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 illustrate our approach by solving a price-setting problem where adjustments are costly, and there are two types of information. One type is freely and continuously available while the other is costly and requires the payment of a lump-sum cost to be observed. Our choice of state variables is key to make the problem tractable. Specifically, we replace the usual state variable in state-dependent problems - the discrepancy between the firm's price and its frictionless optimal level - with its conditional expectation on the firm's information set, and augment the state space with the time elapsed since the last date information was fully factored into the pricing decision ("information date"). This allows us to formulate the price-setting problem as a two-dimensional optimal stopping problem. Our analysis uncovers new insights about price setting. Time dependency in pricing rules arises as a consequence of the build-up of unobserved information. In these circumstances, the inaction range 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 produce an extreme form of inaction: irrespective of the size of the expected price discrepancy, it is never optimal to adjust prior to the information date.