Social Cost of Carbon Under Stochastic Tipping Points

Abstract

Is climate change concerning because of its expected damages, or because of the risk that damages could be very high? Climate damages are uncertain, in particular they depend on whether the accumulation of greenhouse gas emissions will trigger a tipping point. In this article, we investigate how much risk contributes to the Social Cost of Carbon in the presence of a tipping point inducing a higher-damage regime. To do so, we decompose the effect of a tipping point as an increase in expected damages plus a zero-mean risk on damages. First, using a simple analytical model, we show that the social cost of carbon (SCC) is primarily driven by expected damages, while the effect of pure risk is only of second order. Second, in a numerical experiment using a stochastic Integrated Assessment Model, we show that expected damages account for most of the SCC when the tipping point induces a productivity shock lower than 10%, the high end of the range commonly used in the literature. It takes both a large productivity shock and high risk aversion for pure risk to significantly contribute to the SCC. Our analysis suggests that the risk aversion puzzle, which is the usual finding that risk aversion has a surprisingly little effect on the SCC, occurs since the SCC is well estimated using expected damages only. However, we show that the risk aversion puzzle does not hold for large productivity shocks, as pure risk greatly contributes to the SCC in these cases.

1 Introduction

Climate change will induce damages in the future, although their magnitude remains uncertain (Diaz and Moore 2017). A key uncertainty is whether emissions will trigger non-marginal or abrupt changes, often referred to as “tipping points” (Lenton et al. 2008; Alley et al. 2003; Steffen et al. 2018). Examples of large-scale regime shifts include the shutdown of thermohaline circulation, the melting of the Arctic sea-ice or the die back of the Amazonian rainforest. Such shifts could also stem from the limited ability of social and economic systems to cope with climate conditions beyond some threshold. Assessing the present social value of damages from an additional ton of ${\mathrm {CO}}_{2}$released in the atmosphere, i.e. the social cost of carbon (SCC), requires to account for the possibility of triggering such high-impact events as the planet warms.

Early assessments of the SCC relied on deterministic models balancing abatement costs with benefits from avoided damages, assuming they are known to the social planner (Nordhaus 1994). They used a damage function capturing best-guess value of the level of damage for each additional degree of warming. Concerns about catastrophic damages and tipping points have first been addressed in a deterministic fashion. Many studies considered how alternative damage functions, meant to reflect that damages may be more convex or present abrupt jumps, affect the results (Pizer 2003; Dumas and Ha-Duong 2005; Ackerman and Stanton 2012; Wouter Botzen and van den Bergh 2012; Dietz and Stern 2015; Weitzman 2012).

More recently, studies have included in an endogenous fashion different types of climate damages risks, i.e. that damages could differ from their expected value, in integrated assessment models (IAMs). First, some have addressed uncertainty over the damage function by considering a social planner facing a distribution of damage functions rather than a single estimate (see for instance Crost and Traeger 2013). A second type of risks is recurring shocks or volatility in damages which hit the economy, notably as a way to reflect the occurrence of disasters (Pindyck and Wang 2013; Bretschger and Vinogradova 2019). There is an emerging literature on how these risks affect the SCC, also in combination with other uncertain dimensions, such as climate sensitivity or growth (Jensen and Traeger 2016; Lemoine and Rudik 2017; Van Den Bremer and van der Ploeg 2018).

Tipping points belong to another category of risk. They entail that emissions can irreversibly shift the world from a low- to a high-damage regime. Pioneering works on the management of thresholds show how optimal abatement is affected by the risk that pollution may trigger a catastrophic event (Clarke and Reed 1994; Tsur and Zemel 1996). Studies which model tipping points as a stochastic risk in IAMs suggest that accounting for those raises the SCC (Lemoine and Traeger 2014; Cai and Lontzek 2019; Diaz and Keller 2016) or near-term abatement (Keller et al. 2004; Belaia et al. 2014). These studies consider tipping points which induce either a permanent loss of productivity (Belaia et al. 2014; van der Ploeg and de Zeeuw 2018), the destruction of most productive capacities (Bretschger and Vinogradova 2018), or an increased damage regime (Lemoine and Traeger 2014), and provide insightful analysis, notably on how time and risk preferences affect optimal policy.

When considering risk, one naturally wishes to compare how results differ from a risk-free situation only relying on the expected outcome. In financial analysis, this comparison allows to derive the risk premium of an asset. Similarly, for the SCC, the gap between a risky and a risk-free situation indicates how much risk compounds the diminution of social welfare brought by emitting carbon. This is typically done by comparing results with risk to results when all the parameters are set to their expected value (Crost and Traeger 2013; Van Den Bremer and van der Ploeg 2018). However, for tipping points, the standard practice is to compare the SCC with and without a tipping point. Because the additional damage a tipping point could cause does not have a zero mean, comparing cases with and without tipping points conflates the effect of increased expected damages resulting from the introduction of a tipping point, and that from the dispersion of possible damages, i.e., a zero mean risk.

In this paper, we propose a simple method to distinguish the contributions of expected damages and risk to the SCC in the case of a tipping point triggering a shift to a higher-damage regime. We decompose a tipping point on damages as an increase in expected damages plus a (zero-mean) pure risk on damages. We compare the SCC with a tipping point to the SCC under expected damages, i.e. when there is no risk and damages are set at their expected level. First, using a simplified model, we demonstrate that in the case of a stochastic tipping point, the SCC is at first order equal to the SCC under expected damage. Risk introduces a correction that is only of second order, and that is proportional to risk aversion. We then investigate numerically in an IAM the gap between the SCC with a stochastic tipping point and the SCC under expected damages. We introduce the tipping point as a stochastic risk whose hazard rate depends on temperature, leading to a permanent drop in productivity. We analyse how preferences of the social planner (i.e., risk aversion, resistance to intertemporal substitution) and damages from the tipping point affect the comparison. We find that the SCC under expected damages closely approximates the SCC for low values of the productivity shock. Thus, risk plays a minor role when introducing a tipping point: the increase in SCC is due to the increase of expected damages. This result holds as long as the magnitude of the productivity shock is less than 10%. However, risk becomes important with higher productivity shocks and under high risk aversion. The results are also robust when considering a tipping point leading to more convex damage.

Our article contributes to a wider literature on how climate damage risks affect optimal climate policies. Evidence is mixed about whether risk is a fundamental part of optimal abatement, or just a second order correction. On the one hand, some authors argue that hedging against catastrophic outcomes should be the primary driver of abatement (Weitzman 2009; Pindyck 2013; Dietz 2011). However, several studies show that risk has a surprisingly limited effect on the optimal policy—this has been called the risk aversion puzzle. Risk aversion, when disentangled from the elasticity of intertemporal substitution, is found to play a modest role in IAMs (Ackerman et al. 2013), even in the case of a tipping point (Belaia et al. 2014; van der Ploeg 2016). Our contribution isolates the role of risk in the case of a stochastic tipping point, and shows under which assumptions the effect of risk is of second order. In our numerical simulations, risk only matters when there is possible exposure to catastrophic damages, while the SCC is primarily driven by expected damages for moderate productivity shocks triggered by the tipping point. This explains why studies find a modest role of risk aversion on the optimal policy, as productivity shocks usually considered in the literature remain below the 10% threshold. We add to this literature and provide orders of magnitude of the range of shocks and risk aversion it takes for risk aversion to significantly affect the SCC.

Our article also contributes to understanding the effect of different representations for climate damages on optimal policy. Since climate damages are the least-grounded aspect of IAMs (Diaz and Moore 2017; Revesz et al. 2014; Howard 2014) and have a strong impact on the SCC (as large as discounting van den Bijgaart et al. 2016), it is essential to build rigorous methodologies that compare how different representations of damages affect the SCC (Pottier et al. 2015; Guivarch and Pottier 2018). Here, we compare indeed two different settings to represent a tipping point: one where tipping points are represented as a truly stochastic event, one where tipping points are represented as a mere increase in the damage function, so that it reflects expected damages. In other words, we examine how the endogenous introduction of a tipping point in an IAM compares to simply using a more convex function, i.e., the method that was first used to represent catastrophic risks in such models. Our results show in which cases the two methods lead to different outcomes.

We illustrate in Sect. 2 the effects of increased expected damages and pure risk due to a tipping point in a simplified one-period model, and show that the former is the primary driver of the SCC. In Sect. 3, we lay out the stochastic IAM we use and present how we compute the SCC under expected damages. Numerical comparison between the SCC under expected damages and the SCC with a stochastic tipping point are discussed in Sect. 4. Section 5 concludes.

