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Technical Note—Price Trends in a Dynamic Pricing Model with Heterogeneous Customers: A Martingale Perspective

Department of Supply Chain Management and Marketing Sciences, Rutgers, The State University of New Jersey, Newark, New Jersey 07102
Stephen M. Ross School of Business, University of Michigan, Ann Arbor, Michigan 48109
xiaoweix{at}andromeda.rutgers.edu
whopp{at}umich.edu

This note describes probabilistic properties of optimal price sample paths in a dynamic pricing model with a finite horizon and limited stock. We assume that customer arrivals follow a nonhomogeneous Poisson process. We show that if customers' willingness-to-pay increases rapidly over time, then the optimal price process follows a submartingale, which implies an upward price trend. Alternatively, if customers' willingness-to-pay decreases rapidly over time, then the optimal price process follows a supermartingale, which implies a downward price trend.

Subject classifications: probability; stochastic model applications.
History: Received June 2006; revision received December 2008; accepted February 2009.