We study the general problem of Bayesian persuasion (optimal information design) with continuous actions and continuous state space in arbitrary dimensions. First, we show that with a finite signal space, the optimal information design is always given by a partition. Second, we take the limit of...

We study the general problem of information design for a policymaker—a central bank—that communicates its private information (the ``state") to the public. We show that it is optimal for the policymaker to partition the state space into a finite number of ``clusters” and to communicate to...