The Influence of the Specification of Climate Change Damages on the Social Cost of Carbon

Drawing upon climate change damage specifications previously proposed in the literature that the authors have calibrated to a common level of damages at 2.5 C, the authors examine the effect upon the social cost of carbon (SCC) of varying damage specifications in a DICE-like integrated assessment model. They find that SCC estimates are highly sensitive to uncertainty in extrapolating damages to high temperatures at moderate-to-high levels of risk aversion, but only modestly so at low levels of risk aversion. While in the absence of risk aversion, all of the SCC estimates but one agree within a factor of two, with a moderate level of risk aversion included, the differences among estimates grow greatly. For example, one composite damage specification, combining elements of different literature-derived specifications and roughly taking into account calibration uncertainty, yields SCC values 32% higher than the standard quadratic DICE damage function in the absence of risk aversion. With a coefficient of relative risk aversion of 1.4, however, the same uncertain specification yields SCC values almost triple those of the standard function. The authors conclude that failure to consider damages uncertainty and risk aversion jointly can lead to significant underestimation of the SCC.


Introduction
The social cost of carbon (SCC) is a monetized estimate of the change in expected social welfare that results from a marginal change in carbon dioxide (CO 2 ) emissions. More precisely, it is defined as the ratio of the change in expected social welfare from a unit of emissions to the change in expected social welfare from a unit of material consumption in the period of emissions. Its evaluation requires an integrated assessment model (IAM) that couples together a global economic model and a model of the physical climate system. A core component of this coupling is a specification of damages that translates physical climate outcomes (in practice, primarily changes in temperature) into effects on human welfare.
Ideally, damage specifications should be as comprehensive as possible and consistent with the best available results from detailed assessments of vulnerabilities, impacts and adaptation. Because Monte Carlo simulations to estimate the SCC will necessarily sample some low-probability, high-climate sensitivity 1 states of the world, they should also give meaningful results under these extreme conditions. The first pair of objectives is addressed through the calibration of damages within an IAM; the last objective (and the primary focus of this paper) is addressed through the choice of functional forms used to extrapolate damages beyond the calibration range.
The Dyanmic Integrated model of Climate and the Economy (DICE) IAM (Nordhaus and Boyer, 2000;Nordhaus, 2007), as one example, is calibrated against estimates of damages at 2.5 • C warming to agriculture (Darwin et al., 1995), coastal infrastructure (Yohe and Schlesinger, 1998), and health (Murray et al., 1996). It also includes ad hoc estimates of impacts on energy demand, ecosystems, and settlements (Nordhaus, 2007), as well as estimates based on an expert elicitation study of expected damages resulting from potential climate catastrophes (Nordhaus, 1994). In total, DICE estimates non-catastrophic damages equal to 0.6% of GDP 1 Climate sensitivity characterizes the long-term global mean temperature response to a change in CO 2 concentration or other radiative forcings; it is frequently quoted in terms of the equilibrium warming associated with a doubling of atmospheric CO 2 concentrations. The Intergovernmental Panel on Climate Change's Fourth Assessment Report (Forster et al., 2007) estimates that climate sensitivity is most likely 3 • per CO 2 doubling, with a 67% confidence range of 1.5-4.5 • and a difficult to characterize upper tail. substituting these specifications for the default DICE quadratic damage function at four different levels of risk aversion and two different emissions scenarios, while keeping damages at the 2.5 • C calibration point constant. We also consider new, composite, uncertain damage specifications that combine characteristics of the literature-derived specifications. We then return to examine the sensitivity of our results to changes in expected damages at the calibration point. We find that SCC estimates are highly sensitive to uncertainty in extrapolating damages to high temperatures at moderate-to-high levels of risk aversion, but only modestly so at low levels of risk aversion -a result that emphasizes the importance of jointly considering damages uncertainty and risk aversion.

