Cardinality Bundling with Spence–Mirrlees Reservation Prices

We study the pricing of cardinality bundles, where firms set prices that depend only on the size of the purchased bundle, a practice that is increasingly being adopted by industry. The model we study, where consumer choices are discrete, was originally proposed by Hitt and Chen [Hitt L, Chen P (2005) Bundling with customer self-selection: A simple approach to bundling low-marginal-cost goods. Management Sci. 51(10):1481–1493], and it requires that consumers’ preferences obey the Spence–Mirrlees single-crossing property. We correct prior approaches and develop various structural and managerial insights. We develop a fast combinatorial technique to obtain the optimal prices. We extend our analysis to address a quantity discount problem originally proposed in Spence [Spence M (1980) Multi-product quantity-dependent prices and profitability constraints. Rev. Econom. Stud. 47(5):821–841]. We provide examples that demonstrate that the proposed approach of Spence (1980) only identifies local optima without providing guidance on selecting the globally optimal pricing function. Our insights from the discrete model are extended to this context to develop a scheme that provides solutions within an arbitrary prespecified tolerance. Consequently, we also solve the continuous version of the cardinality bundling problem.