The Social Cost of Carbon: Trends, Outliers and Catastrophes

211 estimates of the social cost of carbon are included in a meta-analysis. The results confirm that a lower discount rate implies a higher estimate; and that higher estimates are found in the gray literature. It is also found that there is a downward trend in the economic impact estimates of the climate; that the Stern Review?s estimates of the social cost of carbon is an outlier; and that the right tail of the distribution is fat. There is a fair chance that the annual climate liability exceeds the annual income of many people. --


Introduction
Estimates of the social cost of carbon (dioxide emissions), or the marginal damage cost of climate change are an essential ingredient to any assessment of climate policy. The social cost of carbon (SCC) is a first estimate of the Pigou tax that should be placed on carbon dioxide emissions. Indeed, if the SCC is computed along a trajectory in which the marginal costs of emission reduction equal the SCC, the SCC is the Pigou tax. Few would argue that climate policy should be set by cost-benefit analysis alone, but most economists would feel queasy if climate policy would drift too far from its optimumalthough analysts in other disciplines are less compelled by the branch of utilitarianism that is common in economics. This paper presents a meta-analysis of over 200 estimates of the SCC, and tests three hypotheses: • The Stern Review is an outlier in the literature (Tol 2006).
• The economic estimates of the impact of climate change increase over time (Schneider et al. 2007). • The uncertainty about the social cost of carbon has a fat right tail (Weitzman 2007b).
In Tol (2005), I also presented a meta-analysis of the SCC. There are four reasons for the current update. Firstly, the number of estimates has roughly doubled. Tol (2005) was part of a larger study that led to many new estimates, but other studies were published as well-and my attention was drawn to a handful of estimates I had previously overlooked. See Table A1 for the full list of estimates. 1 Secondly, the Stern Review (Stern et al. 2006) was published, provoking renewed interest in cost-benefit analyses of climate policy (Anderson 2007;Byatt et al. 2006;Carter et al. 2006;Dasgupta 2007;Dietz et al. 2007a, b;Hamid et al. 2007;Mendelsohn 2006;Nordhaus 2007a, b;Spash 2007;Stern and Taylor 2007;Tol 2006;Yohe 2006, 2007a;Yohe 2006;Yohe and Tol 2006;Yohe et al. 2007; note that these are the published papers only-various journals are preparing special issues). The Stern Review also published an estimate of the SCC. Although many newspapers publicised the Stern Review as entirely novel, its estimate is in fact number 211 in chronological order. A number of people argued that the Stern Review is an outlier. This paper formally tests this assertion. Thirdly, the Fourth Assessment Report (AR4) of the Intergovernmental Panel on Climate Change (IPCC) was published (Schneider et al. 2007). It argues that economic estimates of the impact of climate change have become more pessimistic since the previous report of 2001. This paper formally tests this assertion as well. Fourthly, Weitzman (2007b) argues that climate economics has unduly focussed on the middle of the probability distribution, and should have focussed on the tails. This paper supports that argument. Fourthly, I estimate the risk premium and the fraction of people that would be able to afford the estimated carbon tax.
Although there are now over 200 estimates of the SCC, research in this area is still less developed than one would wish. The 200 estimates of the marginal costs of climate change are based on a dozen of estimates of the total costs of climate change (Cline 1992;Fankhauser 1995;Maddison 2003;Mendelsohn et al. 2000;Nordhaus 1991, Data and Methods 211 estimates of the SCC were gathered from 47 studies. 4 See Table A1. The estimates are for different years, but most roughly represent the marginal damage costs for emissions in the year 1995, discounted to 1995, and measured in 1995 US dollars. The studies were grouped in those that were peer-reviewed and those that were not (PR in Table A1). Note that some of the more recent studies are currently under peer-review, but they are counted as gray literature until published. Some studies are based on original estimates of the total costs of climate change, while other studies borrow total costs estimates from other studies (IE in Table A1). Most studies use incremental or marginal calculus to estimate the SCC, as they should, while a few others use average impacts or an unspecified method (ME in Table 1). Some studies assume that climate changes but society does not (i.e., all income elasticities are zero), while other studies include a dynamic model of vulnerability (DM in Table A1). A few studies use entirely _________________________ 2 Note that Nordhaus and Mendelsohn are colleagues; that Fankhauser, Maddison and Tol worked with David Pearce and each other in the formative stages of their careers; and that Rehdanz used to be a PhD student of Maddison and Tol. 3 Some people would argue that climate change is not a marginal change, and that therefore the marginal damage costs of carbon dioxide emissions are an inappropriate measure. This reasoning is incorrect, as any emission reduction has a only a small effect on climate change. The social cost of carbon should be used to inform emission reduction. 4 Most of the estimates of the SCC are along a business as usual scenario of greenhouse gas emissions, but some are along a path of optimal control. Two estimates of the SCC by Stern et al. (2006) were omitted because they are along a path of arbitrary emission reduction.
