 Solution

Equating marginal mitigation cost to the previous marginal damage givesUsing $\dot{T}=\kappa E$ from the cumulative-emissions approximation,Thus $T\propto t^{1/\gamma }$ at late times and $E\propto t^{-(\gamma -1)/\gamma }$. Warming slows but never stops, and cumulative emissions remain unbounded. This follows from a cost model in which zero emissions is infinitely costly and from a marginal-damage rule with constant positive $r-g$.

The approximation eventually fails if cumulative emissions leave the range of constant TCRE, if large damages suppress economic growth, if technological change alters $M_0$, or if fossil resources and mitigation options impose different limits. The slowly varying-temperature approximation also needs $\dot{T}/[T(r-g)]\ll 1$. Thresholds, adaptation, uncertainty and unequal impacts are omitted; the formula is a property of this model rather than a universal policy trajectory.