Résumé
We consider a discrete time, infinite horizon dynamic game of groundwater extraction. A Water Agency charges an extraction cost to water users, and controls the marginal extraction cost so that it depends linearly on total water extraction (through a parameter n) and on rainfall (through parameter m). The water users are selfish and myopic, and the goal of the agency is to give them incentives them so as to, at the same time, improve their total welfare and improve the long-term level of the resource. We look at this problem in several situations for a linear-quadratic model. In the first situation, the parameters n and m are considered to be fixed over time, and the Agency selects the value that maximizes the total discounted welfare of agents. We analyze this solution, from the economic and environmental point of view, as a function of model parameters,including the discount factor that is used. A first result shows that when Water Agency is patient (discount factor tends to 1) optimal marginal extraction cost asks for strategic interactions between agents. In a second situation, we look at the dynamic Stackelberg game where the Agency decides at each time what cost parameter they must announce in order to maximize the welfare function. We present the sensitivity analysis of the solution for a small time horizon, and present a numerical scheme for the infinite-horizon problem.