Tall tales and fat tails: the science and economics of extreme warming

Consideration of Eq. 2 for the equilibrium response to doubling the atmospheric CO ₂ concentrations shows that λ in Eq. 4 can be equated with \(\frac{F_{2\times CO_2}}{S}\) in Eq. 1. We have thus arrived at a formulation that is almost identical to the temperature equation in the DICE model (Eq. 1).¹

The physical basis of Eq. 4 makes it easier to relate different sources of uncertainty in Eq. 1 to the various sources of uncertainty discussed in the physical science literature. We focus here on uncertainty about the climate sensitivity, S, and the heat capacities, C _eff and C _up.

2.2 Effective heat capacity

Examination of Eq. 1 suggests that uncertainty in transient temperature change is dependent not just on climate sensitivity but also on uncertainty in the parameters ξ ₁ and ξ ₃ (and also in \(F_{2\times CO_2}\), although \(F_{2\times CO_2}\) is considered well known). Historically the majority of the heat has remained in the upper oceans (Levitus et al. 2000; Lyman et al. 2010), so we focus our attention on ξ ₁; ξ ₃ only being important for the transfer of heat to the deep oceans.

Comparing Eqs. 1 and 4 shows that \(\xi_1 = \frac{\Delta t}{C_{\rm up}}\). The scientific literature does not provide constraints on C _up directly, but Frame et al. (2005) present uncertainty estimates for effective heat capacity, C _eff, giving 95 % confidence intervals of \((< 0.2~\textrm{GJm}^{-2}\textrm{K}^{-1}, > 1.7~\textrm{GJm}^{-2}\textrm{K}^{-1})\) for the latter half of the 20th century.² A first approximation of ξ ₁ from the observational data would simply be \(\xi_1 = \frac{\Delta t}{C_{\rm eff}}\), but this can be refined using values from the first period of the DICE model to give the implied ratio of C _up to C _eff at the beginning of the 21st century; a value unlikely to be substantially different to that in the latter half of the 20th century and therefore comparable with observations. Equation 5, based on Eqs. 2 and 4, shows this relationship.

$$ \begin{array}{rll} \xi_1 &=& \frac{\Delta t}{C_{\rm up}} \approx \frac{\Delta t \left[ F_{1} - \frac{F_{2 \times CO_2}}{S} T_{1} \right]}{C_{\rm eff} \left[ F_{1} - \frac{F_{2 \times CO_2}}{S} T_{1} - \xi_3 (T_{1}-T^{LO}_{1}) \right]} \label{Auncert} \end{array} $$

(5)

The default DICE value for ξ ₁ is 0.208, which translates into a value for C _eff of \(1.8~ \textrm{GJm}^{-2}\textrm{K}^{-1}\), on the high side of what the observations suggest is likely.³ A natural question to ask, then, is what the consequences would be of assuming a lower heat capacity.

With a lower heat capacity, a given energy input produces more rapid warming, while the equilibrium temperature of the system is not affected. The main consequence, then, is to ‘front-load’ warming. As Fig. 1 illustrates, this front-loading can more than compensate for a lower climate sensitivity in the short term, producing higher temperatures even with a lower climate sensitivity. For a given S, a lower effective heat capacity will tend to lead to faster warming, causing greater damages to occur in the nearer term, when the relieving effect of discounting is least felt. Uncertainty about the value of the effective heat capacity could therefore have important implications for the economic analysis of climate change.

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In this section we empirically address the issues raised above. Our ultimate aim is to investigate how sensitive the economic value of a representative mitigation policy is to uncertainty about the shape of the tail of the climate sensitivity distribution and about the effective heat capacity of the system.

To evaluate the net economic benefits of mitigation, we conduct a typical comparison between a business-as-usual emissions scenario and a scenario in which there is intervention in the economy to abate emissions. The business-as-usual scenario is as standard in DICE-2009, while our mitigation scenario controls emissions so as to prevent the mean atmospheric concentration of CO ₂ from exceeding 500ppm in a stochastic set-up (see supplementary information for details).

