Convex Processes and Hamiltonian Dynamical Systems

Abstract

Many economists have studied optimal growth models of the form

$$\matrix{{\max imize} \hfill & {f_0^\infty e^{ - \rho } U(k(t),\,\,z(t))dt} \hfill \cr {subject\,\,to} \hfill & {k(0) = k_0, \dot k(t) = z(t) - \gamma k(t),} \hfill \cr }$$

(1)

where K is a vector of capital goods, γ is the rate of depreciation, ρ is the discount rate, and U is a continuous concave utility function defined on a closed convex set D in which the pair (k,z) is constrained to lie. The theory of such problems is plagued by technical difficulties caused by the infinite time interval.\The optimality conditions are still not well understood, and there are serious questions about the existence of solutions and even the meaningfulness, in certain cases, of the expression being maximized.

Research sponsored in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under AFOSR grant number 77-0546 at the University of Washington, Seattle.