Cartel Stability in an Exhaustible Resource Model

Abstract

A simple oligopolistic common-pool exhaustible resource game is considered. By analysing punishment strategies, including optimal punishments, it is possible to determine which cartel agreements are implementable in a non-cooperative play of the game. Joint-profit-maximizing allocations are sustainable for sufficiently low discounting, but in general it is shown that no folk theorem exists for this model. In particular, for sufficiently high elasticities of demand, it is shown that optimal punishments are not sufficiently severe to enforce most stationary symmetric extraction paths, thus confirming the hypothesis that sufficient market power is needed for a cartel to be stable. A simple common-property exhaustible resource model in a discrete-time, infinite-horizon framework will be considered. By looking at 'punishments' levelled against firms that deviate from an implicitly agreed extraction path, it is possible to examine the extent to which different allocations can be supported as non-cooperative equilibria. We shall pay particular attention to the question of whether punishments can be severe enough to sustain implicit agreements that maximize joint profits, and to the idea (Pindyck 1979) that cartel stability depends crucially upon market power. This non-cooperative approach to the question of cartel stability has been used by Porter (1983) in a repeated oligopoly model with imperfect information (see also Green and Porter 1984). While we assume perfect observability here, the model is compli- cated by the existence of a state variable in the form of the resource stock; moreover, we shall apply the idea of optimal punishments in this dynamic model. (Abreu et al. (1986, 1990) have looked at general properties of the equilibrium set in the Green-Porter model.) What we find is that, by considering a more general strategy space than that conventionally studied, cooperative extraction rates are sustainable in equilibrium, for low enough discount rates, in contrast to the 'over-extraction' results often obtained. Moreover, in this model the predictions concerning the range of possible sustainable equilibria as the discount rate approaches zero are sharp. While the cooperative solution belongs to the set of equilibria for low discount rates, constant symmetric extraction paths below the maximum possible rate fail to be sustainable in the limit if the elasticity of demand is sufficiently high.' Thus, for low discount rates only extraction rates close to the cooperative rate are sustainable together with 'disagreement equilibria' which involve very high extraction rates. These results are in sharp contrast to those obtained from repeated games where in the limit the range of equilibria is very large. The difference is explained by the existence of a state variable in the form of a declining stock of the resource. These results are also in contrast to what can happen in our model when the demand elasticity is lower: all constant symmetric extraction paths may be sustainable for low enough