  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.
