Two versions of the tragedy of the commons

The commons are a one input-one output production process with increasing marginal cost. In the everage return game, each agent chooses his input contribution and total output is shared in proportion to individual contributions. In the average cost game, each agent chooses his output share and total input cost is shared in proportion to individual demands. The tragedy is that the non cooperative equilibrium results in inefficient overutilization of the technology. We prove formally the tragedy when individual preferences are convex and both goods are normal. This result has not been proved previously on such a general preference domain. We also show that overutilization is less severe in the average cost game than in the average return game.