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This paper deals with stochastic models of an optimal sequential allocation of resources between consumption and production. We obtain the following results by means of the theory of martingales. For a model of resource allocation we establish the dynamic programming equation and show that a supermartingale characterizes the composition of the model and clear the composition of the optimality, representing the sufficient condition for an allocation to be optimal via a martingale. Further, we show the following three results as the application of the above result. (I) For the Kennedy's model we give another proof for the fact on an optimal allocation. (2) For a model with a convex utility function, we represent an optimal allocation via a supermartingale. (3) For a model with a logarithmic utility function, we obtain an explicit optimal allocation. 1. I ntroduct ion In this paper, we consider stochastic models of resource allocation. Resource is sequentially allocated for consumption and production, and utility and resource are sequentially generated. We want to obtain the maximum sum of the expected utilities. The model of resource allocation has been investigated by Beckmann [1], Dynkin and Yushkevich [3], Foldes [4], and Kennedy [5]. The model in section 2 is an extension of the Kennedy's model [5] in which the utility function is a special form. Kennedy's model that we treat in section 3 is an extension of the model in [3]. The model described in [3] is the stochastic version of the deterministic model given by Beckmann [1]. Foldes [4] has obtained necessary and sufficient conditions for an allocation to be optimal under the assumption that the utility function is concave and increasing. © 1989 The Operations Research Society of Japan

AN ANALYSIS OF A STOCHASTIC RESOURCE ALLOCATION MODEL WITH VARIOUS UTILITY FUNCTIONS