that in response to a contingent situation and in an adventure to procure an imagined and wished-for satisfaction may be understood. It is only in a categorial confusion that this enterprise could be made to appear to yield an understanding of the substantive actions and utterances of an agent. (On Human Conduct)

Austrian economics is the economics of people viewed as creative, intelligent agents.

Gene Callahan is studying at the London School of Economics. He is the author of Economics for Real People, from which this article was excerpted. His first novel will be available in June, 2006. Send him mail. See his archive