Determining the Social Cost of Carbon: Under Damage and Climate Sensitivity Uncertainty

Abstract

This article quantifies the impact on optimal climate policy, of both damage elasticity and equilibrium climate sensitivity uncertainty, under separable preferences for risk and intergenerational inequality. The primary findings are as follows. (1) Such preferences can depress the social cost of carbon (SCC) when calibration aims at matching actual economic outcomes, countering the prevailing view that the SCC is greater with separable than with conventional entangled preferences. (2) Damage elasticity uncertainty has larger effects on climate policy than equilibrium climate sensitivity uncertainty, even under high impact tail risk of the latter. (3) Risk aversion decisively strengthens optimal climate policy under joint damage and climate sensitivity uncertainty, than with a single source of uncertainty alone. Indeed, failing to account for the interaction between damage and climate sensitivity uncertainty underestimates the cost of climate change by more than US dollars 1 trillion.

Appendices

Appendix 1: Extra Graphs

See Figs. 4, 5 and 6.

Fig. 4

SCC (in US$), for alternate degrees of intergenerational inequality aversion and risk aversion. Notes: Left panels: simulation with unadjusted pure rate of preference. Right panels: simulations with pure rate of time preference adjusted such that (base year) savings behaviour is consistent across specifications. Top panels: 2040 uncertainty resolution. Bottom panels: 2100 uncertainty resolution

Fig. 5

Response of the SCC to damage elasticity uncertainty (a) and climate sensitivity uncertainty (b), and the interactive effects of damage and climate sensitivity uncertainty (c), for alternate degrees of intergenerational inequality aversion and risk aversion. Notes: Each specification shows the difference between the expected SCC and the deterministic SCC derived from the mean of the ‘uncertain’ damage elasticity and the ‘uncertain’ climate sensitivity. The renormalization of the savings rate is disregarded in the non-baseline specifications

Fig. 6

Response of the SCC to climate sensitivity uncertainty in a counterfactual with higher climate inertia. In DICE, the climate inertia parameter is set to fifty years. In this counterfactual, the climate inertia parameter is set to 20 years. Notes: Each specification shows the difference between the expected SCC and the deterministic SCC. The deterministic SCC is obtained using the mean of the ‘uncertain’ climate sensitivity. Renormalization of the savings rate is pursued for all specifications

Appendix 2: Simplified DICE Model

DICE’s atmospheric temperature Eq. (2), can be reorganized as:

$$\begin{aligned} T_{t+1}=\left( 1-\Delta _{T}\right) T_{t}+z_{1}\mathcal {F}\left( M_{t+1}\right) -z_{2}T_{o,t} \end{aligned}$$

where $\Delta _{T}=z_{1}\nicefrac {\xi }{\zeta }-z_{1}z_{2}$. It is immediate that for $\iota >t$, $T_{\iota }$ increases in climate sensitivity, $\zeta$, at a decreasing rate. DICE sets $z_{2}=0.008624$, which for simplifying the derivation of the SCC can be set zero, yielding:

$$\begin{aligned} T_{t+1}=\left( 1-z_{1}\frac{\xi }{\zeta }\right) T_{t}+z_{1}\mathcal {F}\left( M_{t+1}\right) \end{aligned}$$

The parameter $z_{1}$ defines the inertia in the climate system. DICE2013 sets this value to 0.098, corresponding to a duration of 51 years. In the counterfactual experiment testing the impact of climate inertia, this parameter is set to 0.25.

Furthermore, it is convenient to disregard the upper and deep ocean carbon sinks and only focus on the atmospheric stock (Jensen and Traeger 2016). The simplified stochastic DICE model becomes :

$$\begin{aligned}&V\left( K_{t},M_{t},T_{t}\right) =\max _{C_{t},E_{t}}u_{t}\left( C_{t}\right) +\beta \mathbb {E}_{t}V\left( K_{t+1},M_{t+1},T_{t+1}\right) \end{aligned}$$

s.t.

$$\begin{aligned}&K_{t+1}=\left( 1-\Delta _{K}\right) +F\left( K_{t},T_{t},E_{t}\right) -C_{t}\\&M_{t+1}:=M_{t+1}\left( M_{t},E_{t}\right) ,\quad T_{t+1}:=T_{t+1}\left( M_{t+1},T_{t}\right) \end{aligned}$$

