= Solution
Let $Q$ be annual emissions abatement in tonnes per year. The given marginal abatement cost is
$$
\mathrm{MAC}(Q)=\frac{200}{10^{10}}Q\quad\mathrm{USD\,tonne^{-1}}.
$$
Under the stipulated global, stationary, unconstrained cost-benefit model, the optimal interior abatement equates that marginal cost to the avoided marginal damage of approximately $166.7\,\mathrm{USD}$ per tonne. Thus
$$
\boxed{Q_*\simeq8.33\times10^9\,\mathrm{tonnes\,year^{-1}}.}
$$
Total mitigation expenditure is the area under the marginal-cost curve, not its endpoint value times all abated tonnes:
$$
\boxed{\mathcal C(Q_*)=\int_0^{Q_*}\mathrm{MAC}(Q)\,dQ
=\tfrac12(166.7)Q_*\simeq6.94\times10^{11}\,\mathrm{USD\,year^{-1}}.}
$$
The result also presumes baseline emissions permit this much reduction. Rational policy need not reproduce it: different discounting, damage uncertainty and risk aversion, income weights, omitted ecosystem impacts, adaptation and local air-quality benefits alter valued benefits; technology learning and adjustment constraints alter costs. Separate rational governments may abate less because some benefits accrue abroad, while cooperative policy can internalize those benefits. Future climate or technology changes also undermine a fixed present marginal-damage curve. These are changes in the optimization problem, not evidence of irrational decision-makers.
Back to article page
