OPERATIONS RESEARCH Vol. 56, No. 3, May–June 2008, pp. 630–645 issn 0030-364X  eissn 1526-5463  08  5603  0630 inf orms ® doi 10.1287/opre.1080.0541 ©2008 INFORMS A Static Approximation for Dynamic Demand Substitution with Applications in a Competitive Market Wallace J. Hopp Stephen M. Ross School of Business, University of Michigan, Ann Arbor, Michigan 48109, whopp@umich.edu Xiaowei Xu Department of Supply Chain Management and Marketing Sciences, Rutgers, The State University of New Jersey, Newark, New Jersey 07102, xiaoweix@andromeda.rutgers.edu We propose a static approximation of dynamic demand substitution behavior based on a fluid network model and a service- inventory mapping. This approximation greatly enhances our ability to analyze the interdependent inventory/service, price, and product assortment decisions in noncompetitive and competitive scenarios with demand substitution. We demonstrate that the approximation is well behaved and then apply it to two previously intractable applications. First, we study a price and service competition between single-product retailers. After establishing a unique pure-strategy Nash equilibrium, we find that competition results in lower price, higher demand, and a higher level of inventory. We also observe that the aggregate profit and inventory level increase to positive constants as the number of retailers goes to infinity. Second, we study a duopolistic competition on price, service, and product assortment. We establish a pure-strategy Nash equilibrium for the product assortment competition and identify a condition for uniqueness. We find that competition on both price and product assortment results in lower price and less variety for each competitor, but the total number of products and the aggregate inventory level in a duopoly market are both likely to be higher than in a monopolistic market. Subject classifications : competition; demand substitution; logit demand; pricing; product assortment; service. Area of review : Manufacturing, Service, and Supply Chain Operations. History : Received June 2005; revision received July 2006; accepted July 2006. 1. Introduction and Literature Review Avoiding stockouts is a central concern of inventory man- agement because running out of stock negatively affects the revenue of retailers and manufacturers and can erode customer loyalty to stores and brands. However, despite the large investment in information systems and inventory- tracking technology that has taken place over the past two decades, the worldwide out-of-stock rate remains a surpris- inglyhigh8.3%(Gruenetal.2002).Stockoutsinducesome customers to change their purchase decisions. Gruen et al. (2002) surveyed more than 71,000 customers worldwide and found that 26% of customers encountering a stockout situation substitute brand, 19% substitute another product of the same brand, 31% buy the same product at another store, and 24% choose to delay or cancel their purchase. Because of its prevalence, customer demand substitu- tion is an important consideration for retailer inventory management, pricing, and category management strategies. Unfortunately, it is extremely difficult to accurately model demand substitution behavior, and hence the impact of stockouts on inventory replenishment, price competition, and product assortment is not well understood. In this paper, we provide an approximation of customer demand substitution behavior that greatly facilitates the embed- ding of stockout considerations in inventory, pricing, and product assortment models. Demand substitution arises in two basic forms: firm- driven substitution and customer-driven substitution. Firm- driven substitution usually involves vertically differentiated products/components and often occurs in multiproduct manufacturing systems (see, e.g., Bassok et al. 1999 and Bitran and Dasu 1992). For example, a manufacturer may fill demand for a low-quality product with a high-quality product without charging a higher price. This substitution does not generate customer complaints when all customers agree with a ranking of product quality (e.g., 1G mem- ory chips are better than 512M memory chips). In con- trast to firm-driven substitution, customer-driven substi- tution involves a compensation scheme (e.g., charging a higher price) when a customer switches from one prod- uct to another, and hence does not require a monotonic ranking of products by heterogeneous customer percep- tions. For example, a customer who purchases the higher- priced product when the lower-priced one is out of stock ispracticingcustomer-drivensubstitution.Anotherexample occurs when heterogeneous customers select from a hori- zontally differentiated product line. Because a seller has no 630