Time- and State-Dependent Pricing : A Unified Framework

We develop a tractable unified framework for solving optimal time- and state-dependent price-setting problems. We use it to study models with adjustment costs and infrequent information, entertaining various alternatives for the source and nature of infrequent information. In particular, we consider both models where information is infrequent for exogenous reasons and models where firms face a lump-sum cost to gather and process information. Our choice of state variables is key to make the problems tractable. Specifically, we replace the usual state variable in state-dependent pricing problems – the discrepancy between the firm's price and its frictionless optimal level – with its expectation conditional on the firm's information set, and augment the state space with the time elapsed since the last date when information was fully factored into the pricing decision ("information date"). This allows us to formulate all price-setting problems as two-dimensional optimal stopping problems. Our analysis uncovers new insights about price setting. Despite the presence of menu costs, firms may choose to change prices in the absence of new information. Time dependency in pricing rules arises as a consequence of the build up of unobserved information. In these circumstances, the inaction region changes as a function of the time elapsed since the last information date. When the next information date is known, the presence of menu costs produces an extreme form of inaction: irrespective of the size of the expected price discrepancy, it is never optimal to adjust just prior to the information date