On optimal steady states of n-sector growth models when utility is discounted

This paper contains results on local and global stability of n-sector growth models when utility is discounted mostly for small rates of discount. It is well known that when future utility is not discounted one can prove precise results about optimal steady states (OSS's) under fairly general assumptions. In particular, existence, uniqueness, and turnpike properties have been established by several authors. The counter examples presented by Kurz, Sutherland, and Weitzman, however, show that when utility is discounted, additional assumptions are required to obtain turnpike results. In general, it would be interesting to know how the submanifolds of stability change as δ changes. One hopes that certain conditions on the utility function would be sufficient to “classify” the submanifolds of stability and instability. Such a question is apparently very difficult to answer, but we think that the results obtained here will help in this task.The proof that the turnpike theorem holds for discount factors near one is divided in two parts. First, we prove that optimal paths “visit” neighborhoods of the modified OSS's. Then, we prove that local stability holds for such neighborhoods.In order to show this fact, we must prove that the local “stable manifold” varies continuously with the discount factor. This roundabout method is necessary since our problem is similar to proving uniform continuity with respect to a parameter of solutions of a differential equation in a noncompact interval of time.Other problems analyzed here include uniqueness and continuity of OSS's. We also discuss the relation between the saddle-point property and the local stability of infinite horizon optimal growth paths.