The risk-adjusted carbon price

We use perturbation methods to derive a rule for the optimal risk-adjusted social cost of carbon (SCC) that incorporates the effects of uncertainties associated with climate and the economy from a calibrated DSGE model. We allow for different aversions to risk and intertemporal fluctuations, convex damages, uncertainties in economic growth, atmospheric carbon, climate sensitivity and damages, their correlations, and distributions that are skewed in the longer run to capture long-run climate feedbacks. Our non-certainty-equivalent rule for the SCC incorporates precaution, risk insurance, and climate sensitivity and damage rate hedging effects to deal with future economic and climatic and damage risks.


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The social cost of carbon (SCC) is the Pigouvian tax that internalizes the expected harm of emitting one ton of carbon to the economy, i.e. the expected present discounted value of all future marginal utility losses resulting from emitting one ton of carbon today, converted from utility into dollars today. The risk-adjusted SCC incorporates uncertainties 1 associated with climate and the economy when calculating this tax. If global warming is the only market failure, it is optimal in a decentralized economy to set the price of carbon emissions (e.g. a specific carbon tax or the price in a competitive permit market) to the SCC. To evaluate the SCC, one must know how much of one ton of carbon with 0   the coefficient of relative aversion to 1 We use the terms risk and uncertainty interchangeably.
intergenerational inequality aversion and risk. We then obtain from (1) that 3 (2) * * 2 The discount rate r * in (2)  The literature concerned with finding the optimal risk-adjusted SCC consists of two strands. Numerically, different authors have performed numerical calculations of the optimal SCC under multiple sources of uncertainty, first with Monte-Carlo simulations (e.g., Ackerman and Stanton, 2012;Dietz and Stern, 2015) and more recently by tackling the dynamic programming problem with advanced numerical methods (e.g., Crost and Traeger, 2013;Traeger, 2014a;Jensen and Traeger, 2014;Hambel et al., 2017). 5 Analytically, the literature on discounting under uncertainty and optimal carbon prices typically deals with one uncertainty at a time (e.g., Gollier, 2012;Traeger, 2014b). At the start of this analytical literature, Golosov et al. (2014) obtained a simple rule for the optimal SCC reacting to world GDP only, making bold assumptions including logarithmic utility 6 , which imply that growth uncertainty does not affect the 3 It is easy to allow for richer dynamics in the atmospheric carbon stock. E.g., Joos et al. (2013) Gollier (2012) and Traeger (2014b) for the effect of growth volatility on the discount rate. 5 Lemoine and Traeger (2014;, Lontzek et al. (2015) and Cai et al. (2016) study numerically the optimal SCC in the face of climate tipping risks. Lemoine and Rudik (2017) review recursive numerical assessment and Monte Carlo evaluation of climate policy under uncertainty and discuss learning. 6 They have a discrete-time (decadal) model, assume logarithmic utility, Cobb-Douglas production, 100% depreciation of capital each period, and total factor productivity as an exponential function of the atmospheric carbon stock.
3 SCC (cf. Traeger, 2017). Gerlagh and Liski (2016) also derive a simple rule 7 and examine this in the context of learning about uncertain impacts. Jensen and Traeger (2016) show how the effect of climate sensitivity on the risk premium in the SCC depends on prudence and convexity of marginal damages. Lemoine (2017) decomposed the SCC into different components due to uncertain warming, damages and economic growth. He showed that the sign of the effect of so-called climate betas, representing the normalized covariances of different climatic uncertainties with the rate of economic growth, on the SCC depends on whether relative risk aversion is greater than one or not. 8 In both the decompositions by Jensen and Traeger (2016) and Lemoine (2017) consumption is set exogenously. Very recently, two first steps have been made towards a simple rule for the risk-adjusted SCC in a general equilibrium model. Traeger (2017) transforms an integrated assessment model with a range of climate uncertainties, in which consumption is determined endogenously but with full capital depreciation after one period following Golosov et al. (2014) and the restriction that the model that is linear in states with additively separable controls. Finally, Bretschger and Vinogradova (2018) extend the endogenous growth model of Pindyck and Wang (2013) to allow for Poisson shocks in the capital stock in their analysis of carbon pricing.
Our aim here is to derive a general rule for the optimal SCC that maximizes expected welfare in a Dynamic Stochastic General Equilibrium (DSGE) model.
We allow for uncertainty in projections of the carbon stock, of the impact of the atmospheric carbon stock on temperature, of temperature on damages and of the GDP growth rate, as well as their correlations, and analyse the precautionary, insurance and hedging determinants of the SCC. We allow skewness and uncertainty of the response in temperature resulting from doubling the atmospheric carbon stock, as captured by the climate sensitivity, to rise with time, reflecting key differences in short-term (cf. transient climate response) and long-term (cf. equilibrium climate sensitivity) uncertainty predictions. We allow risk aversion to differ from intergenerational inequality aversion (Kreps and Porteus, 1978;Epstein and Zin, 1989;Duffie and Epstein, 1992). Finally, we allow for general concave or convex relationships between the carbon stock and temperature and between temperature and damages. 9 In doing so, we generalize Golosov et al.'s (2014) rule for non-unitary coefficients of relative risk aversion and intergenerational inequality aversion, more convex damages, uncertainty in the carbon stock, climate sensitivity and damages, and skewness and mean reversion in the distributions governing these variables.
Our DSGE model adapts the endogenous growth model with investment adjustment costs of Pindyck and Wang (2013) to allow for fossil fuel use, climate change and damages. To obtain a simple result akin to (2), we solve our DSGE model using perturbation methods around a known analytical solution path, where the "small" perturbation parameter is the fraction of damages in GDP. 10 By using a power-function transformation of a normal variate displaying a variance that grows in time, 11 we capture the significant right-skew evident in the equilibrium (i.e. long run) climate sensitivity, but not in the transient climate response, whilst capturing the difference in time scales on which these apply, but avoiding the fat tails in Weitzman's (2009) 'dismal theorem'.
We derive three results. Result 1 gives our general expression for the optimal risk-adjusted SCC and can be evaluated by numerical evaluation of a multidimensional integral, avoiding the daunting task of numerically solving the underlying multi-state Hamilton-Jacobi-Bellman equations. For the case of damages proportional to the atmospheric carbon stock and focusing only on the leading-order effects of uncertainty, Result 2 evaluates this rule in closed form.
Result 3 generalizes it for convex dependence of damages on the carbon stock, 9 Hence, the so-called flow damage coefficient  in ( ( ) / ( ))( ( ) / ( )) ( ) D s T s T s S s C s     =  will no longer be constant but depend on the stochastic atmospheric carbon, temperature and damages. 10 Using scaling, we identify the damages ratio as the only "small" non-dimensional quantity. Judd (1996Judd ( , 1998, Judd and Guu (2001) and Binsbergen et al. (2012) use perturbation analysis (e.g. Bender and Orszag, 1999) in discrete time. 11 Specifically, we use an Ornstein-Uhlenbeck process. giving the SCC in the form of one-dimensional deterministic integrals.
Generalizing (2) to recursive preferences, we find that precaution about uncertain growth outcomes implies a lower discount rate and a higher optimal SCC, whilst a risk-insurance term increases the discount rate and curbs the SCC.
If intergenerational inequality aversion exceeds one, the discount rate is adjusted downwards and the SCC upwards with riskier growth prospects. The upward correction to the SCC to allow for temperature uncertainty depends on the combination of the skewness of its equilibrium probability distribution, the convexity of damages, the (non-climatic) risk-adjusted discount rate and, crucially, on the time scale on which the equilibrium distribution is reached.
In our analysis, the three different climatic uncertainties have their own betas, representing their normalized covariances with shocks to the rate of economic growth: the carbon stock beta, the temperature beta and the damage beta with the latter two the most important. If the economy is concentrated in economic sectors that benefit from high (low) temperature, the temperature beta is positive (negative), and we show that the optimal risk-adjusted SCC is lower (higher) provided risk aversion exceeds one, as found by Lemoine (2017). If the economy is concentrated in adaptation industries (e.g. flood defences), shocks to future damages are associated with higher assets returns so the damage beta is positive. We show that, if the coefficient of relative risk aversion exceeds one, the optimal SCC is then reduced, 12 although we note that such capital allocation is rare, especially in the developing world. Finally, we calibrate our model and show how the optimal SCC is quantitatively by the different uncertainties.
Section I presents our model. Sections II derives Result 1. Section III derives Result 2 with Result 3 in Appendix A. After a discussion of our calibration in Section IV, Section V estimates the optimal SCC and the effects of the various uncertainties. Finally, Section VI concludes.

