Supply And Demand Breaking Down Half A Century Ago: The Sonnenschein-Mantel-Debreu Theorem

"[M]ainstream economists [divide] into effective 'castes', with only a tiny but exalted subset of the profession undertaking the detailed mathematical work needed to discover the weaknesses in the theory. The vast majority of economists believe that this high caste, the mathematical economists, did their work properly, and proved that the theory is internally consistent. The caste has indeed done its work properly, but it has proved precisely the opposite: that the theory is consistent only under the most restrictive and specious of assumptions." - Steve Keen, Debunking Economics

1.0 Introduction

Economists like to tell stories about supply and demand, in which a higher price of a good signals that it is more scarce and encourages agents to substitute other goods for the more scarce good. Mainstream economists have known for more than half a century that these stories have no justification in the most rigorous versions of their theory. Their stories are ad hoc and arbitrary.

I have summarized the Cambridge Capital Controversy before. Here I concentrate on the Sonnenschein-Mantel-Debreu (SMD) theorem.

If General Equilibrium Theory (GET) were to have empirical implications, it would restrict what was possible for market behavior. It turns out that, however, supply and demand functions can have almost any shape. No reason exists, in the theory, for equilibria to be unique or stable. As Andreu Mas Colell and his co-authors put it, anything goes.

I rely more on Alan Kirman's presentations than the original papers for the SMD theorem.

2.0 General Equilibrium Theory (GET)

Leon Walras invented GET and set out its canonical problems: the existence of an equilibrium, its uniqueness, and its stability. For the latter, he invented the tatonnement process, an auction in which no transactions are allowed until prices are found in which demand and supplies are equal. The Arrow-Debreu-McKenzie model is the current canonical statement of GET. For purposes of this post, you can consider a pure exchange economy.

Supply and demand are functions. For example, the quantity demanded and supplied of butter are depicted as functions of its price. The difference between demand and supply is an excess demand function.

Expressing the supply and demand of butter as only a function of its price seems inadequate. Should the demand not also depend on the price of margarine? If the price of bread fell and consumers consumed more bread, would not their demand for butter also rise? Would not the supply of bread, and thus the demand for butter, be impacted by decisions of farmers between growing wheat and producing crops for ethanol?

GET attempts to model all these interactions. Households, in a competitive pure exchange economy, are assumed to start with given endowments, with a certain basket of goods. They also are assumed to have preferences among these goods and to face given prices. The households decide how much of each good in their endowment to sell on the market and how much more to buy. In the jargon, they maximize their utility subject to a wealth constraint.

So for any set of prices, the model describes the difference, for each household, between the quantity demanded on the market of each good and their endowment of each good. This is the household's excess demand function. Under certain general and non-restrictive assumptions, individual excess demand functions have certain supposedly intuitive properties. I think the demonstration that demand functions slope down, if substitution effects dominate income effects, applies to the analysis of a household's maximization problem.

Aggregate or market excess demand functions are found by summing over all households. (Aggregate demand, in this sense, is not the aggregate demands in macroeconomics. They are specified for each of thousands of goods, not somehow summed over all goods.) Suppose the market excess demand for some good was positive at some price vector. Then the households would be trying to buy more of that good than exists. This is a disequilibrium.

An equilibrium exists when the prices are such that utility-maximization decisions of the households are mutually consistent. No good exists in which the households want to buy more than the aggregate endowment of that good.

3,0 Characterization of Market Excess Demand Functions

Arrow & Debreu and McKenzie proved that, under fairly general conditions, an equilibrium exists. I am unsure if the first welfare theorem, from GET, is the theoretical justification for claims that an unregulated capitalism can be efficient. Debreu always denied this interpretation, as I understand it. Debreu (1959) provides no attempt to describe how an equilibrium can be achieved. This remains an unsolved problem (see Fisher 1983).

Almost any functions can be excess demand functions. The restrictions are that the functions be continuous, homogeneous of degree zero, and satisfy Walras' law. Also, we only consider the functions bounded an arbitrarily small distance away from zero. That is, the behavior of the function when all prices are zero is not considered.

Homogenity here means only relative, not absolute prices matter. It does not matter if prices are denominated in dimes or dollars, euros or yuan.

Walras law states that if the excess demand for some good is positive, at disequilibrium prices, then some other markets have excess supplies. The disequilibria cancel out, in some sense.

The conclusion is that GET has no empirical implications at the level of markets.

4.0 Failed Attempts at Workarounds

Market excess demand functions can inherit nice properties on individual excess demand functions if all individuals are identical and have homothetic preferences. The latter implies that Engel curves are linear functions. Your relative demands for different goods, for say, chicken or lobster, does not depend on your income.

These assumptions were typical of macroeconomists for a long time after the so-called rational expectations revolution. They talk a lot about micro foundations, but their models lack them. They could not accommodate individuals with different tastes or with tastes that varied in some way with income.

Kirman may have been sensitized to the importance of the SMD theorem by his attempts, with co-authors to relax these assumptions. What happens if individuals have homothetic preferences, but individuals vary among themselves in their preferences? The same class of functions can still be excess demand functions, with the above extremely limited constraints. How about if individuals have identical preferences, but they are not necessarily homothetic? This does not help. Nor does it help to include production.

5.0 Conclusion

Kirman suggests, as I understand it, that part of the problem is that individuals interact in the model only through markets. Maybe some sort of norms or fashions shape preferences to provide some sort of coordination. Or maybe economists should consider broad classes of households as having common preferences. This type of approach is like that of the classical political economists who assumed, for example, that workers consume all their income (they do not have much), capitalists save, and landlords indulge in spending on luxuries.

References

• Gerard Debreu (1959) Theory of Value: An Axiomatic Analysis of Economic Equilibrium.
• Gerard Debreu (1974) Excess demand functions. Journal of Mathematical Economics 1(1): 15-21.
• Franklin M. Fisher (1983) Disequilibrium Foundations of Equilibrium Economics.
• Alan Kirman (1989) The intrinsic limits of modern economic theory: the emperor has no clothes. Economic Journal 99(395): 126-139.
• Alan Kirman (1992). Whom or what does the representative individual represent? Journal of Economic Perspectives 6(2): 117-136.
• Alan Kirman and K. J. Koch (1986) Market excess demand in exchange economies with identical preferences and collinear endowments. Review of Economic Studies 53(3): 457-463.
• Rolf R. Mantel (1974) On the characterization of excess demand functions. Journal of Economic Theory 7(3): 348-353.
• Hugo Sonnenschein (1972) Market excess demand functions. Econometrica 40(3): 549-563.
• Leon Walras(1874/1877) Elements of Pure Economics.