Mathematical economics has several strikes against it underpinning its abject failure to do the job it purported to do.
First off, the formulae themselves are garbage. They are not derived in the way a physicist or an engineer of materials derives his formulae describing real scientific and engineering phenomena. Instead, they are merely hypothetical expressions of hypothetical relationships using either hypothetical or already irrelevant data. And as Henry Hazlitt points out, “the mathematical economists . . . tend to forget that out of a merely hypothetical equation or set of equations they can never pull anything better than a merely hypothetical conclusion.”
Or to put it more simply: Garbage in, garbage out.
The deliberations which result in the formulation of an equation are necessarily of a non-mathematical character [points out Ludwig von Mises]. The formulation of the equation is the consummation of our knowledge; it does not directly enlarge our knowledge. Yet, in mechanics, the equation can render very important practical services. As there exist constant relations between various mechanical elements and as these relations can be ascertained by experiments, it becomes possible to use equations for the solution of definite technological problems. Our modern industrial civilization is mainly an accomplishment of this utilization of the differential equations of physics. No such constant relations exist, however, between economic elements. The equations formulated by mathematical economics remain a useless piece of mental gymnastics and would remain so even it they were to express much more than they really do.Not everything that can be measured is worth measuring. Not everything that is worthwhile is able to be measured. And not everything that pops out at the end of a string of calculations is a reliable predictor of the future. The econometricians make all three errors with their derivations of equations purporting to model systems and behaviour more complex than their ability to model them, and in their results that demonstrate their inability to meet the challenge.
“A sound economic deliberation must never forget these two fundamental principles of the theory of value: First, valuing that results in action always means preferring and setting aside; it never means equivalence or indifference. Second, there is no means of comparing the valuations of different individuals or the valuations of the same individuals at different instants other than by establishing whether or not they arrange the alternatives in question in the same order of preference.”
The aim of the econometricians is to scientise a non-science—as one of their breed once said in the journal Econometrica, the goal is “to help make economics a more or less exact science as was the determination of atomic weights for the development of chemistry.” But economics deals with human beings, not with atoms or molecules—human beings whose individual actions, choices and values are not able to measured in the way atoms and molecules can be, and whose behaviour can be even less predictable than the atomic particles so beloved of physicists.
Take the simple "Lagrangian Multiplier" that we use in basic graduate-school economics to ‘explain’ consumer behavior [suggests William Anderson]. 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.The assumption of precision emerges out of the arithmetic, but it is nowhere in the derivation of the arithmetic. Economic tinkerers over-simplify at the individual level (the bane of non-reality of “partial equlibrium”), while arithmeticising “utility maximisation” in the aggregate, apparently unaware that the aggregation of an error is simply its magnification.
“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.”
To quote Henry Hazlitt again,
Thus, when after consulting the entrails of equations they emerge to chant “market failure,” we should instead realise that the failure that is really evident here is an abject failure of theory.
Second, as Ludwig von Mises pointed out, “mathematics is silent as to causality.” Human action (which is the field that economics studies) involves “the apprehension of causal relations” among entities and existents, and the purposeful actions human beings take to bring those entities and existents into a causal connection with human needs. If it is to be taken seriously, the field of economics must describe and draw out the conclusions of all this human action. Yet mathematics itself is utterly blind to the existence of causality, and inherently confused about the direction of the causal “arrow.”
In short, it scrambles cause and effect—a confusion more than any other that is responsible for the many destructive non-sequiturs of the likes of Keynesian pseudo-analysis, which constantly reverses cause and effect in a manner as destructive of real economic activity as the burning of the nation’s savings. (Just one example will suffice: the absurd assumption that spending, as measured by GDP, is the cause of production rather than its effect—leading to the absurdly destructive notion that, when spending falls, governments must take money from producers to jack up spending in the hope and expectation this will (somehow)increase economic activity. Yet nothing could more successfully suppress it.)
The claim is sometimes made that any relationship, including presumably causal relationships, can be expressed mathematically [notes Roger Garrison]. John Egger's attempt to give plausibility to this claim by offering a far-fetched example is noteworthy because the particular example he chose provides, if only inadvertently, a sound basis for rejecting the claim. Egger translates the old saw ‘absence makes the heart grow fonder’ into f'(x)>0. (The first derivative of fondness with respect to absence is greater than zero.) Tellingly, the word ‘makes,’ which indicates the direction of causality, is unavoidably lost in the translation. The exact same mathematical expression would result from a translation of the converse: ‘growing fonder makes the heart absent.’"The blindness of mathematics to causality, and human relationships, could not be more obvious.
Third, mathematics is blind to real market processes. Because of the very nature of differential equations, the primary tool of the econometric tinkerer, imathematical economics concentrates its attention unduly upon final equilibrium, which as Reisman says, “are all that its differential equations are capable of doing.”
It thus takes attention away from the real-world operation of the profit motive and of the market processes by means of which the economic system continually tends to move toward a state of full and final equilibrium without ever actually achieving such a state...… and young students entering the field of economics never get to learn the real market processes by which human needs and wants are met (or, to paraphrase Bastiat’s famous example, “how Paris is fed!”), and as graduates they come to deny those processes actually exist.
“The effect of the dominance of mathematical economics and of the fact that it ignores market processes has been that all the major principles which explain how prices are actually determined, and which were discovered by the classical economists, have been completely forgotten…”
No wonder they are so ready to describe (and accept) an economy as “a chaotic system.”
Fourth, and finally (for now), the use of mathematical economics has effectively become a means by which entry to the “priesthood” of economics is judged—it has erected (in Reisman’s phrase) “a sort of exclusive ‘Scholar’s Guild’ which, as was the case in the Middle Ages, seeks to shut out all who do not first translate their thoughts into its esoteric language.”
One can say, for example, that the amount of bread people will buy at any given price of bread depends both on the price of bread and on the prices of all other goods in the economic system. Or one can say that the quantity demanded of bread is a mathematical function of all prices in the economic system, and then write out a nonspecific mathematical function using symbolic terminology.Let me conclude with three pieces of advice from three economists who were no mean mathematicians themselves.
“If on merely writes such an equation and stops at this point, all that has taken place is an act of intellectual pretension and snobbery… If, however, one goes further and believes one can actually formulate a specific equation—that, for example, the quantity demanded of bread equals ten thousand divided by half the square of the price of bread minus the price of butter and the average of grocers, then one is led into major errors…
“The belief that an equation could be constructed that would take such changes into account [as the introduction of new goods technologies, and changes in ideas, incomes, populations and valuations] is totally opposed to reality. It is tantamount to a belief in fatalistic determinism and implies, in effect, that a mathematical economist can gain access to a book in which all things past, present and future are written, and then derive it from the corresponding equation. Whatever it may be, such a view is definitely not within the scientific spirit.”
(1) Use mathematics as shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life (5) Burn the mathematics. (6) If you can’t succeed in 4, burn 3. This I do often."
- Alfred Marshall, letter to his friend and colleague Arthur Bowley
Mathematics is ... useful, though not an indispensable adjunct to economic studies; a finish to the training of an economist comparable with a knowledge of the classics as part of a general education…. Not much is gained by the translation into mathematical language."
- Francis Edgeworth, ‘On the Application of Mathematics to Political Economy’
This is not a dispute about heuristic questions, but a controversy concerning the foundations of economics. The mathematical method must be rejected not only on account of its barrenness. It is an entirely vicious method, starting from false assumptions and leading to fallacious inferences. Its syllogisms are not only sterile; they divert the mind from the study of the real problems and distort the relations between the various phenomena.”
- Ludwig von Mises, in ‘Mathematics Versus Economic Logic’