Optimal dynamic regulation of carbon emissions market

In this article, we deal with optimal dynamic carbon emission regulation of a set of firms. On the one hand, the regulator dynamically allocates emission allowances to each firm. On the other hand, firms face idiosyncratic, as well as common, economic shocks on emissions, and they have linear quadratic abatement costs. Firms can trade allowances so as to minimise total expected costs, which arise from abatement, trading and terminal penalty. Using variational methods, we first exhibit in closed-form the market equilibrium in function of the regulator's dynamic allocation. We then solve the Stackelberg game between the regulator and the firms. The result is a closed-form expression of the optimal dynamic allocation policies that allow a desired expected emission reduction. The optimal policy is unique in the presence of market impact. In absence of market impact, the optimal policy is non unique, but all the optimal policies share common properties. The optimal policies are fully responsive, and therefore induce a constant abatement effort and a constant price of allowances. Our results are robust to some extensions, like different penalty functions. We conclude by providing analytical and numerical comparisons of the optimal dynamic policy with three existing policies, namely static allocation, Market Stability Reserve and pure tax mechanisms.