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...

We use the Euler equation to put forward a back-on-the-envelope rule for the global carbon tax based on a two-box carbon cycle with temperature lag, and a constant elasticity of marginal damages with respect to GDP. This tax falls with time impatience and intergenerational inequality aversion...