On the instability of the Edgeworth process

AbstractFollowing Smale's notations let n and l denote respectively the number of agents and commodities in pure exchange barter economy (with no production and no prices). Let u1, u2, … un denote n real valued functions describing the individuals preferences. Assume that the mechanism which describes the adjustment in the quantities traded by the agents at any point in time is given by the Edgeworth process. This dynamic is assumed to take the form of an autonomous ordinary differential equation. The aim of this paper is first to develop necessary conditions for a pure exchange rate mechanism to converge to the set of Pareto points. The proof of this result uses ideas of LaSalle's Theorem. Second, we examine the conditions under which positive results may be obtained as solutions of the Edgeworth mechanism approach the boundary of the sub-manifold of feasible allocations W. Third, we examine the local stability and instability of Pareto points. The instability result is a version of Cetaev's Theorem. These resultsare proved by means of Liapunov Theory which does not require the utility functions to be C2. An example is given which shows that stability does not necessarily imply Pareto-optimality