Rational Inattention When Decisions Take Time

Decisions take time, and the time taken to reach a decision is likely to be informative about the cost of more precise judgments. We formalize this insight using a dynamic model of optimal evidence accumulation. We provide conditions under which the resulting belief dynamics resemble either diffusion processes or processes with large jumps. We then consider the in which discounting effects are small relative to the opportunity cost of time, and show that the state-contingent choice probabilities predicted by our model are identical to those predicted by a static rational inattention model, providing a micro-foundation for such models. In the diffusion case, our model provides a normative foundation for a variant of the drift-diffusion model from mathematical psychology.