Why should we care about Sonnenschein-Mantel-Debreu?

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Lars Pålsson Syll
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Along with the Arrow-Debreu existence theorem and some results on regular economies, SMD (Sonnenschein-Mantel-Debreu) theory fills in many of the gaps we might have in our understanding of general equilibrium theory …

It is also a deeply negative result. SMD theory means that assumptions guaranteeing good behavior at the microeconomic level do not carry over to the aggregate level or to qualitative features of the equilibrium. It has been difficult to make progress on the elaborations of general equilibrium theory that were put forth in Arrow and Hahn 1971 …

Given how sweeping the changes wrought by SMD theory seem to be, it is understand-able that some very broad statements about the character of general equilibrium theory were made. Fifteen years after General Competitive Analysis, Arrow (1986) stated that the hypothesis of rationality had few implications at the aggregate level. Kirman (1989) held that general equilibrium theory could not generate falsifiable propositions, given that almost any set of data seemed consistent with the theory. These views are widely shared. Bliss (1993, 227) wrote that the “near emptiness of general equilibrium theory is a theorem of the theory.” Mas-Colell, Michael Whinston, and Jerry Green (1995) titled a section of their graduate microeconomics textbook “Anything Goes: The Sonnenschein-Mantel-Debreu Theorem.” There was a realization of a similar gap in the foundations of empirical economics. General equilibrium theory “poses some arduous challenges” as a “paradigm for organizing and synthesizing economic data” so that “a widely accepted empirical counterpart to general equilibrium theory remains to be developed” (Hansen and Heckman 1996). This seems to be the now-accepted view thirty years after the advent of SMD theory …

S. Abu Turab Rizvi

And so what? Why should we care about Sonnenschein-Mantel-Debreu?

Because  Sonnenschein-Mantel-Debreu ultimately explains why New Classical, Real Business Cycles, Dynamic Stochastic General Equilibrium (DSGE) and “New Keynesian” microfounded macromodels are such bad substitutes for real macroeconomic analysis!

These models try to describe and analyze complex and heterogeneous real economies with a single rational-expectations-robot-imitation-representative-agent. That is, with something that has absolutely nothing to do with reality. And — worse still — something that is not even amenable to the kind of general equilibrium analysis that they are thought to give a foundation for, since Hugo Sonnenschein (1972) , Rolf Mantel (1976) and Gerard Debreu (1974) unequivocally showed that there did not exist any condition by which assumptions on individuals would guarantee neither stability nor uniqueness of the equlibrium solution.

Opting for cloned representative agents that are all identical is of course not a real solution to the fallacy of composition that the Sonnenschein-Mantel-Debreu theorem points to. Representative agent models are — as I have argued at length here — rather an evasion whereby issues of distribution, coordination, heterogeneity — everything that really defines macroeconomics — are swept under the rug.

Of course, most macroeconomists know that to use a representative agent is a flagrantly illegitimate method of ignoring real aggregation issues. They keep on with their business, nevertheless, just because it significantly simplifies what they are doing. It reminds — not so little — of the drunkard who has lost his keys in some dark place and deliberately chooses to look for them under a neighbouring street light just because it is easier to see there …