Quantity Discount or Small Quantity Premium? Violating the Single-Crossing Condition

This paper analyzes the rationale for a profit-maximizing firm with monopoly power that uses quantity discounts to a point where the additional quantity is sold below its marginal cost. Our model has two consumer types with linear demand functions. Unlike the standard assumption, we assume that their inverse demand functions cross. This will create a violation of the single-crossing condition so that Maskin and Riley’s characterization of nonlinear pricing cannot apply. Thus, we develop algorithms to circumvent the mathematical challenge of finding the optimal price-quantity bundles. We show that selling additional quantities below marginal cost can occur naturally in this framework. The solution indicates that it is the high price of the regular portion for one consumer type - - and not the discount on the additional quantities as commonly perceived - - that induces that consumer type to supersize. For our second result, we consider two consumer types and any linear inverse demand functions. We prove that crossing linear demands is a necessary condition to create the illusion of a great deal.