Uzawa’s transformation and optimal control problems with variable rates of time preference

Uzawa (1968) first introduced a simple and appealing method for reducing problems with variable rates of time preference to single-state systems by transforming the time scale from t to Δ, a utility discount factor. Thistransformation has been used extensively, particularly in models of internationaltrade and finance (e.g., Obstfeld, 1981a, 1981b, 1982, Engel and Kletzer, 1989, and Turnovsky, 1997), where the use of a variable rate of time preference avoids some of the “disturbing implications” drawn from typical open-economy Ramsey models. The purpose of this paper, however, is toshow that Uzawa’s transformation is valid only when the underlying system to be analyzed is autonomous. Unfortunately, except for the simplest control problems, this is rarely the case. In particular, systems with nonautonomous transition equations imply that the correspondence between Δ and t is no longer unique, and thus Uzawa’s transformation is not applicable.