www.economics-ejournal.org arbitrary assumptions about future climate change, while most studies are based on internally consistent scenarios (SC in Table A1). These classifications are used as quality indicators. 5 Specifically, the sum of the values in Table A1 is the "quality" of the study. More recent studies receive a higher weight-publication year minus 1980 over 10-so that age contributes up to one-third of the total quality weight. Many of the studies report multiple estimates. Most of the estimates are sensitivity analyses around a central estimate, and some estimates are only included to (approximately) reproduce an earlier study. The quality weight of a study is distributed over the alternative estimates in that study on the basis of my assessment of what the author thinks are more and less _________________________ 5 Note that the five quality indicators are objective. The aggregation is arbitrary. credible assumptions. Tol (2005) reports a sensitivity analysis, and finds that the results are robust.
The 211 estimates are classified as follows. Most estimates use the Ramsey discount rule-δ = ρ + ηg-but some estimates use a constant consumption discount rate rather than a constant utility discount rate. A few recent studies use a declining discount rate (inspired by Gollier 2002, andWeitzman 2001), a few studies fail to report what discount rate was used, and a few studies include the discount rate in the uncertainty analysis. Some studies use equity weighting (Fankhauser et al. 1997) with the global average income as normalisation (Anthoff et al., forthcoming), but most studies simply add the regional dollar values (for which normalisation is irrelevant; cf. Fankhauser et al. 1998). The discount rate and the age of the study are used to split the sample.
I adjust three alternative kernel density estimators to these data points. Essentially, a kernel density estimator assigns a probability density function to each data point, and the kernel estimator is the weighted sum of these PDFs. As always, the standard choice is the Gaussian distribution. The 211 estimates provide the modes. Only a few of the studies provide an estimate of the uncertainty. Therefore, either the standard deviation or the coefficient of variation is set equal to the sample standard deviation or the sample coefficient of variation. However, the uncertainty in the sample is right-skewed and fattailed. Therefore, the Fisher-Tippett distribution is also used, with the modes equal to the best guesses and the standard deviations equal to the sample standard deviation. The coefficient of variation of the Fisher-Tippett distributed is bounded from above at about 1.7, which is smaller than the sample coefficient variation. However, the Fisher-Tippett distribution is the only distribution that is right-skewed, fat-tailed, and defined on the entire real line. Table 1 shows selected characteristics of the kernel distributions for the whole sample and selected sub-samples. Figure 1 shows the probability density functions. Unsurprisingly, the Fisher-Tippett kernel has fatter tails and therefore higher means and medians than the Gauss kernel. The modes are about the same. Using the Gauss kernel with the sample coefficient of variation rather than the sample standard deviation has mixed effects. The estimates near zero get higher weight, and this pulls the mode and median down. However, the high estimates are spread thinly over a wide range, and this implies fatter tails and a higher mean.

Results
Splitting the sample by discount rate used has the expected effect: A higher discount rate implies a lower estimate of the SCC and a thinner tail. Table 1 also shows that estimates in the peer reviewed literature are lower and less uncertain than estimates in the gray literature. This confirms the findings of Tol (2005).
Splitting the sample by publication date, shows that the estimates of the SCC published before AR2 (Pearce et al. 1995) were larger than the estimates published between AR2 and AR3 (Smith et al. 2001), which in turn were larger that the estimates published since. Note that these differences are not statistically significant if one considers the means and standard deviation. However, the kernel distribution clearly shifts to the left. Therefore, AR4 (Schneider et al. 2007) were incorrect to conclude that www.economics-ejournal.org  the economic estimates of the impact of climate change have increased since 2001. In their words (p. 781): "There is some evidence that initial new market benefits from climate change will peak at a lower magnitude and sooner than was assumed for the TAR, and it is likely that there will be higher damages for larger magnitudes of global mean temperature increases than was estimated in the TAR." It is unclear how Schneider et al. (2007) reached this conclusion, but it is not supported by the data presented here. Then again, impacts of the economic impacts of climate change necessarily lag behind the latest insights from natural scientists, which indeed justify some increase in the alarm about climate change. Perhaps Schneider et al. (2007) speculate on the results of future research in economics, a clear violation of the IPCC mandate. Tol (2005) also finds a downward trend in the estimates of the social cost of carbon. However, the mean estimates of the SCC are higher in this paper than in Tol (2005). This is due to the different treatment of uncertainty. The Gauss/Coefficient of Variation estimates in this paper are methodologically closest to the method used in Tol (2005), and the results are similar. The Gauss/Standard Deviation and particularly the Fisher-Tippett/Standard Deviation estimates put more emphasis on the uncertainty, which leads to higher numbers.  Table 2 reveals no significant differences-the uncertainties are simply too large-but Nordhaus' estimates are lowest, followed by Hope, Tol, and other authors. It is noteworthy that repeat contributors to the literature on the social cost of carbon are less pessimistic about the impact of climate change.
