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Dynamic price competition with fixed capacities

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  • Kalyan Talluri

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  • Víctor Martínez de Albéniz
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    Abstract

    Many revenue management (RM) industries are characterized by (a) fixed capacities in the short term (e.g., hotel rooms, seats on an airline flight), (b) homogeneous products (e.g., two airline flights between the same cities at similar times), and (c) customer purchasing decisions largely influenced by price. Competition in these industries is also very high even with just two or three direct competitors in a market. However, RM competition is not well understood and practically all known implementations of RM software and most published models of RM do not explicitly model competition. For this reason, there has been considerable recent interest and research activity to understand RM competition. 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 (alternately, allocations at a price in quantity-based RM), reacting either to updates in their estimates of market demand, competitor prices, or inventory levels. We first prove existence of a unique subgame-perfect equilibrium for a duopoly. In equilibrium, in each state sellers engage in Bertrand competition, so that the seller with the lowest reservation value ends up selling a unit at a price that is equal to the equilibrium reservation value of the competitor. This structure hence extends the marginal-value concept of bid-price control, used in many RM implementations, to a competitive model. In addition, we show that the seller with the lowest capacity sells all its units first. Furthermore, we extend the results transparently to n firms and perform a number of numerical comparative statics exploiting the uniqueness of the subgame-perfect equilibrium.

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    Bibliographic Info

    Paper provided by Department of Economics and Business, Universitat Pompeu Fabra in its series Economics Working Papers with number 1205.

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    Date of creation: Feb 2010
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    Handle: RePEc:upf:upfgen:1205

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    Web page: http://www.econ.upf.edu/

    Related research

    Keywords: revenue management; bid-prices; subgame-perfect equilibrium.;

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    References

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    1. Dudey, Marc, 1992. "Dynamic Edgeworth-Bertrand Competition," The Quarterly Journal of Economics, MIT Press, vol. 107(4), pages 1461-77, November.
    2. Biglaiser, Gary & Vettas, Nikolaos, 2004. "Dynamic Price Competition with Capacity Constraints and Strategic Buyers," CEPR Discussion Papers 4315, C.E.P.R. Discussion Papers.
    3. Guillermo Gallego & Woonghee Tim Huh & Wanmo Kang & Robert Phillips, 2006. "Price Competition with the Attraction Demand Model: Existence of Unique Equilibrium and Its Stability," Manufacturing & Service Operations Management, INFORMS, vol. 8(4), pages 359-375, June.
    4. Richard E. Levitan & Martin Shubik, 1970. "Price Duopoly and Capacity Constraints," Cowles Foundation Discussion Papers 287, Cowles Foundation for Research in Economics, Yale University.
    5. de Frutos, María-Ángeles & Fabra, Natalia, 2011. "Endogenous capacities and price competition: The role of demand uncertainty," International Journal of Industrial Organization, Elsevier, vol. 29(4), pages 399-411, July.
    6. 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.
    7. Kalyan Talluri, 2003. "On equilibria in duopolies with finite strategy spaces," Economics Working Papers 701, Department of Economics and Business, Universitat Pompeu Fabra.
    8. Morten Hviid, 1990. "Sequential capacity and price choices in a duopoly model with demand uncertainty," Journal of Economics, Springer, vol. 51(2), pages 121-144, June.
    9. Mizuno, Toshihide, 2003. "On the existence of a unique price equilibrium for models of product differentiation," International Journal of Industrial Organization, Elsevier, vol. 21(6), pages 761-793, June.
    10. De Francesco, Massimo A. & Salvadori, Neri, 2008. "Bertrand-Edgeworth games under oligopoly with a complete characterization for the triopoly," MPRA Paper 10767, University Library of Munich, Germany, revised 26 Sep 2008.
    11. Gal-Or, Esther, 1984. "Price Dispersion with Uncertain Demand," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 25(2), pages 441-57, June.
    12. Hirata, Daisuke, 2008. "Bertrand-Edgeworth Equilibrium in Oligopoly," MPRA Paper 7946, University Library of Munich, Germany.
    13. Pradeep K. Chintagunta & Vithala R. Rao, 1996. "Pricing Strategies in a Dynamic Duopoly: A Differential Game Model," Management Science, INFORMS, vol. 42(11), pages 1501-1514, November.
    14. Yuri Levin & Jeff McGill & Mikhail Nediak, 2009. "Dynamic Pricing in the Presence of Strategic Consumers and Oligopolistic Competition," Management Science, INFORMS, vol. 55(1), pages 32-46, January.
    15. Hviid, Morten, 1991. "Capacity constrained duopolies, uncertain demand and non-existence of pure strategy equilibria," European Journal of Political Economy, Elsevier, vol. 7(2), pages 183-190, July.
    16. Osborne, Martin J. & Pitchik, Carolyn, 1986. "Price competition in a capacity-constrained duopoly," Journal of Economic Theory, Elsevier, vol. 38(2), pages 238-260, April.
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