# Social Cost of Carbon Under Stochastic Tipping Points

When does Risk Play a Role?

• Published:

## 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.

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## Notes

1. For notational convenience, we use damage factor $$\varOmega$$ instead of damage function D. The correspondence is simply $$\varOmega =1-D$$.

2. The formula holds for $$\theta <1$$. Otherwise when $$\theta >1$$ utility function is negative, so that $$U_t=-(-(1-\frac{1}{1+\rho })u+\frac{1}{1+\rho }[\mathbb E_t(-U_{t+1})^{1-\gamma }]^\frac{1-\theta }{1-\gamma })^{\frac{1}{1-\theta }}$$.

3. When $$0<\theta <1$$, the recursive formula involves $$u_t-\frac{1}{1+\rho } f(-V_{t+1})$$.

4. A sensitivity test using $$T_{\max }=10$$ is performed in the Appendix. As said above, we also explore in the Appendix the case of a tipping point affecting the convexity of the damage function, rather than its level. Results are similar to those presented in the main text.

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## Acknowledgements

The authors would like to thank two anonymous referees who contributed to improving the paper.

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Correspondence to Nicolas Taconet.

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## 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.

### 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.

### 1.2 Appendix D.2: Parameter $$\rho$$

See Figs. 7 and 8.

### 1.3 Appendix D.3: Parameter $$\theta$$

See Figs. 9 and 10.

### 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.

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Taconet, N., Guivarch, C. & Pottier, A. Social Cost of Carbon Under Stochastic Tipping Points. Environ Resource Econ 78, 709–737 (2021). https://doi.org/10.1007/s10640-021-00549-x