IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v172y2009i1p375-39210.1007-s10479-009-0640-2.html
   My bibliography  Save this article

Learning in nonlinear pricing with unknown utility functions

Author

Listed:
  • Kimmo Berg
  • Harri Ehtamo

Abstract

Optimal screening is one of the basic models of contracting under incomplete information, and we study the problem in a quality pricing application. We present a simple numerical method for solving the pricing problem when the firm has limited information about the buyers’ utility functions. In the method, the firm learns the optimal price schedule as the demand data is collected. We examine what the firm can learn about the preferences by observing the sales, and how the revealed information can be used in adjusting the quality-price bundles to increase the profit. We analyze the properties of the solution and derive the first-order optimality conditions under different assumptions. We show that the problem can be solved by making use of these optimality conditions together with the buyers’ marginal valuations. The firm can estimate the marginal valuations either by offering linear tariffs or by selling test bundles near the current solution. Copyright Springer Science+Business Media, LLC 2009

Suggested Citation

  • Kimmo Berg & Harri Ehtamo, 2009. "Learning in nonlinear pricing with unknown utility functions," Annals of Operations Research, Springer, vol. 172(1), pages 375-392, November.
    • Handle: RePEc:spr:annopr:v:172:y:2009:i:1:p:375-392:10.1007/s10479-009-0640-2
    DOI: 10.1007/s10479-009-0640-2
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10479-009-0640-2
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s10479-009-0640-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. C. Raju & Y. Narahari & K. Ravikumar, 2006. "Learning dynamic prices in electronic retail markets with customer segmentation," Annals of Operations Research, Springer, vol. 143(1), pages 59-75, March.
    2. Franziska Voelckner, 2006. "An empirical comparison of methods for measuring consumers’ willingness to pay," Marketing Letters, Springer, vol. 17(2), pages 137-149, April.
    3. G. Carlier, 2002. "Nonparametric Adverse Selection Problems," Annals of Operations Research, Springer, vol. 114(1), pages 71-82, August.
    4. J. A. Mirrlees, 1971. "An Exploration in the Theory of Optimum Income Taxation," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 38(2), pages 175-208.
    5. Anthony Creane, 2002. "Uncertain Product Quality, Optimal Pricing and Product Development," Annals of Operations Research, Springer, vol. 114(1), pages 83-103, August.
    6. Nisan, Noam & Ronen, Amir, 2001. "Algorithmic Mechanism Design," Games and Economic Behavior, Elsevier, vol. 35(1-2), pages 166-196, April.
    7. Mussa, Michael & Rosen, Sherwin, 1978. "Monopoly and product quality," Journal of Economic Theory, Elsevier, vol. 18(2), pages 301-317, August.
    8. Krishnan S. Anand & Ravi Aron, 2003. "Group Buying on the Web: A Comparison of Price-Discovery Mechanisms," Management Science, INFORMS, vol. 49(11), pages 1546-1562, November.
    9. repec:dau:papers:123456789/6236 is not listed on IDEAS
    10. Wedad Elmaghraby & P{i}nar Keskinocak, 2003. "Dynamic Pricing in the Presence of Inventory Considerations: Research Overview, Current Practices, and Future Directions," Management Science, INFORMS, vol. 49(10), pages 1287-1309, October.
    11. Rasanen, Mika & Ruusunen, Jukka & Hamalainen, Raimo P., 1997. "Optimal tariff design under consumer self-selection," Energy Economics, Elsevier, vol. 19(2), pages 151-167, May.
    12. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    13. A. Michael Spence, 1980. "Multi-Product Quantity-Dependent Prices and Profitability Constraints," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 47(5), pages 821-841.
    14. Armstrong, Mark, 1996. "Multiproduct Nonlinear Pricing," Econometrica, Econometric Society, vol. 64(1), pages 51-75, January.
    15. Jean-Charles Rochet & Philippe Chone, 1998. "Ironing, Sweeping, and Multidimensional Screening," Econometrica, Econometric Society, vol. 66(4), pages 783-826, July.
    16. Eric Maskin & John Riley, 1984. "Monopoly with Incomplete Information," RAND Journal of Economics, The RAND Corporation, vol. 15(2), pages 171-196, Summer.
    17. Christopher Brooks & Robert Gazzale & Rajarshi Das & Jeffrey Kephart & Jeffrey MacKie-Mason & Edmund Durfee, 2002. "Model Selection in an Information Economy : Choosing what to Learn," Department of Economics Working Papers 2002-01, Department of Economics, Williams College.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kimmo Berg & Harri Ehtamo, 2012. "Continuous learning methods in two-buyer pricing problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 75(3), pages 287-304, June.
    2. Braulio Calagua, 2023. "Reducing incentive constraints in bidimensional screening," The Journal of Mechanism and Institution Design, Society for the Promotion of Mechanism and Institution Design, University of York, vol. 8(1), pages 107-150, December.
    3. Mingli Zheng & Chong Wang & Chaozheng Li, 2016. "Insurance Contracts with Adverse Selection When the Insurer Has Ambiguity about the Composition of the Consumers," Annals of Economics and Finance, Society for AEF, vol. 17(1), pages 179-206, May.
    4. Kimmo Berg, 2013. "Complexity of solution structures in nonlinear pricing," Annals of Operations Research, Springer, vol. 206(1), pages 23-37, July.
    5. Pavlin, J. Michael, 2017. "Dual bounds of a service level assignment problem with applications to efficient pricing," European Journal of Operational Research, Elsevier, vol. 262(1), pages 239-250.

