Time- and State-Dependent Pricing: A Unied Framework

We develop a tractable uni…ed framework for solving optimal timeand 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 of information is freely available and ‡ows continuously, while the other type is costly and requires the payment of a lump-sum cost to be obtained. Our choice of state variables is key to make the problem tractable. Speci…cally, we replace the usual state variable in state-dependent pricing problems the discrepancy between the …rm’s price and its frictionless optimal level with its expectation conditional on the …rm’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 the price-setting problem as two-dimensional optimal stopping problems. 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 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. JEL classi…cation codes: E00, E31 Keywords: menu costs, information costs, infrequent information, sticky information, inattentiveness, optimal price setting, state-dependent pricing, time-dependent pricing An earlier working paper version circulated under the title “State-Dependent Pricing under Infrequent Information: A Uni…ed Framework”. For comments and suggestions we thank Mike Golosov, Virgiliu Midrigan, and seminar participants at ESEM 2011, Advances in Macroeconomics Workshop (FGV-Rio), ESWM 2011, SBE 2010, LACEA 2010, INSPER-SP. Emails: [email protected], [email protected], [email protected].