Robustness of a simple rule for the social cost of carbon

The optimal social cost of carbon is in general equilibrium proportional to GDP if utility is logarithmic, production is Cobb-Douglas, depreciation in 100% every period, climate damages as fraction of production decline exponentially with the stock of atmospheric carbon, and fossil fuel extraction does not require capital. The time profile and size of the optimal carbon tax corresponding to this simple formula are not robust to more convex climate damages, smaller elasticities of factor substitution and varying coefficients of relative intergenerational inequality aversion. The optimal timing of energy transitions and the amount of fossil fuel reserves to be locked up in the earth are also not well predicted by this framework. Still, in terms of welfare and global warming the simple formula manages to get quite close to the first best.