The SCC estimate by Stern et al. (2006) is almost an outlier in the entire sample (excluding, of course, the Stern estimate itself). Depending on the kernel density, the Stern estimate lies between the 90 th and the 94 th percentile. Compared to the peerreviewed literature, the Stern estimate lies beyond the 95 th percentile-that is, it is an outlier. The Stern estimate fits in better with estimates that use a low discount rate and were not peer-reviewed-characteristics of the Stern Review-but even in comparison to those studies, Stern et al. (2006) are on the high side. Interestingly, the estimate by Stern et al. (2006) is based on Hope's PAGE model but is an outlier compared to earlier estimates with that model (see Table 2). The Stern estimate also fits in better with the All (211) Hope (48) Nordhaus (8) Tol (112) Other (53) Fisher-Tippett, sample standard deviation older studies. This is no surprise, as the PAGE model (e.g., Hope 2006) is updated only with great delay-that is, after the literature reviews by the IPCC (Pearce et al. 2005;Smith et al. 2001). It does fly in the face, though, of the assertion by Stern et al. (2006) to have used the latest research. Even though the Stern et al. (2006) estimate is almost an outlier, this of course does not mean that it is wrong. However, the most positive verdict in the economic literature on the Stern Review is that it was right for the wrong reasons (Arrow 2007;Weitzman 2007a).

4
Catastrophic Liability Weitzman (2007b) argues that the uncertainty about climate change may be so profound that the expected welfare loss is unbounded. See also Tol (2003) and Tol and Yohe (2007b). Figure 2 has a different take on this. It plots the cumulative kernel density estimate (Fisher-Tippett), and the fraction of the world population for whom the "liability of climate change" (i.e., the SCC times their emissions) exceeds their per capita income. See Tol and Verheyen (2004) for a discussion on liability and impacts of climate change. Figure 2 is based on three rather strong assumptions. Firstly, it assumes that people are liable for their greenhouse gas emissions and compensate the victims. Secondly, Figure 2 only considers the compensation paid but disregards the compensation received. Thirdly, it assumes that liability for emissions does not induce www.economics-ejournal.org greenhouse gas emission reduction. Although clearly unrealistic, Figure 2 does provide insight into the scale of the climate problem. For a rising SCC, first the countries with high emission intensity (CO 2 emissions per gross domestic product) would be "bankrupted"-that is, the annual carbon liability (if paid without reducing emissions) would exceed the annual income for the average person. Using 2002 data, the Ukraine would be the first country to which this would happen. 6 A carbon liability of $418/tC would be too much. 7 The probability that the SCC exceeds $418/tC varies between 5% and 7%. See Table 3. This is a high probability for an "infinite" loss-but such a high liability would trigger emission reduction, other countries may come to the assistance of the Ukraine, and it is unlikely to impose such a high tax in the first place. Table 3 also shows the SCCs that would "bankrupt" 1%, 5%, and 10% of the world population, and the associated probabilities. Obviously, the SCCs are higher, and the probabilities smaller-but there is still a probability of 1-2% that the SCC is larger than $1385/tC, which would "bankrupt" more than 10% of the world population. For all three kernel distributions, there is a positive probability that more than 60% of the world population is "bankrupted". The expected fraction of the world population that goes "bankrupt" lies between 0.6% and 1.1%.
A comparison of Table 3 and Table 1 reveals the probability of "bankruptcy" is dominated by the discount rate, the gray literature, and older studies. Newer studies published in peer-reviewed journals based on high discount rates estimate only a very small chance of a carbon liability in excess of annual income.
