Valuing Unpolluted Natural Resources: A Model of Optimal Consumption and Differential Games

This study investigates the evolution of a common, unpolluted natural resource and the optimal allocation of a consumption good that generates pollution. The resource is spatially distributed across two regions, where it regenerates and may flow between regions at specified rates. In each region, a single agent decides how much of the consumption good to allocate, with consumption directly generating pollution. Using a continuous-time framework, we first formulate an optimal control problem in which a social planner coordinates the two agents’ actions to maximise their joint payoff. We then consider the case where the social planner is removed and the agents act independently in a noncooperative, nonzero-sum differential game, each maximising their own payoff. A key feature of the model, in both cooperative and noncooperative scenarios, is an explicit term in the payoff function reflecting the value of resource conservation. Our analysis provides a comprehensive set of controls and equilibrium strategies, outlining the optimal consumption paths and paving the way for more complex multi-agent models.