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We consider a multiproduct monopoly pricing model. We provide suﬀicient conditions under which the optimal mechanism can be implemented via upgrade pricing—a menu of product bundles that are nested in the strong set order. Our approach exploits duality methods to identify conditions on the distribution of consumer types under which (a) each product is purchased by the same set of buyers as under separate monopoly pricing (though the transfers can be diﬀerent), and (b) these sets are nested. We exhibit two distinct sets of suﬀicient conditions. The ﬁrst set of conditions weakens the monotonicity requirement of types and virtual values but maintains a regularity assumption, i.e., that the product-by-product revenue curves are single-peaked. The second set of conditions establishes the optimality of upgrade pricing for type spaces with monotone marginal rates of substitution (MRS)—the relative preference ratios for any two products are monotone across types. The monotone MRS condition allows us to relax the earlier regularity assumption. Under both sets of conditions, we fully characterize the product bundles and prices that form the optimal upgrade pricing menu. Finally, we show that, if the consumer’s types are monotone, the seller can equivalently post a vector of single-item prices: upgrade pricing and separate pricing are equivalent.
Bergemann, Dirk; Bonatti, Alessandro; Haupt, Andreas; and Smolin, Alex, "The Optimality of Upgrade Pricing" (2021). Cowles Foundation Discussion Papers. 2666.