In this paper, we analyse a linear quadratic multistage game of extraction of a common renewable resource by many players with state de- pendent constraints for exploitation and innite time horizon. We analyse social optimum and Nash equilibrium for feedback information structure and compare the results obtained in both. For Nash equilibria, we obtain a value function that is contrary to intuitions from standard linear quadratic games. We also study introduction of a tax in order to enforce socially optimal be- haviour of the players. Besides, this game constitutes a counterexample to two techniques regarded as standard in computation of Nash equilibrium and/or optimal control.