2 A Simple Model of a Stochastic Tipping Point

We present a simple one-period model of a climate economy to highlight the intuition behind the comparison between the SCC with a stochastic tipping point, and the SCC under expected damages. To that end we will consider successively optimal abatement and the resulting SCC in three situations:

• When there is no tipping point (${\textit{SCC}}^{*}$),

• When there is a stochastic tipping point (${\textit{SCC}}^{t}$),

• When damages are set at their expected level (${\textit{SCC}}^{ed}$).

We show that there is a first-order difference between the SCC with tipping points (${\textit{SCC}}^{t}$) and without ($\textit{SCC}^{*}$), difference that is captured when the SCC is calculated with damages set at their expected level ($\textit{SCC}^{ed}$). We further demonstrate that risk introduces a second-order correction, proportional to risk aversion and to the variance of damages.

Let us consider an economy producing gross output Y. Gross output is reduced because of damage d(E) due to emissions E, and costs c(a) of abatement measures a. Net output is entirely consumed, so that aggregate consumption is given by $C(a,E)=Y-c(a)-d(E)$. With $\sigma$ the (unabated) carbon intensity of production, emissions and abatement are linked by $E=\sigma Y (1-a)$. Let u(C) be the social welfare function that depends only on aggregate consumption, the program of the social planner is:

$$\begin{aligned} \underset{a,E}{\max }\,\,\, u\left( C\left( a,E\right) \right) ,\quad s.t. \;E=\sigma Y (1-a) \end{aligned}$$

(1)

so that optimal abatement satisfies $c'(a^*)=\sigma Y d'(E^*)$ (we star all variables related to the optimal point in this case).

We can define the SCC at the optimal point as the marginal social damages occurring due to an extra emission, measured in consumption units.

$$\begin{aligned} {\textit{SCC}}^{*}=-\frac{\partial _E u(C(a,E))|_{a^*,E^*}}{\partial _C u(C)|_{C^*}}=-\frac{-u'(C^*)d'(E^*)}{u'(C^*)}=d'(E^*) \end{aligned}$$

(2)

Note that (provided the constraints $0\le a \le 1$ are not binding) the equation for optimal abatement can be written as:

$$\begin{aligned} \left. \partial _C u\right| _{C^*}.(-c'(a^*)) +\left. \partial _E u(C(a,E))\right| _{C^*}.(-\sigma Y)=0 \end{aligned}$$

(3)

so that $\textit{SCC}^{*}=\frac{c'(a^*)}{\sigma Y}$: Thus the SCC is also a measure of marginal abatement costs.

2.1 SCC with a Stochastic Tipping Point

We now introduce stochastic tipping points to our setting. There are several states of nature $\omega$, depending on whether or not tipping points have been crossed. The damage function is $d_\omega (E)$ in state $\omega$. The probability of having triggered tipping points, i.e. on being in state $\omega$ depends on aggregate emissions $p_\omega (E)$. A simple case is when there are one tipping point and two states of nature: 1 the pre-tipping point world where the damage function is the $d_1$, and 2 the post tipping point world that leads to a higher damage regime $d_2\ge d_1$. The tipping point induces a jump in damages $d_2-d_1$. The probability of being in the post-tipping world is an increasing function of the quantities of emission released $p_2(E)$. Our framework accommodates for more general formulations, such as multiple points or a single tipping point leading to different post-tipping damage functions.

The social planner maximizes the expected social welfare, where we write $C_\omega (a,E)=Y-c(a)-d_\omega (E)$ the aggregate consumption in state $\omega$:

$$\begin{aligned} \underset{a,E}{\max } \,\,\, {\mathbb {E}}_\omega \left[ u(C_\omega (a,E))\right] , \,\,\,\, s.t. \,\, E=\sigma Y (1-a) \end{aligned}$$

(4)

We note $a^t,E^t$ the optimal abatement and emissions of this program. The SCC is the marginal social damages of an extra emission measured in consumption units, so, in this framework:

$$\begin{aligned} {\textit{SCC}}^{t}=-\frac{\partial _E {\mathbb {E}}_\omega \left[ u(C_\omega (a,E))\right] |_{a^t,E^t}}{{\mathbb {E}}_\omega \left[ u'(C_\omega (a,E))\right] |_{a^t,E^t}} \end{aligned}$$

(5)

Note that for tipping points, the probability of being in a given state of nature is endogenous to the action of the planner, in contrast with exogenous risk (for instance on the value of climate sensitivity). The optimal abatement $a^t$ of this program solves (provided the constraints on it are not binding):

$$\begin{aligned} {\mathbb {E}}_\omega \left[ u'(C_\omega (a,E))\right] |_{a^t,E^t}.(-c'(a^t))+(\partial _E{\mathbb {E}}_\omega \left[ u(C_\omega (a,E))\right] |_{a^t,E^t}).(-\sigma Y)=0 \end{aligned}$$

(6)

so that the SCC is still given by $\frac{c'(a^t)}{\sigma Y}$. Specifying to the case of a single tipping point, we have ${\mathbb {E}}_\omega [u(C_\omega (a,E))]=(1-p_2(E)).u(C_1(a,E))+p_2(E).u(C_2(a,E))$, and the SCC of Eq. (5) can be written as:

$$\begin{aligned} \begin{aligned} {\textit{SCC}}^{t}&= \frac{c'(a^t)}{\sigma Y} = d_1'(E^t)\\&\quad + (d_2'(E^t)-d_1'(E^t)) \frac{p_2(E^t)u'(C_2(a^t))}{{\mathbb {E}}_\omega [u'(C_\omega (a,E))]}\\&\quad + p_2'(E^t)\frac{u(C_1(a^t))-u(C_2(a^t))}{{\mathbb {E}}_\omega [ u'(C_\omega (a,E))]} \end{aligned} \end{aligned}$$

(7)

As in van der Ploeg and de Zeeuw (2019), the previous equation offers a decomposition of the SCC into three terms:

• the marginal damage term, before reaching the tipping point,

• the additional marginal damage in case the tipping point is reached, weighted by the probability of tipping,

• the damage due to a marginal increase in the probability of triggering the tipping point.

Introducing tipping points in this way conflates a level effect and a risk effect. Damages are indeed increased in the same time as risk is introduced, because the effect of the tipping point is not a zero-mean risk on consumption. Indeed, compared to the situation without a tipping point, the mean additional risk of a tipping point is $(1-p(E))(d_1-d)+p(E)(d_2-d)$, which is equal to $p(E)(d_2-d_1)$ when the pre-tipping damage function is the same as the damage function without a tipping point ($d_1=d$). As a consequence, the increase of the SCC found when a tipping point is introduced cannot be attributed to risk alone, it may be simply the effect of higher expected damages. The next subsection will investigate this point.

2.2 SCC Under Expected Damages

To understand the effect of risk induced by a tipping point, we need a way to decompose the SCC so that we can disentangle what comes from expected damages from what is due to a zero-mean risk.

Let us go back to Eq. (5). We consider the expected damage function ${\tilde{d}}(E)={\mathbb {E}}_\omega [d_\omega (E)]$. One can write: $C_\omega (a,E)={\tilde{C}}(a,E)+\epsilon _\omega (E)$ with ${\tilde{C}}(a,E)={\mathbb {E}}_\omega [C_\omega (a,E)]=Y-c(a)-{\tilde{d}}(E)$. This decomposes the effect of tipping point on consumption as an effect on expected damages plus a zero-mean risk $\epsilon _\omega (E)=d_\omega (E)-{\tilde{d}}(E)$. We note $V_\epsilon (E)={\mathbb {E}}_\omega [\epsilon ^2_\omega (E)]$ the variance of damages of the zero-mean risk $\epsilon$.

In the case of a single tipping point, expected damages are ${\tilde{d}}=p_1d_1+p_2d_2=d_1+p_2.(d_2-d_1)$. The risk $\epsilon$ on consumption is $\epsilon _1=-p_2(d_2-d_1)$ in state 1, with probability $1-p_2$, and $\epsilon _2=(1-p_2)(d_2-d_1)$ in state 2 with probability $p_2$ (where all symbols are functions of emissions E). So, compared to a world without tipping points, in which the damage function is the pre-tipping damages function, introducing a tipping point increases expected damages by $p_2.(d_2-d_1)$ and adds risk. Thus, the risk of the tipping point is that damages are less than expected if the world does not tip (state 1, pre-tipping) and that damages are above expectations if the world does tip (state 2, post-tipping). It is a zero-mean risk whose variance is $V_\epsilon (E)=p_2(1-p_2).(d_2-d_1)^2$. The risk is of the same order of magnitude as the jump in damages $d_2-d_1$.

