On characterizing optimality of stochastic competitive processes
In the literature on the theory of intertemporal resource allocation it is known that optimality for infinite-horizon economies can be characterized in terms of two conditions; the first requires intertemporal profit maximization and utility maximization relative to a system of ``competitive'' prices, while the second is a transversality condition, which requires (for undiscounted models) that the value of the capital stock, computed at the competitive prices, be uniformly bounded over time. These results, for stochastic economies, have been proved by \textit{I. Zilcha} [Rev. Econ. Stud. 43, 431-438 (1976; Zbl 0401.90017) and Econometrica 46, 515-525 (1978; Zbl 0387.90025)]. There are two problems with the above characterization of optimality. The first is that the transversality condition involves a limit, so one cannot verify on a period-by-period basis whether or not this condition is being attained. I shall call this the absence of temporal decentralization. The second problem is the absence of informational decentralization, in the sense of \textit{L. Hurwicz} [Definition 10, p. 401, in \textit{K. Arrow} and \textit{L. Hurwicz} (eds.), ``Studies in Resource Allocation Processes'', Cambridge Univ. Press, London (1977)]. In particular, one cannot design a meaningful resource allocation mechanism that will ensure that the transversality condition will be met, but which is constrained so that the rules of behavior of agents at each date depend only upon the partial history at that date. \textit{M. Majumdar} [J. Econ. Theory 45, No.2, 217-227 (1988; Zbl 0647.90017)] provides a condition to replace the transversality condition, which is both temporally and informationally decentralized. The purpose of this paper is to extend their result to handle stochastic economies; in particular to a model of intertemporal resource allocation under uncertainty. The main result of this paper may be described as follows: let (\^x,ŷ,ĉ) be the optimal stationary process and \(\{\) \(\hat p_ t\}\) the corresponding competitive (or supporting) price process (essentially the marginal utilities of consumption). Then a resource allocation process (x,y,c) is optimal if and only if (a) the process is competitive at some prices \(\{p_ t\}\) and (b) at each date t, \(E(p_ t-\hat p_ t)(y_ t-\hat y_ t)\leq 0\).
scientific article; zbMATH DE number 4087385