Optimal growth and impatience: A phase diagram analysis

When the time preference exhibits decreasing marginal impatience, the rate at which the discount function decreases plays a central role in the stability analysis. This paper shows that if the discount function is strictly decreasing, strictly convex and has a uniform bound on its first derivative, then the continuous-time, one-sector optimal growth problem has a unique steady state that exhibits the saddle-point property. Moreover, the phase diagram analysis is geometrically similar to the case of a constant discount rate. The proposed bounded slope assumption is inspired by and in direct contrast with Magill and Nishimura’s “division of countries” example.