    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. Kimmo Berg & Harri Ehtamo, 2012. "Continuous learning methods in two-buyer pricing problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 75(3), pages 287-304, June.
    2. Ehtamo, Harri & Berg, Kimmo & Kitti, Mitri, 2010. "An adjustment scheme for nonlinear pricing problem with two buyers," European Journal of Operational Research, Elsevier, vol. 201(1), pages 259-266, February.
    3. Kimmo Berg, 2013. "Complexity of solution structures in nonlinear pricing," Annals of Operations Research, Springer, vol. 206(1), pages 23-37, July.
    4. Kelvin Shuangjian Zhang, 2019. "Existence in multidimensional screening with general nonlinear preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 67(2), pages 463-485, March.
    5. Kelvin Shuangjian Zhang, 2017. "Existence in Multidimensional Screening with General Nonlinear Preferences," Papers 1710.08549, arXiv.org, revised Dec 2018.
    6. Jakša Cvitanić & Julien Hugonnier, 2022. "Optimal fund menus," Mathematical Finance, Wiley Blackwell, vol. 32(2), pages 455-516, April.
    7. Carlier, Guillaume & Dupuis, Xavier & Rochet, Jean-Charles & Thanassoulis, John, 2024. "A general solution to the quasi linear screening problem," Journal of Mathematical Economics, Elsevier, vol. 114(C).
    8. Pascal Courty & Li Hao, 2000. "Sequential Screening," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 67(4), pages 697-717.
    9. Bing Jing & Roy Radner, 2004. "Nonconvex Production Technology and Price Discrimination," Working Papers 04-04, New York University, Leonard N. Stern School of Business, Department of Economics.
    10. Mark Armstrong, 2016. "Nonlinear Pricing," Annual Review of Economics, Annual Reviews, vol. 8(1), pages 583-614, October.
    11. X. Ruiz del Portal, 2012. "Conditions for incentive compatibility in models with multidimensional allocation functions and one-dimensional types," Review of Economic Design, Springer;Society for Economic Design, vol. 16(4), pages 311-321, December.
    12. Nicol'as Hern'andez Santib'a~nez & Dylan Possamai & Chao Zhou, 2017. "Bank monitoring incentives under moral hazard and adverse selection," Papers 1701.05864, arXiv.org, revised Jan 2019.
    13. Andersson, Tommy, 2004. "Essays on Nonlinear Pricing and Welfare," MPRA Paper 59446, University Library of Munich, Germany.
    14. Figalli, Alessio & Kim, Young-Heon & McCann, Robert J., 2011. "When is multidimensional screening a convex program?," Journal of Economic Theory, Elsevier, vol. 146(2), pages 454-478, March.
    15. Robert J. McCann & Kelvin Shuangjian Zhang, 2023. "A duality and free boundary approach to adverse selection," Papers 2301.07660, arXiv.org, revised Nov 2023.
    16. 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.
    17. Nicolás Hernández Santibáñez & Dylan Possamaï & Chao Zhou, 2017. "Bank monitoring incentives under moral hazard and adverse selection," Working Papers hal-01435460, HAL.
    18. Wong, Kit Pong, 2024. "Optimal nonlinear pricing by a monopoly with smooth ambiguity preferences," International Review of Economics & Finance, Elsevier, vol. 89(PA), pages 594-604.
    19. Dirk Bergemann & Ji Shen & Yun Xu & Edmund M. Yeh, 2012. "Multi-Dimensional Mechanism Design with Limited Information," Levine's Working Paper Archive 786969000000000559, David K. Levine.
    20. Sürücü, Oktay, 2016. "Welfare improving discrimination based on cognitive limitations," Research in Economics, Elsevier, vol. 70(4), pages 608-622.

    Corrections

    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:spr:annopr:v:172:y:2009:i:1:p:375-392:10.1007/s10479-009-0640-2. See general information about how to correct material in RePEc.

    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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.