Discounting and Patience in Optimal Stopping and Control Problems

The optimal stopping time of any pure stopping problem with nonnegative termination value is increasing in “patience,” understood as a partial ordering of discount functions. When utility depends on some Markov state controlled by the agent, the result holds in a “continuation–domain” sense. When utility is increasing in the state, and the state is a one–dimensional diffusion, the result also holds in a “time-domain” sense, and the entire state path is increasing in patience. In all cases, the expected value of the agent is increasing in patience. We extend our concept of patience to study the impact of stochastic discounting and of time–inconsistency on the results and identify, as a by-product, new properties of these environments. For example, we find that a naive agent always experiments longer than a sophisticated one, and introduce an ordering on “partial sophistication.” As an application, the internal rate of return of any endogenously-interrupted project is essentially unique, even when cash flows are stochastic and controlled by the entrepreneur. Other applications include growth theory, bankruptcy decisions, environmental protection, and experimentation. ∗Email addresses: [email protected] and [email protected]. We are grateful for comments from Chiaki Hara, Paul Milgrom, Jean Tirole, and seminar participants at the World Congress of the Econometric Society (2010).