Why the Edgeworth Process Assumption Isn't That Bad

Abstract

A common assumption concerning disequilibrium exchange which has been employed to prove non-tatonnement stability theorems is that such exchange follows an Edgeworth process. One formulation of this assumption is: an exchange takes place if and only if it is feasible at the current set of prices and all parties to the exchange are made better off. This seems to be an innocuous and attractive assumption and so the results it gives rise to on the stability of general equilibrium' appear to be powerful and appealing [3, p. 18]. However, as Fisher [3, p. 19] has recently observed, there is (at least) one drawback: While it is obviously innocuous to assume that individuals will not trade unless they can better themselves by so doing, it is not nearly so simple to assume that trade actually will take place whenever such a situation arises. This is because of the possibility that the only coalitions which can better themselves by mutual trade consist of very large numbers of people. Thus it is possible that there is no mutually advantageous bilateral or trilateral or quadrilateral trade and that the only mutually advantageous trade involves a very complicated swapping of among millions of people. To require as the Edgeworth process assumption does that such a trade must take place is to put very heavy requirements on the dissemination of information and to assume away the costs of coalition formation. Things are not quite that bad though, for as Fisher goes on to report: David Schmeidler has shown (in an unpublished communication) that if there is any mutually advantageous trade there is one involving at most the same number of participants as there are commodities [3, p. 19]. A proof of this theorem is provided below (Theorem 1). However, this result is inadequate to overcome the Fisher criticism of the Edgeworth process assumption. For it is of no help at all when there are more than agents in the economy. Also it shows the sufficiency of bilateral trading (which causal empiricism suggests to be the norm) for only the highly restricted set of economies in which there are just two commodities. It would thus be of interest to know whether there exist interesting assumptions which imply the sufficiency of bilateral trade for the execution of Edgeworth process type exchange (hereafter Edgeworth exchange). It turns out (Theorem 2, below) that under assumptions which are anyway implied by assumptions required for the proof of non-tatonnement stability theorems of the Edgeworth process variety, bilateral trading is sufficient in this sense. Precisely, if utility functions are strictly quasi-concave and if there are unique support hyperplanes to indifference surfaces and if endowments are always strictly positive, then if there's an Edgeworth exchange for some set of agents, there's an Edgeworth exchange for some pairs of agents. Thus the Fisher problem disappears. Edgeworth exchange requires only bilateral trades. The results here will come as no surprise to those familiar with the literature which answers the question: How many agents are required to effect a Pareto improving trade (when there are no price restrictions on the admissible trades)? In that context there exist Schmeidler type results [4, 5, 6] and bilateral results [2, 6, 7, 8]. In effect then all we are