“Economic Theory in the Mathematical Mode,” G. Debreu (1984)

This essay is the Nobel prize speech of Debreu, who is one of the first-ballot Economics Hall of Famers of the 20th century. (I’ll take Marshall, Fisher, Keynes, Ramsey, Hotelling, Hicks, von Neumann, Savage, Nash, Samuelson, Arrow, Becker, Aumann and Myerson. You can see my obvious microeconomic bias: am I missing anyone here? I’m open to the cases for Wald, Friedman, Lucas and Harsanyi…) The bulk of the essay is a description of the development of general equilibrium, from the discovery of existence through to n-replica economies and Sonnenschein-Mantel-Debreu’s disappointing theorem. It is Part II that I find more interesting, though. Here, Debreu discusses why economics has been formalized mathematically (and this is a question which any economist ought be able to answer!)

He gives two main reasons. First, mathematics is an efficient language for unambiguous communication among economists and among other social scientists; we needn’t worry about technical terms being misunderstood when theorems are stated mathematically. Debreu studied mathematics under the influence of Hilbert and the Bourbaki school, so this argument seems taken directly from the analogous argument in math. Second, axiomatization provides “secure bases from which exploration could start in new dimensions” without requiring new researchers to question every statement made by previous researchers; as long as we agree on the axioms, the conclusions are deductions, and can be accepted. This is a very Kuhnian view of social science, with an axiomatic system playing the role of the field’s paradigm.

I agree that these reasons are at the heart of why economists formalize, but I do not think they are very compelling reasons. The problem lies at the heart of the difference between social science and mathematics. In an axiomatic mathematical system, the axioms are by assumption true – there is no sense in which an axiom can be false, since it is merely an abstract statement used to derive implications. That is, if I work on a program in Euclidean space, the very definition of that space means that I accept Euclid’s axioms. Social science is not this way. We know, pace the arguments of Lionel Robbins, that our assumptions are false, though we hope that they are in some sense “good enough” to derive implications (many economists will disagree with that last sentence, and instead argue that economics is a study of abstract relationships, and that deductions from axioms are merely mathematical deductions, but this “overformalization” has been countered many, many times – most famously, by von Neumann in an essay I may discuss on this site at a later date). If axioms are only approximate, though, as in the standard Humean problem of induction, we have no way of knowing whether conclusions will also be approximate; there is no “universal continuity”. I think economists would be better served to think of axiomatization as the formalization of analogies. That is, an axiomatic deduction when axioms are imprecise may tell us nothing about the real world, but it tells us as much as a qualitative analogy, and does so in a formal way that deemphasizes the rhetorical ability of the author. I am working on further results along these lines, so hopefully within the next year I’ll have an expansion of the above idea on these pages.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.121.4168&rep=rep1&type=pdf (it is not obvious from the pdf, but this speech was transcribed in AER 74.3 (1984))

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11 thoughts on ““Economic Theory in the Mathematical Mode,” G. Debreu (1984)

  1. Agent Continuum says:

    Method is career suicide and therefore best left (a) for after retirement; and (b) for people paid 25-50% less than our profession. I don’t want to be depressive or anything…

    I’d also quibble with your list. Nash had some pivotal contributions but is his career comparable to that of the others you listed?

    Re: Sonnenschein-Mantel-Debreu, here’s a post idea. Some people describe it as disappointing (you hear theorists take this view a lot) while some people say it’s no big deal (both groups professors at top 5). How come?! Maybe it’s worth documenting people’s interpretations/take on it.

    Finally, I wouldn’t fret over math. Most of the internal consistency that economic models have comes not from the use of calculus or linear algebra or whatever, but rather from imposing market clearing, rational expectations and so on.

  2. afinetheorem says:

    I agree method is career suicide – I’m interested in these types of papers purely for reasons of personal interest!

    Nash is definitely the weakest on the list…but if you think games are the foundation of modern economics, and if you agree that his solution concept is the dominant one, then I think you have to include him.

