Sunday, May 03, 2009

An Intervention By Kurt Gödel Into Economics

Kurt Gödel attended Karl Menger’s colloquium in Vienna in the 1930s. Sraffians should be interested in this colloquium since Von Neumann, in 1937, presented his classically-inspired economics model to the attendees. (Von Neumann had previously presented an earlier version to a Princeton mathematics seminar.) The later English translation of Von Neumann’s terse article is accompanied by a note from David Champernowne with the following acknowledgement.
"This note is the outcome of conversations with Mr. N. Kaldor, to whom many of the ideas in it are due. I am also indebted to Mr. P. Sraffa of Cambridge and to Mr. Crum of New College, Oxford, for instruction in subjects discussed in this article" -- D. G. Champernowne

Gödel’s published comment, however, was part of a discussion of a marginalist model. Walras’ models of the exchange of several commodities for one another and of production contain equations in which the quantity demanded of each good is a function of prices:
q1 = d1(p1, ..., pn)
qn = dn(p1, ..., pn)
Abraham Wald, in 1934, presented to the colloquium a sort of inverse or dual model, building on a paper from Karl Schlesinger. This model contains equations expressing the prices at which the quantities of commodities are demanded:
p1 = d1(q1, ..., qn)
pn = dn(q1, ..., qn)
I gather this model also contained inequalities, an important development in the theory of general equilibrium. At any rate, Kurt Gödel commented:
"Actually, for each individual entrepreneur the demand also depends on the price of factors of production. One can formulate an appropriate system of equations and investigate whether it is solvable." -- Kurt Gödel
According to John Dawson, Jr., Gödel’s remark is not well-taken; there is no obvious way to introduce prices of factors of production in this "inverse" model with many consumers. By contrast, when Gödel decided to say something about relevatistic physics, his remarks about rotating universes and world-lines traveling into the past, I guess, challenge physicists even decades later.

I stumbled on Gödel’s remark last week by noticing E. Roy Weintraub (1983) referenced in the first volume of Gödel’s collected works and wondering why that should be. I now see that Weintraub also quotes Gödel's remark, and Dawson is concurring with Weintraub.

  • D. G. Champernowne (1945-1946). “A Note on J. v. Neumann’s Article on ‘A Model of Economic Equilibrium’”, Review of Economic Studies, V. 13, N. 1: 10-18
  • S. Feferman et al (editors) (1986). Kurt Gödel: Collected Works: Volume 1: Publications 1929-1936, Oxford University Press
  • J. v. Neumann (1945-1946). “A Model of Economic Equilibrium”, Review of Economic Studies, V. 13, N. 1: 1-9
  • E. Roy Weintraub (1983). “On the Existence of a Competitive Equilibrium: 1930-1954”, Journal of Economic Literature, V. 21, N. 1 (March): 1-39

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