A "quantized" approach to rational inattention

In this paper, I propose a model of rational inattention where the choice variable is a deterministic function of the exogenous variables, and still only a finite amount of information is being used. This holds provided the choice variable is discrete rather than continuous; that is, the mapping from the realization of the exogenous variables to the endogenous ones is piece-wise constant. Thus, limited information is now a source of lumpiness in behavior, rather than a source of noise. A central result is that the mutual information between the exogenous variable and the endogenous one is simply equal to the entropy, in the usual discrete sense, of the endogenous variable. The approach is illustrated with two applications: a general linear-quadratic problem with a uniform distribution, and a simple static model of price-setting where individual price setters face aggregate monetary shocks and idiosyncratic productivity shocks.