On the degree of distortions under second-degree price discrimination

This paper highlights the limitations and applicability of results developed by Chao & Nahata (2015) for nonlinear pricing. Although Chao and Nahata appear to provide necessary and sufficient conditions for general utility functions, we show that one of their results leads only to a restatement of two constraints, and another result may not be valid when consumers can freely dispose of the good. Their model allows for the possibility that higher quantities will have a lower price than smaller quantities. We provide conditions under free disposal that preclude this anomaly. Our analysis suggests that further research on violations of the single-crossing condition should be encouraged.

Nonlinear Pricing of Software with Local Demand Inelasticity

Nonlinear usage-based pricing is applied extensively in software markets. 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 or does not buy at all. It is unlikely to buy licenses for some but not all salespersons. 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 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.

Nonlinear Pricing with Average-Price Bias

Empirical evidence suggests that consumers facing complex nonlinear prices often make choices based on average (not marginal) prices. Given such behavior, we characterize a monopolist’s optimal nonlinear price schedule. In contrast to the textbook setting, nonlinear prices designed for “ average-price bias” distort consumption downward for consumers with the highest marginal utility and typically feature quantity premia rather than quantity discounts. These properties arise because the bias replaces consumer information rents with “curvature rents.” Whether or not a monopolist prefers consumers with average-price bias depends upon underlying preferences and costs. (JEL D11, D21, D42, L12)