Dynamic approaches for the evaluation of the environmental policy efficacy in a nonlinear Cournot duopoly with differentiated goods and emission charges

According to the existing empirical literature the price and the exchanged quantity volatility observed in real-world markets may be explained in terms of the endogenous fluctuations generated by the presence of nonlinearities. We then replace with a sigmoid adaptive best response mechanism, characterized by the presence of two horizontal asymptotes, the linear partial best response mechanism considered in Mamada and Perrings (2020), where the effect produced by quadratic emission charges on the dynamics of a Cournot duopoly model with homogeneous goods was investigated. Due to the sigmoid adaptive mechanism, output variation in each period depends also on current production volume. Moreover, the sigmoid nonlinearity, in addition to being well suited to describe the bounded output variations caused by physical, historical and institutional constraints, makes the model able to generate interesting, non-divergent dynamic outcomes, despite the linearity of the demand function and of marginal costs. Additionally, following the suggestion in Mamada and Perrings (2020), we deal with the more general case of differentiated products. Beyond analytically studying the stability of the unique steady state, coinciding with the Nash equilibrium, and the effect produced by the main parameters on the stability region, we propose two dynamical approaches which allow to evaluate the environmental policy efficacy when the Nash equilibrium is not stable and thus the standard comparative statics technique does not fit for the purpose. In particular, the former approach, which is based on a comparison of emissions for different levels of charges, shows that, also in case the Nash equilibrium is not stable, the considered environmental policy may be effective both with complements and with substitutes. The latter approach, consisting in a comparison of emissions along non-stationary trajectories and along the equilibrium path, in the proposed experiments highlights that emissions are larger along non-stationary trajectories. This gives us the opportunity to show how to act on the level of the asymptotes of the sigmoid adjustment mechanism to reduce output variations, reaching at one time a complete stabilization of the system and limiting pollution.