Optimal dynamic regulation of carbon emissions market: A variational approach

TL;DR

This paper develops a variational approach to derive optimal dynamic regulation policies for carbon emissions markets, accounting for economic shocks, trading, and abatement costs, with closed-form solutions for equilibrium and policies.

Contribution

It introduces a novel variational method to explicitly solve for dynamic emission allowance policies in a Stackelberg game setting, including equilibrium characterization.

Findings

Optimal policies induce constant abatement efforts and allowance prices.

Dynamic allocations outperform static ones due to adjustment costs and uncertainty.

Results are robust to risk aversion and alternative penalty functions.

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.