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Logical Economics vs. Mathematical Economics

Mises Daily: Saturday, February 17, 2001 by

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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 (XVI.5)

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

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

The sterility of this approach to economics 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. What neoclassicals mean by rational behavior is, however, far different than what Austrians mean by it, which is that humans act with the goal of improving their situation.

The neoclassical notion of rationality posits that human behavior should result in the same outcomes as a computer calculating the "best" way to employ certain "input parameters" to achieve "maximum" output. There is a great deal of fiddling around with what the input parameters should be, and exactly how to categorize the maximum output. Many of the challenges to the neoclassical paradigm recommend modifying the current models with some new input parameters, or adjusting what is considered maximum output.

Perhaps an "altruism" parameter would modify the degree of selfishness these 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 to the heart of the matter: Mathematics is not a fruitful means by which to understand human action.

These challenges to the mainstream view have not gone unnoticed. Paul Krugman, a prominent neoclassical economist, writing in Slate, says: "The opponents of mainstream economics… want economics to be what it once was, a field that was comfortable for the basically literary intellectual." (http://slate.msn.com/Dismal/96-10-24/Dismal.asp) His article is very critical of those who wish to capture economics back from the mathematical model-makers.

Basically Krugman's argument reduces to a contention that, once we accept mathematical models as the primary means of discovering economic truth, then there will be elements of the resulting body of economics that are difficult to grasp if we can't grasp the models.

Surely we can agree that someone who is practiced at employing these models will understand them better than someone who isn't. But similarly, someone who has long experience with voodoo spells will understand them better than those who have neglected this study. This has no bearing on whether voodoo spells are a useful way to approach human relations, just as Krugman's argument has no bearing on whether mathematics is the right way to approach economics.

Eugen von Böhm-Bawerk was criticized on similar grounds by George Stigler, who dismissed the Austrian capital theorist by saying: "Böhm-Bawerk was not trained in mathematics." But as Murray Rothbard quipped: "'Training' in mathematics is no more necessary to the realization of its uselessness for and inapplicability to the sciences of human action than, for example, 'training' in agricultural techniques is essential to knowing that they are not applicable on board an ocean liner." (Man, Economy and State)

Mises explained the fundamental gulf between economics and mathematics in Human Action (V.1):

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. As is well know to school children 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 each of the shorter legs equals the square of the longer leg.

Mises' point is that this type of relationship has a fundamentally different nature than do the relationships implied by human action. None of the legs of a triangle "causes" any of the other legs to be a certain length. Neither the Pythagoras's equation nor any of the infinite number of triangles that it describes have any temporal relationship to each other.

Whether or not we believe that mathematical forms have an existence independent of the human mind, once we apprehend this relationship, then the universe of right triangles, along with the relationship of their sides and all other geometric facts about them emerge as the aspects of a completely timeless, ideal form. Although our limited minds must approach these aspects piecemeal, their existence is simultaneous with the very notion "right triangle," and none of these aspects are prior to or stand in a casual relationship to any other aspect of the ideal form.

It is not like that with human action. 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 the least sense of human plans unless we understand that there is a past that, for the human actor, is taken as the soil in which the seeds of action may be sown, there is a present during which the sowing will transpire, and there is a future in which the actor hopes to reap the fruit of his 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.

To attempt to view the economy as if it were an eternal mathematical form creates confusion, and eliminates from view the very phenomena that economics should be describing. For example, Steven Landsburg's intermediate-level microeconomics text book, 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 -- not cause! -- what occurs in the market. We can agree with Landsburg that it is important to distinguish causes from effects. At the same time, we must contend that he has gotten them precisely backwards. Prices and quantities only change as the result of human action. It is the striving of individuals to better their circumstances, in the face of an uncertain future, that drive the market process.

Landsburg is forced into this odd posture because of his desire to model economic events with mathematical equations. These equations cannot take into account creative human decisions based on the categories of cause and effect, before and after. What they describe is world of timeless correlations from which genuine causation is absent. Human intentions play no part in the model, as the model assumes all humans can only accept the current situation as a given. Faced with abandoning his model or eliminating human action from the economy, Landsburg opts for the latter.

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 (with the exception of a socialist central planner) 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 by-product of their actual goals. As Hayek says in Individualism and Economic Order, "That 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 is best shown by examining the familiar list of conditions found in any modern textbook."

In order to make economic theory amenable to a mathematical treatment, neoclassical economists remove the very subject matter of 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 employed in its context and devoid of any sense when used outside of this context." (Human Action, XVI.5)

The subject matter of economics, in the Austrian view, is the causes 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, since, as Mises points out, "No… constant relations exist in the field of human action…" If we determine that last year a ten-cent rise in the price of bread resulted in a two-percent reduction in demand -- in neoclassical terms, we measure the elasticity of demand for bread -- this does not tell us what will happen this year if there is another ten-cent 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 its 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 this news. Once this evaluation is completed, a new risk factor for deal failure can be fed into the model and it will again function reasonably well. The point is that the model cannot capture the change of perception, which is the beginning of the creation of a new plan. And it is precisely this planning which is the subject of economics.

This 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 equilibrium-like phases of markets, when no plans are being created or revised, in other words, when the events that should be of interest to economists, human choices, are absent. It models an "economy" in which no true economic decisions are being made.

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Gene Callahan, who writes frequently for Mises.org, is working on a book called Economics for Real People. See his Archive or send him MAIL