How likely are the Sonnenschein-Mantel-Debreu results?

Question

I have been reading about the SMD results and it's "damning" nature for GET. My understanding of the result is as follows: price changes not only cause a substitution effect but also change the wealth distribution of a society. The latter could, in principle, cause the excess demand function to have any shape (save for homogeneity of degree zero, Walras' Law, boundary conditions, continuity)

My reaction to this is that it seems enormously unlikely. While I'm sure it's possible, it seems to me that if the number of individuals and commodities gets sufficiently large, that "hardly any" economies would actually exhibit this weird behaviour. The wealth redistributions required to get the weird shape would be just too improbable. I am curious if there was some way of defining the space of economies as a measure space, and then saying that the set of economies with weird aggregate excess demand function is of "small" measure (and tending to zero with the number of individuals/commodities). Do mathematical economists agree with this intuition of mine?

This is an interesting question, but I don't really buy that "not having a weird shape" is a good/reasonable criterion for the genericity of a distribution. In fact, my guess is that regularity is the exception rather than the norm, but maybe the CLT might have something to say here. — Theoretical Economist, Sep 15 '18 at 16:21
I'm pretty sure I could answer the question if I were to know what it is. What behavior exactly is supposed to be unlikely? — Michael Greinecker, Sep 16 '18 at 10:21
I was specifically thinking of upward sloping demand as "weird behaviour." — UtilityMaximiser, Sep 16 '18 at 17:06

score 1 · Answer 1 · answered Sep 28 '18 at 01:19

The space of economies can certainly be modeled as a measure space. Perturbing an economy (the parameters of preferences and production functions) generally changes the excess demand function continuously, so if there is one economy with a "weird" excess demand function (however defined), then there is a positive measure of them (open ball around the one weird economy).

As the number of commodities grows, and preferences for each new commodity are randomly drawn from some space, then my intuition is that the probability that at least one commodity has upward sloping demand converges to 1. The probability that all commodities have "weird" demand likely converges to zero.

Empirically, upward sloping demand is rare (examples of inferior goods are rare).

How likely are the Sonnenschein-Mantel-Debreu results?

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