Let $V_{t}\equiv V\left( K_{t},M_{t},T_{t}\right)$

Taking FOC with respect to $C_{t}$ yields:

$$\begin{aligned} 0=\frac{\partial V_{t}}{\partial C_{t}}&=u_{t}^{\prime }\left( C_{t}\right) +\beta \mathbb {E}_{t}\left[ \frac{\partial V_{t+1}}{\partial K_{t+1}}\frac{\partial K_{t+1}}{\partial C_{t}}\right] \\ u_{t}^{\prime }\left( C_{t}\right)&=\beta \mathbb {E}_{t}\frac{\partial V_{t+1}}{\partial K_{t+1}} \end{aligned}$$

and with respect to $E_{t}$:

$$\begin{aligned} 0=\frac{\partial V_{t}}{\partial E_{t}}&=\beta \mathbb {E}_{t}\left[ \frac{\partial V_{t+1}}{\partial K_{t+1}}\frac{\partial K_{t+1}}{\partial E_{t}}+\frac{\partial V_{t+1}}{\partial M_{t+1}}\frac{\partial M_{t+1}}{\partial E_{t}}+\frac{\partial V_{t+1}}{\partial T_{t+1}}\frac{\partial T_{t+1}}{\partial E_{t}}\frac{\partial M_{t+1}}{\partial M_{t+1}}\right] \\&\quad -\frac{\partial F_{t}}{\partial E_{t}}\mathbb {E}_{t}\left[ \frac{\partial V_{t+1}}{\partial K_{t+1}}\right] =\frac{\partial M_{t+1}}{\partial E_{t}}\mathbb {E}_{t}\left[ \frac{\partial V_{t+1}}{\partial M_{t+1}}+\frac{\partial V_{t+1}}{\partial T_{t+1}}\frac{\partial T_{t+1}}{\partial M_{t+1}}\right] \\ \text {SCC}_{t}=\frac{\partial F_{t}}{\partial E_{t}}&=-\frac{\beta }{u_{t}^{\prime }\left( C_{t}\right) }\frac{\partial M_{t+1}}{\partial E_{t}}\mathbb {E}_{t}\left[ \frac{\partial V_{t+1}}{\partial M_{t+1}}+\frac{\partial V_{t+1}}{\partial T_{t+1}}\frac{\partial T_{t+1}}{\partial M_{t+1}}\right] \end{aligned}$$

which is an expression for the SCC.

Finding the quantities $\nicefrac {\partial V_{t+1}}{\partial M_{t+1}}$ and $\nicefrac {\partial V_{t+1}}{\partial T_{t+1}}$ requires differentiating the value function.

First consider $\nicefrac {\partial V_{t+1}}{\partial T_{t+1}}$:

$$\begin{aligned} \frac{\partial V_{t}}{\partial T_{t}}&=\beta \mathbb {E}_{t}\left[ \frac{\partial V_{t+1}}{\partial K_{t+1}}\frac{K_{t+1}}{\partial T_{t}}+\frac{\partial V_{t+1}}{\partial M_{t+1}}\frac{M_{t+1}}{\partial T_{t}}+\frac{\partial V_{t+1}}{\partial T_{t+1}}\frac{\partial T_{t+1}}{\partial T_{t}}\right] \\ \frac{\partial V_{t}}{\partial T_{t}}&=u^{\prime }\left( C_{t}\right) \frac{\partial F_{t}}{\partial T_{t}}+\beta \mathbb {E}_{t}\left[ \frac{\partial V_{t+1}}{\partial T_{t+1}}\frac{\partial T_{t+1}}{\partial T_{t}}\right] \\ \frac{\partial V_{t}}{\partial T_{t}}&=u^{\prime }\left( C_{t}\right) \frac{\partial F_{t}}{\partial T_{t}}+\beta \mathbb {E}_{t}\left[ u^{\prime }\left( C_{t+1}\right) \frac{F_{t+1}}{\partial T_{t+1}}\frac{\partial T_{t+1}}{\partial T_{t}}\right] +\beta ^{2}\mathbb {E}_{t}\left[ \frac{\partial V_{t+2}}{\partial T_{t+2}}\frac{\partial T_{t+2}}{\partial T_{t+1}}\frac{\partial T_{t+1}}{\partial T_{t}}\right] \\ \frac{\partial V_{t}}{\partial T_{t}}&=u^{\prime }\left( C_{t}\right) \frac{\partial F_{t}}{\partial T_{t}}+\sum _{\iota =t+1}^{\infty }\beta ^{\iota -t}\mathbb {E}_{\iota }\left[ u^{\prime }\left( C_{\iota }\right) \frac{\partial F_{\iota }}{\partial T_{\iota }}\prod _{j=t+1}^{\iota }\frac{\partial T_{j}}{\partial T_{j-1}}\right] \\ \frac{\partial V_{t}}{\partial T_{t}}&=\sum _{\iota =t}^{\infty }\beta ^{\iota -t}\mathbb {E}_{\iota }\left[ u^{\prime }\left( C_{\iota }\right) \frac{\partial F_{\iota }}{\partial T_{\iota }}\left( \left. 1\right| _{\iota =t}+\left. \prod _{j=t}^{j=\iota }\frac{\partial T_{j+1}}{\partial T_{j}}\right| _{\iota >t}\right) \right] \\ \frac{\partial V_{t}}{\partial T_{t}}&=\sum _{\iota =t}^{\infty }\beta ^{\iota -t}\mathbb {E}_{\iota }\left[ u^{\prime }\left( C_{\iota }\right) \frac{\partial F_{\iota }}{\partial T_{\iota }}\frac{\partial T_{\iota }}{\partial T_{t}}\right] \end{aligned}$$