We begin by investigating uncertainty about the upper tail of the climate sensitivity pdf. We take the standard deterministic version of DICE-2009 and replace the best guess for the climate sensitivity parameter with a pdf so that Monte Carlo simulations can be performed. We first fit a log-Normal pdf to the IPCC’s expert subjective confidence interval (a mode of 3° C with a density of 0.67 between 2° C and 4.5° C), plotted in both panels of Fig. 2. For convenience, and because this is the upper bound of the IPCC likely interval, we speak of everything above 4.5° C as the tail of the distribution. This pdf has 0.25 density in the tail.

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We then perturb this log-Normal pdf to obtain two further fat-tailed distributions, by shifting probability mass around within the tail while keeping the distribution below 4.5° C fixed. In addition to the tail of the log-Normal itself, which is among the thinnest of fat-tailed distributions, we use the heavier tail of the distribution derived by Roe and Baker (2007). As an intermediate between these two cases, we use the tail of the log-logistic distribution fitted to simulation output from Stainforth et al. (2005).⁴ Note that, because the mass in the tail is fixed, these two distributions will have lower densities than the IPCC-distribution in the lower part of the tail, but higher densities in the upper part of the tail that is typically associated with extreme warming. The resulting pdfs are displayed in panel (b) of Fig. 2.

The Monte Carlo simulations run the DICE model out 400 years into the future for an ensemble of 100,000 climate sensitivities (see supplementary information). They reveal how the distribution of transient temperature trajectories change when we use a different pdf for the climate sensitivity (see Fig. 3). The lower quantiles and the mode are of course fixed by design, but there is a clear fanning out of the upper tail, which increases the mean of the distribution slightly. Comparable distributions can be plotted with and without the mitigation policy. This gives us what we need to conduct a standard, welfarist economic evaluation—i.e. we compute the expected discounted utility of the two emissions scenarios, and the difference between them is the economic value of mitigation. Technically, we measure the change in the stationary equivalent, defined as the difference in welfare between the two constant consumption paths that produce welfare equivalent to the expected values from each of the two emissions scenarios (see supplementary information for details). We use a standard constant relative risk aversion (CRRA) utility function with a coefficient of relative risk aversion of 1.5, and a pure rate of time preference of 1.5 %, both default values in DICE-2009. The results are reported in Table 1.

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Table 1

Value of 500 ppm policy

Climate sensitivity	Increase in stationary equivalent (%)
distribution	Nordhaus damages	High damages
IPCC AR4	0.31	0.47
Stainforth et al. (2005)	0.34	0.75
Roe and Baker (2007)	0.35	76.70

Reading first the column entitled ‘Nordhaus damages’, which describes the results for the standard version of DICE-2009, notice that the shape of the tail does not appear to matter much for the value of the policy. Going from the thinnest (IPCC) to the fattest tail (Roe & Baker) increases the economic value of the policy by only 0.04 percentage points. But these results are for Nordhaus’ specification of the damage function, which has been criticized for being too sanguine about the economic impact of extreme warming (Weitzman 2012; Ackerman et al. 2010), the very scenario that interests us most. The damage function is an especially disputed and speculative element of any IAM because there are no data to constrain it at high temperatures. This makes it difficult to adjudicate between these positions on empirical grounds, though one might perhaps think it unreasonable that Nordhaus’ default function implies ‘only’ the equivalent of a 17 % loss of global GDP for a temperature increase of 10° C above the pre-industrial level, and less than a 50 % loss for a temperature increase of 20° C. One cannot help but wonder what tall tales need be imagined to account for such survival scenarios. Nordhaus’ damage function effectively assumes that catastrophic climate change is impossible.

It seems reasonable, then, for us to perform our experiments with an alternative damage function too. In Nordhaus’ specification, climate change damages increase as a quadratic function of temperature. To account for the potentially catastrophic consequences of extreme warming, others have suggested employing a more convex damage function (Weitzman 2012; Ackerman et al. 2010). We achieved this by adding a higher-order term, which returns very similar results to Nordhaus’ damage function for small temperature increases but much higher damages for large temperature changes. Our higher-order term corresponds to that used in equation 9 Dietz and Asheim (2012), with the coefficient on the higher-order term assuming the value 0.082 (the mean value in Dietz and Asheim 2012). Others have proposed even more convex damage functions (Weitzman 2012).

