Procrastination, self-imposed deadlines and other commitment devices

In this paper we model a decision maker who must exert costly effort to complete a single task by a fixed deadline. Effort costs evolve stochastically in continuous time. The decision maker optimally waits to exert effort until costs are less than a given threshold, the solution to an optimal stopping time problem. We derive the solution to this model for three cases: (1) exponential decision makers, (2) Naive hyperbolic discounters and (3) sophisticated hyperbolic discounters. Absent deadlines, we show that sophisticated hyperbolic decision makers behave as if they were time consistent but instead have a smaller reward for completing the task, while naifs never complete the task. In the presence of deadlines, sophisticated decision makers may, counterintuitively, have a threshold which is decreasing as they approach the deadline. An extensive numerical study shows that, unlike exponential or naifs who always prefer longer deadlines, sophisticated decision makers will often self-impose a binding deadline as a form of commitment, while naive decision makers will not, and we show how this varies with changes in underlying cost, preference and self-control parameters.