We consider a renewable resource being exploited in common by firms that compete both in the output market and in the exploitation of the resource. We show that the introduction of the slightest cost differentiation among the firms can have a drastic effect on the nature of the equilibria that may be expected as compared to the identical cost case. To do this, we take as a benchmark case a Markov Perfect Nash Equilibrium that exists with identical cost firms, with the property that the firms play a linear strategy up to some endogenously determined threshold level of the stock and the static Cournot equilibrium strategy beyond that threshold. Having shown that an equilibrium of that nature is not sustainable with asymmetric cost, we fully characterize a Markov Perfect Nash Equilibrium of the differential game for that case.