Pareto Optimality in Non-Convex Economies
This article uses the concept of cone of interior displacements, which extends the notion of differentiability, to set up a characterization of Pareto optima in non-convex economies. A general theorem asserting that a Pareto optimum is a PA equilibrium is given and specifications are discussed. It is finally argued that the usual formulation of the doctrine of marginal cost pricingas a doctrine for achieving Pareto optimal states in a non-convex decentralized economy has unsatisfactory logical basis, and a way of defining a minimum degree of centralization inherent to non-convex economies is suggested. THE MAIN RESULTS of the economic theory of allocation of resources rest upon assumptions of convexity: convexity of production sets, and convexity of preferences. The relevance of these assumptions is often doubtful; even if in a many consumer economy the classical statements can be extended to the case of nonconvex preferences (this idea, pointed out first by Farrell [13] and Rothenberg [24], was developed in the general framework of economics with a continuum of agents as introduced by Aumann [4]; see W. Hildenbrand [18 and 19]), the indivisibilities arising in production are often large and create non-convexities that cannot be overlooked. Furthermore, non-convexities may arise with externalities (see Baumol [5], Kolm [20], and Starrett [25]), exchange of information (see Radner [23]), or stock markets (see Dreze [10]). Even if one must take the risk of producing less elegant results, a relevant economic theory cannot ignore non-convexities.