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Learning in nonlinear pricing with unknown utility functions

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  • 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
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    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. Nisan, Noam & Ronen, Amir, 2001. "Algorithmic Mechanism Design," Games and Economic Behavior, Elsevier, vol. 35(1-2), pages 166-196, April.
    3. 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.
    4. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, number 9780195102680.
    5. 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.
    6. Mussa, Michael & Rosen, Sherwin, 1978. "Monopoly and product quality," Journal of Economic Theory, Elsevier, vol. 18(2), pages 301-317, August.
    7. Franziska Voelckner, 2006. "An empirical comparison of methods for measuring consumers’ willingness to pay," Marketing Letters, Springer, vol. 17(2), pages 137-149, April.
    8. Armstrong, Mark, 1996. "Multiproduct Nonlinear Pricing," Econometrica, Econometric Society, vol. 64(1), pages 51-75, January.
    9. 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.
    10. Jean-Charles Rochet & Philippe Chone, 1998. "Ironing, Sweeping, and Multidimensional Screening," Econometrica, Econometric Society, vol. 66(4), pages 783-826, July.
    11. 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.
    12. Eric Maskin & John Riley, 1984. "Monopoly with Incomplete Information," RAND Journal of Economics, The RAND Corporation, vol. 15(2), pages 171-196, Summer.
    13. G. Carlier, 2002. "Nonparametric Adverse Selection Problems," Annals of Operations Research, Springer, vol. 114(1), pages 71-82, August.
    14. repec:dau:papers:123456789/6236 is not listed on IDEAS
    15. 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.
    16. Anthony Creane, 2002. "Uncertain Product Quality, Optimal Pricing and Product Development," Annals of Operations Research, Springer, vol. 114(1), pages 83-103, August.
    17. 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.
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    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.

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