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Dynamic Price Competition with Fixed Capacities

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  • Kalyan Talluri
  • 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 right), (b) homogeneous products (e.g., two airline rights 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 itevolves,sellersmaychangetheirpricesatanytime. 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 ton 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 Barcelona Graduate School of Economics in its series Working Papers with number 425.

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    Date of creation: Feb 2010
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    Handle: RePEc:bge:wpaper:425

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    Related research

    Keywords: Revenue management; bid-prices; subgame-perfect equilibrium;

    References

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    1. 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.
    2. 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.
    3. 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.
    4. 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.
    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. 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.
    7. Dudey, Marc, 1992. "Dynamic Edgeworth-Bertrand Competition," The Quarterly Journal of Economics, MIT Press, vol. 107(4), pages 1461-77, November.
    8. Hirata, Daisuke, 2008. "Bertrand-Edgeworth Equilibrium in Oligopoly," MPRA Paper 7946, University Library of Munich, Germany.
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