Nonlinear Pricing of Software with Local Demand Inelasticity

Abstract

Nonlinear usage-based pricing is applied extensively to price software products. Different from other products, software customers often cannot vary their required usage volume, a property we label local demand inelasticity. For instance, a client firm that needs a salesforce automation software either buys one user license for every salesperson in its organization or does not buy at all. It is unlikely to buy licenses for some salespersons, but not the others. This demand feature violates a critical assumption of the standard nonlinear pricing literature that consumers are flexible with their usage volume, and the utility that consumers derive from using the product changes smoothly with their usage volume. Consequently, standard pricing solutions are inapplicable to many software products. This paper studies the optimal nonlinear usage-based pricing of software when customers demand is locally inelastic. We demonstrate that this unique demand feature necessitates a fundamental reformulation of the traditional nonlinear pricing problem. We provide this reformulation and characterize the solution to a complicated nonlinear pricing problem with discontinuous and inelastic individual demand functions, virtually no restriction on demand distribution, and no single-crossing restriction on utility functions. We show that under a weak ordering condition of customer types, this complex pricing problem can be decomposed into a set of much simpler pricing problems with known solutions. Our pricing solution is applicable to a broad range of demand systems including the very popular normal and exponential distributions in prior literature. Managerially, our solution is based on parameters that are easily measurable in practice: the required consumption volume and value from this consumption for each customer. In contrast, typical nonlinear pricing problems require one to specify a complete utility function for every customer drawn from a continuum of customer types. Therefore, our solution likely makes empirical demand estimation and the real-world use of the analytical solution more viable. We discuss the characteristics of the optimal nonlinear pricing schedule and its welfare implications at the end.