IDEAS home Printed from https://ideas.repec.org/a/kap/revind/v67y2025i2d10.1007_s11151-025-10013-5.html
   My bibliography  Save this article

Random Pricing: Bertrand Competition with Uncontested Consumers

Author

Listed:
  • Severin Lenhard

    (Universität St. Gallen
    Secretariat of the Swiss Competition Commission)

Abstract

Two firms offer a homogeneous product and compete in prices. Consumers are homogeneous yet differ in their access to the firms. Three groups exist: Consumers who have access to both firms and can compare prices; and consumers who can access only one of the two firms and are thus uncontested. The group sizes may differ: One firm’s uncontested consumer base may be zero. No pure strategy equilibrium exists. In the mixed equilibrium, firms randomize on the same continuum of prices; in expectation, the firm with the larger consumer base plays a higher price and has a higher expected demand. We study: the firms’ incentive to invest in demand; effects of price discrimination; and collusion. Moreover, we extend the model to more than two firms and discuss mergers.

Suggested Citation

  • Severin Lenhard, 2025. "Random Pricing: Bertrand Competition with Uncontested Consumers," Review of Industrial Organization, Springer;The Industrial Organization Society, vol. 67(2), pages 191-208, August.
    • Handle: RePEc:kap:revind:v:67:y:2025:i:2:d:10.1007_s11151-025-10013-5
    DOI: 10.1007/s11151-025-10013-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11151-025-10013-5
    File Function: Abstract
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s11151-025-10013-5?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

    for a different version of it.