As the right-most column of Table 1 shows, with our ‘high’ damage function the precise shape of the tail matters hugely for the value of the policy. In going from the thinnest to the fattest tail, the economic value of the policy jumps from increasing consumption by a mere 0.47 %, to increasing it by fully 76.7 %. This clearly illustrates the role of the tail in economic analysis of climate change. Thus, provided we do not exclude the possibility of a climatic catastrophe, as Nordhaus’ damage function effectively does, economic assessments of climate policy are highly sensitive to what is assumed about the precise shape of the tail of the climate sensitivity distribution.

Until now, the discussion of uncertainty about extreme warming has focused exclusively on the climate sensitivity. This reflects the primary focus in the economics and physical climate science literatures. As we discussed in Section 2, however, transient temperature changes are also strongly influenced by the effective heat capacity of the system. In particular, uncertainty about the heat capacity has a strong impact on our ability to forecast nearer term transient temperatures. A lower heat capacity front-loads warming, and consequently alters the time profile of economic benefits associated with emissions abatement.

DICE implicitly assumes an effective heat capacity of \(1.8~\textrm{GJm}^{-2}\textrm{K}^{-1}\) (see Section 2), which is on the high side of what is considered plausible (Frame et al. 2005), and a higher effective heat capacity suppresses large temperature increases in the short term. This is visible when we plot transient temperature change for heat capacities of approximately \(1.2~\textrm{GJm}^{-2}\textrm{K}^{-1}\) and \(0.6~\textrm{GJm}^{-2}\textrm{K}^{-1}\) (see Fig. 4), values closer to the median and lower end of the distribution in Frame et al. (2005).⁵

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The results in Tables 1 and 2 are best understood in the context of the temperature changes necessary to produce catastrophic economic damages. The economic criterion used to value future consumption assumes the willingness to pay is extremely high to avoid an outcome where consumption is extremely low. Any future periods with near-zero consumption will therefore completely dominate the calculation of the value of the policy, however far off in the future they are. A policy that can avert or even postpone such extreme warming will appear immensely valuable. With Nordhaus’ damage function, damages rise very slowly as temperatures rise. Because of thermal inertia, there is no value of the climate sensitivity (or reasonable combination of climate sensitivity and effective heat capacity) that can produce sufficient temperature change to drive economic output close to zero over the next 400 years. Consequently, uncertainty about the tail shape is not all that important (for completeness, Table 2 is replicated with Nordhaus’ damage function in the supplementary materials). With our high damage function, on the other hand, damages rise faster with temperatures. As a consequence, it is possible to reach sufficiently extreme temperatures under the business-as-usual scenario within the modeling horizon, but not with the 500ppm policy in place. With a sufficiently fat tail of the climate sensitivity pdf, therefore, our sample of climate sensitivities is likely to include some high values that result in catastrophic warming under business-as-usual. When there is the possibility of catastrophic warming, then, the value of the policy derives almost exclusively from its ability to compress down the upper tail of the distribution of transient temperature changes. The figures in Table 2 therefore closely correspond to the effects of the mitigation policy on the behaviour of the higher quantiles of the distribution, as shown in Fig. 5.

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The role of the heat capacity can also be understood in these terms. Firstly, as discussed earlier, a lower heat capacity front-loads warming, thus increasing the value of mitigation even in moderate warming scenarios. Secondly with a lower effective heat capacity, a lower climate sensitivity will be sufficient to produce extreme warming within the period of analysis. We are thus more likely to have extreme warming even with a thinner tail of the climate sensitivity distribution. This is why the red lines keep shifting higher and higher as we go from right to left in Fig. 5, and the value of mitigation derives primarily from averting these more and more catastrophic possibilities. This is also why the economic value of the policy increases so rapidly as we go from right to left in Table 2.