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Dynamic Price Competition with Fixed Capacities

  • Victor Martínez-de-Albéniz


    (IESE Business School, University of Navarra, 08034 Barcelona, Spain)

  • Kalyan Talluri


    (ICREA and Universitat Pompeu Fabra, 08005 Barcelona, Spain)

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    In this paper, we study price competition for an oligopoly in a dynamic setting, where each of the sellers has a fixed number of units available for sale over a fixed number of periods. Demand is stochastic, and depending on how it evolves, sellers may change their prices at any time. This reflects the fact that firms constantly, and almost costlessly, change their prices, reacting to updates in their estimates of market demand, competitor prices, or inventory levels. In a setting with demand uncertainty, we show that there is a unique subgame-perfect equilibrium for a duopoly, in which all states sellers engage in Bertrand competition and the seller with the lower equilibrium reservation value sells a unit at a price equal to the competitor's equilibrium reservation value. This structure therefore extends the marginal-value concept of bid-price control, used in many revenue management implementations, to a competitive model. We give a closed-form solution to the equilibrium price paths for a duopoly and extend all the results to an n-firm oligopoly. We then study extensions to multiple customer types, uncertain valuations, and differentiated products. This paper was accepted by Martin Lariviere, operations management.

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    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 57 (2011)
    Issue (Month): 6 (June)
    Pages: 1078-1093

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    Handle: RePEc:inm:ormnsc:v:57:y:2011:i:6:p:1078-1093
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    1. Partha Dasgupta & Eric Maskin, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," Review of Economic Studies, Oxford University Press, vol. 53(1), pages 1-26.
    2. De Francesco, Massimo A. & Salvadori, Neri, 2008. "Bertrand-Edgeworth games under oligopoly with a complete characterization for the triopoly," MPRA Paper 8634, University Library of Munich, Germany.
    3. Partha Dasgupta & Eric Maskin, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, II: Applications," Review of Economic Studies, Oxford University Press, vol. 53(1), pages 27-41.
    4. Guillermo Gallego & Garrett van Ryzin, 1994. "Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons," Management Science, INFORMS, vol. 40(8), pages 999-1020, August.
    5. Osborne, Martin J. & Pitchik, Carolyn, 1983. "Price Competition in a Capacity-Constrained Duopoly," Working Papers 83-08, C.V. Starr Center for Applied Economics, New York University.
    6. Levitan, Richard & Shubik, Martin, 1972. "Price Duopoly and Capacity Constraints," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 13(1), pages 111-22, February.
    7. Hirata Daisuke, 2009. "Asymmetric Bertrand-Edgeworth Oligopoly and Mergers," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 9(1), pages 1-25, July.
    8. Beth Allen & Martin Hellwig, 1986. "Bertrand-Edgeworth Oligopoly in Large Markets," Review of Economic Studies, Oxford University Press, vol. 53(2), pages 175-204.
    9. Biglaiser, Gary & Vettas, Nikolaos, 2004. "Dynamic Price Competition with Capacity Constraints and Strategic Buyers," CEPR Discussion Papers 4315, C.E.P.R. Discussion Papers.
    10. Carl Davidson & Raymond Deneckere, 1986. "Long-Run Competition in Capacity, Short-Run Competition in Price, and the Cournot Model," RAND Journal of Economics, The RAND Corporation, vol. 17(3), pages 404-415, Autumn.
    11. Lin, Kyle Y. & Sibdari, Soheil Y., 2009. "Dynamic price competition with discrete customer choices," European Journal of Operational Research, Elsevier, vol. 197(3), pages 969-980, September.
    12. David M. Kreps & Jose A. Scheinkman, 1983. "Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes," Bell Journal of Economics, The RAND Corporation, vol. 14(2), pages 326-337, Autumn.
    13. Marc Dudey, 1992. "Dynamic Edgeworth-Bertrand Competition," The Quarterly Journal of Economics, Oxford University Press, vol. 107(4), pages 1461-1477.
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