Tacit Collusion in a Dynamic Duopoly with Indivisible Production and Cumulative Capacity Constraints

This paper studies a dynamic, quantity setting duopoly game characterized as follows: Each firm produces an indivisible output over a potentially infinite horizon, facing the constraint that its cumulative production cannot exceed an initially given bound. The environment is otherwise stationary; the remaining productive capacities of the firms at any moment are common knowledge; the firms choose production plans contingent on these capacities which are mutual best responses in every contingency. The resulting Markov Perfect Equilibria are analyzed using a two—dimensional backward induction, and compared with the equilibria which emerge when precommitment to time paths of output is possible. It is shown that the ability to precommit can be disadvantageous; that collusion in Markov Equilibrium is facilitated by the symmetrical placement of the firms; and that having greater capacity confers basic strategic advantage on a firm by enabling it to credibly threaten future production. The model solves an open problem in the theory of exhaustible resource economics by imposing subgame perfection in a resource oligopoly with independent stocks. It also formalizes the intuition that, when indivisibilities are important, tacit coordination of plans so as to avoid destructive competition is facilitated by establishing a convention of "taking turns" — that is, a self—enforcing norm of mutual, alternate forebearance. * The author thanks Ariel Halperin, Eric Maskin, Robert Rosenthal and John Vickers for helpful discussions; and the Bradley Foundation, the Olin Foundation, and the Center for Energy Policy Research, MIT, for research support. ** This paper was written while the author was in residence as a Visiting Scholar at the Department of Economics, MIT.