For $\nicefrac {\partial V_{t+1}}{\partial M_{t+1}}$

$$\begin{aligned} \frac{\partial V_{t}}{\partial M_{t}}&=\beta \frac{\partial M_{t+1}}{\partial M_{t}}\mathbb {E}_{t}\left[ \frac{\partial V_{t+1}}{\partial M_{t+1}}+\frac{\partial V_{t+1}}{\partial T_{t+1}}\frac{\partial T_{t+1}}{\partial M_{t+1}}\right] \\ \frac{\partial V_{t}}{\partial M_{t}}&=\beta \frac{\partial M_{t+1}}{\partial M_{t}}\mathbb {E}_{t}\left[ \frac{\partial V_{t+1}}{\partial M_{t+1}}\right] +\beta \frac{\partial M_{t+1}}{\partial M_{t}}\mathbb {E}_{t}\left[ \frac{\partial V_{t+1}}{\partial T_{t+1}}\frac{\partial T_{t+1}}{\partial M_{t+1}}\right] \\ \frac{\partial V_{t}}{\partial M_{t}}&=\beta ^{2}\frac{\partial M_{t+1}}{\partial M_{t}}\mathbb {E}_{t}\left[ \frac{\partial M_{t+2}}{\partial M_{t+1}}\mathbb {E}_{t+1}\left[ \frac{\partial V_{t+2}}{\partial M_{t+2}}\right. \right. \\&\quad \left. \left. +\frac{\partial V_{t+2}}{\partial T_{t+2}}\frac{\partial T_{t+2}}{\partial M_{t+2}}\right] \right] +\beta \frac{\partial M_{t+1}}{\partial M_{t}}\mathbb {E}_{t}\left[ \frac{\partial V_{t+1}}{\partial T_{t+1}}\frac{\partial T_{t+1}}{\partial M_{t+1}}\right] \\ \frac{\partial V_{t}}{\partial M_{t}}&=\sum _{\tau =t+1}^{\infty }\beta ^{\tau -t}\mathbb {E}_{t}\left[ \frac{\partial V_{\tau }}{\partial T_{\tau }}\frac{\partial T_{\tau }}{\partial M_{\tau }}\left( \prod _{j=t}^{j=\tau -1}\frac{\partial M_{j+1}}{\partial M_{j}}\right) \right] \end{aligned}$$

