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 in the context of a dynamic rational inattention (RI) model. Under standard conditions on the flow cost of information in our discrete-time model, we obtain a tractable model in the continuous-time limit. We next provide conditions under which the resulting belief dynamics resemble either diffusion processes or processes with large jumps. We then demonstrate that the state-contingent choice probabilities predicted by our model are identical to those predicted by a static RI model, providing a micro-foundation for such models. In the diffusion case, our model provides a normative foundation for a variant of the DDM models studied in mathematical psychology.