Dynamic Price Competition with Fixed Capacities

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of competition, Dudey (1992), somewhat surprisingly, shows that the dynamic version of BertrandEdgeworth duopoly competition with a known and fixed market size has a unique subgame-perfect equilibrium. The work of Dudey (1992) has some bearing on RM as (i) in many RM situations, such as an airline market served by low-cost carriers or a car rental market offering comparable vehicles, price is the main consideration for customers and there is very little brand or product differentiation; (ii) a dynamic model of competition reflects operational RM price-setting behavior better; and (iii) it holds the promise that by modeling the dynamics of competition, one can obtain better predictions of equilibrium behavior and thereby get some managerial insight into the nature of RM competition. It therefore appears to be the right starting point to study competition in RM, with the objective of gradually relaxing some of the assumptions of his model and making it more relevant to RM. In this paper, we study a dynamic model of price competition between sellers who offer an identical resource with limited capacities and a stochastic market demand. Our model, though stylized and simplified, is intended to reflect the competitive considerations in RM. We generalize and simplify Dudey’s (1992) results, showing that under a more general model, the equilibrium behavior of a duopoly (that we later generalize to n firms) can be characterized by a simple “competitive bid-price” type of rule: each firm calculates a competitive (equilibrium)
Victor Martínez-de-Albéniz
IESE Business School, University of Navarra, 08034 Barcelona, Spain, valbeniz@
Kalyan Talluri
ICREA and Universitat Pompeu Fabra, 08005 Barcelona, Spain, kalyan.talluri@
Martínez-de-Albéniz and Talluri: Dynamic Price Competition with Fixed Capacities
Management Science 57(6), pp. 1078–1093, © 2011 INFORMS
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reservation value, and in each period, the firm with the lower reservation value makes the sale at a price equal to the other firm’s reservation value. This is the unique subgame-perfect equilibrium. This interpretation is appealing, as many RM systems use the concept of bid-price control: they sell the resource at prices higher than a threshold value that is a function of remaining capacity and time. In that respect, our results show how competition influences the bid-price mechanism: in equilibrium, even when both firms are aware of the valuation of the customer, they do not use the marginal value to set a threshold value (as in bid-price control), but use the competitor’s equilibrium reservation value to determine (i) whether to sell or not and (ii) the actual price at which to sell. Our results suggest that as time passes (without arrivals), equilibrium prices decrease and jump up when a unit is sold. Note that this behavior is similar to the one obtained in Gallego and van Ryzin (1994) in a setting where the arrival rate of customers depends on price. In this setting, prices decrease over time because the seller should increase per-period demand and match it to the existing capacity when there is less time to sell. In contrast, in our model, prices decrease because, even though per-period demand cannot be increased and valuations are fixed and known, the competitive intensity to sell the existing capacity is stronger with shorter time and pushes prices lower. The price patterns here and in Gallego and van Ryzin (1994) hence share many similarities, because both share the same fundamental pressure to sell the same capacity in less time. The main difference is that in our model, strategic interactions play an important role. Our analysis also highlights the critical role of capacity asymmetries in determining the degree of competition and the profits obtained. When capacities are unequal, the lower-capacity firm always has a lower reservation value, sells first, and continues to sell until it exhausts capacity. The higher-capacity firm has no choice but to wait until the other firm sells out and thus profits by charging monopoly price for the residual demand. Nevertheless, its competitive presence forces the lower-capacity firm to price much lower than the monopoly price—even when both firms know that the customers are price insensitive. The degree of profits will depend on the overall market size. The rest of the paper is organized as follows. In §2, we review the relevant literature from economics and operations management. We then present our model in §3. In §4, we develop the main analytical results for a duopoly and an oligopoly with n firms; we then extend the results to uncertain valuations with capacities of one each, different customer types, and differentiated products. In §5, we discuss some of the implications of our results. Finally, in §6, we conclude