Let us assume that the risk $\epsilon$ is small and make a Taylor-expansion at the second-order in formula (5):

$$\begin{aligned} SCC^{t}&=-\frac{\left. \partial _E {\mathbb {E}}_\omega \left[ u({\tilde{C}}(a,E)) +u'({\tilde{C}}(a,E))\epsilon _\omega (E)+u''({\tilde{C}}(a,E))\frac{\epsilon ^2_\omega (E)}{2}\right] \right| _{a^t,E^t}}{\left. {\mathbb {E}}_\omega \left[ u'(C_\omega (a,E))\right] \right| _{a^t,E^t}} \end{aligned}$$

(8)

$$\begin{aligned}&=\frac{ u'({\tilde{C}}(a^t,E^t)){\tilde{d}}'(E^t)}{{\mathbb {E}}_\omega \left[ u'(C_\omega (a,E))\right] |_{a^t,E^t}} -\frac{\left. \partial _E \left( u''({\tilde{C}}(a,E)) \frac{V_\epsilon (E)}{2}\right) \right| _{a^t,E^t}}{{\mathbb {E}}_\omega \left[ u'(C_\omega (a,E))\right] |_{a^t,E^t}} \end{aligned}$$

(9)

We show in Appendix A that we can make a Taylor expansion also of the denominator and, after reordering, we obtain the following decomposition of the SCC (ignoring higher-than-second-order terms):

$$\begin{aligned} {\textit{SCC}}^{t}={\tilde{d}}'(E^t) +\gamma ({\tilde{C}}(a^t,E^t))\frac{{\tilde{C}}(a^t,E^t)}{2} \left. \partial _E V_\epsilon (E)\right| _{E^t} \end{aligned}$$

(10)

The SCC in the presence of a tipping point is thus shown to be the sum of marginal expected damages plus a second-order correction that is proportional to the Arrow–Pratt measure of relative risk aversion $\gamma (C)=-u''(C)/(u'(C).C)$ of the utility u, and to the marginal increase in the variance of risk $\partial _E V_\epsilon (E)$. When the effect of tipping point is split into an increase in expected damage plus a zero-mean risk, this decomposition shows that changes in expected damages drive the effect on the SCC whereas the zero-mean risk introduces only a second-order correction.

To make this statement more precise, we introduce the SCC under expected damages, that is the SCC in a deterministic model similar to Eq. (1) but with the damage function replaced by the expected damages of the tipping point ${\tilde{d}}$. We note $a^{ed},E^{ed}$ the optimal abatement and emissions of this program under expected damages. Thus the SCC under expected damages $\textit{SCC}^{ed}$ is given by:

$$\begin{aligned} {\textit{SCC}}^{ed}=\frac{c'(a^{ed})}{\sigma Y} = {\tilde{d}}'(E^{ed}) \end{aligned}$$

(11)

We have the following proposition:

Proposition 1

The difference between the SCC without a tipping point $(\textit{SCC}^{*})$ and the SCC with a tipping point $(\textit{SCC}^{t})$ is proportional to the difference between marginal expected damages and marginal damage in the pre-tipping point state of the world. It is first-order in the magnitude of risk induced by the tipping point.

On the contrary, the difference between the SCC under expected damages ($\textit{SCC}^{ed}$) and the SCC with a tipping point ($\textit{SCC}^{t}$) is only second-order. It is proportional to risk aversion and the marginal increase in the variance of risk induced by the tipping point.

Proof

See Appendix B$\square$

To summarize, we have demonstrated that introducing a tipping point raises the SCC, compared to a SCC without a tipping point, where the damage function is given by the pre-tipping point damage function. This first order difference is explained by an increase in expected damages. Whereas comparing the SCC with and without a tipping point introduces a first-order correction in the SCC, the correction is only of second order when we compare the SCC with a tipping point and the SCC under expected damages.

This means first that it is misleading to compare the SCC with and without stochastic tipping point, and second that the increase due to the tipping point is, in this simple model, a matter of expected level rather than risk. The SCC under expected damages captures most of the value of the SCC, as long as the damage shocks are small. Risk introduces a correction that increases with risk aversion. Interestingly, this result holds under rather general conditions. We did not assume a specific form for the change in damages induced by the tipping point. Hence, it applies generally, and in particular, in the case of one tipping point, it applies when the tipping point induces a jump in damage ($d_2-d_1$ is constant) or more convex damage, i.e. higher marginal damage ($d_2-d_1$ is an increasing function of temperature).

The key insight of this simple one-period model is that we can think of the SCC as being composed of two parts: SCC under expected damages and a risk premium proportional to risk aversion. For small tipping point damages, the SCC under expected damages will be close to the actual SCC. In the next section, we define the SCC under expected damages for an intertemporal model and use an IAM to explore numerically the gap between both SCC.

3 Contributions of Expected Damages versus Risk in a Stochastic IAM

We now use an IAM to numerically compare the SCC under a stochastic tipping point and the SCC under expected damages, in order to quantify the contribution of pure risk to the SCC. This allows us to examine whether the intuition from the simple model holds in a multi-period framework, and also for larger damages. We present in Sect. 3.1 the climate-economy model and in Sect. 3.2 the social welfare functions we use. We then explain how we construct in this model the SCC under expected damages (Sect. 3.3), and the values we explore for the parameters of the model (Sect. 3.4).

3.1 The Climate-Economy Model

An IAM is meant to capture the main crossed interactions between the economy and the climate system. On the one hand, the economy, depending on growth, mitigation policies and technological choices, produces greenhouse gas emissions which interfere with the climate system. On the other, these changes in the climate system cause damages to the economy. An IAM allows to derive an optimal emission path from the point of view of a social planner balancing the costs of mitigation and damages of climate change, and to calculate the social value of intertemporal marginal damages caused by emissions—the SCC.

We use a classical DICE-like model, building on the Ramsey–Cass–Koopmans framework (Guivarch and Pottier 2018). The economy produces a single good in quantity $Q_t$ using two factors, capital $K_t$ and labour $L_t$ through a Cobb-Douglas function. The productivity is affected by climate change via a damage factor¹$\varOmega _t$ that depends on temperature $T_t$, so that final production $Y_t$ writes:

$$\begin{aligned} Y_t=\varOmega (T_t)A_t K_t^{\alpha } L_t^{1-\alpha } \end{aligned}$$

(12)

Production induces emissions, which can be mitigated at a certain cost. The social planner trades off consumption, mitigation costs (which represent a share $\varLambda _t$ of production), and investment in capital (share $s_t$ of production)

$$\begin{aligned} C_t= & {} Y_t(1-\varLambda _t-s_t) \end{aligned}$$

(13)

$$\begin{aligned} \varLambda _{t}= & {} \theta _1(t)\mu _t^{\theta _2} \end{aligned}$$

(14)

$$\begin{aligned} K_{t+1}-K_t= & {} -\delta K_t+Y_t s_t \end{aligned}$$

(15)

where $\delta$ is capital depreciation, and $\mu _t$ the abatement rate. $\theta _1(t)$ measures total mitigation costs and decreases exogenously due to technical progress.

The difference with DICE equations concerns the climate system. It has been shown that DICE’s climate model implies a lag between $\mathrm {CO}_2$ emissions and warming that is too long, i.e., the temperature rises too slowly in response to emissions (National Academies of Sciences 2016; Mattauch et al. 2019) DICE’s climate model is also inconsistent at long timescales (Glotter et al. 2014). We adopt a simple linear formula linking temperature change to cumulative $\mathrm {CO}_2$ emissions, as in Guivarch and Pottier (2018) and Dietz and Venmans (2019). Indeed, the ratio of global temperature increase to cumulative emissions has been shown to be almost independent of time and of emission pathways in simulations of the response to a range of emission scenarios with climate models, as well as in observations (Matthews et al. 2009; Zickfeld et al. 2009, 2013; Gillett et al. 2013; Collins et al. 2013). There are physical explanations to this near-linear dependence between warming and cumulative carbon emissions, due to the compensating effects of oceanic uptake of heat and carbon (Solomon et al. 2009; Goodwin et al. 2015; MacDougall and Friedlingstein 2015). There are also limitations to such a simple climate representation, for instance Leduc et al. (2015) have shown that the linear relationship between temperature change and cumulative emissions is no longer valid for high emission pathways such as the RCP 8.5.

Our equation for temperature change is thus:

$$\begin{aligned} T_t=\beta (CE_0+\sum _{s=0}^{t} E_s) \end{aligned}$$

(16)

where $T_t$ is the global temperature increase at time t, $CE_0$ is cumulated emissions up to the first period of the model and $E_s$ the emissions at time s. The current stock of cumulated emissions is $S_t=CE_0+\sum _{s=0}^t E_s$.

$$\begin{aligned} E_t=\sigma _tY_t(1-\mu _t) \end{aligned}$$

(17)

where $\sigma _t$ is the carbon content of production that decreases exogenously over time, and $\mu _t$ the abatement rate.