I. A DSGE Model of Global Warming and the Economy
We start from the DSGE model with endogenous AK growth of Pindyck and Wang (2013) and add fossil fuel use as a production factor. Fossil fuel use gives rise to global warming and damages to output. The coefficient of relative risk aversion,  = CRRA  0, may differ from the coefficient of relative intergenerational inequality aversion, IIA = 1/EIS =   0, where EIS is the elasticity of intertemporal substitution. We use the continuous-time version of recursive preferences (Duffie and Epstein, 1992), where the recursive aggregator ( , ) f C J depends on consumption C and the value function where Y is aggregate production, F fossil fuel use, and b the production cost of fossil fuel. Fossil fuel is supplied inelastically at fixed cost.
The final goods production function is and 13 With AK growth, shocks to the capital stock and productivity are equivalent. To avoid an extra state, we introduce volatility directly in the capital dynamics (cf. Pindyck and Wang, 2013). 14 For ease of presentation, we first introduce the separate evolution equations for the four stochastic variables before introducing the covariance matrix of these four state variables.
is total factor productivity. Damages as share of pre-damages aggregate output D increase in global mean temperature relative to preindustrial temperature T. We use the power-function specification to temperature ( 0 T   ). From (8), total factor productivity and aggregate output fall in the carbon stock and the shocks to climate sensitivity and damages:  19 Although , E  and  can formally become negative with finite probabilities due to their Gaussian distributions, we will show in section V that these probabilities are negligibly small. To avoid a formally ill-defined problem, we use truncated distributions in (6), (10a) and (10b) and in doing so place (negligibly small) probability atoms at zero values of the states. We subsequently ignore these atoms in the derivation of the asymptotic solutions for the optimal SCC presented in Result 1, 2 and 3. 20 How long it takes for an exponentially growing quantity to rise by a factor e = 2.72. For all three uncertain climate processes E ,  and  , the uncertainties are exogenously given and cannot be learned in our model. Fundamentally, both statistical (or aleatoric) uncertainty and systemic (or epistemological) uncertainty play a role but cannot always be separated. 21 For all three processes, we use in our calibration the most high-level or 'consensus' range of uncertainty estimates available, which also do not make this distinction (see section IV). For example, the 'consensus' uncertainty range for the climate sensitivity (e.g. IPCC, 2014, AR5, Chapter 12, Box 12.2) captures both statistical uncertainty in individual climate models and epistemological uncertainty arising from different climate models. The climate sensitivity uncertainty we examine is this aggregate measure of uncertainty, and similarly for the carbon stock and damage ratio.
Combining (4), (6) and (10), we have one truncated multi-variate Ornstein-Uhlenbeck process: use the terms 'carbon price' and SCC interchangeably and denote these by P.
There is no closed-form analytical solution to the stochastic dynamic optimal control problem (15). Solving numerically by approximating the value function and its derivatives in 5-dimensional space (time and the four states) is challenging due to the curse of dimensionality and does not yield analytical insight into the stochastic drivers of the optimal SCC. Instead, we derive an approximate solution using perturbation methods. We first examine the system for small parameter(s) (see Appendix B), then perform asymptotic expansions to leading-order in the thus identified small parameter, namely the share of climate damages in total GDP, To solve our problem, we perform a perturbation expansion in the small parameter around a base solution for which 0 = and the analytical solution is known. At each order n of the expansion, the problem is linear in the value function () , n J but remains fully nonlinear in the states, thus retaining riskaversion and prudence properties without approximation. Mathematically, at each order n, the problem is of the form where L is a linear differential operator in the states and the nonlinear forcing  is formed from products or derivatives of lower-order solutions (in n), so that the order of the 13 forcing thus obtained (from products or derivatives) is also ( ).

n O
We use the following truncated series solution up to n = 1 and thus restrict our attention to zeroth-and first-order terms in only, as denoted by the superscripts 23 , , , , , ( , , ) and similarly for F and C. The parameter appears both as small parameter of the series solution and as the multiple-scales parameter in front of the dependence on damages. We let total factor productivity be a slowly-varying power-law function of the climate-related variables , E  and :  higher derivatives required to model strong variation are thus ignored at leading order.
The zeroth-order value function in (18) inherits the properties of the production function (9). Our consistent leading-order estimate of the optimal SCC from the zeroth and first-order value function is thus changes to the economy resulting not from climate-induced changes to the marginal productivity of capital (as captured by (0) J 's slow dependence the climate-related states), but from direct damages to the economy arising from the three climate-related states. Combining the zeroth-and first-order solutions, we obtain the following result (corresponding to (C3.19) in Appendix C).