Taxonomy of damage specifications
The damage specifications that appear in peer-reviewed literature can be characterized based on a number of factors. Damages can affect output, utility, and/or capital, and they can be either unbounded or bounded through a rational or exponential mapping. Most specifications represent damages as functions of temperature increase over preindustrial levels, while others take into account additional climate parameters, such as rate of warming or absolute CO 2 concentration. Rate terms reflect damages to which society can adapt when warming slows, while certain biogeochemical impacts (notably, CO 2 fertilization and ocean acidification) are controlled by concentration rather than level of warming. The damage functions are typically power functions of these parameters. Some damage functions attempt to treat uncertain catastrophic damages in an explicit, stochastic fashion, whereas others fold their expected impacts into deterministic terms. IAMs can also take into account damages to environmental goods that do not become more abundant with increasing material consumption, either by including within the damage function terms that increase with material consumption, or through a modification of the production or utility functions that effectively achieves the same end. Income elasticity of damages can also reflect greater adaptive capacity at higher income levels.
The damage functions evaluated in this study are summarized in Table 1 and discussed below. Table 1 also summarizes the abbreviations used to refer to these damage functions in the remainder of the paper. Figures 1 and 2 show damages as a function of temperature with different specifications; Figures 3 and 4 provide an alternate perspective, showing the implications for consumption in the reference scenario of different damage functions.
In this paper, we focus on damage specifications that can be implemented in a one-region, DICE-type model, in order to highlight the sensitivity of SCC results to choices about functional form. Table 2 provides parallel descriptors for damage specifications that appear in cost-benefit IAMs other than DICE. In the context of multi-region models like PAGE and FUND, other authors (e.g., Hope, 2008;Anthoff et al., 2009) have dealt with questions related to how to combine social costs experienced in different regions with different levels of vulnerability; addressing these issues in the context of this paper would expand the scope too broadly. Like DICE and PAGE and unlike FUND, we do not model separate economic sectors with separate damage functions; such a choice would similarly obscure our focus, which is on how different choices about extrapolation affect the SCC. Nonetheless, the core elements of the functional forms used in all the sectors in FUND do appear -albeit within in a single sector -in the specifications we examine.

The nature of damages
In principle, damages from climate change can negatively impact three different terms in an economic model: utility U, output Y , and capital K. Damages to utility represent negative impacts to the ability of individuals to benefit from material consumption; damages to output represent negative impacts on the ability to make productive use of capital; and damages to capital are simply that. (Fankhauser and Tol (2005) describe a similar division, between non-market impacts, market impacts, and accelerated capital depreciation; they also note the possibility of damages to population via health impacts.) Following the standard DICE formalism but keeping the savings rate fixed, we employ a Solow production model. In the absence of climate damages, www.economics-ejournal.org 5 Figure 1: Expected climate damages for different damage functions, calibrated such that they all agree with the default DICE damage function for 2.5 • C warming. Damage functions are defined in Table 1. In (a), rate of temperature increase and level of material consumption are set at the calibration levels (the values achieved at 2.5 • C warming in the reference scenario with damages D and 3 • C/CO 2 doubling climate sensitivity); in (b), they are set to twice these levels. In (a), heavy dashed grey lines indicates the lowest 5th percentile value and the highest 95th percentile value across all damage functions, excluding Xau, Xaau and Xbu. In (a), SP, Wa, Ad, Xa, Xaa and Xb match D and so are not shown. In (b), only D and functions with rate or consumption sensitivity are shown. Specifications are labeled with primes in the legend to highlight that they are shown for non-calibration levels of warming rate and consumption. Xb and Xbu match Xaa and Xaau, respectively.
www.economics-ejournal.org 8 where C denotes per capita material consumption, U C represents the utility of a representative agent based upon his or her material consumption, A denotes total factor productivity, L represents population, γ is the elasticity of capital in the production function, s is the savings rate, d is the depreciation rate, and η is the elasticity of the marginal utility of consumption. Note that climate change impacts on labor productivity, another possible pathway for damages, would manifest as decreases in A and are therefore equivalent to impacts on output in this formalism.
