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The Optimality of Upgrade Pricing

Author

Listed:
  • Dirk Bergemann
  • Alessandro Bonatti
  • Andreas Haupt
  • Alex Smolin

Abstract

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 is given by a weak version of monotonicity of types and virtual values, while maintaining 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.

Suggested Citation

  • Dirk Bergemann & Alessandro Bonatti & Andreas Haupt & Alex Smolin, 2021. "The Optimality of Upgrade Pricing," Papers 2107.10323, arXiv.org, revised Dec 2021.
    • Handle: RePEc:arx:papers:2107.10323
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    References listed on IDEAS

    as
    1. Hart, Sergiu & Nisan, Noam, 2017. "Approximate revenue maximization with multiple items," Journal of Economic Theory, Elsevier, vol. 172(C), pages 313-347.
    2. Bikhchandani, Sushil & Mishra, Debasis, 2022. "Selling two identical objects," Journal of Economic Theory, Elsevier, vol. 200(C).
    3. Pavlov Gregory, 2011. "Optimal Mechanism for Selling Two Goods," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 11(1), pages 1-35, February.
    4. Bergemann, Dirk & Yeh, Edmund & Zhang, Jinkun, 2021. "Nonlinear pricing with finite information," Games and Economic Behavior, Elsevier, vol. 130(C), pages 62-84.
    5. Roger B. Myerson, 1981. "Optimal Auction Design," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 58-73, February.
    6. William James Adams & Janet L. Yellen, 1976. "Commodity Bundling and the Burden of Monopoly," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 90(3), pages 475-498.
    7. Justin P. Johnson & David P. Myatt, 2003. "Multiproduct Quality Competition: Fighting Brands and Product Line Pruning," American Economic Review, American Economic Association, vol. 93(3), pages 748-774, June.
    8. Manelli, Alejandro M. & Vincent, Daniel R., 2006. "Bundling as an optimal selling mechanism for a multiple-good monopolist," Journal of Economic Theory, Elsevier, vol. 127(1), pages 1-35, March.
    9. Glenn Ellison, 2005. "A Model of Add-On Pricing," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 120(2), pages 585-637.
    10. R. Preston McAfee & John McMillan & Michael D. Whinston, 1989. "Multiproduct Monopoly, Commodity Bundling, and Correlation of Values," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 104(2), pages 371-383.
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    Cited by:

    1. Frank Yang, 2021. "Costly Multidimensional Screening," Papers 2109.00487, arXiv.org, revised Sep 2025.

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    More about this item

    JEL classification:

    • D42 - Microeconomics - - Market Structure, Pricing, and Design - - - Monopoly
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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