Result 1:
The optimal risk-adjusted SCC is: The term in (20)

III. A Closed-Form Rule for The Optimal Risk-Adjusted SCC
To obtain a closed-form solution for the optimal SCC in Result 1, we consider only leading-order terms in the climatic and damage uncertainties 2 E  , 2 ,   2   and their covariance terms (including with the capital stock). Appendix E then shows that the five-dimensional integral in Result 1 can be explicitly evaluated except for one time integral, and we obtain Result 3 given in Appendix A. For ease of exposition, we present in Result 2 below the special case of proportional damages ( 0 ET  = ) also examined by Golosov et al. (2014), in which marginal damages do not depend on the carbon stock, and we further assume the temperature and damage ratio are at their steady-state values ( 00 ,     == ). 26 The investment and growth rates of GDP are given to leading-order by their values without climate policy (cf. (C7)).
Implicitly, we get from the Euler and capital accumulation equations This requires five-dimensional numerical integration over the probability space corresponding to the four states and with respect to time. If the processes are independent, the integrals over the probability space of states can be evaluated independently. 28 In Appendix G we examine the accuracy of Results 2 and 3 by comparing with Result 1.

Result 2:
If 00 ,     == and 0, ET  = the optimal SCC is g  term in * r ). Higher economic growth also implies growing damages and a lower (growth-corrected) discount rate (the (0) g − term in * r ), which increases the optimal SCC. Economic growth thus depresses the SCC if 1.
  Higher economic activity (Y) and flow-damage coefficient () also push up the SCC. 29 In contrast to exogenous Ramsey growth models such as Golosov et al. (2014) and , our rate of economic growth g (0) is endogenous (see footnote 26). Hence, there are indirect effects on the optimal SCC via the growth rate g (0) . For example, the direct effect of a higher rate of pure time preference  is to lower the SCC and the indirect effect is to raise the SCC as economic growth is lowered (for 1   ). Together, the effect of a higher rate of pure time preference on the discount rate is always positive * (and thus always negative on the SCC). Although the optimal SCC does not depend directly on the share of fossil fuel in value added, the cost of fossil fuel, adjustment costs or the depreciation rate of physical capital, it does depend on adjustment costs and the depreciation rate via their effect on the endogenous rate of economic growth, which we treat as fixed in the analysis below. Furthermore, Ramsey growth models with an exogenous long-run growth rate include a second time scale associated with economic convergence, which will typically be faster than the climatic time scales. We conjecture that our formula for the optimal SCC derived in an AK growth model will therefore be a good approximation to the optimal SCC for a Ramsey growth model. A small fraction of emissions that stays forever in the atmosphere (  ) and fast decay of atmospheric carbon (higher  ) curb the SCC.

A. Economic growth uncertainty and the climate beta
Including economic, but not climatic uncertainty, Result 2 gives 0 * / ( ) The estimate of future economic growth is thus cut to take account of its uncertain nature, especially if risk aversion  is high. When 1   and rising affluence dominates the effect of growing damages, growth uncertainty cuts the discount rate and pushes up the risk-adjusted SCC. We rewrite the risk-adjusted discount rate as where we recover the first three terms of the introductory example. The prudence term depresses the discount rate and pushes up the SCC (cf. Leland, 1968;Kimball, 1990). This effect increases in the coefficient of relative prudence 1,  + risk aversion , and economic growth uncertainty. The insurance term stems from the perfect correlation between damages and GDP (damages are proportional to GDP). The insurance term acts to increase the optimal discount and reduce the optimal SCC, reflecting that positive shocks to damages are associated with positive shocks to GDP and are thus less harmful to welfare.
This corresponds to a built-in climate beta of one. 30 For 1   the prudence term dominates the insurance term, so growth uncertainty curbs the discount rate and boosts the optimal SCC, and vice versa for 1.
  If utility is logarithmic, 1  == and * , r  = so economic growth (or asset return) uncertainty does not affect the optimal SCC, as in Golosov et al. (2014). 31 It is instructive to consider the risk-adjusted discount rate when damages are not proportional to GDP:  is the elasticity of damages with respect to GDP and we only consider the case .

 = 32
The growing damages term indicates that due to the direct effect of a lower D  and the further stochastic reduction to the expected growth rate of damages (compared with 1 D  = ), the discount rate is higher and the SCC smaller for a given (0) 0 g  . The prudence term is unaffected. The selfinsurance effect now depends on the built-in beta of 0.
where  is the fraction of emissions that goes into the temporary reservoir. 32 In our model, we only consider the case of proportional damages. We have derived (22) in a similar, ad-hoc, fashion to the introductory example by assuming ( )( ) A similar expression is derived by Svenssen and Traeger (2014) and Dietz et al. (2018). Rewriting (22), the risk-adjusted discount rate becomes corresponding to Proposition 1 in Dietz et al. (2018). 33 This follows from 22 * / 1/ 2 0 , which is generally true.

B. Climate and damage uncertainties
The term in (21) is the climate sensitivity risk correction and depends on and convex dependence of damages on temperature ( 0 T   ). The climate sensitivity uncertainty correction is positive and larger for a more convex damage function, a more skewed climate sensitivity distribution with high uncertainty (   ), smaller discount rate ( * r ), and faster carbon decay rate ( ).
The damage rate risk correction (21) is zero if the distribution of the damage ratio is not skewed ( 0   = ). A rightskewed distribution requires an upward-correction of the SCC, more so if damages are more uncertain. In both cases, when keeping the steady-state fixed, increasing the rates of mean reversion   and   increases the risk corrections, as the near future becomes more uncertain.

