Simultaneous and Sequential Abatement in a Differentiated Duopoly Under Linear and Quadratic Cost

In this paper, we examine the sequential versus simultaneous abatement choices of firms in a differentiated duopoly where the production decisions of firms lead to environmental damage. We consider two cases — one, where the production cost is linear and two, where the production cost is quadratic. Introducing product differentiation and quadratic cost yields nuanced results. First, considering linear production cost, we find that when the cost of abatement technology is high, the leader firm free rides on the follower firm by lowering the amount of abatement. Second, when the degree of product differentiation is very low, the abatement of the leader in the sequential case is lower than the abatement in the simultaneous case. For the follower, the opposite holds. Furthermore, the total abatement under the sequential case is lower than in the simultaneous case. The above findings continue to hold under quadratic cost. However, the comparison between sequential and simultaneous cases depends on the values of the cost of production and the degree of product differentiation. We then consider an alternative game in which the regulator can pre-commit and choose the tax before the firms make their abatement choices. In this case, the abatement investments are equal regardless of whether the firms move simultaneously or sequentially. Finally, we find that, under some parametric configurations, the total abatement when the regulator cannot pre-commit is greater than that when the regulator can pre-commit.