Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S25 / 3

Put $\vartheta =T-T_0$ and assume $C>0$, $\lambda >0$. The linear differential equation for a constant forcing step isMultiplying by the integrating factor $e^{\lambda t/C}$ and integrating from zero to $t$ givesThe response initially grows at rate $F_0/C$ and tends to the equilibrium warming $F_0/\lambda$. Heat capacity per unit area and radiative-feedback units make $k_T$ an inverse time.

For logarithmic forcing $F=a\ln (C_a/C_{a0})$, where $C_a$ denotes atmospheric carbon-dioxide concentration rather than heat capacity, $\dot{F}=(a/C_a){\dot{C}}_a$. Approximately constant $\alpha$ in $\dot{F}=\alpha E$ therefore requires a roughly constant conversion from emissions to persistent atmospheric burden and a concentration range small enough to hold $a/C_a$ fixed. Carbon removal must be negligible on the modeled horizon, or be replaced by a fixed retained fraction; a realistic multiscale carbon-cycle response generally violates the permanent-step approximation.

A one-tonne pulse at time $t_e$ is $E(t)=\delta (t-t_e)$ in tonne-normalized units. It gives a persistent forcing step $F=\alpha \Theta (t-t_e)$. Convolving that step with the thermal response yieldsThere is no instantaneous temperature jump for finite $C$, although forcing jumps. Both persistence and constant coefficients are deliberate simplifications, rather than an assertion that all emitted carbon dioxide remains airborne forever.

Insert the pulse response into $D=\beta (T-T_0)$ and write $u=t^{\prime }-t_e$. For a positive discount rate $k_D$,ThereforeWith $k_D=0.03\mathrm{yea}{\mathrm{r}}^{-1}$, $k_T=0.05\mathrm{yea}{\mathrm{r}}^{-1}$ and the supplied one-tonne normalization of $A$,This is valued at the time of emission because the discount factor is relative to that date. Valuing an emission at a future date from an earlier reference date would require an additional discount factor. The infinite-horizon integral converges for $k_D>0$ under the assumed persistent forcing.

Let $Q$ be annual emissions abatement in tonnes per year. The given marginal abatement cost isUnder the stipulated global, stationary, unconstrained cost-benefit model, the optimal interior abatement equates that marginal cost to the avoided marginal damage of approximately $166.7\mathrm{USD}$ per tonne. ThusTotal mitigation expenditure is the area under the marginal-cost curve, not its endpoint value times all abated tonnes:The result also presumes baseline emissions permit this much reduction. Rational policy need not reproduce it: different discounting, damage uncertainty and risk aversion, income weights, omitted ecosystem impacts, adaptation and local air-quality benefits alter valued benefits; technology learning and adjustment constraints alter costs. Separate rational governments may abate less because some benefits accrue abroad, while cooperative policy can internalize those benefits. Future climate or technology changes also undermine a fixed present marginal-damage curve. These are changes in the optimization problem, not evidence of irrational decision-makers.