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Article Excerpt
1. Introduction
Pricing is one of the most fundamental but also most difficult decisions that firms have to make. An important reason is that businesses today are facing a generation of increasingly sophisticated customers. Whether firms adopt the most powerful category pricing software or the most data-intensive revenue management system, customers are becoming extremely adept at finding the best deals. Readers of this article may even have some personal experiences that they are proud to share. According to SmartMoney magazine, there is constantly a "cat-and-mouse game between retailers, who hope to charge full price for everything, and shoppers, who wait for a sale" (see Kadet 2004, pp. 90-91). The zero-sum nature of pricing makes this inevitable. Firms are constantly improving their pricing strategies to collect as much revenue as possible, and customers are constantly modifying their purchase plans to pay as little as possible.
Recently, a major electronics retailer, Best Buy, has expressed some strong opinions about this kind of strategic customer behavior. In an article that appeared on the front page of the Wall Street Journal, Best Buy Chief Executive Officer Brad Anderson openly labels some customers as "devils" (see McWilliams 2004). According to Anderson, these are the customers who wait for markdowns, respond to promotions, and apply for rebates. In contrast, Anderson also describes the "angels" as the customers who snap up high-end gadgets without hesitation. Best Buy estimates that approximately 100 million out of its 500 million customer visits each year are "undesirable." Although these pejorative labels have attracted criticism (see Queenan 2005), Best Buy has implemented customer relationship management programs to better distinguish the "angels" from the "devils" (see Arndorfer and Creamer 2005). Apart from Best Buy, many other firms are also beginning to recognize that revenues are lost when customers wait for sales. Retailers, such as Bloomingdale's, Ann Taylor, Gap, and Home Depot, are turning to price optimization software instead of blindly slashing prices toward the end of the selling season (see Schlosser 2004).
Although the importance of strategic customer behavior is recognized by many, its implications on intertemporal pricing strategies have not been widely studied. Therefore, in this paper, we set out with two main research questions. First, we would like to understand the interplay between the firm's pricing strategies and customers' purchasing strategies. In this environment, should prices increase or decrease (or stay fixed) over time? What is the optimal timing and extent of the markups and/or markdowns? For the customers, how should they determine whether to wait for a sale, if the product may be sold out by then? We would like to model and understand these strategic interactions. Next, our second research question concerns the impact of strategic customer behavior. When customers delay purchases and wait for sales, does this behavior hurt or benefit the seller, and in what ways do these effects arise?
In our model, there is a monopolist who sells a finite inventory over a finite time horizon. The seller may charge different prices over time, and may also practice rationing by fulfilling only a portion of market demand. There is a continuous inflow of customers arriving into the market. If they are unwilling to purchase the product immediately, they may leave the system, or may wait for more attractive purchase opportunities in the future. Although prices may fall, there is also the possibility that the product will become unavailable. Furthermore, customers incur waiting costs. Within this environment, the seller seeks to maximize revenue, and customers wish to maximize individual utility.
Heterogeneity plays a key role in our model. We allow the customers to vary along two dimensions: they may have different valuations for the firm's product and may have different degrees of patience (i.e., different waiting costs). Heterogeneous valuations imply that dynamic pricing is worthwhile because there is an opportunity to practice intertemporal price discrimination. Heterogenous waiting costs allow us to capture a wide variety of customer behavior. At one extreme, when waiting costs are infinitely large, customers are myopic and make a one-time buy-or-exit decision upon arrival; on the other hand, customers with finite waiting costs may delay their purchases strategically. Existing models consider customer populations that are either purely myopic or purely strategic, whereas we allow for arbitrary combinations of both. With these two dimensions of heterogeneity, we believe that we can capture a good representation of reality.
This paper generates two novel insights. First, we demonstrate that customer heterogeneity is a main driver determining the structure of optimal pricing policies. In particular, the two dimensions of heterogeneity (valuation and patience) jointly determine whether prices should increase or decrease over time. Markdowns such as end-of-season sales are preferred when low-valuation customers are sufficiently patient to wait for sales, while high-valuation customers are sufficiently impatient to buy early at higher prices. On the other hand, markups are preferred when high-valuation customers are more patient than low-valuation customers: this discourages inefficient waiting and also captures surplus from high-valuation customers who miss the initial low prices. In this way, our analysis endogenizes the structure of pricing policies and captures both markups and markdowns within a single framework.
Another managerial insight from this work is that strategic customer behavior may actually benefit the seller. This finding counters common intuition that customers who practice intertemporal substitution hurt seller revenues when they are able to find lower prices. While this is true, we point out another effect that to the best of our knowledge has not been identified before. That is, in an environment of scarcity (i.e., finite inventory), low-valuation customers who wait for discounts end up competing for availability with their high-valuation counterparts. This threat of stock-outs discourages the high-valuation customers from waiting and also increases their willingness to pay. Therefore, scarcity and heterogeneity interacts in a subtle way that puts strategic customer behavior in a new light.
