Optimal Carbon Abatement in a Stochastic Equilibrium Model

Model architecture — three building blocks

A fully stochastic integrated assessment model

All key variables — CO₂ concentration, global temperature, and world GDP — evolve as stochastic processes. This allows state-dependent optimal policies and model-based confidence bands, going well beyond deterministic IAMs such as DICE.

①

Carbon dioxide model

CO₂ concentration follows a GBM driven by business-as-usual emissions. Society reduces the drift via an abatement policy α, which maps linearly to the DICE emission control rate ε. Natural carbon sinks absorb CO₂ at a declining rate as they saturate.

②

Climate model with feedback

Temperature follows the log-CO₂ relation augmented by Brownian noise and a self-exciting jump process. The jumps capture delayed feedback loops (e.g. ice-albedo) and generate the empirically documented right-skewed distribution of equilibrium climate sensitivity.

③

Production economy

An AK model with Hayashi-style quadratic adjustment costs and Epstein-Zin preferences. Climate damages reduce either the level of GDP (standard DICE approach) or its growth rate (consistent with Dell, Jones & Olken). Abatement expenditures directly reduce output.

Key equations

Stochastic dynamics of the model

CO₂ dynamics

\[dM_t = M_t\bigl[(g_m(t)-\alpha_t)\,dt + \sigma_m\,dW^m_t\bigr]\]

$\alpha_t$: abatement policy (beyond BAU); $g_m(t)$: BAU drift calibrated to RCP 8.5; $\sigma_m = 0.0078$ estimated from Mauna Loa data.

Global warming process

\[dT_t = \frac{M_t\eta_\tau}{\bar{M}_t}(g_m-\alpha_t)\,dt + \frac{M_t\sigma_\tau}{\bar{M}_t}\,d\tilde{W}^\tau_t + \theta_\tau(T_t)\,dN^\tau_t\]

Self-exciting jump process $N^\tau$ with intensity $\pi_\tau(T_t)$ captures delayed feedback loops. Generates right-skewed equilibrium climate sensitivity (skewness 0.92).

GDP dynamics — growth rate impact

\[dY_t = Y_t\bigl[(g(t,\chi_t)-\kappa(t,\varepsilon_t)-\zeta_d T^n_t)\,dt + \sigma_k\,d\tilde{W}^k_t\bigr]\]

$\kappa(t,\varepsilon_t)=A\,a(t)\varepsilon_t^b$: abatement cost (convex, $b=2.8$); $\zeta_d T^n_t$: temperature damage to growth rate; $\chi_t=C_t/Y_t$: consumption rate.

Key results

What the model reveals about the social cost of carbon

Growth vs. level impact: first-order differences

A growth rate impact always yields a higher SCC than a level impact at the same calibration. Under the empirical Dell-Jones-Olken specification, the SCC is approximately four times larger than under (L-N), reflecting the permanent and irreversible nature of growth-rate damages.

Climate uncertainty is the dominant risk driver

Contributions from GDP and CO₂ volatility to the SCC are negligible (<1%). Temperature jump risk (feedback loops) accounts for 25–206% additional SCC above a deterministic baseline. Combining diffusion and jump temperature risk produces a super-additive amplification effect.

EIS dominates; risk aversion only matters for convex damages

The elasticity of intertemporal substitution (EIS) has a first-order effect on optimal abatement and SCC across all specifications. Risk aversion is nearly irrelevant for moderate linear damages but becomes quantitatively important when damage functions are steep and convex (Weitzman calibrations).

State-dependent optimal policy

Because all shocks are modeled explicitly, optimal abatement reacts to realized temperature paths. Along high-temperature paths, the SCC rises far above DICE estimates. Along low-temperature paths, society reduces abatement and raises consumption — a flexibility ignored by deterministic models.

Damage specifications

Five calibrations — two damage types, three empirical anchors

The paper systematically compares level and growth rate damages across three empirical calibrations: Nordhaus (DICE), Weitzman, and Dell-Jones-Olken. Initial SCC values ($/tCO₂, year 2015) are shown below.

(L-N)

Level · Nordhaus

$10.63

$D_N(T)=\tfrac{1}{1+0.00266T^2}$

(L-W)

Level · Weitzman

$18.07

Fat-tailed at high $T$

(G-N)

Growth · Nordhaus

$11.12

$\zeta_d=0.00026,\ n=1$

(G-W)

Growth · Weitzman

$42.86

$\zeta_d=0.000075,\ n=3.25$

(G-DJO)

Growth · Dell et al.

$38.73

$\zeta_d=0.00137,\ n=1$

Model	SCC 2015	SCC 2055	SCC 2095	Temp 2095 (°C)	Emission ctrl 2095
(G-N) growth · Nordhaus	$11.12	$50.67	$171.21	2.4°C	82%
(L-N) level · Nordhaus	$10.63	$58.37	$183.03	2.3°C	83%
(G-W) growth · Weitzman	$42.86	$145.93	$188.20	1.4°C	87%
(L-W) level · Weitzman	$18.07	$93.19	$189.00	1.8°C	85%
(G-DJO) Dell-Jones-Olken	$38.73	$127.85	$207.78	1.3°C	100%

Why it matters

Implications for climate policy, carbon pricing, and IAM design

Carbon tax design

The optimal carbon tax is state-dependent. Policy makers who rely on deterministic IAM estimates systematically underreact along high-temperature paths. The model provides a principled framework for setting an adaptive carbon price that rises with realized warming.

Damage specification is first-order

Whether temperature hits GDP levels or growth rates changes the SCC by a factor of up to four. The empirical evidence (Dell et al. 2009, 2012; Burke et al. 2015) favors a growth rate impact — supporting substantially more ambitious abatement than DICE implies.

Fat-tailed climate risk must be priced

Climate feedback loops generate right-skewed temperature distributions and interact with economic risk in a super-additive way. Stress tests that treat climate scenarios as deterministic miss the dominant source of SCC uncertainty identified in this paper.

Calibration of preferences matters

Most deterministic IAMs entangle EIS and risk aversion via CRRA utility. This paper shows that EIS, not risk aversion, is the key preference parameter — and that Epstein-Zin utility is essential to separate the two, especially for empirically severe damage calibrations.

Optimal carbon abatement in a stochasticequilibrium model with climate change

Optimal carbon abatement in a stochastic
equilibrium model with climate change