Finally, Table 3 shows the risk premium of the SCC for the average person on Earth. The risk premia vary between 15% and 27%-for the average. For over 60% of the world population, the risk premium is infinite. This confirms Weitzman's (2007b) claim that climate policy analysis is dominated by the tails of the distribution. It also highlights that climate is an equity problem.
The results in Figure 2 and Table 3 omit emission reduction, but a simple scaling of the results gives a crude estimate of the effect of emission reduction. According to Table 3, a carbon liability of $440/tC would equal 100% of income for 1% of the world population. If a liability of $880/tC would reduce emissions by 50%, then $880/tC would equal 100% of income for 1% of the world population. However, in the shortterm, emission reductions will be far less than 50%.
The results change when we recast the numbers as a "carbon tax" rather than a "carbon liability". The price of carbon is the same, but the revenue flows to the government and is presumably recycled as lower taxes and higher benefits. A carbon tax of $440/tC (or $880/tC with a 50% emission reduction) would imply that 1% of the world population would pay 100% of its income in carbon taxes, and would live of government benefits.

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6 Note that although richer people within a country tend to emit more greenhouse gases than do poorer people, but that the emissions distribution typically is less skewed than the income distribution. This implies that the poor in the Ukraine would be hit harder by a carbon liability than would the rich. 7 Note that a country like the Ukraine is not particularly vulnerable to climate change, and would therefore receive less compensation than it would pay.
www.economics-ejournal.org a Fisher-Tippett with sample standard deviation; Gauss with sample standard deviation; Gauss with sample coefficient of variation. b Also shown are the SCC that triggers the first bankruptcy and its exceedance probabilities; the expected fraction of the population that faces "bankruptcy" (exp); and the risk premium (RP).

Discussion and Conclusion
This paper presents an update of an earlier meta-analysis (Tol 2005) of the social cost of carbon. Using more data and more advanced statistical analysis, this paper confirms the findings of Tol (2005) that estimates of the social cost of carbon are driven to a large extent by the choice of the discount rate and equity weights; and that the more pessimistic estimates have not been subject to peer review. This paper also offers four new results. Firstly, there is a downward trend in the estimates of the social cost of carbon-even if the IPCC (Schneider et al. 2007) would like to believe the opposite. Secondly, the Stern Review (Stern et al. 2006) is an outlier-and its impact estimates are pessimistic even when compared to other studies in the gray literature and other estimates that use low discount rates. Thirdly, the uncertainty about the social cost of carbon is so large that the tails of the distribution may dominate the conclusions (Weitzman 2007b)-even though many of the high estimates have not been peerreviewed and use unacceptably low discount rates. Fourthly, if everyone were to pay a carbon price equal to the social cost of carbon (but not reduce emissions much, as they cannot in the short-term), there is a fair chance that annual taxes would exceed annual income for many people. If the carbon price is a liability, this would imply bankruptcy. If the carbon price is a tax, this would imply complete collectivisation of the economy. There are three implications. Firstly, greenhouse gas emission reduction today is justified. Even the most conservative assumption lead to positive estimates of the social cost of carbon (cf. Table 1) and the Pigou tax is thus greater than zero. Yohe et al. (2007) argue that there is reason to reduce greenhouse gas emissions further than recommended by cost-benefit analysis. The median of the Fisher-Tippett kernel density for peer-reviewed estimates with a 3% pure rate of time preference and without equity weights, is $20/tC. This compares to a price of carbon permits of $160/tC in the www.economics-ejournal.org European Union, 8 and a zero price in most of the rest of the world. The case for intensification of climate policy outside the EU can be made with conservative assumptions. One does not have to rely on speculation as in Schneider et al. (2007) or dodgy analysis as in Stern et al. (2006). At the same time, current EU climate policy seems to fail the cost-benefit test unless one puts a heavy emphasis on risk and uses a very low discount rate (cf. Tol, 2007, for a more detailed discussion). Secondly, the uncertainty is so large that a considerable risk premium is warranted. With the conservative assumptions above, the mean equals $23/tC and the certainty-equivalent $25/tC. More importantly, there is a 1% probability that the social cost of carbon is greater than $78/tC. This number rapidly increases if we use a lower discount rate-as may well be appropriate for a problem with such a long time horizon-and if we allow for the possibility that there is some truth in the scare-mongering of the gray literature. Thirdly, more research is needed into the economic impacts of climate change-to eliminate that part of the uncertainty that is due to lack of study, and to separate the truly scary impacts from the scare-mongering. Papers often conclude with a call for more research, and often this is a call for funding for the authors or a justification for further papers by the authors. In this case, however, quality research by newcomers in the field would be particularly welcome.