Investment Timing with Incomplete Information and Multiple Means of Learning

Abstract

We consider a firm that can use one of several costly learning modes to dynamically reduce uncertainty about the unknown value of a project. Each learning mode incurs cost at a particular rate and provides information of a particular quality. In addition to dynamic decisions about its learning mode, the firm must decide when to stop learning and either invest or abandon the project. Using a continuous-time Bayesian framework, and assuming a binary prior distribution for the project’s unknown value, we solve both the discounted and undiscounted versions of this problem. In the undiscounted case, the optimal learning policy is to choose the mode that has the smallest cost per signal quality. When the discount rate is strictly positive, we prove that an optimal learning and investment policy can be summarized by a small number of critical values, and the firm only uses learning modes that lie on a certain convex envelope in cost-rate-versus-signal-quality space. We extend our analysis to consider a firm that can choose multiple learning modes simultaneously, which requires the analysis of both investment timing and dynamic subset selection decisions. We solve both the discounted and undiscounted versions of this problem, and explicitly identify sets of learning modes that are used under the optimal policy.

Keywords: optimal control, optimal stopping, subset selection, Bayesian sequential hypothesis testing, costly learning

JEL Classification: C12, D81, D83