Optimal policies for environmental assets under spatial heterogeneity and global awareness

The aim of this paper is to formulate and study a stochastic model for the management of environmental assets in a geographical context where in each place the local authorities take their policy decisions maximizing their own welfare, hence not cooperating each other. A key feature of our model is that the welfare depends not only on the local environmental asset, but also on the global one, making the problem much more interesting but technically much more complex to study, since strategic interaction among players arise. We study the problem first from the $N$-players game perspective and find open and closed loop Nash equilibria in explicit form. We also study the convergence of the $N$-players game (when $n\to +\infty$) to a suitable Mean Field Game whose unique equilibrium is exactly the limit of both the open and closed loop Nash equilibria found above, hence supporting their meaning for the game. Then we solve explicitly the problem from the cooperative perspective of the social planner and compare its solution to the equilibria of the $N$-players game. Moreover we find the Pigouvian tax which aligns the decentralized closed loop equilibrium to the social optimum.