In our central estimates, we model the tipping point as a stochastic process with an endogenous hazard rate $h_t$, leading to a permanent productivity shock, in line with Cai and Lontzek (2019), van der Ploeg and de Zeeuw (2018) and Belaia et al. (2014). Such a change in the damage function can potentially apply to a large range of tipping points inducing larger damages than expected. It can be direct impact on the economy, either caused by the melting of ice caps, leading to severe sea-level rise; a slowing down of thermohaline circulation; or a social tipping point beyond which adaptation is no longer possible. Before the tipping point, the damage factor is:

$$\begin{aligned} \varOmega _1(T)=\frac{1}{1+\pi T^2} \end{aligned}$$

(18)

In our central case, once the tipping point has been reached, the damage factor writes:

$$\begin{aligned} \varOmega _2(T)=\frac{1-J}{1+\pi T^2} \end{aligned}$$

(19)

J is the magnitude of the productivity shock, ranging from 0 to 1. In this representation, the tipping point increases the level of damages, but it does not affect the convexity of the damage function. Indeed, damages go approximatively from $\approx \pi T^2$ to $\approx \pi T^2+J$. Uncertainty about the convexity of damages may lead to greater changes than uncertainty about the level of damages (Crost and Traeger 2013), so we also consider in the Annex the case of a tipping point increasing marginal damages.

At each period, the tipping point occurs with a hazard rate $h_t$ which depends on the temperature level. We assume that the location of the tipping point is unknown. The initial prior is that the tipping point is uniformly distributed between $T_{\min }$ and $T_{\max }$. At each time $t-1$ with temperature $T_{t-1}$, the social planner learns whether the tipping point has been reached or not, as in Lemoine and Traeger (2014). If it has not been reached, this means that it can only occur when temperature is above $T_{t-1}$, so that the social planner updates prior for the next period. Hence the probability to reach the tipping point at t conditional to not reaching it at $t-1$ is given by:

$$\begin{aligned} h_t(T_t,T_{t-1})={\left\{ \begin{array}{ll} 0 &{} \text{ if } T_t\le T_{t-1} \text{ or } T_t \le T_{\min }\\ \frac{T_{t}-\max (T_{\min },T_{t-1})}{T_{\max }-\max (T_{\min },T_{t-1})} &{} \text{ if } T_t> T_{t-1} \text{ and } T_{\min } \le T_t \le T_{\max }\\ 1 &{} \text{ if } T_t > T_{t-1} \text{ and } T_t\ge T_{\max }\\ \end{array}\right. } \end{aligned}$$

(20)

Note that the marginal hazard rate tends to increase (i.e., $\partial _2\partial _1 h_t\ge 0$), as experienced temperatures increase. This setting differs from the representation chosen in Cai and Lontzek (2019) and van der Ploeg and de Zeeuw (2018), in which the hazard rate depends solely on current temperature with no learning. In that case the tipping point is therefore unavoidable in the long term. In our setting however, the tipping point is avoided with certainty if temperature stabilizes below $T_{\min }$; it can be avoided—but with no certainty—if temperature stabilizes between $T_{\min }$ and $T_{\max }$, and it is triggered with certainty if $T_{\max }$ is exceeded. This representation does not consider possible processes where a tipping point would be triggered with some lag.

3.2 Social Welfare Functions

We study two types of social welfare functions: expected utilitarianism with Constant Relative Risk Aversion (CRRA), and an Epstein–Zin social welfare function. In the CRRA representation, time and risk preferences are embedded in a single parameter, elasticity of marginal utility, which conflates the resistance to intertemporal substitution and the risk aversion. However, the resistance to intertemporal substitution and risk aversion can have opposite effects in the presence of risk (Ha-Duong and Treich 2004): while the former favours the consumption of present generations, the latter encourages more abatement in the present in order to lower the risk of triggering the tipping point. For this reason, we also apply Epstein–Zin preferences, which disentangle intertemporal substitution and risk aversion.

Welfare after time t, $U_t$, is defined recursively:

• For CRRA preferences

  $$\begin{aligned} U_{t}=\left( 1-\frac{1}{1+\rho }\right) u_t+\frac{1}{1+\rho } {\mathbb {E}}[U_{t+1}] \end{aligned}$$

  (21)

  where $\rho$ is the pure time preference rate, and utility at each time step is given by:

  $$\begin{aligned} u_t=L_t \frac{{(C_t/L_t)}^{1-\eta }}{1-\eta } \end{aligned}$$

  (22)

  $\eta$ is the elasticity of marginal utility.

  So that we can define Bellman functions as follows:

  $$\begin{aligned} V_{t}(x_t)=\underset{y_t}{\max }\left[ u(x_t,y_t)+\frac{1}{1+\rho } {\mathbb {E}}[V_{t+1}(G(x_t,y_t))]\right] \end{aligned}$$

  (23)

  where $x_t=(S_t,K_t)$ are state variables, $y_t=(\mu _t,s_t)$ are control variables, and $x_{t+1}=G(x_t,y_t)$ is the transfer function.

• For Epstein Zin preferences:²

  $$\begin{aligned} U_{t}= & {} \left( \left( 1-\frac{1}{1+\rho }\right) u_t+\frac{1}{1+\rho } {\mathbb {E}}[U_{t+1}^{1-\gamma }]^\frac{1-\theta }{1-\gamma }\right) ^\frac{1}{1-\theta } \end{aligned}$$

  (24)
  $$\begin{aligned} u_t= & {} L_t \frac{{(C_t/L_t)}^{1-\theta }}{1-\theta } \end{aligned}$$

  (25)

  For the sake of clarity, we use different notations in the Epstein–Zin case. We denote $\theta$ the resistance to intertemporal substitution (the inverse of the elasticity of intertemporal substitution), and $\gamma$ the risk aversion parameter.

  We can define Bellman functions in order to solve this dynamic program: $V_t=\frac{U_t^{1-\theta }}{1-\frac{1}{1+\rho }}$.

  $$\begin{aligned} V_{t}(x_t)=\underset{y_t}{\max }[u(x_t,y_t)+\frac{1}{1+\rho } f(V_{t+1}(G(x_t,y_t)))] \end{aligned}$$

  (26)

  f accounts for the decision maker’s attitude towards the risk of reaching a tipping point.³$f(V_{t+1})=[{\mathbb {E}}(V_{t+1}^\frac{1-\gamma }{1-\theta })]^\frac{1-\theta }{1-\gamma }$. It is the same formula as for CRRA preferences, in which $f={\mathbb {E}}$.

Using dynamic programming, we first approximate Bellman functions in the post-tipping world, and then in the pre-tipping world using expectations over the temperature at which the tipping point occurs.

3.3 Comparing the SCC for a Stochastic Tipping Point and the SCC Under Expected Damages

If S is the stock of emissions (* denotes that control variables $y_0$ are optimally chosen given $x_0$.), the SCC (at initial time) under a stochastic tipping point writes:

$$\begin{aligned} {\textit{SCC}}=- \frac{1}{1+\rho } \frac{\partial _S \mathbb E[V_{1}]|_{x_{1}}}{\partial _C V_0|_{(x_0,y_0^*)}} \end{aligned}$$

(27)

We use a modified damage factor $\varOmega _{ed}(T)$ to calculate the SCC under expected damages. This modified damage factor represents the expected damage factor given the prior on the temperature at which the tipping point occurs. Let us note p(T) the prior probability of having reached the tipping point at temperature T. The expected damage factor writes:

$$\begin{aligned} \varOmega _{ed}(T)=(1-p(T)) \varOmega _1 (T)+p(T) \varOmega _2 (T) \end{aligned}$$

(28)

Damages at a given temperature are set at their expected level given the prior knowledge on the temperature at which the tipping point occurs. Figure 1 shows the resulting expected damages for two productivity shocks ($J=10{\%}$ and $J=50{\%}$). Even though the tipping point only affects the level of damages, the expected damage function is more convex than the pre-tipping point damage function. Indeed, whereas the tipping point changes the damage function from $\approx \pi T^2$ to $\approx \pi T^2 +J$, the expected damage function is $\approx \pi T^2 + p(T)J$, to be compared to $\approx \pi T^2$ without a tipping point.