(For our numerical calculations, we keep s fixed at 20% of output and, following DICE, set γ = 0.3 and d = 10% per year. The time series of A and L are exogenously specified by the scenario.) As discussed at greater length in section 2.4, Sterner and Persson (2008) introduce a utility function for a representative agent that is dependent upon both material consumption and consumption of ecological goods. Effectively, their model can be viewed as substituting effective consumption per capita C eff for the standard material consumption per capita C in the utility function: where E is the per capita consumption of ecological goods, β is the fraction of utility accounted for by consumption of ecological goods, and σ is the elasticity of substitution between material and ecological goods. The following discussion employs this formalism for generality. We represent damage functions as the composite of two functions, a function D with an arbitrary form and a function Ω that can map D into a bounded interval. Early versions of DICE (e.g., Nordhaus, 1992) used D directly as the damage function; that is, Ω(D) = D, so that output net of damages was given by www.economics-ejournal.org 11 This representation can at sufficiently high temperatures give rise to negative net output. To avoid this consequence, Nordhaus and Boyer (2000) bound damages at 100% of GDP by the rational mapping Weitzman (2009b) (our damage function We) suggests bounding damages using an exponential mapping rather than a rational mapping. This mapping matches the standard DICE mapping at low-to-moderate temperatures but approaches 100% of GDP more rapidly at higher temperatures ( Figure 5). As indicated in Table 1, all of the damage specifications in our analysis except We employ a rational bounding mapping.
Let f U , f Y , f K represent the initial fractional contribution of damages to utility, output and capital to reducing utility ( f U + f Y + f K = 1). Then damages Ω can be distributed: where E 0 is the supply of ecological goods in the absence of climate change damages and K , Y , E , and C eff represent respectively capital, output, consumption of ecological goods, and effective consumption after adjustment for damages. Even if calibrated thus, so that the immediate impact on utility is the same, damages to utility, output, and capital have moderately different long-term implications. Damages to utility impact well-being but not the growth of the material economy. Damages to output leave current capital untouched but reduce investment and therefore future capital and output. Damages to capital will produce the same investment reduction as damages to output, while also impacting current capital. Consider the effect of a short but very severe shock to each of these factors: a severe shock to utility will temporarily make people very unhappy; a severe shock to output will cause a recoverable depression; and a severe shock to capital will require a protracted period of rebuilding. For the same damage function, the SCC will therefore be higher the greater the proportion of damages accruing to capital, and lower the greater the proportion accruing to utility, as can be seen in Figure  6. In this example, using the standard DICE damage function, a constant climate sensitivity of 3 • /CO 2 doubling and a flat 3% discount rate, damages applied purely to output yield an SCC in 2015 of $33/tonne CO 2 , damages applied purely to utility yield an SCC of $25 ton/CO 2 , and damages applied purely to capital yield an SCC of $41/tonne CO 2 .
Most of the damage specifications in the literature impact output, while Weitzman (2009a) simplifies his discussion by focusing on utility damages, and the  (2005) construct a simple climate-economic model for the purpose of examining the multiple pathways by which climate change can cause economic damages; this paper is the only example we are aware of examining direct capital impacts in a standard climate change IAM.

Polynomial functions of temperature
As noted in the introduction, climate damages in DICE are a quadratic function of global average temperature increase over preindustrial levels (D(T ) = aT b , b = 2). This expression for D is employed in our damage specification D with a rational mapping (i.e., D is the standard DICE damage function); our damage specification We, following Weitzman (2009b)'s suggestion, uses the same D with an exponential mapping. A number of IAMs adopt similar polynomial forms to DICE. The Model for Estimating the Regional and Global Effects (MERGE) (Manne et al., 1995) adopts the DICE quadratic function for market damages and applies a separate willingness to pay (WTP) function for non-market damages. PAGE (Hope, 2006) models economic, non-economic, and catastrophic ("discontinuity") damages separately for eight geographic regions, all as power functions in a form also similar to DICE, but with the exponent treated as uncertain, ranging from 1 to 3, instead of being fixed at 2 as in DICE. Ackerman et al. (2010) (our damage specification ASB) introduces greater uncertainty in the damages exponent by modifying DICE, treating b as a random variable with a triangular distribution with mode 2, minimum 1, and maximum 5. Along similar lines, Mastrandrea and Schneider (2001) adjust b through iteration with a simple model of the Atlantic Meridional Overturning Circulation (AMOC) to incorporate the expected damages associated with decline or collapse of AMOC.

