Controlling polluting firms: Nash and Stackelberg strategies

In this paper we model the conflict between the group of polluting firms of a country and the social planer of the same country which attempts to control the volume of emissions generated during the production process. Both players of the game have their own control policies which are the rate of emissions on behalf the polluting firms and the rate of pollution control (e.g. abatement or taxation) on behalf the home country. The common state variable of the model is the number of the polluting firms, which is better to minimized through the country’s control policy, but beneficial to maximized on the polluters’ side. From the game theoretic point of view the model setup is very simple and belongs in to the special class of differential games also called state separable differential games. An important property for these games is that the open-loop Nash equilibrium coincides with the Markovian (closed-loop) equilibrium and in the case of hierarchical moves the analytical solutions are easy 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 analytical expressions of the controls for both players as well as the steady state stock of the polluting firms. A sensitivity analysis of the crucial variables of the model 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 play as the leader of the hierarchical move game.