    I think the SMD post would be interesting, but perhaps I should leave it for you! “Topic creep” is really common in blogs, and I want to restrict this one solely to discussing research. If you wrote a post on SMD, I’d be glad to send over my thoughts on why it’s a disappointing result (my understanding is that Debreu thought it was thus, also).

    Finally, I’m fine with the math to the extent that it’s simply a translation of logical statements. The problem (which I don’t think is terribly serious, but I do think economists should know why) is not internal consistency per se, but that for axiomatic systems in social science, internal consistency is insufficient for anything useful to be said about markets in the real world. To the extent that theory is useful, the theory must represent real world objects. Axioms in economic models are always false; if 99% of people have rational expectations 99% of the time, that doesn’t mean that some RE model is 99% accurate in predicting the real world. As to whether theory is still useful even given this problem, I say yes, but the reasons why we should say yes are 1) nontrivial philosophically, and 2) utterly rejected by essentially every other social science, which means that economists ought pause and take stock before blindly mathematizing.

  3. jenizaro says:

    “I think economists would be better served to think of axiomatization as the formalization of analogies. That is, an axiomatic deduction when axioms are imprecise may tell us nothing about the real world, but it tells us as much as a qualitative analogy, and does so in a formal way that deemphasizes the rhetorical ability of the author.”

    This is basically the same as Rubinstein’s idea of economic models being stories or fables that help to think clearly about a given problem.

    With regard to the use of axiomatization in Mathematics versus Economics, the difference in my opinion is that Mathematics’ goal is abstraction, or the delineation of the most general possible structures. Economics formalization is just a precise language.

  4. Vincent says:

    As a current first year grad student, I would be interested in hearing you say more about this “method is career suicide”.

    Thanks!

  5. afinetheorem says:

    Jenizaro…I agree more or less completely. I think the Rubinstein interpretation is right as well. If you look at the literature, though, I think you’ll see than only micro theorists are thinking of economic models in this way, unfortunately.

    Vincent…just watch the job market. Unlike in some other social sciences, methodology is essentially never the point of a top job market paper, many departments have no one specializing in the field, and even bigshots have difficulty publishing papers in this area. That said, it’s still important!

  6. dsquared says:

    I think Aumann and Becker are both much weaker claims than Nash and to select Savage over Wald as your representative of the journeyman mathematicians-statisticians hanging around in the 50s is bordering on perverse ;-). Modigliani, surely?

  7. afinetheorem says:

    Dan, Modigliani is surely justified, but my knowledge of finance is so limited that I barely know the difference between a stock and a bond…I’m probably not his best judge!

    Actually, I see Aumann as one part Nash and one part Wald: he had one giant idea (agreeing to disagree, and how to model knowledge), and a bunch of papers that presaged work done twenty years later.

    Becker has to be on just for influence, right? If Samuelson formalized economics, then Becker got economists to consider social phenomena beyond the strict economy. It seems like half the papers in top journals are on the “Becker domain”: crime, addiction, the family, education, etc.

  8. Robert says:

    Should Goodwin, Kaldor, Kalecki, Pasinetti, Joan Robinson, and Sraffa be in the running?

    • dsquared says:

      I was presuming that the reference to the Baseball Hall of Fame meant that we were only talking about Americans popular in America.

  9. k says:

    I thought immediately of the Rubinstein argument as well.

    The other (similar) argument comes from McCloskey’s “Rhetoric of Economics” where she argues that most of the debate that carries on in economics stems from the failure to realize that economists are all (in one way or another) using an analogy to explain their point of view on any particular outcome of economic interest.

    These analogies are not meant to be exactly in line with the real world; but the better analogies capture the essence of any behaviour of interest. The problem I guess is to find the most appropriate analogies.

    Being creatures of habit, economists find it hard to get out of thinking along certain analogies that come to dominate.

  10. finfin says:

    I don’t quite understand the part about axiomatization (sp?) as analogy. Do you think you could formalize the argument?

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