Polynomial functions of temperature and rate of warming
A number of functions are dependent on the trajectory of temperature increase. Damages in the Carbon Emissions Trajectory Assessment (CETA) model (Peck and Teisberg, 1992), for instance, depend on the decadal rate of temperature change. Lempert et al. (2000) (our damage specification L) develop a damage function intended to capture the impacts of climate variability. Their function includes a term with the five-year running average of temperature for variability to which society and ecosystems can adapt on the timescale of several years, and another term with 30-year running averages for variability to which society and ecosystems adapt over longer timescales: www.economics-ejournal.org 15 where T 5 and T 30 are five-and thirty-year running averages of temperature. (Since our model has ten-year time steps, T = T 5 , so we set a 2 to zero in our implementation of this function.)

Polynomial functions of temperature and consumption or income
Damage functions are typically expressed as a fractional loss of output, but some work has explored damage specifications in which effective damages at a given temperature increase more rapidly than output. These specifications represent damages to environmental goods (for example, clean air and water or ecological diversity) for which material goods are imperfectly substitutable. As material consumption increases, the relative price of the fixed supply of environmental goods also increases, and so the welfare impacts of damages to this supply become more severe. As mentioned previously, Sterner and Persson (2008) model these imperfectly substitutable goods directly. Following their numerical example, our damage specification SP sets β = 0.1 and σ = 0.5 in equation 5. Weitzman (2009a) presents an "additive" specification of damages, which he contrasts with the standard "multiplicative" specification. His terminology derives from the appearance of the utility function with the two different specifications when η is equal to 2. In equation 4, with η = 2 and all damages applied to utility, If Ω is a rational mapping, this is what Weitzman (2009a) calls a multiplicative specification. The additive specification is given by this is equivalent to www.economics-ejournal.org to and to the standard formalism if D = DC is substituted for D. The additive specification (our damage specification Wa) is thus equivalent to making relative damages a function of material consumption as well as temperature: (We employ b = 2 for Wa). Weitzman argues that this alternative form is as plausible as its standard "multiplicative" equivalent on a priori grounds. He also shows that shows that the Sterner-Persson utility function possesses the same properties as a utility function with both additive and multiplicative damage terms. Note that -due to the relative price effects Sterner and Persson (2008) discuss -the introduction of multiple, non-substitutable utility-enhancing goods has important implications for SCC calculations. The SCC is defined as the ratio of the change in expected welfare from a unit of emissions to the change in expected welfare from a unit of material consumption in the period of emissions. If E is declining over time -or even increasing slower than C -the relative value of a unit of material consumption is declining over time. This effect will appear in the denominator of the SCC, but it will appear more strongly in the numerator, where a stream of damages is being inflicted on a world with lower utility than a world where C and E were perfectly substitutable. For values of η less than one, lower utility worlds make a smaller contribution to welfare than higher utility worlds, so the net effect will be to decrease the social cost of carbon. In the calculations below where we employ a Sterner-Persson utility function, we therefore present results calculated in terms of equivalent first-period material consumption: essentially, converting damages from current dollars to constant dollars. In constant dollars, incorporating an imperfectly substitutable, non-increasing good will increase the SCC. The conversion to constant dollars is given by: (For reference, we also report current dollar values without this conversion in Tables 4 and 5.) As noted in the introduction, many of the damage functions in FUND include per capita income as a term, in order to reflect greater adaptive capacity in higher income societies (Anthoff and Tol, 2010). Some sectoral damages in FUND are modulated by terms such as (y/y 0 ) ε , where y is per capita GDP, y 0 is per capita GDP in a reference year, and ε is an income elasticity. Since in our formalism consumption is a fixed fraction of output, this can be viewed as equivalent to a damage function of the form where ε < 0 reflects increasing adaptive capacity in higher consumption societies and ε > 0 reflects effects of the sort discussed by Sterner and Persson and by Weitzman. In FUND, ε ranges from -2 for health impacts, to between -0.2 and -0.3 for water, energy, and agricultural impacts, to 0 for sea level rise impacts. Based on a weighted average of income elasticities across sectors in FUND, we construct damage specification Ad with ε = −0.4 (and the DICE standard b = 2).

