Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S25 / 3

The transient climate response to cumulative carbon emissions is $\boxed{\kappa \simeq \Delta T/\mathcal{E}}$, where $\mathcal{E}$ is cumulative emitted carbon dioxide in a stated mass unit. Here the unit is tonnes of $\mathrm{C}{\mathrm{O}}_2$, rather than tonnes of carbon. Increasing concentration weakens the incremental radiative forcing per added atmospheric tonne because forcing is approximately logarithmic. Over a substantial range this is partly offset by a weakening of carbon sinks and by the changing ratio between transient warming and forcing as ocean heat uptake evolves. These approximate compensations make warming nearly proportional to cumulative emissions; no exact cancellation follows from the logarithmic forcing law alone. The approximation is range- and pathway-dependent, and is not a theorem valid for arbitrary emissions histories.

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.

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.

With $\gamma =2$ and $T(t+s)=T(t)+\dot{T}s$, the marginal damage is $2\kappa D_{0}W(t)e^{gs}[T+\dot{T}s]$. Integrating the constant and linear terms givesThe prefactor is $2(0.5\times 10^{-12})(0.003)(90\times 10^{12})=0.27$ 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 $\boxed{S=81\mathrm{USD}\mathrm{per}\mathrm{tonne}\mathrm{C}{\mathrm{O}}_2}$. 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 $\boxed{S_{\mathrm{domestic}}=0.756\mathrm{USD}\mathrm{per}\mathrm{tonne}\mathrm{C}{\mathrm{O}}_2}$.

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.