Mathematics and Economic Analysis
While living near Greenville, South Carolina, a couple of years ago, I attended a dinner that featured the late Sherwin Rosen of the University of Chicago as its speaker. Rosen, who had been a colleague of F.A. Hayek's at Chicago, spoke that evening on Austrian economics--or, at least, he attempted to talk about Austrian economics. In a short time, he had clearly demonstrated that he understood very little about that discipline, and he wound up giving a caricature of Austrianism.
During the question-and-answer session that followed, someone asked about the effectiveness of organizations like the Ludwig von Mises Institute. Rosen replied that the LVMI really was not particularly effective in economic circles, especially since it represents Austrian economics, something that, according to Rosen, "fails the market test."
Many of us who have received mainstream training in economics but call ourselves Austrians (or at least fellow travelers) have heard this "market test" criticism time and again. The gist of the charge is as follows: The economics profession over the years has accepted those ideas that work and has rejected ideas that do not work. Whatever is "good" in Austrian economics has already been incorporated into the greater body of economic literature, so there is no need for a special division of "Austrian" economics.
Mainstream economists are especially critical of Austrians for their lack of desire to incorporate mathematics in general, and multivariable calculus in particular, into their economic analysis. The criticism goes something like this: It does not matter whether or not mathematics is the most appropriate tool to describe economic human action. Since the majority of people in the economics profession use math for their work, it has passed the "market test" and, therefore, is the correct tool to use. In the vernacular, everybody uses math because everybody uses it. One hundred thousand economists cannot be wrong, so the belief goes.
Given this "standard" approach, it is not surprising, then, that the "top" economic journals look more like a collection of applied mathematics than anything resembling economics. People who do well in math are more likely to be published in the "best" journals than people who are not as adept in that area. Because publication in these journals is the key to tenure and promotion in the top-flight economics departments, one can see quickly that some people who may have a good understanding of economics but either are not math whizzes or choose not to use math, will not be placed in these most coveted slots, and, thus, are relegated to the "outer darkness" of smaller and less prestigious institutions. (I do not know if there is more "wailing and gnashing of teeth" in the "bigs" than in the "smalls.")
It is not surprising, then, that many in the mainstream claim that Austrians disapprove of the extensive use of mathematics in economic analysis because they are unable to do math themselves. My guess is that most folks, including myself, who can survive a modern economics graduate program, are at least competent if not proficient in mathematics, so I am not sure that charge can stick.
Furthermore, the most vociferous critics of math in economics, Murray N. Rothbard and Ludwig von Mises, were both well versed in mathematics, Rothbard entering Columbia University at age 16 as a statistics major. Rothbard and Mises did not build mathematical theoretical models because they believed it was inappropriate for economic analysis, not because they lacked mathematical competence.
What is the problem, then, of using math for economics, and why are Austrians opposed to such a methodology? In a word, math is not an appropriate tool to describe human action. As Mises and Rothbard often pointed out, one cannot quantify human action. This does not mean that people do not engage in activity in which mathematics is not important, but rather that we cannot accurately use math to describe how humans behave.
Take the simple "Lagrangian Multiplier" that we use in basic graduate-school economics to "explain" consumer behavior. Here, economists construct an equation in which one’s utility depends upon, say, goods "x" and "y." The ability to accumulate such goods is constrained by one’s income and the prices paid for the goods.
In determining the "optimal" state that the consumer can enjoy, one uses tools of multivariable calculus to reach a point where "equilibrium" is reached. At that point, the marginal utility of good "x" divided by the price of good "x" is equal to the marginal utility of good "y" over the price of that good. (I have not done the mathematical work on this page for obvious reasons.)
The problem here is that this "solution" is nonsense. Utility (or consumer satisfaction) cannot be measured in cardinal terms. There is no way to take a cardinal measure of someone’s satisfaction. I can say that I like chocolate more than vanilla, but I cannot put that preference in cardinal numbers. An attempt to do so is nothing short of an exercise in fraud.
When asked why they engage in such activities, economists usually admit that they cannot take cardinal measures of individual utility, nor can they compare the utility of one individual to another in cardinal terms. However, they then do it anyway, saying that while their activities are technically wrong, they pass the "market test" in economic analysis. People, they add, will "act as though they are engaging in measurement of cardinal utility," even if they really are not. In other words, even if something is not true, we pretend that it is true and act accordingly.
In The Failure of the New Economics, Henry Hazlitt declares
. . . 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. (p. 99)
In using mathematics as the main tool for advancing economic thought, economists must operate on the assumption that human action adheres to a constant mathematical formula. While, as Rothbard and Mises note, that might be appropriate for the physical sciences, it is not appropriate when describing how humans behave.
This is not to say that the use of logic is inappropriate in economic science. Indeed, the Austrian methodology draws heavily upon logical inferences in deducing economic science from the initial observation that people act. Furthermore, mathematics is a branch of logic.
However, while one can make logical deductions in observations of human action, it is impossible to make those deductions with the precision that mathematical reasoning requires. It is one thing to say (again) that I like chocolate better than vanilla; it is quite another to claim that a bit of chocolate gives me 6.7 "utils" of pleasure and that the same amount of vanilla yields only 4.2 "utils." The former way of "measuring" my preferences is entirely appropriate. That latter method, however, is not only inappropriate, but also downright silly. Yet, that is what the economic "mainstream" claims passes the "market test." One can only be thankful that mainstream economists are not in charge of building bridges, highways, and other structures.
 In their defense, some economists say they are not actually measuring marginal utility (which is immeasurable in cardinal terms), but rather are taking a measurement of a rate of commodity substitution. However, that, too, is nonsense, since it tells us absolutely nothing about consumer behavior and preferences.