Uncertainty in functional form
Several damages specifications use uncertainty in functional form to represent more explicitly impacts from low-probability catastrophic events. For example, PAGE (Hope, 2006) allows for a discontinuity that causes damages equal to 5-20% of GDP. The probability of such a discontinuity increases at a linear rate with increasing temperature. Keller et al. (2004) (our damage specification K) model damages from the collapse of the AMOC, assumed to vary uniformly between 0 and 3 percent of global GDP, by adding a term for damages above a certain threshold.
Other modelers represent states of the world that experience catastrophic damages with higher values for the exponent b in the damage function. Azar and Lindgren (2003) (our damage specification AL) model two uncertain future states of the world: a high-probability state in which damages follow the quadratic form of DICE, and a low-probability but catastrophic state in which damages are a quartic function of global average surface temperature change. Similarly, as mentioned previously, Ackerman et al. (2010) (ASB) treat the exponent b of temperature in the damage function as uncertain; higher values of b cause damages to increase more rapidly with temperature and represent states of the world more subject to catastrophic climate change.

Disaggregation of damages
We introduce six new damage specifications that combine elements of the literature specifications discussed above to progressively disaggregate the standard DICE damage function.
Whereas the standard DICE damage function treats catastrophic damages through their expected value, specification Xa follows K and PAGE in separating the damage function into a deterministic gradual damages term and an uncertain threshold damages term: where m(T ) is a multiplier equal to 1 at low-to-moderate values of T , T c is the uncertain temperature threshold for a catastrophe, and a 3 (set equal to 10) controls the warming range over which the catastrophe occurs. To calibrate, we note that Nordhaus (2007) estimates that risk-neutral expected damages from catastrophic climate change are responsible for 66% of total damages (0.61% of GDP) at the 2.5 • C calibration point and that Nordhaus and Boyer (2000) defines a catastrophe as causing 30% loss of global GDP indefinitely. Using these constraints to set a 1 and a 2 , we then calibrate the distribution of T c and the value of m(T ) so that expected damages match their standard DICE values. Damage specification Xaa builds upon Xa by allowing adaptation to gradual impacts; as in Ad, the income elasticity of damages ε is set equal to -0.4. Damage specification Xb distributes the damages in Xaa between output, capital, and Sterner-Persson-type environmental goods, as described in section 2.1. Nordhaus (2007) attributes about one-quarter of non-catastrophic damages at 2.5 • C to his "ecosystems and settlements" sector; we assume that about two-thirds of those damages are to environmental goods, and therefore infer that about 15% of total consumption is accounted for by environmental goods (i.e,. we set β in equation 5 to 0.15). Assuming that another third of this sector constitutes damages to capital, and that one third of damages due to coastal impacts are also to capital, we infer that another 15% of damages occur via capital losses. Again following Sterner and Persson's numerical example, we set σ in equation 5 to 0.5. Damage specifications Xau, Xaau and Xbu ( Figure 2) add uncertainty to, respectively, Xa, Xaa and Xb. Distributions of key parameters for these specifications are summarized in Table 3. We let b be log normally distributed with a mean of 2.0 and a geometric standard deviation of 1.4 (i.e., a 95% uncertainty range spanning a factor of two). We let expected damages at 2.5 • C be log normally distributed with a mean of 1.77% of global GDP (matching DICE) and a geometric standard deviation of 2. We keep gradual damages at one-third of expected damages. The damage resulting from a catastrophe is triangularly distributed, with a mean of 30% of global GDP and a range of 15% to 45%. For Xaau and Xbu, the income elasticity of damages is triangularly distributed with a mean of -0.4 and a range of -0.8 to 0. For Xbu, the elasticity of environmental goods in the utility function is triangularly distributed between 0.5 and 1.0, with a mean of 0.75. Both the share of environmental goods in the utility function and the fraction of damages accruing through capital impacts are triangularly distributed with means of 0.15 and ranges of 0 to 0.3.
