Some Results on the Uniqueness of Steady States in Multisector Models of Optimum Growth When Future Utilities Are Discounted*

IT IS WELL KNOWN that in multisector models of optimal growth that optimal paths converge to a unique steady state when future utilities are not discounted. See Gale [8], McKenzie [18], and Brock [3] for results of this type. Then Sutherland [32], in the case when future utilities are discounted, produced examples of multiple steady states in Gale's [8] model. Thus the qualitative behavior of optimal growth when future utilities are discounted may be quite different than the Ramsey case when future utilities are not discounted. Also Kurz [11] and Liviatan and Samuelson [13] have produced multipLe equilibria in the discounted case by introducing wealth effects and joint production effects respectively. What we shall do here is give conditions for several multisector models that give uniqueness of steady states. In the case of no joint production and one primary factor of production we use a nonsubstitution theorem to determine relative prices and the choice of technique independently of the utility function. We then assume normality of the utility function and show that there is at most one steady state for that case. We examine a model with no joint production, no primary factor and no depreciation of capital. In this case we show that summability of the utility function and a type of "normality" condition on each production function yields uniqueness of steady state. New techniques have to be invented to handle this case. The techniques may be independently interesting. A very general model that handles cases of joint production is examined in Section 3. All that is required here is that the steady state equations can be written in the form G(k, o) = 0 where p is the discount. It turns out that if