Optimal dynamic regulation of carbon emissions market: A variational approach

Abstract

We consider the problem of reducing the carbon emissions of a set of firms over a finite horizon. A regulator dynamically allocates emission allowances to each firm. Firms face idiosyncratic as well as common economic shocks on emissions, and have linear quadratic abatement costs. Firms can trade allowances so to minimise total expected costs, from abatement and trading plus a quadratic terminal penalty. Using variational methods, we exhibit in closed-form the market equilibrium in function of regulator's dynamic allocation. We then solve the Stackelberg game between the regulator and the firms. Again, we obtain a closed-form expression of the dynamic allocation policies that allow a desired expected emission reduction. Optimal policies are not unique but share common properties. Surprisingly, all optimal policies induce a constant abatement effort and a constant price of allowances. Dynamic allocations outperform static ones because of adjustment costs and uncertainty, in particular given the presence of common shocks. Our results are robust to some extensions, like risk aversion of firms or different penalty functions.

Green Veblen effect: Sustainability in pollution management

“…The insights of Proposition 1 differ, but remain consistent with extant academic literature (Li, 2013(Li, , 2014(Li, , 2016. For instance, the study by Aïd and Biagini (2021) revealed that the optimal abatement effort increases with the tax rate and the marginal pollution damage. This finding aligns with the results presented in (11a), where the abatement effort, u, increases with the tax rate, τ , and the degradation of pollution, δ. Additionally, the survey of Chen et al (2020) found that product prices increase with the marginal pollution damage.…”