Some results on the control of polluting firms according to dynamic Nash and Stackelberg patterns

In this paper we model the conflict between the group of polluting firms in a country and any social planner in the same country who attempts to control the volume of emissions generated during the production process. Both players of the game have their own control policies, i.e., the rate of emissions on behalf of the polluting firms and the rate of pollution control (e.g., pollution abatement or environmental taxation) on behalf of the home country. The common state variable of the model is the number of polluting firms, which aims to be minimized via the country's control policy, but on the polluters' side it is beneficial to be maximized. Regarding the game model, its setup belongs to the special class of differential games, which are called 'state separable differential games'. An important property of these games is that the open-loop Nash equilibrium coincides with the Markovian (closed-loop) equilibrium and, in the case of hierarchical moves, analytical solutions are easily obtained. The game proposed here is analyzed for both types of equilibrium, i.e., Nash and Stackelberg. In the simultaneous move game (i.e., the Nash game) we find the equilibrium's analytical expressions of the controls for both players, as well as the stationary value of the stock of polluting firms. A sensitivity analysis of the model's crucial variables takes place. In the hierarchical move game (i.e., the Stackelberg game) we find the equilibrium values of the controls, as well as of the state variable. As a result, a comparison between the two types of equilibrium for the game takes place. The analysis of the comparison reveals that the conflict is more intensive (since both controls have greater values) for the case in which the polluting firms act as the leader in the hierarchical move game.