Optimal Dynamic Regulation of Carbon Emissions Market: A Variational Approach

30 Pages Posted: 22 Mar 2021

See all articles by René Aïd

René Aïd

Université Paris-Dauphine

Sara Biagini

LUISS University

Date Written: February 24, 2021


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.

Keywords: Stochastic optimization, environmental economics, cap and trade, linear quadratic problem, variational methods, market equilibrium, social cost minimisation

JEL Classification: C62, E63, H23, Q52, Q58

Suggested Citation

Aid, Rene and Biagini, Sara, Optimal Dynamic Regulation of Carbon Emissions Market: A Variational Approach (February 24, 2021). Université Paris-Dauphine Research Paper No. 3792384, Available at SSRN: https://ssrn.com/abstract=3792384 or http://dx.doi.org/10.2139/ssrn.3792384

Rene Aid (Contact Author)

Université Paris-Dauphine ( email )

Place du Maréchal de Lattre de Tassigny
Paris, 75016

Sara Biagini

LUISS University ( email )

Viale Romania 32
Rome, 00197

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