Market Excess Demand in Exchange Economies with Identical Preferences and Collinear Endowments

Abstract

In this paper we show that up to an arbitrary small neighbourhood of the boundary of the set of prices any given excess demand function can be considered as the excess demand of an economy with any large enough but finite number of individuals having identical preferences. Moreover, the individual endowments can be chosen collinear and in such a way that they yield any arbitrary price independent distribution of relative wealth. One of the most striking results in economic theory is that developed by Sonnenschein (1972,1973) and perfected by Debreu (1974). The question posed by Sonnenschein and answered in the affirmative was, given any excess demand function f is there a finite number of consumers satisfying all the standard assumptions on preferences whose excess demands sum to f? Debreu showed that indeed one only needs 1 consumers where 1 is the number of goods in the economy question. This result may be considered as extremely negative since it suggests that the standard assumptions on consumer preferences imply no restrictions on the class of excess demand functions that may be observed. Indeed, the hope that, by restricting even further our assumptions on individuals, we may restrict the class of market excess demand functions seems vain. Mantel (1974) has shown for example that even if one requires consumers to have homothetic preferences Debreu's result still holds. A standard objection to this type of result is that, in constructing the economy which has f as its excess demand function, one may be obliged to choose individuals with very different characteristics. If then one restricts the dispersion of preferences or endowments in some way one might reduce the class of possible excess demand functions. It has also been suggested that it may be that the initial resources of the individuals have to be chosen so that the income distribution for the economy is highly price dependent. Hence to confine one's attention to economies with a fixed price-independent income distribution might severely restrict the class of resultant excess demand functions. Taking these points one at a time.