Three Reasons to Price Carbon Under Uncertainty

Three channels of climate damage

①

Productivity losses

Warming reduces total factor productivity through a damage function that is linear in temperature. This is the standard approach in integrated assessment models such as DICE.

②

Recurring disasters

The frequency of climate-related disasters rises linearly with temperature. Each event destroys a random share of the capital stock. Following Karydas & Xepapadeas (2022), who document this relationship empirically using data from EM-DAT.

③

Tipping points

A directed Markov chain governs irreversible shifts in the transient climate response to cumulative emissions (TCRE), which jumps permanently after tipping. Follows the gradual cascading tipping framework of Cai & Lontzek (2019).

The model — key equations

Capital dynamics

\[dK_t=\Bigl[I_t-\delta K_t-\tfrac{1}{2}\phi\tfrac{I_t^2}{K_t}\Bigr]dt+K_t\sigma\,dW_t^K-K_{t-}\ell_e\,dN_t^e-K_{t-}\ell_c\,dN_t^c\]

$I_t$: investment; $\delta$: depreciation; $\phi$: adjustment cost. Brownian term $\sigma\,dW^K$ captures normal shocks. $N^e$: macroeconomic disaster (constant rate $\lambda_e$); $N^c$: climate disaster (temperature-dependent rate $\lambda^c_t$). Recovery rates $Z_i=1-\ell_i$ follow a power distribution.

Production & output use

\[Y_t = A_t\, K_t^\alpha F_t^{1-\alpha}, \qquad Y_t = C_t + I_t + b F_t\]

Cobb-Douglas production with capital $K_t$, fossil fuel $F_t$, and TFP $A_t$ (which depends on temperature). Share $1-\alpha=4.3\%$ is the fossil fuel share in value added; $b$ is the unit cost of fossil fuel. Output is split between consumption $C_t$, investment $I_t$, and fossil fuel cost $bF_t$. In equilibrium the model collapses to an endogenous AK growth model with $Y_t = B_t K_t$.

Epstein-Zin preferences

\[J_t=\mathbb{E}_t\!\int_t^\infty f(C_s,J_s)\,ds\] \[f(C_t,J_t)=\frac{1}{1-\eta}\frac{C_t^{1-\eta}-\rho\bigl[(1-\gamma)J_t\bigr]^{\!\frac{1-\eta}{1-\gamma}}}{\bigl[(1-\gamma)J_t\bigr]^{\!\frac{1-\eta}{1-\gamma}-1}}\]

Separates risk aversion $\gamma$ from the inverse elasticity of intertemporal substitution $\eta$. Reduces to CRRA utility when $\gamma=\eta$. Calibrated to match equity premium and risk-free rate.

Figure 1 (paper). Gradual climate tipping Markov chain: the system transitions from pre-tipping ($H=1$) through intermediate states to an absorbing post-tipping state ($H=\mathcal{H}$) where $\chi_{\rm post}>\chi_{\rm pre}$, permanently amplifying warming per unit of cumulative emissions.

The simple rule — Results 1 & 2

Optimal SCC: perturbation approximation

Result 1 — without tipping risk

\[P^{R1}=\left[D_{1T}+\lambda^c_{1T}\,\frac{\mathbb{E}\!\left[1-Z_c^{1-\gamma}\right]}{1-\gamma}\,\frac{q^{(0)}}{B^{(0)}}\right]\frac{\chi\,Y^{(0)}}{r^\star}\]

Adjusted discount rate: $\displaystyle r^\star=\rho+(\eta-1)g^{(0)}-\frac{1}{2}\gamma\sigma^2-\frac{\lambda_e}{1-\gamma}\,\mathbb{E}\!\left[1-Z_e^{1-\gamma}\right]-\frac{\lambda^c}{1-\gamma}\,\mathbb{E}\!\left[1-Z_c^{1-\gamma}\right]$

Disaster term

$\lambda^c_{1T}$: marginal increase in disaster frequency per °C. $\mathbb{E}[1-Z_c^{1-\gamma}]/(1-\gamma)$: risk-adjusted expected capital loss per event, increasing in $\gamma$. Multiplied by $q^{(0)}/B^{(0)}=q^{(0)}K/Y^{(0)}$ to convert to output units.

TFP term

$D_{1T}$: marginal effect of 1°C on damage ratio $D$. Both terms multiplied by TCRE $\chi$ and output $Y^{(0)}$, discounted at $r^\star$. Setting $\lambda^c_{1T}=0$ recovers van den Bremer & van der Ploeg (2021).