C. Hedging: temperature beta and damage beta
We rewrite the term in Result 2 that corrects for correlations between climate and damage risks, on the one hand, and economic risks, on the other hand, as  denote the temperature beta and damage beta, respectively. These betas measure the normalized covariance with shocks to the rate of economic growth analogously to the beta in asset 20 pricing theory (e.g. Lucas, 1978;Breeden, 1979). 34 The sign of (23) depends on whether relative risk aversion  exceeds one or not, i.e. on whether the climate hedging effect dominates the offsetting effect due to growing damages. 35 We will first discuss the hedging effects, corresponding to the terms implies that asset returns in industries producing, for example, agricultural products, heating systems or winter garments are low in future states of nature in which temperature is high. It is then optimal to hedge these investments more by raising the SCC. If the economy is dominated by industries whose returns benefit from higher temperature (e.g. air conditioning), the temperature beta is positive, and it becomes optimal to have a lower SCC. The adjustment is large if risk aversion is high, climate sensitivity is more uncertain and skew, damages are more convex, and the climate sensitivity beta is large (high , , , , non-zero even for a symmetric climate sensitivity distribution and a linear dependence of the damage ratio on temperature (i.e. 0 T   = ).
A negative damage beta K  implies that asset returns in industries will be low in future states of nature in which damages are high, over and above the effect of the built-in climate beta. It is likely to be negative, especially in vulnerable areas (e.g. investments in flood prone regions), justifying a higher SCC. Economies dominated by industries that make money from climate damage (e.g. water engineering) have a positive damage beta ( 0 K   ) and should price carbon less vigorously. The adjustment is large if risk aversion is high, the damage ratio has high uncertainty and skewness (high ,,     ) and is non-zero even for a symmetric damage ratio distribution (i.e. 0   = ).
34 Consistent with our perturbation scheme, the volatility of GDP is given to leading order by the volatility of the capital stock neglecting the effect of climate damages and thus the carbon stock, climate sensitivity and damage uncertainties. 35 Lemoine (2017) calls these the risk insurance and risk exposure effects, respectively.

D. Correlation between temperature and damage ratio risks
The term * CC captures the effect of correlation between temperature and damage ratio uncertainty on the SCC. This is positive if high temperature is associated with disproportionally high damages (e.g. extreme events such as hurricanes and fires as far as they are not captured by the convex dependence of damages on temperature), in which case the optimal SCC is higher. Risk aversion  plays no role, since there is no possibility of hedging the returns on assets.

E. Result 3
To derive Result 2, we have made two important simplifying assumptions: damages are proportional to the carbon stock ( 0 ET  = ) and the mean of the climate sensitivity parameter (a proxy for temperature) is at its equilibrium 36 Two further climate-beta effects have been suggested in the literature. First, Sandsmark and Vennemo (2007) only have one stochastic parameter, i.e. the loss of GDP for a given temperature, and additive damages (not proportional to GDP, so 0 D  = ). In this setup high future damages are associated with low levels of future aggregate consumption, and a large benefit from mitigating future climate change. The corresponding beta is thus negative. It relies on the product of the change in marginal utility due to damages and marginal damages themselves, is thus 2 () O in our perturbation scheme and too small to be included. Second, Nordhaus (2011) argues on basis of simulations with the RICE-11 integrated assessment model that "those states in which the global temperature increase is particularly high are also ones in which we are on average richer in the future", suggesting a positive beta. In the asymptotic approach framework of the paper,it does not feature in our correction factors, since it requires the integration of a Geometric Brownian Motion (for K), when solving the differential equation for the carbon stock, which cannot conveniently be done in closed form.
Crucially, if 0 ET  = , this effect is zero as marginal damages are no longer proportional to the carbon stock E and enhanced uncertainty of this term due to uncertain new emissions does not contribute to the optimal SCC. For the case 0 ET   , we examine this effect by numerically solving the stochastic differential equations and the integral in Result 1 and find it to be small (see Appendix G).

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value from the outset ( 0  = ), which are relaxed in Result 3 (see Appendix A).
As a result of the first, the adjustment to the SCC for carbon stock uncertainty (A4) is zero in Result 2. Generally, this adjustment is negative as marginal typically remain concave even for convex damages , marginal damages will not be constant but increase with future emissions, resulting in a higher SCC, as captured by multiplicative correction factors in the form of single-variable deterministic integrals in Result 3. As a result of the second assumption, the mean temperature response at initial times and thus the SCC is overestimated by Result 2, but, due to the multiplicative correction factors, this is not the case for Result 3. and a risk premium of 2 K  = 6.4% per year. In line with the specification in equation (6), we assume the global ratio of CO2 emissions to GDP declines at a rate of 2.0% per year, which matches recent data. 37 Following Nordhaus (2017), we use world GDP at PPP of 116 trillion US dollars in 2015. Table 1 gives details for investment, depreciation and the cost of fossil fuel.

A. Carbon stock uncertainty
To calibrate our 1-box model for carbon stock dynamics (6), we use the 17 linear impulse response functions from the survey in Joos et al. (2013) and find  = 0.65 and  = 0.35%/year. 38 We use the 90% confidence range 794-1149 ppmv in 2100 predicted by simulations for the high temperature scenario RCP 8.5 (Chapter 12.4.8.1, IPCC, 2014 AR5) to calibrate E  = 13 ppmv/year 1/2 . Fig. 1a shows the impulse response function for our 1-box model and Fig. 1b shows the stock of atmospheric carbon, including 95%-confidence bounds. 39 Fig. 1 shows that our simple 1-box model compares well with the 4-box model fitted to the same data by Aengenheyster et al. (2018) and the 2-box model of Golosov et al. 38 It is possible to estimate these values from historical data too (see Appendix F.3). 24 (2014). 40,41,42 Our confidence bands are much wider than those obtained from Joos et al. (2013) 43 and still much wider than the uncertainty range obtained from historical data, 44 suggesting that model uncertainty far exceeds any inherent variability. Nevertheless, we will show in section V that even with our high value of E  , the correction to the optimal SCC is small for  ( 0) St== {0.85, 0.15}  401 ppm, ignoring its third box for carbon that decays within the first decadal period. 41 We set the initial atmospheric carbon concentration to 0 S = 401 ppm of CO2 (May 2015), corresponding to 0.854 TtC or 3.13 TtCO2, and the preindustrial atmospheric carbon concentration to 280 ppm CO2, 0.596 TtC or 2.19 TtCO2, so that 0 E = 121 ppm CO2, 0.258 TtC or 0.94 TtCO2. Updated and historical values can be found online at http://www.esrl.noaa.gov/gmd/ccgg/trends/global.html. 42 Although the impulse response function is less well captured by our 1-box model, this must be time integrated (after discounting) to evaluate the SCC. Agreement of the time path of the atmospheric stock ( Figure 1b) -2004, 1900-2004 and 1959-2004. This large variation of volatility with time suggest that historical volatility in the atmospheric carbon concentrations is better described by an Arithmetic Brownian Motion, as in (6).

