We analyze a problem of revealed preference given state-dependent stochastic choice data in which the payoff to a decision maker (DM) only depends on their beliefs about posterior means. Often, the DM must also learn about or pay attention to the state; in applied work on this subject, a convenient assumption is that the costs of such learning are linearly dependent in the distribution over posterior means:

A seller offers a buyer a schedule of transfers and associated product qualities. After observing this schedule, the buyer chooses a flexible costly signal about his type. We show it is without loss to focus on a class of mechanisms that compensate the buyer for his learning costs. Focusing attention on these mechanisms allows one to employ techniques from both the information design and mechanism design toolkits.