MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 13, No. 1, Winter 2011, pp. 89–107 issn 1523-4614  eissn 1526-5498  11  1301  0089 inf orms ® doi 10.1287/msom.1100.0306 ©2011 INFORMS Strategic Capacity Rationing when Customers Learn Qian Liu Industrial Engineering and Logistics Management Department, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, qianliu@ust.hk Garrett van Ryzin Graduate School of Business, Columbia University, New York, New York 10027, gjv1@columbia.edu C onsider a firm that sells products over repeated seasons, each of which includes a full-price period and a markdownperiod.Thefirmmaydeliberatelyunderstockproductsinthemarkdownperiodtoinducehigh- value customers to purchase early at full price. Customers cannot perfectly anticipate availability. Instead, they use observed past capacities to form capacity expectations according to a heuristic smoothing rule. Based on theirexpectationsofcapacity,customersdecidetobuyeitherinthefull-priceperiodorinthemarkdownperiod. We embed this customer learning process in a dynamic program of the firm’s capacity choices over time. One mainresultdemonstratestheexistenceofamonotoneoptimalpathofcustomers’expectations,whichconverges toeitherarationingequilibriumoralow-price-onlyequilibrium.Further,thereexistsacriticalvalueofcapacity expectation such that the market converges to a rationing equilibrium if customers’ initial expectations are less thanthatcriticalvalue;otherwise,alow-price-onlyequilibriumisthelimitingoutcome.Theseresultsshowhow firms can be stuck with unprofitable selling strategies from incumbent customer expectations. We also examine numerically how this critical value is affected by the firm’s discount factor and customers’ learning speed and risk aversion. Last, we show that the equilibrium under adaptive learning converges to that under rational expectations as the firm’s discount factor approaches one. Key words : consumer behavior; pricing and revenue management; dynamic programming History : Received: September 4, 2007; accepted: May 1, 2010. Published online in Articles in Advance October 13, 2010. 1. Introduction Faced with dynamic pricing, customers have an incentive to strategize the timing of their purchases and attempt to buy only when prices are low. One way for a firm to thwart this behavior is to delib- erately create shortages to induce customers to buy early at higher prices. An earlier paper (Liu and van Ryzin 2008) addresses how a firm should opti- mallybalancethebenefitofcapacityrationingagainst theopportunitycostoflostsales.Itshowsthatalarge high-value customer segment, large price differences overtime,andriskaversionamongcustomersalltend to make rationing an optimal strategy. A key assumption made in Liu and van Ryzin (2008) is that customers have rational expectations 1 about the firm’s capacity choice; that is, customers perfectly anticipate availability. The concept that cus- tomers rationally predict future product availability has been adopted in several recent papers on strate- gic customer behavior, for example, Aviv and Pazgal 1 Noteweusetheterm rational expectations torefertothecasewhere customers’ expectations about the firm’s capacity are rational; that is, based on all available information and correctly anticipating the firm’s optimal stocking decision. Although it is related, we do not use the term in the sense it is used in macroeconomics. (2008),CachonandSwinney(2009),Elmaghrabyetal. (2008), Liu and van Ryzin (2008), Su (2007, 2008), and Jerath et al. (2007). Yet in reality, customers may only learn about a firm’s strategy through repeat experiences. This is the situation this paper explores; we drop the rational expectations assumption and assume instead that customers adaptively learn over time.Togiveaconcreteexampleofthisphenomenon, wehavehearddepartmentstoremanagerslamentthe fact that they have, in effect, “trained customers to buy on sale” as a result of a longstanding practice of frequent promotional sales and end-of-season mark- downs. Stores would like to regain the credibility of selling at full price but worry that without promo- tions or markdowns, customers may refuse to buy at all.Insuchasituation,shouldtheyattempttochange customers’ expectations by restricting markdowns or the availability of goods? Or is it simply too difficult and costly to do so? What is the optimal response? Thesearethemainquestionsweaddressinthispaper. The fundamental question of how customers form expectationsofthefutureisaddressedbyadaptation- level theory; see Helson (1964), Sterman (1987), and Rubinstein (1998) for an overview. Experimen- tal and empirical evidence support the hypothesis 89