B. Climate sensitivity uncertainty
We calibrate our temperature model (7) and (10a) 47 We take the mean of these distributions and fit our model to the first two moments of the TCR (mean and variance) and the first three moments of the ECS (mean, variance and skewness), as well as an initial temperature of 0 T = 0.89°C above preindustrial. 48,49 Table 2 shows that we match these moments well, and Table 3

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SCC rise as the atmospheric carbon stock rises with continued emissions (for 0 ET   ) and the climate sensitivity rises to equilibrium. For comparison, Golosov et al. (2014, p. 67-68) have a constant value of  = 2.4% GDP/TtC, which includes an upward adjustment for tipping risk.

A. Market-versus. ethics-based calibration
Using Result 3 and the calibration in Table 1, if one does not allow for the lags in reaching the ECS (by setting   →), the optimal risk-adjusted SCC is considerably increased (cf. column 3), as the large uncertainties associated with the ECS are then experienced instantly. The optimal SCC of $6.6/tCO2 is low, since it is based on market rates of return.

B. Volatility from asset returns vs. GDP
The most important drawback of our AK model is that asset returns (capital) and GDP growth have the same volatility (see also the discussion in Pindyck and Wang (2013)), while the former is empirically much greater. 56 Table 5 shows that the mark-up for economic risk drops dramatically if volatility of 54 E.g. Gollier (2018) relies on ethical arguments to use a zero or much lower discount rates than derived from asset market returns. To analyse this problem, the government should maximize expected welfare using low ethicallymotivated discount rates subject to the constraints of the decentralized market economy with a lower discount rate. The optimal carbon price will then typically fall short of the social cost of carbon (Belfiori, 2017;Barrage, 2018). 55 Kelly and Tan (2015) find that the mass of tail uncertainty in the climate sensitivity is curbed quickly even though overall learning is slow, because observations near the mean are evidence against fall tails. Bayesian learning curbs emissions by 50% instead of 38% without. Once the mass of the tail diminishes, remaining uncertainty is largely irrelevant for optimal emissions policy. Our formula for the optimal SCC shows that, once learning has removed tail uncertainty and skewness, in the distribution of the climate sensitivity, the SCC is increased by less. 56 Ramsey-or Solow-type models in which consumption is a concave function of capital display a smaller relative volatility of consumption than of capital. Furthermore, a Ramsey-type model would introduce a second timescale to the problem (of economic convergence), which is likely to be fast compared to climatic timescales and, for purposes of calculating the optimal SCC, can probably be ignored.
GDP growth rates instead of asset returns is used. 57 Due to the higher riskadjusted discount rate (0) r , the mark-ups for climate sensitivity and damage ratio uncertainty and the risk-adjusted SCC are considerably reduced. With asset return volatility, an increase in RRA 58 from 4.3 to 6.0 depresses the discount rate (0) r from 2.9% to 2.3% per year and pushes up the risk-adjusted SCC to $92.2/tCO2, corresponding to a total risk mark-up of 705%, whereas with GDP volatility this effect is negligibly small. With asset return volatility, an increase in IIA from 1.5 to 2.0 also pushes down the discount rate (0) r to 2.3% per year and the risk-adjusted SCC up to $87.2/tCO2. With GDP volatility, a similar increase in IIA instead increases the discount rate (0) r (from 4.5% to 5.5% per year), pushes down the deterministic SCC from $11.5 to $8.1/tCO2 and the risk-adjusted SCC from $14.6 to $10.2/tCO2. Summarizing, the effect of RRA on the risk-adjusted SCC depends crucially on the magnitude of economic volatility and is very substantial for asset return volatility but negligibly small for GDP growth volatility. More IIA substantially boosts the risk-adjusted SCC for asset return volatility, 59 but decreases for GDP growth volatility. This accords with Crost and Traeger (2013), Ackerman et al. 57 Historical data for the growth rate of world GDP for 1961-2015 imply K  = 1.5 %/year 1/2 , which we use here. 58 In this section, we will use the short-hands RRA and IIA to denote relative risk aversion ( RRA  = ) and intergenerational inequality aversion( IIA  = ), respectively.

31
(2013) and Hambel et al. (2017), who all use uncertainty based on GDP. 60 C. Convexity of the damage function Table 6 considers the effect of our convex damage function ( ET  = 0.28) on the SCC. Generally, the SCC is larger due to larger damages for higher temperatures (cf. Fig. 3b), which is felt more strongly for lower discount rates. 61 A small mark-up for carbon stock uncertainty is now required, which is negative due to the concavity of marginal damages for ET  = 0.28 (cf. (A4), Result 3). The climate sensitivity risk mark-up increases considerably due to the more convex damages-temperature relationship ( T  = 1.0 vs. 0.56). If we consider the highly convex damage function of Ackerman and Stanton (2012) (henceforth AS12), also shown in Fig. 3b with damages rapidly increasing above 1°C, we obtain an even larger deterministic SCC of $77.2/tCO2, a climate sensitivity risk mark-up of 61% and a total risk-adjusted SCC of $140.8/tCO2. 62,63 60 With GDP growth volatility, it is possible to use an even lower ethics-based value of impatience of  = 0.1%/year without negative discount rates and unbounded value of the SCC, which we will use below. 61 This effect more than compensates the higher effective discount rate due to atmospheric decay of carbon in the case of convex damages (cf.  Fig. 3b. We retain the distribution for  and the value of   for convex damages given in Table 1. 63 As an alternative to our multiplicative uncertainty, Crost and Traeger (2014) Table 7 examines the effect of the different climate betas. If the elasticity of damages with respect to world GDP (the built-in climate beta) is reduced from 1 to D = 0.8, two effects take hold: damage shocks are no longer fully insured, depressing the risk-adjusted discount rate (self-insurance term in (22)) and pushing up the SCC, and damages now grow less rapidly than GDP, pushing up the discount rate (growing damages term in (22)) and depressing the SCC. Table 7 shows that the former effect dominates when economic volatility is based on asset returns, and the latter when it is based on GDP growth. 12%/ year 1/2 ), and for  = 0.1%/year in the case of GDP growth volatility ( K  = 1.5%/ year 1/2 ).

