On the Role of Separability Assumptions in Determining Impatience Implications

Abstract

The impatience implications of continuous time utility indicators are interesting to the extent that they differ from the discrete time results. The class of tFaditional integral utility indicators are considered and impatience implications are shown to depend on the dif- ferent convergence implications of the continuous time case. The stronger separability assumptions of continuous time utility indicators allow a weakening of compactness assumptions often required to demonstrate impatience. presence of impatience. Specific separability assumptions were invoked by Koopmans (8) and Koopmans, Diamond, and Williamson (9) in order to demonstrate the presence of impatience in problems involving choice over an infinite program horizon. From a paper by Diamond (4) one,can infer much of the relationship between separability assumptions and impatience implications. Diamond employed several intertemporal non-complementary assumptions to demonstrate eventual impatience for a case in which the consumption space was not compact in the topology of the norm. The use of non-complementary axioms seems justifijable as their economic implications are straightforward while those of compactness assumptions are not immediately obvious.2 Moreover the natural extension of Diamond's first axiom to all time periods yields a condition equivalent to the independence assumption employed by Debreu (3) in representing preferences by an additive function. Consequently, this paper analyzes separable utility indicators directly for impatience implications; the analysis considers the continuous time case as it subsumes the discrete time analog. However the discrete time case will be discussed in order to facilitate analogy construction.