 Solution

An additional tonne changes eventual temperature by approximately $\delta T=\kappa$. Linearizing the damage function gives $\delta L(t^{\prime })=\kappa \gamma D_{0}W(t^{\prime })T(t^{\prime }{)}^{\gamma -1}$. Put $s=t^{\prime }-t$, $W(t+s)=W(t)e^{gs}$, and hold $T$ approximately fixed across the effective discount horizon. The discounted integral issoIf $r\le g$, this infinite-horizon approximation diverges; a finite result then needs different growth, damages or discounting assumptions. It also treats the marginal warming from the extra tonne as effectively persistent over the relevant horizon.