Fig. 1

Comparison between the two approaches. If the stochastic tipping point is triggered, the damage function jumps from pre-tipping point level (black curve) to a post-tipping point level (the green curve for a productivity shock of 10%, the red curve for a shock of 50%). The expected damage function of a tipping point is the prior expected damage level at each temperature (dashed curves), given that the tipping point is uniformly distributed between $T_{{\textit{min}}}=0.8\,^{\circ }\mathrm{C}$ and $T_{{\textit{max}}}=7\,^{\circ }\mathrm{C}$. The sextic damage function proposed in Weitzman (2012) is pictured for comparison. (Color figure online)

For each computation of the SCC for a stochastic tipping $\textit{SCC}$, we can compute the corresponding SCC under expected damages, noted $\textit{SCC}^{ed}$, which is the SCC of a deterministic run with damages set at their expected level. The SCC in a stochastic setting represents the full effect of the tipping point, whereas we take $\textit{SCC}^{ed}$ as representing the level effect of the tipping point, absent risk. We will analyse the ratio $\textit{SCC}^{ed}/{\textit{SCC}}$, which is the part of the SCC that is explained by expected damages. Its complement is the part of the SCC that is due to a pure risk effect, purged of any level effect.

3.4 Calibration of the Parameters

We summarize in Table 1 the values and range explored for the parameters of interest. We use typical ranges of possible values for parameters related to attitudes towards risk and time. The pure rate of time preference is set at 1.5% (Nordhaus 2008),—a lower value of 0.5% is explored in the Appendix. The elasticity of marginal utility ranges from 0.8 to 3 in the CRRA case. In the Epstein–Zin case, $\theta$ is set at 0.8, and we perform a sensitivity analysis in the Appendix. Considering the range used in the literature (Ackerman et al. 2013; Crost and Traeger 2013; Jensen and Traeger 2014; Cai and Lontzek 2019), we explore risk aversion ($\gamma$) from 0.5 to 20.

For the parameters describing the tipping point, most of the published literature use impacts under a 10% decrease of productivity. Nevertheless, we acknowledge that the impacts of such a phenomenon are difficult to quantify and could be very large, so we explore a much larger window for the productivity shock J, from 0 to 50%. The temperature at which the tipping point would occur is uncertain, and we assume it is distributed between current temperature ($T_{\min }=0.8$) and $T_{\max }=7\,^{\circ }\text {C}$ (Lemoine and Traeger 2014 consider the upper bound for the temperature threshold between 3 and 9 $^{\circ }$C). This means for instance that there is a 19% probability of triggering a tipping point between 0.8 and 2 $^{\circ }$C.⁴

Table 1 Main parameters for the stochastic tipping point numerical exercise. All other parameters are calibrated according to Guivarch and Pottier (2018)

4 Results

We present the comparison between the SCC under a stochastic tipping point and the SCC under expected damages, first with CRRA preferences, then with Epstein–Zin preferences where risk aversion and resistance to intertemporal substitution differ. We finally discuss the significance of our results and lessons that can be drawn for the risk aversion puzzle.

4.1 With CRRA Preferences

We compute the SCC for a stochastic tipping point for different values of the elasticity of marginal utility ($\eta$) and productivity shocks (J) in the range specified in Sect. 3.4, as well as the SCC under expected damages $\textit{SCC}^{ed}$.

One striking result is that the ratio $\textit{SCC}^{ed}/{\textit{SCC}}$ is very close to one for low productivity shocks and low risk aversion (see Fig. 2b), meaning that most of the SCC stems from expected damages enhanced by the tipping point rather than from the risk on damages the tipping point introduces. As J increases, aversion to the risk of high damages make the SCC rise faster than the SCC under expected damages, so that the ratio decreases.

The share of SCC due to expected damages also decreases as the elasticity of marginal utility $\eta$ increases. The role of the elasticity of marginal utility $\eta$ on the SCC is a priori ambiguous. Indeed, CRRA preferences conflate intertemporal trade-offs and risk aversion, and $\eta$ has opposing effects on the SCC. On the one hand, a higher $\eta$ favours present consumption relative to future consumption of wealthier generations (intertemporal substitution), which decreases the SCC. On the other hand, it encourages mitigation to reduce the risk induced by reaching a tipping point (risk aversion), which increases the SCC. In the case of $\textit{SCC}^{ed}$, only the intertemporal substitution effect is at play, not the countervailing risk aversion effect. Thus the SCC under expected damages ($\textit{SCC}^{ed}$) decreases faster with $\eta$ than the $\textit{SCC}$ does, and the ratio $\textit{SCC}^{ed}/SCC$ decreases with $\eta$. Note that in the stochastic case, the intertemporal substitution effect outweighs the risk aversion effect: for a given productivity shock J, the SCC decreases when $\eta$ increases (Fig. 2a).

Though SCC varies by more than an order of magnitude with the ranges of shocks and preferences explored here, most of this variation is explained by expected damages. For example, at $\eta =2$, introducing a tipping point with a shock of $J=10{\%}$ trebles the SCC from 34 to 103 $\$/\mathrm {tCO}_2$. However, this increase is not related to risk but to the simple fact that expected damages have increased—indeed, $\textit{SCC}^{ed}$ is 97 $\$/\mathrm {tCO}_2$. A mere 6% of the SCC is due to a pure risk effect.

Numerically, it takes both a high productivity shock and a high elasticity of marginal utility for $\textit{SCC}^{ed}$ to significantly underestimate $\textit{SCC}$. Expected damages explain more than 90% of the SCC, as long as the productivity shock is inferior to 10%, whatever the value of risk aversion in the range explored here. Only with productivity shocks higher than 40%, jointly with an elasticity of marginal utility higher than 2, does risk contribute to around half of the SCC.

The same pattern is found with a lower pure time preference rate ($\rho$). Though a lower $\rho$ significantly raises the level of the SCC, it does so to the same extent in the stochastic case and under expected damages, so that the part of the SCC explained by expected damages is similar in the case of a lower $\rho$ (see Fig. 7 in the Appendix).

Fig. 2

Comparison of the Social Cost of Carbon to a risk-free SCC under expected damages for CRRA preferences: Heatmap of the Social Cost of Carbon (a in US$2005) and the share of its value that can be explained by expected damages (b ratio $\textit{SCC}^{ed}/{\textit{SCC}}$). The closer the ratio to 100%, the less risk plays a role. 90% of the SCC comes from expected damages for shocks lower than 10%, while it takes shocks greater than 40% for risk to explain at most half of the SCC

Fig. 3

Comparison of the Social Cost of Carbon to a risk-free SCC under expected damages for CRRA preferences: Heatmap of the Social Cost of Carbon (a in US$2005) and the share of its value that can be explained by expected damages (b ratio $\textit{SCC}^{ed}/SCC$). The ratio decreases with risk aversion and with the magnitude of the shock, but remains above 90% for productivity shocks lower than 10%, unless risk aversion is greater than 10

4.2 With Epstein–Zin Preferences

We perform the same exercise using Epstein–Zin preferences, i.e., disentangling preferences for risk and intertemporal substitution. We set the resistance to intertemporal substitution ($\theta$) at 0.8, and present the results when the risk aversion $\gamma$ and the productivity shock J vary in Fig. 3. The corresponding graphs for different values of resistance to intertemporal substitution and pure rate of time preference can be found in the Appendix as a sensitivity check, as well as alternative values for $T_{\max }$.

The part of the SCC explained by expected damages has a similar pattern as in the case of CRRA preferences. As expected, the ratio $\textit{SCC}^{ed}/{\textit{SCC}}$ decreases with risk aversion $\gamma$ and productivity shock J. Values are somewhat similar to the case with CRRA preferences (with a correspondence $\eta \sim \gamma$) but as we explore a much larger range in risk aversion, the part explained by expected damages can become much lower. Interestingly, the SCC does not significantly increase with risk aversion for low values of the productivity shock (see Fig. 3a), hence the low horizontal gradient below $J=10{\%}$. However, for higher productivity shocks, the SCC shows a high sensitivity to risk aversion, and a higher $\gamma$ leads to SCC orders of magnitude greater. This suggests that risk aversion only matters when the economy is exposed to catastrophic risks.

For instance, for a productivity shock equal to 10%, 90% of the SCC is explained by expected damages up to a risk aversion of 4 (as in the CRRA case), but the part explained is only 60% when $\gamma =15$. Productivity shocks higher than 25%, combined with risk aversion higher than 5, lead to the ratio $\textit{SCC}^{ed}/{\textit{SCC}}$ below 50%. For a productivity shock equal to 40% and a risk aversion parameter equal to 5, the SCC under expected damages only makes 20% of the SCC.

