IDEAS home Printed from
   My bibliography  Save this paper

An Empirical Analysis of Optimal Nonlinear Pricing


  • Soheil Ghili
  • Russ Yoon


In "continuous choice" settings, consumers decide not only on whether to purchase a product, but also on how much to purchase. As a result, firms should optimize a full price schedule rather than a single price point. This paper provides a methodology to empirically estimate the optimal schedule under multi-dimensional consumer heterogeneity. We apply our method to novel data from an educational-services firm that contains purchase-size information not only for deals that materialized, but also for potential deals that eventually failed. We show that the optimal second-degree price discrimination (i.e., optimal nonlinear tariff) improves the firm's profit upon linear pricing by about 7.9%. That said, this second-degree price discrimination scheme only recovers 7.4% of the gap between the profitability of linear pricing (i.e., no price discrimination) and that of infeasible first degree price discrimination. We also conduct several further counterfactual analyses (i) comparing the role of demand- v.s. cost-side factors in shaping the optimal price schedule, (ii) examining third-degree price discrimination, and (iii) empirically quantifying the magnitude by which incentive-compatibility constraints impact the optimal pricing and profits.

Suggested Citation

  • Soheil Ghili & Russ Yoon, 2023. "An Empirical Analysis of Optimal Nonlinear Pricing," Papers 2302.11643,
  • Handle: RePEc:arx:papers:2302.11643

