Characterization by prices of optimal programs under uncertainty

A central problem in the theory of economic growth is to characterize all the ‘optimal decisions’, usually by a system of competitive or efficiency prices. We work in a multi-sector economy with production and consumption overtime. Our model is a generalization of Radner’s (1973) model similar to that presented by Dana (1973). A central planner has to make decisions, at the very beginning, concerning consumption and production at each date according to some optimality criterion. A deterministic model can be regarded as an approximation to a more general model. The planner may face uncertainty about, e.g., the weather, natural resources and their locations, technological progress, labor, prices at external markets, etc. In this work we show that when the assumptions of the model are loosened to include uncertainty, the qualitative results are not radically different. We assume that there is uncertainty about the production possibilities and the (exogenous) supply of resources which are essential to production. This uncertainty, at each date, is represented by the occurrence of some ‘state of the environment’ which is independent of the planner’s decisions. Also these uncertainties arise similarly in successive planning periods. The set of all possible ‘states’, at each period, is a complete separable metric space. Hence if the ‘state’ is, for instance, a vector in an Euclidean space, then each component describes the situation of some element which effects production possibilities. The planner’s decision on consumption and investment in production, at each epoch, is done in a way that will maximize the expected sum of discounted utilities. Suppose that a certain program is optimal according to the planner’s evaluation: Is there a price system such that decentralized producers, which maximize their profits, will produce this program, and decentralized consumers, which