= Solution
With $\gamma=2$ and $T(t+s)=T(t)+\dot T s$, the marginal <damage> is $2\kappa D_0W(t)e^{gs}[T+\dot T s]$. Integrating the constant and linear terms gives
$$
\boxed{S(t)=2\kappa D_0W(t)\left[\frac{T}{r-g}+\frac{\dot T}{(r-g)^2}\right],\qquad r>g.}
$$
The prefactor is $2(0.5\times10^{-12})(0.003)(90\times10^{12})=0.27$ in the corresponding dollar–temperature–time units. For the historical scenario with $r=0.03$ and $g=0.02$, the bracket is $1/0.01+0.02/0.01^2=300$, so $\boxed{S=81\,\mathrm{USD\ per\ tonne\ CO_2}}$. For $r=0.07$, $r-g=0.05$, giving global $S=0.27(20+8)=7.56$ dollars per tonne. Restricting impacts to ten per cent gives $\boxed{S_{\rm domestic}=0.756\,\mathrm{USD\ per\ tonne\ CO_2}}$.
The factor of about 107 reduction comes from both the higher <discount rate> and the geographical restriction; it is not a change in climate physics. The administration descriptions are those of the 2019 question, not claims about present policy. With a net discount rate of only one per cent, a century-scale horizon contributes substantially, making the linear warming and exponential income extrapolations consequential assumptions.
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