Collusion in an investment game
If collusion was often considered in a market facing uncertainty (Bag-well and Staiger (1997)), or imperfect information (Athey and Bagwell (2008), Harrington and Skrzypacz (2010)), the relationship between collusion and investment is less known. That is the purpose of what follows. This work studies a dynamic game in discrete time with in nite periods. In each period rms make two decisions, investment (or disinvestment) in production capacity and the quantities they produce. Companies can choose to increase or reduce capacity. The irreversibility of decisions is modeled by the difference between purchase price and sale of building (when the gap is zero, the decisions are totally reversible). In each period rms are competing in Cournot, the quantities produced are of course limited by production capacity. The model is presented in section 1.3. In comparison with the Account, Jenny and Rey (2003), capacity is endogenous, modi ed in each period, and the game of competition is a game of Cournot competition while Account Jenny and Rey (2003) are interested in a game competition in Bertrand-Edgeworth. In comparison with Boyer, Lasserre and Moreaux (2010), demand is not random, so there is no uncertainty and the equilibrium concept used is far less restrictive than the Markov equilibrium. Production capacities are not discrete and are not irreversible. These papers are presented in section 1.1 and 1.2. To de ne the collusion, it is necessary to determine a non-collusive equilibrium. In a repeated game, this benchmark equilibrium is constituted by the repetition of the equilibrium of the one shot game. In a stochastic game, as here, we can not implement this solution. We must therefore de ne a reference equilibrium. This point is develloped in section 2.1. In section 2.2 and 2.3 we prove the existence and the unicity of this benchmark equilibrium. If we discretize the game (ie the actions of the players belong to a nite space, which can be chosen in nitesimally large), section 3.1 presents a folk theorem (the proof uses the result of Horner, Sugaya , Old and Takahashi (2010)). This theorem tells us that when the discount rate tends to 1, the set of equilibrium payoff vectors tends to the set of equilibrium payoff vectors of the in nitely repeated Cournot game (without cost or production capacity). The theorem is therefore a borderline result, which gives an equivalence between this game and the Cournot game (in nitely repeated) when players are in nitely patient. Finally, section 3.2 studies conditions for the existence of a speci c collusive equilibrium (the Grim-Trigger equilibrium in capacities).
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