The same holds for the sensitivity checks we explore in the Appendix, i.e., for alternative values of the pure time preference rate ($\rho =0.5$) and of the elasticity of intertemporal substitution ($\theta =0.5,\ 1.5$). A decrease in the elasticity of substitution (a higher $\theta$) tends to decrease the SCC, but it does not affect how much the SCC is explained by expected damages. Indeed, $\theta$ plays a similar role in both deterministic and stochastic settings, as it governs the trade-off between future and present consumption. This is a strong indication that our construction of the expected damages has correctly isolated the level effect of the tipping point. It is graphically confirmed with panel b of Fig. 4 in Appendix, where the iso-lines for the ratio $\textit{SCC}^{ed}/{\textit{SCC}}$ are almost flat in the $\theta$ direction. For the same reason, changes in the pure time preference rate ($\rho$) or using a higher temperature threshold $T_{\max }$ do not affect much the shape or position of the contours of the ratio. Even when considering a tipping point affecting the convexity of the damage function, it takes both very high post-tipping point marginal damages and high risk aversion for the SCC to deviate from the value given by expected damages.

4.3 Significance of our Findings and Discussion of the Risk Aversion Puzzle

A productivity shock of 10% is in the higher range of those typically considered in the literature. For instance, Lontzek et al. (2015), with a similar framework, do not consider shocks above $J=10{\%}$. Belaia et al. (2014) only considers productivity shocks below 4.5% when thermohaline circulation collapses. Other modelling choices, in Lemoine and Traeger (2014), assume a tipping point induces a change from a quadratic to a sextic damage function, i.e., Weitzman’s damage function that relies on an expert panel explicitly considering physical tipping points. At 4 $^{\circ }$C, this corresponds to a change of damage factor from $\varOmega _1 =0.96$ to $\varOmega _2 =0.91$ (so equivalent to a productivity shock of 5%). Our results thus suggest that the increase of SCC found in studies considering tipping points is mostly due to an increase in expected damages, rather than to the risk itself.

Identifying that most of the SCC in the presence of a small tipping point is due to expected damages and not risk sheds light on the so-called risk aversion puzzle. It has been found in previous work that risk aversion has a surprisingly little effect on the SCC (Ackerman et al. 2013), even in the case of a tipping point (Belaia et al. 2014). When risk aversion is higher, the SCC does not change much. This seems counter-intuitive, as risk aversion, especially after it has been disentangled from the elasticity of substitution in Epstein–Zin preferences, is expected to significantly increase the SCC.

The simple model of Sect. 2 reveals why this may be the case. Indeed, the SCC is equal to the SCC under expected damages plus a risk premium proportional to risk aversion. Moreover, the smaller the tipping point, the smaller the risk premium. For instance, for a productivity shock $J=5{\%}$ and a risk aversion $\gamma =1$, the SCC under expected damages represents 99.5% of the value of the SCC. Based on the simplified model, increasing $\gamma$ from 1 to 20 increases the SCC from a base $100=99.5 +0.5$ to $109.5=99.5+20\times 0.5$, i.e., an increase of less than 10%. However, for a productivity shock $J=50{\%}$, the SCC under expected damages represents around 70% of the SCC at $\gamma =1$. Setting $\gamma =20$ increases the SCC from a base $100=70+30$ to $670=70+20\times 30$. In our numerical results, the effects of risk turns out to be more than proportional to risk aversion, but the pattern is as described by the simple model of Sect. 2.

While the numerical results explain away the risk aversion puzzle for small shocks, they also show that this puzzle does not always hold. We quantify the magnitude of the shock it takes for risk aversion to play a role. Shocks leading to a minimum of a 10% drop in productivity are necessary for risk aversion to impact the SCC.

Only few studies explore the possibility of large shocks, which are required for risk aversion to significantly affect the SCC: for instance, Dietz (2011) explores damages as large as 90% of consumption; Méjean et al. (2020) explore possible extinction, though with a very low probability. In such cases, risk aversion is expected to play a significant role.

5 Conclusion

When considering climate damages, one might wonder whether it is their expected level or their possible dispersion (i.e., risk) that warrants undertaking mitigation actions. This question has been studied for many types of risks, for instance regarding climate sensitivity or other critical aspects of the climate-economy system, but has not been applied to tipping points in damage functions.

Our research fills this gap. We model a tipping point as an endogenous risk, with a hazard rate increasing with temperature, leading to a permanent productivity drop. First, we demonstrate in a simple setting that the SCC with a stochastic tipping point is at the first order the SCC with damages set at their expected level. We thus isolate the effect of the increased expected damages brought by the introduction of a tipping point, so that the difference between the two methods can be attributed to the sole effect of risk. Numerically, when risk aversion is equal to resistance to intertemporal substitution (CRRA preferences), the SCC under expected damages closely approximates the SCC for a low productivity shock and a low risk aversion. Even when disentangling resistance to intertemporal substitution and risk aversion, using Epstein–Zin preferences, the share of SCC attributable to risk remains limited. For both Social Welfare Functions, risk contributes less than 10% to the SCC, as long as shocks remain below 10% of production and risk aversion is below 10. We test the robustness of our results with a number of sensitivity runs regarding the parametrization of the tipping point and preferences. In particular, the results are qualitatively unchanged when considering a tipping point that leads to more convex damages rather than higher damages.

Providing realistic values for the damage shocks triggered by tipping points is beyond the scope of the article, but we nevertheless provide orders of magnitude for the type of damages it may take for risk to play a role. This sheds light on the risk aversion puzzle. We show that risk aversion only plays a role when the economy is exposed to very high post-tipping damages. When low post-tipping point impacts are considered, the SCC is sensitive to expected damages, so that risk aversion plays a moderate role, in particular compared to time preferences. Thus, the low level of possible damages considered in the literature explains the risk aversion puzzle. However, this stresses once again that the shape of the damage function is central in IAMs. In particular, provided that the risk of triggering a tipping point increases with temperature, expected damages are more convex when considering tipping points. Thus, simply using more convex damage functions in a deterministic fashion may be a good proxy to determine the SCC, as was done for instance by Pizer (2003), Ackerman and Stanton (2012), Wouter Botzen and van den Bergh (2012), Dietz and Stern (2015) and Weitzman (2012).

These results can be extended to other situations. For instance, multiple tipping points (Lemoine and Traeger 2016b) are equivalent to a single tipping point with compounding damages. Our approach could be applied to this case, provided that these combined effects are accounted for (i.e., including the increased probability of triggering another tipping point if one is reached). Damages affecting growth can also result into a higher SCC (Moore and Diaz 2015), but they would make the comparison with expected damages more difficult, because of the different time profiles of damages, with losses of potential growth adding up each year (Guivarch and Pottier 2018). Thus, we leave for future research the question of whether the risk effect plays a greater role for a tipping point affecting growth.

Because the role of preferences is critical to assess the optimal strategy under uncertainty, a direct extension of our work would be to study how the comparison between methods is affected by alternative preferences for the social planner. Epstein–Zin preferences have recently received a lot of attention in this literature, but they may violate first-order stochastic dominance (Bommier and Le Grand 2014). Bommier et al. (2015) propose another class of preferences, which could be used in the case of catastrophic risks. The social planner may also be ambiguity averse to differing worldviews about tipping points (Berger et al. 2016; Lemoine and Traeger 2016a). Though we have focused here on how uncertainty about the tipping points affects optimal policy, this uncertainty could be combined with other uncertainties at stake, such as climate sensitivity or the volality of growth (Cai and Lontzek 2019; Van Den Bremer and van der Ploeg 2018). Finally, even if the damage function is the weakest point of IAMs, they also embed a number of assumptions about the climate-economy system that may be regarded as simplistic and questionable (Pindyck 2013).

Deterministic approaches using best-guess expected damages (together with sensitivity analyses) are currently used to set a value for the SCC for policy evaluations (IAWG 2010), and benefit from a lower computational burden than a fully fledged stochastic model. Knowing when deterministic approaches can be used as a good proxy for computing the SCC under risk can guide policy-making. Our results show that the SCC is primarily driven by the expected level of damages, when the shock induced by a potential tipping point remains lower than 10% or so. In that case, the effects of tipping points are well captured by updating the damage function typically used to a more convex function, reflecting the increasing probability to trigger a tipping point as the Earth warms.

