Rational Inattention with Continuous Time

We derive, from first principles, continuous time models of rational inattention. We begin by describing dynamic discrete time models, and stating conditions on the cost of information in such models. Using these conditions, we characterize the cost of a small amount of information, which allows us to derive tractable and parsimonious models for the continuous time limit. We then provide conditions under which the resulting belief dynamics will resemble diffusion processes, and when the beliefs will involve large jumps. There is an extensive literature in psychology, neuroscience, and behavioral economics on continuous time models of belief dynamics; our results can be viewed as a sort of micro-foundation for models in this literature.