Sales and Price Guarantees under Markovian Pricing

Abstract

Firms often vary product prices over time to price discriminate customers. In response, customers may delay purchases to obtain the product at a more favorable price. We consider a model in which a firm interacts with short-lived customers over an infinite time horizon. Customers differ in their valuations and lifetime. A proportion (type I customers) leave immediately with or without a purchase, whereas the remainder (type II customers) are willing to wait for a time that is exponentially distributed before making a purchase or leaving. The firm adopts a Markovian pricing strategy. We show that the optimal pricing policy is either static pricing or high/low pricing with flash sales, where the firm charges a high price all the time, except for occasional price drops. Moreover, customer heterogeneity affects the profitability of Markovian pricing. Specifically, when type II customers are more likely to have low valuations, the firm is more likely to offer high/low pricing. To mitigate customers' waiting behavior, some firms offer price guarantees that refund customers the price difference in the event of a markdown within a given time after product purchase. We show that offering price guarantees is not optimal when all customers take advantage of them. When only type II customers take advantage, offering price guarantees can improve the firm's profit because it enables price discrimination between high- and low-valuation type II customers. Moreover, the firm offers sale prices less often under price guarantees. Perhaps surprisingly, offering price guarantees can decrease the aggregate customer surplus.