Appendices

Appendix A: Proof of Equation 10

Recall that we note $V_\epsilon (E)={\mathbb {E}}_\omega [\epsilon ^2_\omega (E)]$ the variance of damages of the zero-mean risk $\epsilon$, which is of second order in $|\epsilon |$ (a norm of the risk $\epsilon$). Let us start with Taylor expansion at second order of the term evaluated in the denominators of Eq. (8):

$$\begin{aligned}&{\mathbb {E}}_\omega \left[ u'(C_\omega (a,E))\right] \nonumber \\&\quad ={\mathbb {E}}_\omega \left[ u'({\tilde{C}}(a,E))+u''({\tilde{C}}(a,E))\epsilon _\omega (a,E)+u'''({\tilde{C}}(a,E))\frac{\epsilon ^2_\omega (E)}{2}+o(|\epsilon |^2)\right] \end{aligned}$$

(29)

$$\begin{aligned}&\quad =u'({\tilde{C}}(a,E)) +u'''({\tilde{C}}(a,E)){\mathbb {E}}_\omega \left[ \frac{\epsilon ^2_\omega (E)}{2}\right] +o(|\epsilon |^2) \end{aligned}$$

(30)

$$\begin{aligned}&\quad =u'({\tilde{C}}(a,E)) +u'''({\tilde{C}}(a,E))\frac{V_\epsilon (E)}{2}+o(|\epsilon |^2) \end{aligned}$$

(31)

So the denominator is finally:

$$\begin{aligned} \left. {\mathbb {E}}_\omega \left[ u'(C_\omega (a,E))\right] \right| _{a^t,E^t} =u'({\tilde{C}}(a^t,E^t)) +u'''({\tilde{C}}(a^t,E^t))\frac{V_\epsilon (E^t)}{2}+o(|\epsilon |^2) \end{aligned}$$

(32)

Let us go back to Eq. (8). It is a sum of two terms, the first one is of zero-order, whereas a second term is of second order. In this second term, we can simply replace the denominator by its zero-order approximation as any correction would induce terms with orders higher than 2. For the first term of the sum, we have to keep the full Taylor expansion of the denominator. We then reorder terms of the Taylor expansion.

$$\begin{aligned} SCC^{t}&=\frac{ u'({\tilde{C}}(a^t,E^t)){\tilde{d}}'(E^t)}{{\mathbb {E}}_\omega \left[ u'(C_\omega (a,E))\right] |_{a^t,E^t}} -\frac{\left. \partial _E \left( u''({\tilde{C}}(a,E)) \frac{V_\epsilon (E)}{2}\right) \right| _{a^t,E^t}}{{\mathbb {E}}_\omega \left[ u'(C_\omega (a,E))\right] |_{a^t,E^t}}+o(|\epsilon |^2) \end{aligned}$$

(33)

$$\begin{aligned}&=\frac{ u'({\tilde{C}}(a^t,E^t)){\tilde{d}}'(E^t)}{u'({\tilde{C}}(a^t,E^t)) +u'''({\tilde{C}}(a^t,E^t))\frac{V_\epsilon (E^t)}{2}} -\frac{\left. \partial _E \left( u''({\tilde{C}}(a,E)) \frac{V_\epsilon (E)}{2}\right) \right| _{a^t,E^t}}{u'({\tilde{C}}(a^t,E^t))}+o(|\epsilon |^2) \end{aligned}$$

(34)

$$\begin{aligned}&=\frac{ {\tilde{d}}'(E^t)}{1 +\frac{u'''}{u'}\left( {\tilde{C}}(a^t,E^t)\right) \frac{V_\epsilon (E^t)}{2}} \nonumber \\&\quad -\frac{ u''({\tilde{C}}(a^t,E^t)) \left. \partial _E\left( \frac{V_\epsilon (E)}{2}\right) \right| _{E^t} -u'''({\tilde{C}}(a^t,E^t)){\tilde{d}}'(E^t)\frac{V_\epsilon (E^t)}{2}}{u'({\tilde{C}}(a^t,E^t))}+o(|\epsilon |^2)\end{aligned}$$

(35)

$$\begin{aligned}&={\tilde{d}}'(E^t)\left( 1 -\frac{u'''}{u'}\left( {\tilde{C}}(a^t,E^t)\right) \frac{V_\epsilon (E^t)}{2}\right) -\frac{ u''({\tilde{C}}(a^t,E^t)) \left. \partial _E\left( \frac{V_\epsilon (E)}{2}\right) \right| _{E^t}}{u'({\tilde{C}}(a^t,E^t))} \nonumber \\&\quad +\frac{u'''({\tilde{C}}(a^t,E^t)){\tilde{d}}'(E^t)\frac{V_\epsilon (E^t)}{2}}{u'({\tilde{C}}(a^t,E^t))}+o(|\epsilon |^2)\end{aligned}$$

(36)

$$\begin{aligned}&={\tilde{d}}'(E^t) -\frac{ u''({\tilde{C}}(a^t,E^t)) \left. \partial _E\left( \frac{V_\epsilon (E)}{2}\right) \right| _{E^t}}{u'({\tilde{C}}(a^t,E^t))}+o(|\epsilon |^2) \end{aligned}$$

(37)

$$\begin{aligned}&={\tilde{d}}'(E^t) +\gamma ({\tilde{C}}(a^t,E^t))\frac{{\tilde{C}}(a^t,E^t)}{2} \left. \partial _E V_\epsilon (E)\right| _{E^t}+o(|\epsilon |^2) \end{aligned}$$

(38)

At the last line, we have introduced the Arrow–Pratt measure of relative risk aversion $\gamma (C)=-u''(C)/(u'(C).C)$ of the utility u to get Eq. (10).

Appendix B: Proof of Proposition 1

We are first interested in the difference between SCC with a tipping point $\textit{SCC}^{t}$, given by Eq. (5) and SCC without a tipping point $\textit{SCC}^{*}$, given by (2), with damage function d egal to damage function in the pre-tipping state of the world (we call 1 this state).

The proof is a little bit more complicated than just comparing the Eqs. (2), (10) and (11). Indeed, one has to take into account not only that there are additional terms but also that these are not evaluated at the same point. This is because the planner reacts to additional damages terms and thus the optimal emissions change accordingly (it is respectively $E^*, E^t, E^{ed}$).

We have, at first order in the magnitude of risk $\epsilon$, thanks to (10):

$$\begin{aligned} {\textit{SCC}}^{t}-{\textit{SCC}}^{*}={\tilde{d}}'(E^t)-d(E^*)+o(|\epsilon |) \end{aligned}$$

(39)

The optimal abatement and emission levels solve:

$$\begin{aligned} \frac{c'(a^*)}{\sigma Y}&=d'(E^*) \end{aligned}$$

(40)

$$\begin{aligned} \frac{c'(a^t)}{\sigma Y}&={\tilde{d}}'(E^t)+o(|\epsilon |) \end{aligned}$$

(41)

Let us write $a^t=a*+h$, then $E^t=E*-\sigma Y h$ and assume that h is at first-order in $|\epsilon |$. We make a Taylor-expansion of the last line in h:

$$\begin{aligned} \frac{c'(a^*)}{\sigma Y}+\frac{c''(a^*)}{\sigma Y}h={\tilde{d}}'(E^*)-{\tilde{d}}''(E^*)\sigma Y h+o(|\epsilon |) \end{aligned}$$

(42)

Hence

$$\begin{aligned} \left( \frac{c''(a^*)}{\sigma Y} +{\tilde{d}}''(E^*)\sigma Y \right) h={\tilde{d}}'(E^*) - d'(E^*)+o(|\epsilon |) \end{aligned}$$

(43)

The right hand side is simply $-\epsilon _1'(E^*)$, the marginal increase in risk $\epsilon$ in state 1. Thus h is correctly at first-order in $|\epsilon |$ and the difference between the SCCs is given by:

$$\begin{aligned} {\textit{SCC}}^{t}-{\textit{SCC}}^{*}&={\tilde{d}}'(E^*)-d'(E^*)-d''(E^*)\sigma Y h+o(|\epsilon |) \end{aligned}$$

(44)

$$\begin{aligned}&= \left( {\tilde{d}}'(E^*)-d'(E^*)\right) \frac{\frac{c''(a^*)}{\sigma Y}}{\frac{c''(a^*)}{\sigma Y} +{\tilde{d}}''(E^*)\sigma Y}+o(|\epsilon |) \end{aligned}$$

(45)

$$\begin{aligned}&= -\epsilon _1'(E^*)\frac{\frac{c''(a^*)}{\sigma Y}}{\frac{c''(a^*)}{\sigma Y} +{\tilde{d}}''(E^*)\sigma Y}+o(|\epsilon |) \end{aligned}$$

(46)

This proves our first claim. We proceed similarly for the second. By definition and thanks to (10),

$$\begin{aligned} {\textit{SCC}}^{t}-{\textit{SCC}}^{ed}={\tilde{d}}'(E^t)+\gamma ({\tilde{C}}(a^t,E^t))\frac{{\tilde{C}}(a^t,E^t)}{2} \left. \partial _E V_\epsilon (E)\right| _{E^t}-{\tilde{d}}'(E^{ed})+o(|\epsilon |^2) \end{aligned}$$

