Wednesday, June 06, 2007

Saari Extends Sonnenschein-Mantel-Debreu Results

I recently found out that one can download Saari (1995). (Hat tip to Ars Mathematica.)

Before going into advanced mathematics, Saari notes a theoretically unjustified popular portrayal of economics :
"On the evening news and talk shows, in the newspapers, and during political debate we hear about the powerful moderating force of the market which, if just left alone, would steadily drive prices toward an equilibrium with the desired balance between demand and supply. The way this story is invoked to influence government and even health policies highlights its imporant critical role. But, is it true? I have no idea whether Adam Smith's invisible hand holds for the 'real world', but, then, no one else does either. This is because, even though this story is used to influence national policy, no mathematical theory exists to justify it. Quite the contrary... "
Saari considers dynamic processes in which the Walrasian auctioneer lowers prices in which supply exceeds demand and raises prices in which demand exceeds supply. The auctioneer knows about offers in all markets, and no trades are allowed unti equilibrium is achieved, if ever.

Saari does not only consider processes in which the rate at which the auctioneer changes prices is proportional to the discrepancy between demand and supply. He generalizes the problem to consider all processes which depend on both this discrepancy and the rate of change with respect to price at which agents change their offers. He finds that even when their is a stable equilibrium for some such process, some agents can upset this stability by withholding one commodity from the market.

I have thought about documenting - after having written about all the Sraffa examples I care to treat - Scarf's example of a non-stable equilibrium. Saari makes clear many more interesting examples exist. I don't know whether I am bright enough to find specific numeric examples.

I'm interested in the Arrow-Debreu model of intertemporal equilibrium, with production. Two sorts of dynamics exist in this model. Saari is talking about a tâtonnement process before the beginning of time. A question exists about how relative prices change over time in the cleared forward markets. I think one can explore the implications of Sraffa effects on the latter dynamics. I think it is still an open question whether Sraffa effects have implications for tâtonnement processes, although Mandler gives good arguments in the negative.

For Radek: I note that some editor has complained that the Wikipedia entry on General Equilibrium uses too much technical jargon. I don't know what to do about this. Compare and contrast the level of jargon in the entry on "Null set". I don't know what decides the level that is acceptable; perhaps the laity expects to be able to follow economics and not to be able to follow advanced mathematics.
  • Donald Saari (1995). "Mathematical Complexity of Simple Economics", Notices of the AMS, V. 42, N. 2 (Feb): 222-230

1 comment:

YouNotSneaky! said...

Thanks for the note - I removed the tag and requested comment. I conciously tried to make that article as non-technical as possible, but there's only so far as it can go.

Also, in the Saari article:
It now is trivial to dismiss the Smith
story simply by choosing a vector field of the kind
illustrated in Figure 2a with a lone, unstable

I think if you impose the relatively weak condition that prices get "bounced back" at the boundaries of the simplex (as p --> 0, excess demand increases sufficiently fast) + strict convexity than at least one equilibrium is stable. So in the case of unique equilibrium, it's always stable. Saari's example (u=xy) violates this condition. In this case the simple tatonnement process is
dp = v[(px-y)/(1+p^2)] --> unstable (one can draw it) but which is neg. but finite at p = 0 (p being relative price of good y, x being total endowment of x and likewise for y). The tougher case is where they get bounced back but you get cycling around the equilibrium which is in Scarf. I'm not sure about this one right now. Anyways, reading that Scarf paper makes me realize where my grad school prof got her homework problems from.