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Dynamic Price Competition with Fixed Capacities

Author

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  • Kalyan Talluri
  • Victor Martínez-de-Albéniz

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 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.

Suggested Citation

  • Kalyan Talluri & Victor Martínez-de-Albéniz, 2010. "Dynamic Price Competition with Fixed Capacities," Working Papers 425, Barcelona Graduate School of Economics.
  • Handle: RePEc:bge:wpaper:425
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    References listed on IDEAS

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    1. 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.
    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, 1986. "Price competition in a capacity-constrained duopoly," Journal of Economic Theory, Elsevier, vol. 38(2), pages 238-260, April.
    4. 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.
    5. Marc Dudey, 1992. "Dynamic Edgeworth-Bertrand Competition," The Quarterly Journal of Economics, Oxford University Press, vol. 107(4), pages 1461-1477.
    6. 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.
    7. Hirata, Daisuke, 2008. "Bertrand-Edgeworth Equilibrium in Oligopoly," MPRA Paper 7946, University Library of Munich, Germany.
    8. 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.
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