Extended Oligopolies with Pollution Penalties and Rewards - Akio Matsumoto; Ferenc Szidarovszky; Hirokazu Takizawa (Discrete Dynamics in Nature and Society)
An extended n-firm oligopoly with product differentiation is considered. It is assumed that the government selects an emission standard for the industry and based on the output and technology of each firm it selects a maximum allowed amount of emission for each firm. If the actual amount is higher than the allowed maximum, then the firm has to pay a constant multiple of the excess to the government; otherwise it is rewarded similarly based on the saved emission amount. The existence of the unique interior equilibrium is first proved, and then the effect of the level of penalty or reward and that of the emission standard on the industry output and therefore on the total emission level is also examined. Time delay is introduced into the penalties the firms have to pay and into the rewards the firms receive. In analyzing the stability of the equilibrium both discrete and continuous time scales are considered. For mathematical simplicity the case of symmetric firms is analyzed. In the discrete case the various values of the delay length are examined. The equilibrium is stable if either the total industry output is sufficiently large or the common speed of adjustment of the firms is sufficiently small. In the continuous case, either the equilibrium is always stable or stability occurs if the delay is sufficiently small and at the critical value Hopf bifurcation occurs.