All-Pay Oligopolies: Price Competition with Unobservable Inventory Choices*
All-Pay Oligopolies: Price Competition with Unobservable Inventory Choices* Joao Montez Nicolas Schutz November 13, 2020 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. Keywords: Oligopoly, inventories, production in advance, all-pay contests. 1 Introduction Most retail markets are characterized by production in advance, as each store chooses not only its price but also its inventory level, sourced from suppliers and made readily available *We thank the Editor, Christian Hellwig, three anonymous referees, Simon Anderson, Heski Bar-Isaac, Pierre Boyer, Mathijs Janssen, Bruno Jullien, S´ebastienMitraille, Volker Nocke, Hans-Theo Normann, Martin Peitz, Alexander Rasch, R´egisRenault, Markus Reisinger, Patrick Rey, Armin Schmutzler, Sandro Shelegia, Andre Speit, Yossi Spiegel, Roland Strausz, Thomas Tr¨oger,Thomas von Ungern-Sternberg, Ralph Winter, as well as many seminar and conference participants for helpful comments and suggestions.
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