The remainder of this paper is organized as follows. Section 2 provides a literature review. Section 3 describes the model. Section 4 develops some structural properties, uses them to characterize an upper bound on seller revenues, and shows that this upper bound can be attained. Section 5 presents the main results, characterizing the seller's optimal policy. Section 6 examines the seller's revenue, consumer surplus, and social welfare under the optimal regime. The long-run problem of selecting an optimal initial stocking quantity is analyzed in [section]7. Finally, [section]8 offers concluding remarks. All proofs are presented in the online appendix (provided in the e-companion). (1)
2. Literature Review
The three main issues examined in this paper are: (i) strategic customer behavior, (ii) price dynamics, and (iii) limited capacity. There are several streams of related literature, each addressing different subsets of these issues.
The revenue management literature on dynamic pricing of finite inventories is closely related to our work. This stream of papers focuses on price dynamics and limited capacity (the primary question is how to set prices as a function of remaining inventory). However, strategic customer behavior is absent from the earlier models. The first papers were by Gallego and van Ryzin (1994, 1997). They model customer arrivals using Poisson processes and formulate the dynamic pricing problem as an intensity control problem; their 1997 paper generalizes the basic model to a network (multiproduct) setting. Federgruen and Heching (1999) combine pricing with inventory decisions. Feng and Gallego (1995, 2000) make a practically justified restriction: they consider a discrete menu of prices and policies involving at most one price change. Feng and Xiao (2000a, b) extend this to policies involving multiple and reversible (nonmonotonic) price changes. Similarly, Bitran and Mondschein (1997) consider periodic pricing policies that modify prices only at pre-specified times. Zhao and Zheng (2000) study dynamic pricing in more general situations with time inhomogeneous customer arrivals. For surveys of this literature, readers are referred to Bitran and Caldentey (2003), Elmaghraby and Keskinocak (2003), and McAfee and te Velde (2005). For a comprehensive coverage of revenue management, readers are referred to the book by Talluri and van Ryzin (2005). A common approach in this literature is to determine optimal prices dynamically by considering the option value of unsold units. The result is that optimal price paths are decreasing over time (on average) because the option value of unsold units decreases as the deadline approaches. However, this result requires the assumption that demand is exogenous and independent across time. This no longer applies in the current work because strategic purchase delays in our model imply that demand may spill over into the future. As a result, we obtain optimal price schedules that may both increase or decrease over time.
Recent papers in revenue management have begun to examine customer behavior more closely. However, the focus is on how customers choose between substitute products offered by the firm (rather than on intertemporal demand substitution). Talluri and van Ryzin (2004) use discrete choice models to describe how customers, in the context of airlines, choose among the set of fare classes offered; van Ryzin and Liu (2004) extends this analysis to the network setting. Shumsky and Zhang (2004) consider demand substitution, via upgrading, when inventory has been depleted. Netessine et al. (2006) consider cross-selling (i.e., offering customers a choice between their requested product and a package containing the requested product as well as another product). Cooper et al. (2006) show that neglecting substitution across products can lead to a spiral-down effect, in which the capacity allocation policy systematically performs worse and worse as the forecasting-optimization process continues. Zhang and Cooper (2005a) analyze a capacity allocation model with customer choice over parallel flights, and they extend this analysis (Zhang and Cooper 2005b) to incorporate pricing decisions. Maglaras and Meissner (2006) show that the dynamic pricing problem when customers choose between multiple products can be reduced to an equivalent one-dimensional problem. In all these papers, customers choose what to buy, whereas in our work, customers choose when to buy. Both aspects of strategic customer behavior are important, and they are addressed using different modeling techniques.
The papers on dynamic pricing of finite inventories that are most closely related to ours involve intertemporal demand. These papers explicitly model customers' decisions regarding when to buy. Aviv and Pazgal (2003) assume that there is a single price reduction at a fixed time. Elmaghraby et al. (2004) also focus on markdown mechanisms, and customers with multi-unit demands choose how many units to purchase at each price step. These two papers explicitly assume that prices should decrease over time. On the contrary, Gallien (2006) demonstrates that optimal prices should increase over time, but his results apply to the infinite-horizon case; for the simpler case where customers do not wait, Arnold and Lippman (2001) and Das Varma and Vettas (2001) obtain similar conclusions. Unlike these papers, we consider a finite horizon, and we find that markups and markdowns may be optimal under different situations.
Instead of pricing decisions, Liu and van Ryzin (2005) consider quantity decisions in a two-period capacity rationing model with strategic customers. They assume that the seller pre-commits to prices in both periods. In another paper, Xu and Hopp (2004) use a martingale approach to endogenize strategic customer...
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