All-Pay Oligopolies: Price Competition with Unobservable Inventory Choices

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Abstract
We study production in advance in a setting where firms first source inventories that remain unobservable to rivals, and then simultaneously set prices. In the unique equilibrium, each firm occasionally holds a sale relative to its reference price, resulting in firms sometimes being left with unsold inventory. In the limit as inventory costs become fully recoverable, the equilibrium converges to an equilibrium of the game where firms only choose prices and produce to order—the associated Bertrand game (examples of which include fully asymmetric clearinghouse models). Thus, away from that limit, our work generalizes Bertrand-type equilibria to production in advance, and challenges the commonly held view associating production in advance with Cournot outcomes. The analysis involves, as an intermediate step, mapping the price-inventory game into an asymmetric all-pay contest with outside options and non-monotonic winning and losing functions. We apply our framework to public policy towards information sharing, mergers, cartels, and taxation.

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