(47)

The optimal abatement and emission levels solve:

$$\begin{aligned} \frac{c'(a^{ed})}{\sigma Y}&={\tilde{d}}'(E^{ed}) \end{aligned}$$

(48)

$$\begin{aligned} \frac{c'(a^t)}{\sigma Y}&={\tilde{d}}'(E^t)+\gamma ({\tilde{C}}(a^t,E^t))\frac{{\tilde{C}}(a^t,E^t)}{2} \left. \partial _E V_\epsilon (E)\right| _{E^t}+o(|\epsilon |^2) \end{aligned}$$

(49)

Let us write $a^t=a^{ed}+g$, then $E^t=E^{ed}-\sigma Y g$ and assume that g is at second-order in $|\epsilon |$. We make a Taylor-expansion of the last line in g:

$$\begin{aligned}&\frac{c'(a^{ed})}{\sigma Y}+\frac{c''(a^{ed})}{\sigma Y}h \nonumber \\&\quad = {\tilde{d}}'(E^{ed})-{\tilde{d}}''(E^{ed})\sigma Y g+\gamma ({\tilde{C}}(a^t,E^t))\frac{{\tilde{C}}(a^t,E^t)}{2} \left. \partial _E V_\epsilon (E)\right| _{E^t} +o(|\epsilon |^2) \end{aligned}$$

(50)

Hence:

$$\begin{aligned} \left( \frac{c''(a^{ed})}{\sigma Y} +{\tilde{d}}''(E^{ed})\sigma Y \right) g=\gamma ({\tilde{C}}(a^t,E^t))\frac{{\tilde{C}}(a^t,E^t)}{2} \left. \partial _E V_\epsilon (E)\right| _{E^t}+o(|\epsilon |^2) \end{aligned}$$

(51)

Thus g is correctly at second-order in $|\epsilon |$ (as $\left. \partial _E V_\epsilon (E)\right| _{E^t})$ is) and given by:

$$\begin{aligned} g&=\frac{\gamma ({\tilde{C}}(a^t,E^t))\frac{{\tilde{C}}(a^t,E^t)}{2} \left. \partial _E V_\epsilon (E)\right| _{E^t}}{\frac{c''(a^{ed})}{\sigma Y} +{\tilde{d}}''(E^{ed})\sigma Y }+o(|\epsilon |^2) \end{aligned}$$

(52)

$$\begin{aligned}&=\frac{\gamma ({\tilde{C}}(a^{ed},E^{ed}))\frac{{\tilde{C}}(a^{ed},E^{ed})}{2} \left. \partial _E V_\epsilon (E)\right| _{E^{ed}}}{\frac{c''(a^{ed})}{\sigma Y} +{\tilde{d}}''(E^{ed})\sigma Y }+o(|\epsilon |^2) \end{aligned}$$

(53)

The difference between the SCCs is given by:

$$\begin{aligned} {\textit{SCC}}^{t}-{\textit{SCC}}^{ed}&={\tilde{d}}'(E^{ed})-{\tilde{d}}''(E^{ed})\sigma Y g+\gamma ({\tilde{C}}(a^{ed},E^{ed}))\frac{{\tilde{C}}(a^{ed},E^{ed})}{2} \left. \partial _E V_\epsilon (E)\right| _{E^{ed}} \nonumber \\&\quad -{\tilde{d}}'(E^{ed})+o(|\epsilon |^2) \end{aligned}$$

(54)

$$\begin{aligned}&=\gamma ({\tilde{C}}(a^{ed},E^{ed}))\frac{{\tilde{C}}(a^{ed},E^{ed})}{2} \left. \partial _E V_\epsilon (E)\right| _{E^{ed}}\frac{\frac{c''(a^{ed})}{\sigma Y}}{\frac{c''(a^{ed})}{\sigma Y} +{\tilde{d}}''(E^{ed})\sigma Y} \end{aligned}$$

(55)

Appendix C: Additional Graph: Sensitivity to Resistance to Intertemporal Substitution

Two parameters are involved in welfare evaluation at each time step: risk aversion ($\gamma$) and resistance to intertemporal substitution ($\theta$). In the main text, we have analyzed the influence of risk aversion combined with the value of the shock. On Fig. 4, we display how resistance to intertemporal substitution affects the comparison of determinsitic and stochastic methods. A change in $\theta$ does not affect the share of the SCC explained by expected damages, the contour lines on the graph are horizontal.

Fig. 4

Epstein–Zin preference: influence of the resistance to intertemporal substitution $\theta$. Heatmap of the Social Cost of Carbon (a in US$2005) and the share of its value that can be explained by expected damages (b ratio $\textit{SCC}_{ed}/SCC$), for different values of resistance to intertemporal substitution ($\theta$) and productivity shocks (J). Horizontal lines in b. indicate that preferences for intertemporal substitution does not affect the comparison between SCC and SCC under expected damage

Appendix D: Robustness Checks

We perform a sensitivity analysis on several parameters of the model:

• The maximum temperature threshold for the tipping point $T_{\max }$. We look at $T_{\max }=10$ instead of 7.

• Pure rate of time preference $\rho$. We run the model for lower $\rho$ (0.5%)

• Elasticity of intertemporal substitution ($1/\theta$) in the Epstein–Zin case. We consider $\theta =0.5$ and $\theta =1.5$.

The graphs show that the shapes of the curves are not affected by a change in these parameters, and our finding that most of the SCC is still explained by expected damages as long as the shock remain under 10%.

1.1 Appendix D.1: Parameter $T_{\max }$

See Figs. 5 and 6.

Fig. 5

Sensitivity analysis for $T_{\max }=10\,^{\circ }\mathrm{C}$ (CRRA preferences). Heatmap of the Social Cost of Carbon (a in US$2005) and the share of its value that can be explained by expected damages (b ratio $\textit{SCC}_{ed}/{\textit{SCC}}$)

Fig. 6

Sensitivity analysis for $T_{\max }=10\,^{\circ }\mathrm{C}$ (Epstein–Zin preferences). Heatmap of the Social Cost of Carbon (a in US$2005) and the share of its value that can be explained by expected damages (b ratio $\textit{SCC}_{ed}/{\textit{SCC}}$)

1.2 Appendix D.2: Parameter $\rho$

See Figs. 7 and 8.

Fig. 7

Sensitivity analysis for $\rho =0.5{\%}$ (CRRA preferences). Social Cost of Carbon under Epstein–Zin preferences. Heatmap of the Social Cost of Carbon (a in US$2005) and the share of its value that can be explained by expected damages (b ratio $\textit{SCC}_{ed}/{\textit{SCC}}$)

Fig. 8

Sensitivity analysis for $\rho =0.5{\%}$ (Epstein–Zin preferences). Heatmap of the Social Cost of Carbon (a in US$2005) and the share of its value that can be explained by expected damages (b ratio $\textit{SCC}_{ed}/{\textit{SCC}}$)

1.3 Appendix D.3: Parameter $\theta$

See Figs. 9 and 10.

Fig. 9

Sensitivity analysis for $\theta =0.5$ (Epstein–Zin preferences). Heatmap of the Social Cost of Carbon (a in US$2005) and the share of its value that can be explained by expected damages (b ratio $\textit{SCC}_{ed}/{\textit{SCC}}$)

Fig. 10

Sensitivity analysis for $\theta =1.5$ (Epstein–Zin preferences). Heatmap of the Social Cost of Carbon (a in US$2005) and the share of its value that can be explained by expected damages (b ratio $\textit{SCC}_{ed}/{\textit{SCC}}$)

1.4 Appendix D.4: A Tipping Point Affecting the Convexity of Damages

We plot the same graphs when the tipping point affects the convexity of the damage function. Thus, the tipping point increases marginal damages rather solely damage level. We assume that the coefficient $\pi$ in the damage factor $\varOmega =\frac{1}{1+\pi T^2}$ can jump from its initial value ($\pi _1=0.00028$) to a higher value $\pi _2$ (see Fig. 11 for an illustration of the effect on the damage function). Results are shown in Fig. 12.

Fig. 11

Comparison between stochastic damage function and expected damages approaches, for a tipping point which affects $\pi$

Fig. 12

Sensitivity analysis for a tipping point affecting the convexity of the damage function (Epstein–Zin). Heatmap of the Social Cost of Carbon (a in US$2005) and the share of its value that can be explained by expected damages (b ratio $\textit{SCC}_{ed}/{\textit{SCC}}$)