Optimizing Service Operations With Price- and Density-Dependent Demand: A Copula-Based Approach

Exclusivity is a key component of certain service offerings such as elite members-only social or country clubs. Indeed, such services may only appeal to customers if they limit the crowd density, that is, the maximal number of customers granted access to the service. We consider a service provider, who sets its crowd density cap in addition to its price. Customers are heterogeneous in both their willingness to pay (WTP) for the service and their density tolerance. Each customer purchases the service only if the price is below her WTP and the density is below her tolerance. Importantly, customers’ WTP and density tolerance may be statistically dependent, and we develop a novel, copula-based framework to model the dependence structure of these two dimensions of heterogeneity. We analytically characterize the provider’s optimal price and density decisions. As long as the customer population is not too severely sensitive to density: (i) the provider optimally serves all segments of the market and sets the price and density so that the price, not the density, is the determining factor for customers (from a particular subpopulation) on the margin between purchasing and not and (ii) as customers’ valuations and density tolerances become more positively dependent, the provider earns higher revenue and optimally increases its service density. By contrast, the optimal price is quasi-convex but not necessarily increasing. On the other hand, with severely density-sensitive customers, it may be optimal to partially cover the market and to make density the determining factor in customer decisions. Our findings offer prescriptive guidelines for service operations in the presence of density-sensitive demand and also provide an explanation for the failed versus successful practices of prominent private clubs.