Temperature, abatement and SRM
When the concentration of greenhouse gases increases in the atmosphere it alters the balance between incoming solar radiation and outgoing terrestrial radiation, resulting in an increase in the mean global temperature of Earth. Radiative forcing describes how the radiation balance is altered by human activity. Radiative forcing, R, is a function of the concentration of CO2 in the atmosphere, S, relative to the preindustrial level, S
0:
$$ R=\beta ~\ln\big(S/S_0\big) $$
(1)
where, according to the IPCC (2007), β = 5.35 watts-per-meter-squared [Wm − 2]. Abatement, which we denote by A, refers to measures that reduce the concentration level of CO2 in the atmosphere. In particular, assume that S = S
BAU
− A, where S
BAU
is the business as usual concentration of CO2 in the atmosphere measured in parts per million [ppm].
Changes in mean global temperature, ΔT—measured in °C—are defined as a linear function of radiative forcing, R:
$$ \Delta T=\lambda R $$
(2)
where λ is the climate sensitivity parameter with units °C m2/W.
When SRM is introduced in the model, the relation between CO2 concentrations and temperature is altered. We measure SRM, G, in terms of its radiative forcing potential and, since temperature change is a linear function of radiative forcing, Eq. 2 can be written as:
$$ \Delta T(A,G)=\lambda \left(\beta ~{\rm Ln}\left(\frac{S_{BAU}-A}{S_0}\right)- G\right) $$
(3)
Economic damages
We represent total climate damages as the sum of impacts from three different sources: temperature, SRM and uncompensated CO2 damages (e.g. ocean acidification). Following Nordhaus (2008), we assume temperature damages are quadratic. Following Brander et al. (2009), damages from ocean acidification are also quadratic on the concentration of CO2. We assume that SRM damages are also a quadratic function of the total level of SRM.Footnote 1 To be able to compare the different sources of impacts, we express damages in terms of reductions in economic output. Thus, total damages are given by:
$$D(A,G)=\eta_S(S_{BAU}-A)^2+\eta_T\lambda^2\left(\Delta T(A,G)\right)^2+\eta_GG^2 $$
(4)
where \(\eta_S(S_{BAU}-A)^2\) are the damages caused by ocean acidification and other uncompensated damages from CO2, \(\eta_T\lambda^2\left(\Delta T(A,G)\right)^2\) are damages caused by temperature changes, and \(\eta_GG^2\) are the damages caused by the side-effects of SRM. In Eq. 4, when A equals S
BAU
and G equals zero, damages are zero. However, when A is less that S
BAU
, damages are always positive, showing the inability of SRM to perfectly compensate for greenhouse gas driven climate change (see bottom panel in Fig. 2). That is, although technically SRM can reduce temperature changes to zero, it may do so at the expense of other economic damages.
Implementation costs
We assume that abatement costs are increasing and convex. In particular, following Nordhaus (2008), we have:
$$\Lambda(A)= K_A A^{\alpha} $$
(5)
where K
A
has units [$/ppm] and α = 2.8.
Following Keith and Dowlatabadi (1992) we assume that SRM costs are linear and given by
$$\Gamma(G)= K_G G $$
(6)
where K
G
has units [$/(Wm − 2)].
Total social costs are the sum of the implementation costs, given by Eqs. 5 and 6, and the economic damages given by Eq. 4. The optimal policy consist of the level of abatement and the level of SRM that minimize total social costs.
Calibration
We use the year 2100 as our planning horizon, a common target in the analysis of climate change policy.Footnote 2 To calibrate our model, we use the projected costs and damages in 2100 reported by the DICE-2007 model (Dynamic Integrated Model of Climate and the Economy) (Nordhaus 2008). We complete the information needed for our calibration using data from the IPCC (2007) and publications related to the costs of SRM. The information given below is, unless otherwise noted, from Nordhaus (2008). The assumptions and calibrated values are summarized in Table 1.
Table 1 Calibration of model
We calibrate costs and damages as percentages of global GDP, when we report dollar values we assume global GDP to be around $50 trillion per year (World Bank, World Development Indicators). Although not relevant for our study, incorporating discounting is simple. For example if we assume a discount rate of 1%, the yearly GDP value would be equivalent to $33 trillion. If we assume a discount rate of 7%, yearly GDP would be $7 trillion. Economic growth is equally easy to introduce. Introducing economic growth at a rate of 2.5% will yield a yearly GDP value of $200 trillion. Considering a discount rate close to the rate of economic growth would leave the yearly value of GDP at around $50 trillion.
There is insufficient information to allow us to quantify the risks of SRM, η
G
, with any confidence, so we treat them parametrically. In Section 3 we analyze optimal policy as a function of η
G
and in Section 4 we introduce uncertainty and learning on η
G
.