These damage specifications are intended to be illustrative, and the choices of variance are admittedly somewhat arbitrary -indeed, necessarily so, given the absence of any basis in the literature for calibrating them. Lacking other guideposts, we have chosen to rely primarily on factors of two. As an intuitive check of the reasonableness of our choices, we can examine their implications for consumption paths. The expected consumption path for Xbu (Figure 4) is very close to that calculated using D; the median consumption path for Xbu is higher than for D and the 5th percentile is considerably lower, reflecting the explicit representation of low-probability, high-impact states of the world through both the catastrophic damages term and the uncertain exponent. In Xaau and Xbu, there is an˜1.8% probability in the reference scenario that our 24th century descendants will have lower equivalent material consumption per capita than we do, and an˜0.8% chance that they will be eking out a subsistence existence, with a equivalent material consumption per capita of less than $2/person/day. It is our judgement that this distribution is a reasonable guess.  www.economics-ejournal.org 22 a stabilization scenario calculated to yield a 50% chance of limiting warming to 2.5 • C (Figure 7). 3 We replace the default DICE damage function with the damage functions listed in Table 1, calibrated such that all the functions are in agreement on climate damages at 2.5 • C. Rate-or consumption-dependent functions are calibrated at the same warming rate (0.37 • /decade) and per capita consumption level ($12.6 thousand/person/year) as is reached in the reference scenario with the standard DICE damage specification D and a climate sensitivity of 3 • C per CO 2 doubling. For our primary calculations, we set the calibration point to agree with the default DICE 2007 calibration (1.77% of GDP loss at 2.5 • C). We employ a lower bound to effective consumption of $500/person/year. (See the discussion in Weitzman, 2009b, on bounding damages with a parameter akin to the value of a statistical life.) 4 Following the US government analysis, we treat climate sensitivity as an uncertain parameter with a Roe and Baker (2007) distribution, truncated at 10 • C per CO 2 doubling and calibrated such that the median value is 3 • C per CO 2 doubling and the 67% range is approximately 2 to 4.5 • C per CO 2 doubling. We take one thousand samples evenly from the distribution, and we use Latin hypercube sampling when considering more than one uncertain parameter. (For specifications Xau, Xaau and Xbu, with a larger number of random variables, we use five thousand samples).
For the reference scenario, the standard DICE damage function D, and a flat 3% discount rate, our model yields a 2010 SCC of $37/tonne CO 2 . This value which can be compared to the value of $29/tonne calculated with DICE for the same year, scenario, and discount rate in the U.S. government analysis. The slight difference is largely attributable to a change in the calibration of the climate and carbon cycle transfer coefficients between DICE 2007, used in the U.S. government analysis, and DICE 2010, whose values we employ.

Discounting and risk aversion
The social cost of carbon (called the "certainty-equivalent SCC" by Newbold et al., 2010) is defined as the change in expected welfare from a unit emission of carbon dioxide in a given year, normalized to change in expected welfare from a unit of consumption in the same year. Note that it is distinct from "deterministic SCC" values, which are ratios of the corresponding changes in welfare, conditioned upon a specific state of the world. It makes sense to talk about a probability distribution for the latter, but not for the former. As shown by Newbold et al. (2010), the expected deterministic SCC is equal to the SCC when the denominator is known with certainty (e.g., in the first time step), but not generally. In the presentation of our results, we show both the SCC and the distribution of deterministic SCC values.
In the isoelastic utility function employed by DICE and other IAMs, the elasticity of the marginal utility of consumption η serves as both the coefficient of relative risk aversion and the inverse of the intertemporal elasticity of substitution. The value of η and the pure rate of time preference ρ are related to the discount rate r and the growth rate of (effective) per capita consumption g via Ramsey's rule: 5 Increasing risk aversion while holding the pure rate of time preference ρ constant will therefore also increase the discount rate.
