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 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 demonstrate 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.