In recent months there has been a discussion both in the traditional media and in the blogosphere about why orthodox macroeconomics failed to predict or explain the financial crisis and the subsequent Great Recession. Some of that discussion focused around Paul Krugman’s criticism that economics mistook (mathematical) beauty for truth. Subsequently, there was a further discussion about the role of mathematics in economics.
Of course, this is a big topic. My task here is only to investigate, by means of a simple example, three claims made for the superiority of mathematics over ordinary (natural) language.
This example comes from a very interesting article, “Austrian Marginalism and Mathematical Economics” by Karl Menger. Karl Menger was the mathematician son of Carl Menger, one of the three pioneers of the marginal revolution and the founder of the Austrian school of economics. (The article is a chapter in J. R. Hicks and W. Weber, Carl Menger and the Austrian School of Economics.)
Karl Menger evaluates some claims by mathematical economists in the context of the Principle of Diminishing Marginal Utility. He states the idea in words:
“For each good, the utility of a larger quantity is greater (or at any rate not less) than that of a smaller quantity, whereas the marginal utility of the larger quantity is less (or at any rate not greater) than that of the smaller.”
(For our purposes here let us disregard the question of the cardinal measurement of utility.)
Compare this to the standard formulations in terms of a twice-differentiable utility function.
U=f (q) where f’(q)> or = 0 and f”(q)< or = 0.
Some economists claim that the mathematical formulation is: (1) more general, (2) more explicit and (3) more precise.
Is it more general?
No. Actually, the verbal formulation is “more general since it is valid even if there are places where the function does not admit a second derivative and its graph has no curvature, whereas at such places the mathematical formulation fails to assert anything.”
Does the mathematical formulation make the assumptions underlying the Principle more explicit?
There us a lot of ambiguity in this question. Do the proponents of this view mean the assumptions underlying the verbal formulation or do we mean the assumptions underlying the mathematical formulation? Clearly the meaning must be the mathematical formulation.
Here again the answer is no. The assumptions of continuity and differentiability of the utility function are made “as though these properties were matters of course, whereas they are nothing but prerequisites for the application of classical [mathematical] analysis…” Far from bringing the tacit assumptions of the verbal formulation to the fore, the mathematical adds new assumptions in an almost casual and therefore implicit manner.
Is the mathematical formulation more precise?
No. Precision must be defined relative to what the analyst desires to express. Each of these formulations is equally precise. The only difference between the two formulations is that the mathematical is restricted to cases in which the “functions … are differentiable, and therefore have tangents (which from an economic point of view are not more plausible than curvature)…”
Having said all of this, there is an additional consideration that goes beyond those considered by Karl Menger. This is the very strange “essentialist” idea that the true Principle of Diminishing Marginal Utility is the mathematical one. Consider this nuance: If an economist says the virtue of the mathematical approach is that it makes the assumptions underlying the Principle more explicit, he is begging the question. This is because he is not referring to anything that exists independently of his own formulation.
Could he really be saying that the mathematical formulation of the Principle is more useful? This is probably what the mathematical economist means. But when words like “useful” are being bandied about, we must have a pragmatic meaning in mind. Useful for what? For making more elaborate mathematical models? Obviously. But why do we want them? What are they useful for?
At this point we come to some fundamental issues. I cannot treat them all here. However, there is the often-ignored three hundred pound gorilla in the room.
We have data; we have implications of a model. The data have been processed and massaged to fit the general needs of the theoretical approaches. The implications of the model are mostly abstract; they must be interpreted as corresponding to phenomena that are observed in everyday life.
The most important issue is what counts as an explanation? More exactly, what shall we accept as an explanation? (This is the gorilla.) What we shall accept is a decision.
Some of us are implicitly asking for an explanation that avoids or minimizes or de-emphasizes assumptions that are purely mathematical. As Karl Menger says, “…they are nothing but prerequisites for the application of classical analysis and not based on facts.” In other words, they are for the convenience of applying the method.
There is a certain self-contained quality to al this. (I shall not say circularity.) Mathematical theory transforms the phenomena of everyday life (Schutz and Luckmann’s “life-world”) into another form and then explains those data.
I hope the reader can see that in itself there is nothing “wrong” with this. However, it is a different endeavor. It is not doing the same thing as economists of old did in ordinary language but doing it better or more generally or more explicitly or more precisely.
It is doing something else. It is changing the subject to fit a method.
Is there not a role for answering different questions with the precision (and so forth) most appropriate to them?
Yes, this is just what Aristotle famously says in his Nicomachaen Ethics:
“Our discussion will be adequate if it has as much clearness as the subject-matter admits of, for precision is not to be sought for alike in all discussions, any more than in all the products of the crafts.” (Book I, Chap. 3.)