Identifying Present Bias from the Timing of Choices

A (partially naive) quasi-hyperbolic discounter repeatedly chooses whether to complete a task. Her net benefits of task completion are drawn independently between periods from a time-invariant distribution. We show that the probability of completing the task conditional on not having done so earlier increases towards the deadline. Conversely, we establish nonidentifiability by proving that for any time-preference parameters and any dataset with such (weakly increasing) task-completion probabilities, there exists a stationary payoff distribution that rationalizes the agent's behavior if she is either sophisticated or fully naive. Additionally, we provide sharp partial identification for the case of observable continuation values.