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 ="">
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.”