Rather than disaggregating discount rates into values of η and ρ along lines of equation 29, the U.S. government SCC analysis used flat discount rates of 2.5%, 3% and 5% per year. Because the growth rates in that analysis's reference scenarios (as in our scenarios) decline to zero between 2100 and 2300, the choice of flat discount rates is strictly consistent only with η = 0. (This choice is consistent with the U.S. government analysis's risk neutrality, but inconsistent with its description of a plausible range for η as being between 0.5 and 4.) In our analysis, we vary η and ρ such that, in the reference scenario in the absence of climate change damages, the average discount rate r over 2015-2115 remains fixed at the U.S. government analysis's middle value of 3% per year. We view this 3% value as an "observed" risk-free interest rate specified by the scenario, with which any decomposition into η and ρ must be consistent. (We do not address the separate scenario-design question of whether the 3% rate is a reasonable choice.) We choose four different values of η (0, 1.0, 1.4 and 2.0), which, in light of the average 2.03% annual growth rate between 2015 and 2115, lead to selection of values of ρ of 3.0%, 0.95%, 0.14% and -1.1% per year, respectively. 6 These choices allow us to investigate the implications for the SCC of different values of the elasticity of the marginal utility of consumption while retaining consistency with the initial risk-free interest rates specified in the U.S. government analysis. Within the constraints of the isoelastic utility function, however, it is not readily possible to isolate the two different effects of increasing η -increasing risk aversion and increasing the weight placed on damages in the slower growing 22nd and 23rd centuries. Caution should therefore be used in attributing the differences between SCCs calculated at different values of η solely to risk aversion; for a single pair of η and ρ, however, the differences in SCCs for damage functions giving rise to nearly the same expected consumption path with different levels of uncertainty (e.g., D vs. Xa vs. Xau) can be so attributed.

Results and discussion
The consequences of changing the damages specification depend strongly on η (Table 4, Figure 8). For the risk neutral case (η = 0), specifications with similar expected damages (Figure 1) yield similar results. Thus D, We, K, L and Xa cluster in the range of $34-$39/tonne CO 2 emitted in 2015. Because of the modestly (AL, Xau) or highly (ASB) uncertain exponent in their damage functions, AL, Xau and ASB are associated with higher SCC values of $42, $44, and $62/tonne, respectively. Wa is also high, at $49/tonne, reflecting the increasing severity of damages under this specification as material consumption increases. As noted previously, due to relative price effects, employing a Sterner-Persson damage function decreases the SCC ($26 and $23/tonne for SP and Xb). When these SCCs are corrected for price effects, they roughly double. The incorporation of adaptation through a negative income elasticity of damages modestly reduces the SCC ($30 for Ad vs. $37 for D, $33 for Xaa vs. $35 for Xa, and $42 for Xaau vs. $44 for Xau). (In all the specifications in our analysis, the costs of adaptation are implicitly included in the damages.) When risk aversion is taken into account, the SCC depends not only on expected damages but also on the variability of damages. Since we keep the average discount rate over 2015-2115 constant while increasing risk aversion, SCC values from deterministic damage functions with an approximately quadratic form increase only modestly. This increase is due in part to the lowering of the discount rate in the 22nd and 23rd century, when the reference consumption growth rate slows, and in part to the risk associated with an uncertain climate sensitivity. Damages with D, for example, increase by about 40% as η goes from 0 to 1.4. The increase in damages with Ad, which lets adaptive capacity increase with income and so exhibits a smaller spread, is less (22%). The exponentially-mapped specification We exhibits a slightly larger response (increasing by 49%) than D, because of the greater damages it yields in high climate sensitivity states of the world. Although Wa is also deterministic, it responds more severely because of its high damages at high levels of material consumption, increasing by 165% to $130/tonne. SCCs associated with uncertain damage specifications also grow rapidly with risk aversion (e.g., by 63%, 114% and 120% for Xa, ASB and AL). The degree of this effect depends upon the variability of damages. The most uncertain damage η and ρ chosen to maintain an average discount rate of 3% per year over 2015-2115.
www.economics-ejournal.org 27 conomics: The Open-Access, Open-Assessment E-Journal functions (Xau, Xaau and Xbu) exhibit this effect most dramatically, with the SCC more than tripling from a risk-neutral $42-$48/tonne with η = 0 to $130-$151/tonne at η = 1.4, as an increasing amount of weight is placed on the small number of states of the world in which the future is poorer than the present. As η increases, the current-dollar SCC penalty associated with adopting a Sterner-Persson utility function also disappears. At η = 1, for example, currentdollar SCC values from D and SP (denoted SP* in the tables ) are identical. This is because the future in the Sterner-Persson world is poorer in effective consumption terms; at η = 0, the future therefore contributes less to welfare, while at η = 1, this effect vanishes because marginal utility is proportional to utility. At η > 1, poorer states of the world contribute more than proportionally to expected social welfare, so current-dollar SCC values from SP exceed their counterparts from D.