D. Correlated risk and climate betas
Taking economic volatility based on GDP growth, the SCC drops from $40.1 to $28.1/tCO2 as the temperature beta K   , which measures correlation between temperature and GDP, is increased from its minimum to its maximum value ( K   from -1 to 1). Similarly, the SCC drops from $36.5 to $31.7/tCO2 as the damage beta K  , which measures correlation between damages and GDP, is 2.45 at 4  C), which is much higher than our (constant) value of 0.29, especially at higher temperatures. Figure 3a indicates that this alternative gives a damage ratio distribution that is also more uncertain (wider confidence bands) at temperatures higher than 3  C or 4  C compared to proportional damages. Both the higher skewness and higher uncertainty push up the optimal SCC for low discount rates, but this effect is like our case of convex damages (Fig. 3b).
increased from its minimum to its maximum value ( K  from -1 to 1). Finally, if we vary  from -1 to 1, the SCC increases from $29.2 to $39.0/tCO2, with the largest value corresponding to the case when future climate sensitivity shocks are perfectly (positively) correlated with future damage ratio shocks. 64

E. Comparison with other calibrations
In Table 8, we evaluate the optimal risk-adjusted SCC for different calibrations in the literature. Golosov et al. (2014) uses proportional damages, logarithmic utility (IIA = RRA = 1), and  = 1.5% per year, which gives a riskadjusted discount rate (0) r of 3.5% per year. With logarithmic utility, neither the expected rate of growth nor uncertainty about the future rate of growth influences the optimal SCC. Gollier (2012) uses RRA = IIA = 2 and  = 0, giving a risk-adjusted discount rate (0) r of 2.5% or 4.0% per year and a riskadjusted SCC is $62.6 or $18.5/tCO2 for economic volatility based on asset markets and GDP growth, respectively. 65 The discount rate is only substantially lowered for asset return uncertainty; asset return uncertainty depresses the discount rates and pushes up the risk-adjusted SCC as IIA exceeds one. r of 3.9% or 4.4%/year for economic volatility based on asset markets and GDP growth, respectively. Correspondingly, we obtain a lower deterministic SCC of $11.9/tCO2 and a lower risk-adjusted SCC of $19.1 or $15.1/tCO2 corresponding to lower total risk mark-ups of 60% or 27% for economic volatility based on asset markets and GDP growth, respectively. Finally, the last column of Table 8 uses IIA = RRA = 1.45 and a very low rate of time preference of  = 0.1%/year corresponding to a discount rate (0) r of 2.5% per year (for GDP-based economic volatility) and uses AS12 damages, which reflects the choice of low discount rate and convexity of damages used by Stern (2007). This gives very high values for the deterministic SCC of $87 and the risk-adjusted SCC of $165.2 per tCO2.

VI. Concluding Remarks
We have derived a tractable rule for the optimal risk-adjusted SCC under The optimal SCC decreases in intergenerational inequality aversion if trend economic growth corrected for its uncertainty is positive but increases in risk aversion if economic growth (or asset returns) are volatile. If damages are proportional to GDP, there is a built-in climate beta of one. This self-insurance effect depresses the optimal SCC. If the elasticity of damages with respect to GDP is less than one, there is less potential for self-insurance, which pushes up the SCC, and damages grow less rapidly, which pushes down the SCC. The first effect dominates if economic volatility is derived from asset returns, but the second effect dominates if volatility is derived from GDP growth. Uncertain climate sensitivity increases the SCC significantly, especially due to the skewness of the equilibrium climate sensitivity distribution, further enhanced by the convex dependence of damages on temperature. The magnitude of this mark-up depends crucially on the time scale on which it arises, and the much larger and more skew equilibrium climate sensitivity only plays a role for lower ethics-based discount rates. There is some evidence that the distribution damage ratio is right-skewed with an increase in the optimal SCC as a result.
Our rule for the optimal SCC also allows for correlated risks. If relative risk aversion exceeds one, what we call the hedging effects dominate the offsetting effects resulting from damages being proportional to GDP. It is then optimal to hedge and raise the SCC if the temperature beta is negative. This occurs when asset returns are high in future states of nature in which temperature is low (e.g. industries producing agricultural products, heating systems or winter garments).
If risk aversion exceeds one, we also show that the optimal SCC is higher if the damage ratio beta is negative. This occurs when asset returns are high in future states of nature in which the damage ratio is lower than expected, which is typical, except for in adaptation industries (e.g. industries building flood defences). If risk aversion equals one, correlated risks do not affect the SCC, except for through the correlation between temperature and damage ratio risk.
We have found that the role of climate sensitivity uncertainty relies crucially on the time scale on which the large uncertainty and skewness associated with the ECS arise, a time scale that is not well understood.
Instead of the TCR and the ECS, the so-called transient climate response to cumulative emissions (TCRE, e.g. ) is gaining traction. Although its uncertainty has not yet been as thoroughly studied as the TCR and the ECS, the absence of inherent time scales makes the TCRE useful for calculating the SCC needed to keep temperature below a cap.
Future research should be aimed at models that can have ethics-based discounting for policy makers but market-based discounting for the private 36 sector and that are general enough to distinguish volatility of equity returns and GDP growth. We have abstracted from long-run risk in economic growth (Bansal and Yaron, 2004) and a downward-sloping term structure resulting from mean reversion in economic growth (Gollier and Mahul, 2017). 66 Models that include these three aspects should give more robust estimates of temperature and damage betas. Other challenges are to allow for compound Poisson shocks to temperature and damages (cf. Hambel et al., 2018;Bretschger and Vinogradova, 2018;Bansal et al., 2016), positive feedbacks such as the CO2 absorption capacity of the oceans declining with temperature (Millar et al., 2016), the timing of climatic uncertainty, the risk of tipping points (e.g., Lemoine andTraeger, 2014, 2016;Lontzek et al., 2016;Cai et al., 2016;van der Ploeg and de Zeeuw, 2018), which may further increase the optimal SCC. (

