A Clarification and a New Proof of the Certainty Equivalence Theorem
Alan I. Duchan · International Economic Review · 1974
SIMON AND THEIL'S CERTAINTY EQUIVALENCE THEOREM [9, 10] states that for a certain class of stochastic control models, the optimal first period decision can be obtained by replacing all stochastic variables by their expected values and then finding the optimal decision for the resulting deterministic model. One of the assumptions needed for the theorem to hold is that the decision maker can not affect the probability distribution of the stochastic elements in the model. In the literature on the theorem, attempts at making this rather informal statement more precise have been misleading. Because the theorem has been widely used in applied studies and has stimulated many theoretical papers, it is important that the assumption in question be clarified. The main purpose of this paper is to show that the most recent formal statement of the assumption [12, (130)] is stronger than needed. Indeed, were the conditions given in [12] necessary ones, many applied studies which use the theorem would be in error (e.g., [2, 3]). In order to demonstrate that the certainty equivalence theorem holds under a weaker condition than the one given in [12], we prove the theorem by a method different from Simon and Theil's. Besides showing exactly what formal assumption is needed, our proof serves another purpose-out of it falls a computationally convenient expression for the optimal decision. In the next section, we review the Theil-Simon model after which we look at Theil's discussion of the above assumption. A new, formal statement of the assumption is presented in Section 3 together with our proof of the certainty equivalence theorem. Finally, we compare, from the point of view of computational ease and generality, various representations of the optimal strategy.