Discount rate $r^\star$

Corrects for growth $g^{(0)}$, normal uncertainty $\sigma^2$, and both disaster processes. Decomposes into time preference, affluence, prudence, and insurance effects. Valid when damage and disaster parameters are small relative to growth.

Result 2 — with tipping risk (recursive)

\[P^{R2}(E,H)=P^{R1}(E,H)\;+\;h_E(E,H)\,\frac{1}{r^\star}\,\frac{K\,q^{(0)}}{\gamma-1}\left[\frac{\psi(E,H{+}1)}{\psi^{(0)}(E,H)}-1\right]\;+\;h(E,H)\,\frac{1}{r^\star}\left[P^{R2}(E,H{+}1)\,\frac{\psi(E,H{+}1)}{\psi^{(0)}(E,H)}-P^{R1}(E,H)\right]\]

Evaluated recursively from the post-tipping state $\mathcal{H}$ backwards. The second term (marginal hazard effect, MHE) prices the welfare loss from a marginal increase in tipping risk with temperature. The third term (differential welfare impact, DWI) reprices carbon to account for the anticipated future upward jump in TCRE and resulting higher damages. Both are discounted at the pre-tip $r^\star$. Terminal condition: $\psi(E,\mathcal{H})=\psi^{(0)}(E,\mathcal{H})$.

Quantitative results — Tables 2 & 3 (paper)

SCC by discount rate and externality

The rule is compared to a full numerical finite-difference solution. All figures below are taken directly from Tables 2 and 3 of the paper.

Market-based · r* = 5.3%/yr

$39 / tCO₂

Rule vs. Numerical: −0.24% error

TFP: $13.92 (35.6%)
Disasters: $24.04 (61.5%)
Tipping: $1.15 (2.9%)

Ethics-based · r* = 3%/yr

$90 / tCO₂

Rule vs. Numerical: −0.97% error

TFP: $24.67 (27.5%)
Disasters: $60.03 (66.9%)
Tipping: $4.97 (5.5%)

Ethics-based · r* = 2%/yr

$172 / tCO₂

Rule vs. Numerical: −2.46% error

TFP: $36.93 (22.0%)
Disasters: $116.30 (69.2%)
Tipping: $14.94 (8.9%)

TFP damages

Climate disasters

Tipping risk

(bars show rule shares)

SCC by externality & discount rate (Table 3, paper)

Stacked bars show rule values for each component at three discount rates. Disasters dominate at all rates; their absolute contribution grows strongly as $r^\star$ falls.

Why it matters

Practical relevance for policy, finance, and research

Carbon pricing policy

At $172/tCO₂ (r*=2%), the optimal SCC is 3.4× the current US benchmark of $51. The disaster channel — largely ignored in standard IAMs — drives 70% of the total. Omitting it leads to severely misleading results.

Transparent communication

Unlike black-box numerical models, the analytical rule shows exactly how risk aversion $\gamma$, discount rate $\rho$, TCRE $\chi$, and disaster parameters each shift the optimal carbon price — essential for policy dialogue.

Climate risk & asset pricing

Calibrated to match equity premium (6.5%/yr) and risk-free rate (0.8%/yr) with moderate risk aversion ($\gamma\approx 5.3$). Direct link to pricing climate-exposed assets, green bonds, and catastrophe insurance.

Robustness across IAMs

Accurate across multi-sector models, technical progress, stochastic tipping, and exponential disaster intensities — a reliable benchmark for comparing divergent SCC estimates in the integrated assessment literature.

Robustness

Rule accuracy in alternative models

The rule is tested outside the model for which it was derived. Errors stay below 1% for the market-based calibration across all endogenous-growth extensions. The rule is less accurate for the Solow-Swan exogenous growth model.

Model	Rule r* = 5.3%	Error	Rule r* = 2%	Error
Benchmark model	$39.11	+0.64%	$168.17	−2.46%
Technical progress in fossil fuel	$39.11	+1.06%	$168.17	−4.65%
Two-sector model	$39.11	−0.86%	$168.17	+6.32%
Exponential disaster intensity	$46.23	−0.94%	$207.59	−5.09%
Stochastic climate system	$39.11	+0.64%	$168.17	−2.65%
More general tipping model	$39.03	+0.59%	$165.46	−2.78%

Source: Table 4, paper. Rule applied outside the realm for which it was derived.

Three reasons to price carbonunder uncertainty

Three reasons to price carbon
under uncertainty