Recall that

$$\begin{aligned} \text {SCC}_{t}&=-\frac{\beta }{u_{t}^{\prime }\left( C_{t}\right) }\frac{\partial M_{t+1}}{\partial E_{t}}\mathbb {E}_{t}\left[ \frac{\partial V_{t+1}}{\partial M_{t+1}}+\frac{\partial V_{t+1}}{\partial T_{t+1}}\frac{\partial T_{t+1}}{\partial M_{t+1}}\right] \\ \text {SCC}_{t}&=-\frac{1}{u_{t}^{\prime }\left( C_{t}\right) }\frac{\partial M_{t+1}}{\partial E_{t}}\left( \frac{\partial V_{t}}{\partial M_{t}}\right) \left( \frac{\partial M_{t+1}}{\partial M_{t}}\right) ^{-1}\\ \text {SCC}_{t}&=-\frac{1}{u_{t}^{\prime }\left( C_{t}\right) }\frac{\partial M_{t+1}}{\partial E_{t}}\left( \frac{\partial M_{t+1}}{\partial M_{t}}\right) ^{-1}\sum _{\tau =t+1}^{\infty }\beta ^{\tau -t}\mathbb {E}_{t}\left[ \frac{\partial V_{\tau }}{\partial T_{\tau }}\frac{\partial T_{\tau }}{\partial M_{\tau }}\left( \prod _{j=t}^{j=\tau -1}\frac{\partial M_{j+1}}{\partial M_{j}}\right) \right] \\ \text {SCC}_{t}&=-\frac{1}{u_{t}^{\prime }\left( C_{t}\right) }\frac{\partial M_{t+1}}{\partial E_{t}}\left( \frac{\partial M_{t+1}}{\partial M_{t}}\right) ^{-1}\sum _{\tau =t+1}^{\infty }\beta ^{\tau -t}\mathbb {E}_{t}\\&\quad \left[ \left( \sum _{\iota =\tau }^{\infty }\beta ^{\iota -\tau }\mathbb {E}_{\iota }\left[ u^{\prime }\left( C_{\iota }\right) \frac{\partial F_{\iota }}{\partial T_{\iota }}\frac{\partial T_{\iota }}{\partial T_{\tau }}\right] \right) \frac{\partial T_{\tau }}{\partial M_{\tau }}\left( \prod _{j=t}^{j=\tau -1}\frac{\partial M_{j+1}}{\partial M_{j}}\right) \right] \\ \text {SCC}_{t}&=-\frac{1}{u_{t}^{\prime }\left( C_{t}\right) }\frac{\partial M_{t+1}}{\partial E_{t}}\left( \frac{\partial M_{t+1}}{\partial M_{t}}\right) ^{-1}\mathbb {E}_{t}\\&\quad \left[ \left( \sum _{\tau =t+1}^{\infty }\sum _{\iota =\tau }^{\infty }\beta ^{\iota -t}\mathbb {E}_{\iota }\left[ u^{\prime }\left( C_{\iota }\right) \frac{\partial F_{\iota }}{\partial T_{\iota }}\frac{\partial T_{\iota }}{\partial T_{\tau }}\right] \right) \frac{\partial T_{\tau }}{\partial M_{\tau }}\left( \prod _{j=t}^{j=\tau -1}\frac{\partial M_{j+1}}{\partial M_{j}}\right) \right] \\ \text {SCC}_{t}&=-\frac{1}{u_{t}^{\prime }\left( C_{t}\right) }\frac{\partial M_{t+1}}{\partial E_{t}}\left( \frac{\partial M_{t+1}}{\partial M_{t}}\right) ^{-1}\mathbb {E}_{t}\\&\quad \left[ \left( \sum _{\tau =t+1}^{\infty }\sum _{\iota =\tau }^{\infty }\beta ^{\iota -t}\mathbb {E}_{\iota }\left[ u^{\prime }\left( C_{\iota }\right) \frac{\partial F_{\iota }}{\partial T_{\iota }}\frac{\partial T_{\iota }}{\partial T_{\tau }}\right] \right) \frac{\partial T_{\tau }}{\partial M_{\tau }}\frac{\partial M_{\tau }}{\partial M_{t}}\right] \\ \text {SCC}_{t}&=-\frac{1}{u_{t}^{\prime }\left( C_{t}\right) }\frac{\partial M_{t+1}}{\partial E_{t}}\mathbb {E}_{t}\left[ \sum _{\tau =t+1}^{\infty }\sum _{\iota =\tau }^{\infty }\beta ^{\iota -t}\mathbb {E}_{\iota }\left[ u^{\prime }\left( C_{\iota }\right) \frac{\partial F_{\iota }}{\partial T_{\iota }}\frac{\partial T_{\iota }}{\partial T_{\tau }}\right] \frac{\partial T_{\tau }}{\partial M_{\tau }}\frac{\partial M_{\tau }}{\partial M_{t+1}}\right] \\ \text {SCC}_{t}&=-\frac{1}{u_{t}^{\prime }\left( C_{t}\right) }\frac{\partial M_{t+1}}{\partial E_{t}}\mathbb {E}_{t}\left[ \sum _{\tau =t+1}^{\infty }\sum _{\iota =\tau }^{\infty }\beta ^{\iota -t}u^{\prime }\left( C_{\iota }\right) \frac{\partial F_{\iota }}{\partial T_{\iota }}\frac{\partial T_{\iota }}{\partial T_{\tau }}\frac{\partial T_{\tau }}{\partial M_{\tau }}\frac{\partial M_{\tau }}{\partial M_{t+1}}\right] \end{aligned}$$

Letting $\frac{\partial M_{t+1}}{\partial E_{t}}=1$ gives the expression in the text.