Nonlinear Pricing Of Software With Local Demand Inelasticity

Nonlinear usage-based pricing is applied extensively in software markets. Different from other products, customers of software products usually cannot vary their required usage volume, a property we label local demand inelasticity. For instance, a client firm that needs a sales force 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 their valuation changes smoothly with usage volume. Consequently, standard nonlinear 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. This unique demand feature necessitates a new approach to solve the nonlinear pricing problem. We provide the solution to a complicated nonlinear pricing problem with discontinuous and inelastic individual demand functions, with virtually no restriction on demand distribution, and no single-crossing restriction on valuation 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 subproblems with known solutions. Our pricing solution is easily implementable and applicable to a broad range of demand systems, including those described by the families of exponential and normal distributions. Moreover, local demand inelasticity has a critical impact on key efficiency results. Although in standard nonlinear pricing models, the optimal pricing schedule typically involves distortion (deviation from the first best) at all but one point, this is no longer the case with local demand inelasticity. We characterize the conditions under which the optimal nonlinear pricing strategy involves quantity discounts and compare nonlinear usage-based pricing with flat-fee (for unlimited usage) pricing strategies.