  ) (see Appendix E for the derivation). The resulting
Result 3 includes additional correction factors, which can be evaluated as simple, onedimensional integrals. The only additional assumption is that the future atmospheric carbon stock does not inherit any of the uncertainty from new emissions through their dependence on the stochastic capital stock (cf. (E2.3)), which is associated with only a very small error, as discussed in Appendix G.

Result 3:
The leading-order optimal SCC is: to the zeroth order of A2 approximation. We refer to the  -terms in (A2) as uncertainty adjustments. We distinguish two types of correction factors, for 0 ET   and for 0 ,   which can be linearly combined, for example: . We will discuss the uncertainty adjustments below. The correction factors are given in (E3.4)-(E3.5) in Appendix E.
The adjustments for uncertainty in the carbon stock, climate sensitivity, the damage ratio and the interaction between the two, which are now multiplied by their respective correction factors, are given by The adjustments for correlated climate and economic risk is Similarly, all the adjustments are corrected by their respective correction factors to take this delayed temperature increase into account.

Appendix B: Transformation to Non-Dimensional Form (For Online Publication)
We define the non-dimensional variables . We define to be the growth rate of the economy without additional climate change, 0 g   and The HJB equation (16)  (1 ) The resulting non-dimensional expressions are , , dˆˆˆ, an Damages and total factor productivity become where the damage fraction D D  is already non-dimensional, * * 1 The first-order conditions of (B2) with respect to Ĉ and F are, respectively, where we have defined the optimal SCC in non-dimensional terms as and use (B7) to write the production function as

Appendix C: Derivation of Zeroth-Order Solution (For Online Publication)
In non-dimensional terms, the truncated series solutions for the value function and the forward-looking control variables (18) is ,, ( O the Hamilton-Jacobi-Bellman equation (B2) can be written as where we have substituted for the forward-looking variables Ĉ and F at (1) O from (B6) and (B7) and we have used i is the (constant) optimally chosen investment rate. Equation (C2) has a power-law solution of the form (0) 1 0 , J K   − = and following some manipulation we obtain From the first-order condition (B6), we obtain (C5) q the price of capital in consumption terms. 1 We can thus write the value function (C4) as Substituting in forF from (B7) and for Ŷ from (B9), we obtain from ˆˆ: In equilibrium, the marginal propensity to consume (0) (0)ĉ q equals the expected return on investment (0) r minus the growth rate of the economy (0) . g In turn, the expected return on investment equals the sum of the risk-free rate (0) rf r and the risk premium (0) . r  Hence, and with a risk premium of (0) 2K r  = in the absence of any climate risk at zeroth-order, the risk-free rate is: Although (   .ˆˆˆˆˆˆK We begin by integrating the multi-variate Ornstein-Uhlenbeck process (D1.1), including only terms at zeroth order, so that the coefficients are constant, and a closed-form solution is available. Specifically, where we have relied on the solution for K from the zeroth-order problem (cf. (C8)). The slow dependence of productivity Â on the states Ê ,  and  can be neglected when

D.2. Evolution equations for K and Ê
We consider the expected evolution equations of the states K and Ê at () O and (1) O , respectively. At this order, we have for the expected evolution of ˆ: where the first identity makes use of , IC =− since production net of fossil fuel costs is unaffected by the SCC in our formulation: The identity in (D2.2) relies on the Cobb-Douglas nature of the production function. The third identity in (D2.1) follows from a Taylor-series expansion of ˆ, C given by (B6), with respect to the small parameter (about 0 = ): Noting that (1) (1) ,îc =− we can rearrange this linear equation to give which is used in the third identity in (D2.1). For ˆ,

D.3. The Hamilton-Jacobi-Bellman equation
Substituting for the forward-looking variables Ĉ from (B6) and F from (B7), the Hamilton-Jacobi-Bellman equation (B2) becomes at ( ) : where we have used the identity ˆˆk Substituting from (D3.2), two of the terms in (D3.1) simplify to Using (D3.5), (D3.1) can be rewritten as a forced equation: (1) ( where the forcing is defined as To obtain derivatives of the zeroth-order value function with respect to Ê ,  and ˆ,  we first differentiate with respect to the marginal productivity of capital (0) mpk , r which depends on these three variables (via the chain rule of differentiation). From (C6), we obtain: Since the investment rate is implicitly defined, we get from (C7) by implicit differentiation: Combining (D3.8) and (D3.9), we obtain (D3.10) Using the chain rule of differentiation, we find the individual terms that contribute to the forcing (D3.7): and similarly for derivatives with respect to  and ˆ,  as well as cross-derivatives. From the zeroth-order solution where have used the following short-hands The scaled forcing is defined by 4 (D3.17)

Appendix E: Leading-Order Effects of Uncertainty (For Online Publication)
Assuming that the future atmospheric carbon stock does not inherit any of the uncertainty from new emissions through their dependence on the stochastic capital stock (assumption I), examining only the leading-order effects of uncertainty (assumption II), setting the initial value of the damage ratio but not of the climate sensitivity parameter at its steady-state (ˆ1 ) ( t   = , 0 ) (t   = ) (assumption III), this appendix derives closed-form solutions for the optimal risk-adjusted SCC based on Result 1. In doing so, we derive Result 3 (and its special case Result 2 for 0 ET  = ).

