Optimally Selling Heterogeneous Objects by Quantity

Abstract

I study the optimality of selling K heterogeneous objects to a buyer with private values through a quantity menu: the seller posts a price for each number of objects, and the buyer chooses a quantity, pays its price, and selects any objects up to that quantity. I propose a pricing method that aggregates critical values for second differences of quantity values (the highest values attainable at each quantity). Critical values can be determined separately when each second difference's conditional inverse-hazard ratio depends only on that difference. For additive values, I show that the resulting menu is optimal among all incentive-compatible and individually rational mechanisms under explicit distributional and support conditions, including ex ante symmetry across object labels. A separate result applies the same pricing method to obtain an optimal menu for nonadditive bundle values without ex ante symmetry, assuming free disposal, independence between the quantity-value vector and the vector of bundle shortfalls from their respective quantity values, and corresponding distributional and support conditions. For an arbitrary number of objects, applications yield explicit optimal menus that involve quantity discounts, sparse quantity options, or two-part tariffs. These results differ from those establishing the optimality of pure bundling in Haghpanah and Hartline (2021), nested bundling in Yang (2025), separate sales in Kwak (2026), and core-peripheral bundling in Zhong (2026).