    References listed on IDEAS

    as
    1. Michael R. Baye & Dan Kovenock & Casper G. Vries, 1996. "The all-pay auction with complete information," Springer Books, in: Roger D. Congleton & Arye L. Hillman & Kai A. Konrad (ed.), 40 Years of Research on Rent Seeking 1, pages 209-223, Springer.
    2. Joao Montez & Nicolas Schutz, 2021. "All-Pay Oligopolies: Price Competition with Unobservable Inventory Choices [Extremal Equilibria of Oligopolistic Supergames]," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 88(5), pages 2407-2438.
    3. Mark Armstrong & John Vickers, 2022. "Patterns of Competitive Interaction," Econometrica, Econometric Society, vol. 90(1), pages 153-191, January.
    4. Huber, Martin & Imhof, David, 2019. "Machine learning with screens for detecting bid-rigging cartels," International Journal of Industrial Organization, Elsevier, vol. 65(C), pages 277-301.
    5. David Autor & David Dorn & Lawrence F Katz & Christina Patterson & John Van Reenen, 2020. "The Fall of the Labor Share and the Rise of Superstar Firms [“Automation and New Tasks: How Technology Displaces and Reinstates Labor”]," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 135(2), pages 645-709.
    6. Ganesh Iyer & David Soberman & J. Miguel Villas-Boas, 2005. "The Targeting of Advertising," Marketing Science, INFORMS, vol. 24(3), pages 461-476, May.
    7. Abrantes-Metz, Rosa M. & Froeb, Luke M. & Geweke, John & Taylor, Christopher T., 2006. "A variance screen for collusion," International Journal of Industrial Organization, Elsevier, vol. 24(3), pages 467-486, May.
    8. Lenhard, Severin, 2022. "Imperfect Competition with Costly Disposal," VfS Annual Conference 2022 (Basel): Big Data in Economics 264038, Verein für Socialpolitik / German Economic Association.
    9. Narasimhan, Chakravarthi, 1988. "Competitive Promotional Strategies," The Journal of Business, University of Chicago Press, vol. 61(4), pages 427-449, October.
    10. Ofer H Azar, 2015. "A Linear City Model with Asymmetric Consumer Distribution," PLOS ONE, Public Library of Science, vol. 10(6), pages 1-13, June.
    11. Rothschild, R., 1999. "Cartel stability when costs are heterogeneous," International Journal of Industrial Organization, Elsevier, vol. 17(5), pages 717-734, July.
    12. Arye L. Hillman & John G. Riley, 1989. "Politically Contestable Rents And Transfers," Economics and Politics, Wiley Blackwell, vol. 1(1), pages 17-39, March.
    13. Jan De Loecker & Jan Eeckhout & Gabriel Unger, 2020. "The Rise of Market Power and the Macroeconomic Implications [“Econometric Tools for Analyzing Market Outcomes”]," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 135(2), pages 561-644.
    14. Lenhard, Severin, 2022. "Imperfect competition with costly disposal," International Journal of Industrial Organization, Elsevier, vol. 82(C).
    15. Abreu, Dilip, 1986. "Extremal equilibria of oligopolistic supergames," Journal of Economic Theory, Elsevier, vol. 39(1), pages 191-225, June.
    16. Mariano Tappata, 2009. "Rockets and feathers: Understanding asymmetric pricing," RAND Journal of Economics, RAND Corporation, vol. 40(4), pages 673-687, December.
    17. Stahl, Dale O, II, 1989. "Oligopolistic Pricing with Sequential Consumer Search," American Economic Review, American Economic Association, vol. 79(4), pages 700-712, September.
    18. Dieter Pennerstorfer & Philipp Schmidt‐Dengler & Nicolas Schutz & Christoph Weiss & Biliana Yontcheva, 2020. "Information And Price Dispersion: Theory And Evidence," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 61(2), pages 871-899, May.
    19. Harrington, Joseph Jr., 1989. "Collusion among asymmetric firms: The case of different discount factors," International Journal of Industrial Organization, Elsevier, vol. 7(2), pages 289-307, June.
    20. Shilony, Yuval, 1977. "Mixed pricing in oligopoly," Journal of Economic Theory, Elsevier, vol. 14(2), pages 373-388, April.
    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. Johannes Johnen & David Ronayne, 2021. "The only Dance in Town: Unique Equilibrium in a Generalized Model of Price Competition," Journal of Industrial Economics, Wiley Blackwell, vol. 69(3), pages 595-614, September.
    2. Anderson, Simon & Baik, Alicia & Larson, Nathan, 2015. "Personalized pricing and advertising: An asymmetric equilibrium analysis," Games and Economic Behavior, Elsevier, vol. 92(C), pages 53-73.
    3. Avi Weiss & Joshua Sherman, 2014. "An Empirical Analysis of Search Costs and Price Dispersion," Working Papers 2014-06, Bar-Ilan University, Department of Economics.
    4. repec:cdl:compol:qt9pt7p9bm is not listed on IDEAS
    5. Hakan Yilmazkuday, 2017. "Geographical dispersion of consumer search behaviour," Applied Economics, Taylor & Francis Journals, vol. 49(57), pages 5740-5752, December.
    6. Michael R. Baye & John Morgan & Patrick Scholten, 2004. "Price Dispersion In The Small And In The Large: Evidence From An Internet Price Comparison Site," Journal of Industrial Economics, Wiley Blackwell, vol. 52(4), pages 463-496, December.
    7. Joshua Sherman & Avi Weiss, 2017. "On Fruitful And Futile Tests Of The Relationship Between Search And Price Dispersion," Economic Inquiry, Western Economic Association International, vol. 55(4), pages 1898-1918, October.
    8. Michael D. Noel & Hongjie Qiang, 2023. "Missing Price Information and Its Impact on Equilibrium Price Dispersion: Evidence From Gasoline Signboards," Journal of Industrial Economics, Wiley Blackwell, vol. 71(3), pages 814-854, September.
    9. Kai A. Konrad, 2023. "The Geoeconomics of Trade Infrastructure and the Innovation Competition between China and the US," Working Papers tax-mpg-rps-2023-14, Max Planck Institute for Tax Law and Public Finance.
    10. Dirk Bergemann & Benjamin Brooks & Stephen Morris, 2021. "Search, Information, and Prices," Journal of Political Economy, University of Chicago Press, vol. 129(8), pages 2275-2319.
    11. J. Amegashie, 2011. "Incomplete property rights and overinvestment," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 37(1), pages 81-95, June.
    12. Rosa-Branca Esteves & Joana Resende, 2016. "Competitive Targeted Advertising with Price Discrimination," Marketing Science, INFORMS, vol. 35(4), pages 576-587, July.
    13. Brito, Duarte & Ribeiro, Ricardo & Vasconcelos, Helder, 2018. "Quantifying the coordinated effects of partial horizontal acquisitions," European Economic Review, Elsevier, vol. 110(C), pages 108-149.
    14. Michael R. Baye & J. Rupert J. Gatti & Paul Kattuman & John Morgan, 2009. "Clicks, Discontinuities, and Firm Demand Online," Journal of Economics & Management Strategy, Wiley Blackwell, vol. 18(4), pages 935-975, December.
    15. Michael R. Baye & John Morgan, 2009. "Brand and Price Advertising in Online Markets," Management Science, INFORMS, vol. 55(7), pages 1139-1151, July.
    16. Banks, Jeffrey & Moorthy, Sridhar, 1999. "A model of price promotions with consumer search," International Journal of Industrial Organization, Elsevier, vol. 17(3), pages 371-398, April.
    17. Shelegia, Sandro & Wilson, Christopher, 2022. "Costly Participation and Default Allocations in All-Pay Contests," CEPR Discussion Papers 17611, C.E.P.R. Discussion Papers.
    18. Garrod, Luke & Li, Ruochen & Russo, Antonio & Wilson, Chris M, 2024. "Understanding Cost Pass-Through when Prices are Dispersed," MPRA Paper 123285, University Library of Munich, Germany.
    19. Chris M. Wilson & Sandro Shelegia, 2016. "A Generalized Model of Sales," Working Papers 934, Barcelona School of Economics.
    20. David P. Myatt, 2019. "A Theory of Stable Price Dispersion," Economics Series Working Papers 873, University of Oxford, Department of Economics.
    21. Alan Gelder & Dan Kovenock & Brian Roberson, 2022. "All-pay auctions with ties," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 74(4), pages 1183-1231, November.

    More about this item

    Keywords

    ;
    ;
    ;

    JEL classification:

    • D21 - Microeconomics - - Production and Organizations - - - Firm Behavior: Theory
    • L11 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Production, Pricing, and Market Structure; Size Distribution of Firms
    • L13 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Oligopoly and Other Imperfect Markets

    Statistics

    Access and download statistics

    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:kap:revind:v:67:y:2025:i:2:d:10.1007_s11151-025-10013-5. 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.