E.1. Carbon stock dynamics
The expected value of the carbon stock is governed by the differential equation (D2.5) with solution We use alternative star symbols A19 to denote rates corrected for atmospheric carbon stock decay. To leading order, we have for the terms involving the carbon stock: where we let the subscript on Using (E2.2)-(E2.5), we now consider the terms in the forcing (D3.17) consecutively and let the subscript indices correspond to the sequence of terms in (D3.17) (left to right).
To consider the covariance terms in the forcing (D3.17), we also expand in ( ) and only consider deviations from the zeroth-order mean consistent with our search for leading-order terms only. The following terms arise:       1ˆÊ where elements of the covariance matrix have been substituted from (D1.5).

E.3. Leading-order solution
Combining all the leading-order terms in the forcing equation (       Equation (E3.1) together with (E3.2)-(E3.5) gives the optimal SCC. We do not explicitly give the correction factors for the correlation terms involving carbon stock uncertainty.

F.1. Asset returns, risk aversion and intertemporal substitution
We follow the calibration of Pindyck and Wang (2013), but ignore the effect of catastrophic shocks. 5,6 Using monthly asset data from the S&P 500 for the period 1947-2008, we obtain an annual return on assets (capital gains plus dividends) of (0) r = 7.2%/year with annual volatility of K  = 12%. For a return on safe assets of 0.80%/year based on the annualized monthly return on 3-months T-bills, we obtain a risk premium of

F.2. Productivity, fossil fuel, adjustment costs and the depreciation rate
To calibrate total factor productivity, we consider the production function in the absence of climate damage that can be obtained by setting 0 P = (i.e. at zeroth order), namely (cf. (B9)). Pindyck and Wang (2013) use empirical estimates of the physical, human and intangible capital stocks and find 5 Pindyck and Wang (2013) use Poisson shocks to capture small risks of large disasters (cf. Barro, 2016) and thus match skewness and kurtosis of asset returns. These shocks are responsible for approximately 1%-point of the risk premium. 6 The alternative is to calibrate our AK model to the observed volatility of consumption or output (cf. Gollier, 2012), which are generally much less volatile than capital (asset returns). Because the volatilities of capital, consumption and output are equal to the volatility of capital in an AK model, this alternative calibration gives a much lower volatility and, consequently, a higher coefficient of relative risk aversion to match the equity premium (see also the discussion in Pindyck and Wang, 2013). Historical data for the growth rate of world GDP for 1961-2015 imply a volatility of K  = 1.5%/year 1/2 and thus a much higher value of risk aversion of  = 2.8 2 10  for an equity premium of 6.4%/year. Kocherlota (1996)

F.3. Atmospheric carbon stock and uncertainty
Here we calibrate our carbon stock model (6) to the Law Dome Ice Core 2000-year data set and historical emissions. The first column of Figure F1 shows maximum-likelihood estimates, from which it is evident that estimates displaying a certain linear relationship between  and  are of comparable likelihood. 9 7 We estimate the share of energy costs from data for energy use and energy costs from BP Statistical Review of World Energy 2017. Data for emissions are obtained from the same source available online at https://www.bp.com/en/global/corporate/energy-economics/statistical-review-of-world-energy.html. Our estimate of energy costs as a percentage of GDP is in good agreement with data from the U.S. Energy Information Administration available online at https://www.eia.gov/totalenergy/data/annual/showtext. php?t=ptb0105. 8 This is in line with Caselli and Feyrer (2007), who estimate annual marginal products of capital of 8.5% for rich countries and 6.9% for poor countries, and an observed annual risk premium of 5-7%. They use a depreciation rate of 6.0% to calculate the capital stock from investment, include the share of reproducible capital rather than the share of total capital, account for differences in prices between capital and consumption goods and correct for inflation. 9 Annual data from the Law Dome firn and ice core records and the Cape Grim record are available online at ftp://ftp.ncdc.noaa.gov/pub/data/paleo/icecore/antarctica/law/law2006.txt.This data is based on spline fits to different dataset with different spline windows across time reflecting changes in the temporal resolution of the data. The discrete nature of the fitted data is evident for the early years. Annual carbon emissions from fossil fuel consumption and cement production are available online at http://cdiac.ornl.gov/trends/emis/tre_glob_2013.html.

FIGURE F1. HISTORICAL ATMOSPHERIC CARBON STOCK CALIBRATION
These loci of maximum likelihood are shown separately in Figure F2, with the overall maximum denoted by a red circle and corresponding values given in Table F1. The remaining columns in Figure F1 show the predicted and observed rate of change of the A31 atmospheric carbon stock (second column), the predicted and observed atmospheric carbon stock (third column) and the remaining variability (fourth column). 10  Table F1 shows volatility as percentage of the initial carbon stock, from which we note that the stochastic carbon stock correction to the optimal SCC will be tiny if estimated from historical emissions.

Appendix G: Accuracy of Results 3 (For Online Publication)
Result 1 is evaluated numerically by discretization in time before evaluating the expectation operator numerically exactly and summing up the discounted contributions of every time step. Whereas the stochastic processes for  and  are autonomous, the stochastic process for K remains autonomous in Result 1, and all three have (independent) probability distributions available in closed form, the probability distribution of E at any time period in the future must combine all uncertain emissions (proportional to K ) before that time. As the time integral of a Geometric Brownian motion does not have a closedform solution, we update the probability distribution function of E every time step with the stochastic emissions and the decay in that period according to the differential equation for E and project on a fixed grid for E to enable transfer of the probability density function between time periods. Of course, the validity of Result 1 itself still relies on the parameter being small. Consistent with our perturbation scheme, all our optimal riskadjusted carbon prices in Results 1 and 3 are evaluated along the business-as-usual path for which 0 P = . We assess the accuracy of Result 3 for a number of the calibrations examined in section V. By choosing the grid size to be sufficiently small and the grid to be sufficiently large in each case, we ensure that discretization errors associated with Result 1 are negligible. Two factors determine the accuracy of using Result 3 instead of Result 1. First, in Result 3 we ignore any uncertainty in the atmospheric carbon stock that arises because of the uncertain nature of future economic growth and thus of future emissions.
For our base case calibration with proportional damages ( 0 ET  = ), the stochastic nature of E does not lead to a change in the SCC. Second, in Result 3 we only consider leadingorder terms in the climate sensitivity uncertainty.

A36
We can confirm from Table G1 that the combined effect of these two errors is sufficiently small to be ignored for all practical purposes. As expected, it is larger for low discount rates, higher economic volatility and convex damages.