    Download full text from publisher

    File URL:
    File Function: Latest version
    Download Restriction: no

    References listed on IDEAS

    1. Yao Luo & Isabelle Perrigne & Quang Vuong, 2018. "Structural Analysis of Nonlinear Pricing," Journal of Political Economy, University of Chicago Press, vol. 126(6), pages 2523-2568.
    2. Schmalensee, Richard, 1981. "Output and Welfare Implications of Monopolistic Third-Degree Price Discrimination," American Economic Review, American Economic Association, vol. 71(1), pages 242-247, March.
    3. Igal Hendel & Aviv Nevo, 2013. "Intertemporal Price Discrimination in Storable Goods Markets," American Economic Review, American Economic Association, vol. 103(7), pages 2722-2751, December.
    4. Mussa, Michael & Rosen, Sherwin, 1978. "Monopoly and product quality," Journal of Economic Theory, Elsevier, vol. 18(2), pages 301-317, August.
    5. Raghuram Iyengar & Kamel Jedidi, 2012. "A Conjoint Model of Quantity Discounts," Marketing Science, INFORMS, vol. 31(2), pages 334-350, March.
    6. Varian, Hal R, 1985. "Price Discrimination and Social Welfare," American Economic Review, American Economic Association, vol. 75(4), pages 870-875, September.
    7. Phillip Leslie, 2004. "Price Discrimination in Broadway Theater," RAND Journal of Economics, The RAND Corporation, vol. 35(3), pages 520-541, Autumn.
    8. Eric T. Anderson & James D. Dana, Jr., 2009. "When Is Price Discrimination Profitable?," Management Science, INFORMS, vol. 55(6), pages 980-989, June.
    9. Kadiyali, Vrinda & Vilcassim, Naufel J & Chintagunta, Pradeep K, 1996. "Empirical Analysis of Competitive Product Line Pricing Decisions: Lead, Follow, or Move Together?," The Journal of Business, University of Chicago Press, vol. 69(4), pages 459-487, October.
    10. Frank Verboven, 2002. "Quality-Based Price Discrimination and Tax Incidence: Evidence from Gasoline and Diesel Cars," RAND Journal of Economics, The RAND Corporation, vol. 33(2), pages 275-297, Summer.
    11. Nima Haghpanah & Jason Hartline, 2021. "When Is Pure Bundling Optimal? [Commodity Bundling and the Burden of Monopoly]," Review of Economic Studies, Oxford University Press, vol. 88(3), pages 1127-1156.
    12. Michaela Draganska & Dipak C. Jain, 2006. "Consumer Preferences and Product-Line Pricing Strategies: An Empirical Analysis," Marketing Science, INFORMS, vol. 25(2), pages 164-174, 03-04.
    13. Gabriel Carroll, 2017. "Robustness and Separation in Multidimensional Screening," Econometrica, Econometric Society, vol. 85, pages 453-488, March.
    14. Jean-Charles Rochet & Lars A. Stole, 2002. "Nonlinear Pricing with Random Participation," Review of Economic Studies, Oxford University Press, vol. 69(1), pages 277-311.
    15. Timothy Derdenger & Vineet Kumar, 2013. "The Dynamic Effects of Bundling as a Product Strategy," Marketing Science, INFORMS, vol. 32(6), pages 827-859, November.
    16. Aviv Nevo & John L. Turner & Jonathan W. Williams, 2016. "Usageā€Based Pricing and Demand for Residential Broadband," Econometrica, Econometric Society, vol. 84, pages 411-443, March.
    17. Armstrong, Mark, 1996. "Multiproduct Nonlinear Pricing," Econometrica, Econometric Society, vol. 64(1), pages 51-75, January.
    18. Eric Maskin & John Riley, 1984. "Monopoly with Incomplete Information," RAND Journal of Economics, The RAND Corporation, vol. 15(2), pages 171-196, Summer.
    19. Peter C. Reiss & Matthew W. White, 2005. "Household Electricity Demand, Revisited," Review of Economic Studies, Oxford University Press, vol. 72(3), pages 853-883.
    20. Devanur, Nikhil R. & Haghpanah, Nima & Psomas, Alexandros, 2020. "Optimal multi-unit mechanisms with private demands," Games and Economic Behavior, Elsevier, vol. 121(C), pages 482-505.
    21. Peter Cohen & Robert Hahn & Jonathan Hall & Steven Levitt & Robert Metcalfe, 2016. "Using Big Data to Estimate Consumer Surplus: The Case of Uber," NBER Working Papers 22627, National Bureau of Economic Research, Inc.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Mark Armstrong, 2016. "Nonlinear Pricing," Annual Review of Economics, Annual Reviews, vol. 8(1), pages 583-614, October.
    2. Stole, Lars A., 2007. "Price Discrimination and Competition," Handbook of Industrial Organization, in: Mark Armstrong & Robert Porter (ed.), Handbook of Industrial Organization, edition 1, volume 3, chapter 34, pages 2221-2299, Elsevier.
    3. Anderson, Simon P. & Celik, Levent, 2015. "Product line design," Journal of Economic Theory, Elsevier, vol. 157(C), pages 517-526.
    4. Dirk Bergemann & Benjamin Brooks & Stephen Morris, 2015. "The Limits of Price Discrimination," American Economic Review, American Economic Association, vol. 105(3), pages 921-957, March.
    5. Crawford, Gregory S & Shum, Matthew, 2007. "Monopoly Quality Degradation and Regulation in Cable Television," Journal of Law and Economics, University of Chicago Press, vol. 50(1), pages 181-219, February.
    6. Atanu Lahiri & Rajiv M. Dewan & Marshall Freimer, 2013. "Pricing of Wireless Services: Service Pricing vs. Traffic Pricing," Information Systems Research, INFORMS, vol. 24(2), pages 418-435, June.
    7. Bikhchandani, Sushil & Mishra, Debasis, 2022. "Selling two identical objects," Journal of Economic Theory, Elsevier, vol. 200(C).
    8. Han, Jun & Weber, Thomas A., 2023. "Price discrimination with robust beliefs," European Journal of Operational Research, Elsevier, vol. 306(2), pages 795-809.
    9. Tang, Yao & Chen, Rachel R. & Guan, Xu, 2021. "Daily-deal market with consumer retention: Price discrimination or quality differentiation," Omega, Elsevier, vol. 102(C).
    10. Preyas S. Desai & Devavrat Purohit & Bo Zhou, 2018. "Allowing Consumers to Bundle Themselves: The Profitability of Family Plans," Marketing Science, INFORMS, vol. 37(6), pages 953-969, November.
    11. McAfee, R. Preston, 2007. "Pricing Damaged Goods," Economics - The Open-Access, Open-Assessment E-Journal (2007-2020), Kiel Institute for the World Economy (IfW Kiel), vol. 1, pages 1-19.
    12. Ke-Wei Huang, 2009. "Optimal criteria for selecting price discrimination metrics when buyers have log-normally distributed willingness-to-pay," Quantitative Marketing and Economics (QME), Springer, vol. 7(3), pages 321-341, September.
    13. Marco Cornia & Kristopher S. Gerardi & Adam Hale Shapiro, 2012. "Price Dispersion Over the Business Cycle: Evidence from the Airline Industry," Journal of Industrial Economics, Wiley Blackwell, vol. 60(3), pages 347-373, September.
    14. Diego Escobari & Nicholas G. Rupp & Joseph Meskey, 2019. "An Analysis of Dynamic Price Discrimination in Airlines," Southern Economic Journal, John Wiley & Sons, vol. 85(3), pages 639-662, January.
    15. Guillermo Marshall, 2020. "Search and Wholesale Price Discrimination," RAND Journal of Economics, RAND Corporation, vol. 51(2), pages 346-374, June.
    16. Etro, Federico, 2016. "Research in economics and industrial organization," Research in Economics, Elsevier, vol. 70(4), pages 511-517.
    17. Diego Escobari & Manuel A. Hernandez, 2019. "Separating Between Unobserved Consumer Types: Evidence From Airlines," Economic Inquiry, Western Economic Association International, vol. 57(2), pages 1215-1230, April.
    18. Renato Gomes & Alessandro Pavan, 2011. "Price Discrimination in Many-to-Many Matching Markets," Discussion Papers 1540, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    19. Roach, Travis, 2019. "Market power and second degree price discrimination in retail gasoline markets," Energy Economics, Elsevier, vol. 84(C).
    20. repec:cwl:cwldpp:1896rrr is not listed on IDEAS
    21. Yossi Spiegel & Simon Wilkie, 2000. "Optimal Multiproduct Nonlinear Pricing with Correlated Consumer Types," Discussion Papers 1299, Northwestern University, Center for Mathematical Studies in Economics and Management Science.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:2302.11643. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.