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On the dynamic stability of a price dispersion model using gradient dynamics

Author

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  • Rabanal, Jean Paul
  • Lee, Dongwook
Abstract
This paper studies the evolutionary stability of a unique Nash equilibrium in a price dispersion model (Burdett and Judd, 1983) using gradient dynamics. The numerical solution to the partial differential equation that governs the evolution of prices shows that the stationary equilibrium is not a Nash Equilibrium and that it differs from the cyclical behavior predicted by another family of dynamics, such as replicator and logit, in a continuous action space.

Suggested Citation

  • Rabanal, Jean Paul & Lee, Dongwook, 2017. "On the dynamic stability of a price dispersion model using gradient dynamics," Journal of Economic Dynamics and Control, Elsevier, vol. 84(C), pages 32-42.
    • Handle: RePEc:eee:dyncon:v:84:y:2017:i:c:p:32-42
    DOI: 10.1016/j.jedc.2017.08.007
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    References listed on IDEAS

    as
    1. Cheung, Man-Wah, 2016. "Imitative dynamics for games with continuous strategy space," Games and Economic Behavior, Elsevier, vol. 99(C), pages 206-223.
    2. repec:bla:jindec:v:49:y:2001:i:4:p:541-58 is not listed on IDEAS
    3. Friedman, Daniel & Ostrov, Daniel N., 2013. "Evolutionary dynamics over continuous action spaces for population games that arise from symmetric two-player games," Journal of Economic Theory, Elsevier, vol. 148(2), pages 743-777.
    4. Glenn Ellison & Sara Fisher Ellison, 2009. "Search, Obfuscation, and Price Elasticities on the Internet," Econometrica, Econometric Society, vol. 77(2), pages 427-452, March.
    5. Ed Hopkins & Robert M. Seymour, 2002. "The Stability of Price Dispersion under Seller and Consumer Learning," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 43(4), pages 1157-1190, November.
    6. Fernando Louge & Frank Riedel, 2012. "Evolutionary Stability in First Price Auctions," Dynamic Games and Applications, Springer, vol. 2(1), pages 110-128, March.
    7. Burdett, Kenneth & Judd, Kenneth L, 1983. "Equilibrium Price Dispersion," Econometrica, Econometric Society, vol. 51(4), pages 955-969, July.
    8. Lahkar, Ratul & Riedel, Frank, 2015. "The logit dynamic for games with continuous strategy sets," Games and Economic Behavior, Elsevier, vol. 91(C), pages 268-282.
    9. Ratul, Lahkar, 2011. "The dynamic instability of dispersed price equilibria," Journal of Economic Theory, Elsevier, vol. 146(5), pages 1796-1827, September.
    10. 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.
    11. Michael D. Smith & Erik Brynjolfsson, 2001. "Consumer Decision-making at an Internet Shopbot: Brand Still Matters," NBER Chapters, in: E-commerce, pages 541-558, National Bureau of Economic Research, Inc.
    12. Cason, Timothy N. & Friedman, Daniel & Wagener, Florian, 2005. "The dynamics of price dispersion, or Edgeworth variations," Journal of Economic Dynamics and Control, Elsevier, vol. 29(4), pages 801-822, April.
    13. Simon P. Anderson & Jacob K. Goeree & Charles A. Holt, 2004. "Noisy Directional Learning and the Logit Equilibrium," Scandinavian Journal of Economics, Wiley Blackwell, vol. 106(3), pages 581-602, October.
    14. Jean Paul Rabanal, 2017. "On the Evolution of Continuous Types Under Replicator and Gradient Dynamics: Two Examples," Dynamic Games and Applications, Springer, vol. 7(1), pages 76-92, March.
    15. Chakrabarti, Anindya S. & Lahkar, Ratul, 2017. "Productivity dispersion and output fluctuations: An evolutionary model," Journal of Economic Behavior & Organization, Elsevier, vol. 137(C), pages 339-360.
    16. Liran Einav & Theresa Kuchler & Jonathan Levin & Neel Sundaresan, 2015. "Assessing Sale Strategies in Online Markets Using Matched Listings," American Economic Journal: Microeconomics, American Economic Association, vol. 7(2), pages 215-247, May.
    17. Friedman, Daniel & Ostrov, Daniel N., 2010. "Gradient dynamics in population games: Some basic results," Journal of Mathematical Economics, Elsevier, vol. 46(5), pages 691-707, September.
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    Cited by:

    1. Ma, Junhai & Si, Fengshan & Zhang, Qin & Wang, Zongxian, 2024. "Impact of extended warranty service on product pricing in online direct retailers' competitive market," Energy Economics, Elsevier, vol. 129(C).

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    More about this item

    Keywords

    Population games; Evolutionary dynamics; Gradient dynamics; Price dispersion;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • L11 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Production, Pricing, and Market Structure; Size Distribution of Firms

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