In general, SCC values calculated off the stabilization scenario are moderately lower than SCC values calculated off the reference scenario (Table 5, Figure 9). Employing fat-tailed specifications like ASB ($73/tonne at η = 1.4, vs. $139/tonne with the reference scenario) magnifies this reduction, as for most climate sensitivities, the stabilization scenario avoids the high-temperature portion of the damage function, where the consequences of higher exponents are felt most strongly. Wa exhibits the converse effect: with this damage specification, the SCC is higher under the stabilization scenario than under the reference scenario ($150/tonne vs. $130/tonne at η = 1.4). Later in the 21st century, other damage specifications, including both D and Xbu in the risk neutral case, exhibit a similar phenomenon ( Figure 10). These examples provide a concrete illustration of the concerns about non-convex damage functions raised by Baumol (1972) in the context of the social cost of pollution generically and by Kopp and Mignone (2012) in the context of the SCC. With non-convex damage functions, emissions and growing consumption in the reference scenario can carry temperature well over an inflection point beyond which the marginal damages associated with additional warming start decreasing. In this situation, applying the SCC calculated off the reference path as a Pigouvian tax will drive emissions toward a local optimum but not necessarily to the globally optimal level. For a given damage function, adjusting damages at the 2.5 • C calibration point downward has a roughly linear effect on the SCC (Figure 11 50%, while doubling them has a similar but slightly muted effect, increasing SCC values by about 50%-80%. This muting is due to the upper bound on the damage functions; higher calibration damages bring the damage functions closer to the bound, and thus into non-convexity, at lower temperatures. The muting effect can be seen most clearly with Wa, which, as noted previously, is well in the non-convex range of behavior under the central calibration. The SCC calculated under Wa decreases by about 30% in response to a halving of 2.5 • C calibration damages and increases by about 30% in response to a doubling of calibration damages.

Conclusions and next steps
Our analysis highlights the importance of jointly considering risk aversion and uncertainty in damages when estimating the SCC. Incorporating uncertainty into damages while assuming risk neutrality will minimize the effects of this modification; incorporating risk aversion while using a certain damage function will do likewise.
For a certain damage specification but uncertain climate sensitivity, increasing risk aversion (while adjusting the pure rate of time preference to maintain exogenously specified average discount rates) modestly increases the SCC. For an uncertain damage specification -even one that yields the same expected damages -increasing risk aversion can greatly increase the SCC. Our composite uncertain damage specifications (Xau, Xaau and Xbu) -which yield nearly the same expected future consumption path as the standard DICE damage specification D but also acknowledge a small chance that climate change could make the future less well off than the world today -yield SCC values that go from being just 14-32% higher than the SCC calculated with the standard DICE damage specification D with no risk aversion to nearly triple it (150-190% higher) with a coefficient of relative risk aversion of 1.4.
Because we employed a standard isoelastic utility function, we could not isolate the effects of increasing risk aversion from the effects of a declining intertemporal elasticity of substitution. Fundamentally, this is because the isoelastic utility function requires the elasticity of intertemporal substitution and the level of risk aversion to be controlled by a single elasticity of the marginal utility of consumption (η). Yet empirical psychological work (e.g., Atkinson et al., 2009) indicates that individuals are not equally averse to risk and to intertemporal inequality; future work should therefore examine the joint sensitivity of the SCC to damage specification and risk aversion in models with utility functions that do not require these parameters to be coupled (Traeger, 2009).
In addition to the damage function's form, its calibration also matters. The risk-neutral SCC scales approximately linearly with damages at the calibration point. If the uncertainty in this calibration is large -as seems likely, given that FUND estimates non-catastrophic damages of about 0.9% to 1.6% of GDP at 2.5 • C warming while DICE estimates non-catastrophic damages of 0.6%, and that both the likelihood of climate catastrophes and their economic consequences are poorly characterized -then both form and calibration can have effects on the SCC of similar magnitude.
Both calibration of the damage function and identification of a suitable form can be advanced through both empirical and modeling work. Emerging retrospective analyses (e.g., Lobell et al., 2011) can help characterize damages for 0.8 • C of warming realized to date more accurately, while modeling economic impacts at levels of warming significantly higher than 2.5 • C (e.g., New et al., 2011) can advance the construction of damage functions beyond the stage of fitting a curve to two points.