JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica. This paper treats dynamic optimization problems from the point of view of programming in infinite dimensional spaces. Sufficient conditions (of a dominant diagonal nature) for smooth dependence of optimal paths on initial conditions and other parameters are given. It is also shown that the smooth dependence and the so called "turnpike property" are very closely related. This relationship is used to prove a turnpike theorem for a model that may be interpreted as a multisectoral model of optimal growth with positive discounting and to show that the turnpike property is kept under "small" perturbations.
INTRODUCTION
SINCE FRANK RAMSEY'S 1928 ARTICLE [14], many dynamic economic problems have been stated in terms of maximization of a discounted sum of stationary utility subject to stationary constraints. Among the applications of this type of model are the study of optimal programs of economic growth, of accumulation of capital stocks by firms, of optimal human capital accumulation, of optimal advertising, etc. A large abstract literature developed studying the existence, uniqueness, and asymptotic behavior of solutions. The study of the effect of changes in the parameters of the model on solutions has, however, been restricted either to very particular models or to stationary or "balanced" solutions.This "comparison of solutions" problem has been dealt with extensively in the context of static models, where one usually uses the implicit function theorem to show that solutions depend differentiably on the parameters. This technique depends on the existence of the inverse of a certain matrix. This existence problem can usually be solved by using certain curvature properties of the model. Furthermore, in certain instances one can sign the elements of the inverse of the relevant matrix to compare solutions.When dealing with a dynamic model, one can study the problem in a similar way, by considering it as one of "programming" in an infinite dimensional space a set-up that was introduced in economics by the pioneering work of Hurwicz [7]. A solution to (1.1) is just a sequence of real numbers {kt}r=0 with ko = ko. In many economic problems, it makes sense to assume that the optimal solution is bounded. Hence, if one considers the space of all bounded sequences B, one can restate (1.1) as max v(ko, k), ko = ko kGB where 00 v(ko, k) = 8 8tV(kt, kt+1), k = {kt}t2=1 t=O Here v takes bounded sequences in the real line. The space of bounded sequences can be given a "norm" Ilki |= suptlktl, where IktI denotes, say, the Euclidean norm in R , such that it makes sense to talk about smooth, i....
