Chapter 35: Integrated Assessment Models

Climate-Economy Modeling, DICE, and the Social Cost of Carbon

“The social cost of carbon is the most important price in the world. It should reflect the present discounted value of all future damages from one additional ton of CO₂.” — Nicholas Stern

Cross-reference: Principles Ch. 37 (climate change and macroeconomics, DICE model, social cost of carbon, Stern vs. Nordhaus); Ch. 40 (capstone: climate policy) [P:Ch.37, P:Ch.40]

35.1 The Integrated Assessment Modeling Framework¶

An Integrated Assessment Model (IAM) combines an economic model of production, consumption, and investment with a physical climate model of emissions, atmospheric concentrations, and temperature — solving for the socially optimal emission path and the corresponding social cost of carbon (SCC).

The DICE model (Dynamic Integrated model of Climate and the Economy; Nordhaus, 1992, 2008, 2018) is the most widely used IAM. It is the quantitative underpinning of U.S. federal regulatory policy — the SCC from DICE variants is used by the EPA to value carbon regulations.

Mathematically, DICE is a continuous-time optimal control problem: the social planner chooses the abatement fraction $\mu(t)$ (emissions reduction relative to business-as-usual) to maximize global welfare subject to economic and climate dynamics. The optimal abatement path and the SCC emerge from the Hamiltonian conditions of Chapter 11.

35.2 The DICE Model Structure¶

35.2.1 Economic Module¶

The economy produces output $Y(t)$ using capital $K(t)$ and exogenous labor $L(t)$ :

Y(t) = \Omega(T_{AT})\cdot[1-\Lambda(\mu(t))]\cdot A(t)K(t)^\gamma(A_L(t)L(t))^{1-\gamma},

(1)

where:

$\Omega(T_{AT}) = 1/(1 + \pi_1 T_{AT} + \pi_2 T_{AT}^2)$ is the damage function — output fraction lost due to temperature increase $T_{AT}$ above pre-industrial. Nordhaus calibrates $\pi_1 \approx 0$ , $\pi_2 \approx 0.00236$ .
$\Lambda(\mu) = \theta_1\mu^{\theta_2}$ is the abatement cost function — output fraction spent on emission reduction.
$A(t)$ is total factor productivity (exogenous).

Capital accumulation: $\dot{K} = Y(t) - C(t) - \delta_K K(t)$ .

Emissions: $E(t) = \sigma(t)(1-\mu(t))Y(t)$ , where $\sigma(t)$ is emission intensity (declining exogenously with energy efficiency).

Welfare objective:

\max_{\mu(\cdot),C(\cdot)}\int_0^\infty e^{-\rho t}L(t)\frac{[C(t)/L(t)]^{1-\sigma}-1}{1-\sigma}\,dt.

(2)

35.2.2 Climate Module¶

Carbon cycle: Three carbon reservoirs (atmosphere, upper ocean, lower ocean) exchange carbon with flow rates $\phi_{ij}$ :

\dot{M}_{AT} = E(t) - \phi_{10}M_{AT} + \phi_{01}M_{UO}, \qquad \text{etc.}

(3)

This is a linear system $\dot{\mathbf{M}} = \Phi\mathbf{M} + \mathbf{e}(t)$ (where $\mathbf{e}$ contains $E(t)$ in the first entry).

Temperature dynamics:

C_{AT}\dot{T}_{AT} = F(M_{AT}) - \lambda T_{AT} - \frac{C_{AT}}{C_{OC}}(T_{AT} - T_{OC}),

(4)

C_{OC}\dot{T}_{OC} = \frac{C_{AT}}{C_{OC}}(T_{AT} - T_{OC}),

(5)

where $F(M_{AT}) = \eta\log_2(M_{AT}/M_{AT,1750})$ is the radiative forcing (logarithm of atmospheric carbon concentration ratio).

The SCC is derived formally from the Hamiltonian of the DICE optimal control problem. The key insight is that the SCC is the shadow price of atmospheric carbon — exactly the costate variable of the Hamiltonian, analogous to Tobin’s $q$ in Chapter 13.

Setting up the Hamiltonian: The state variables are $(K, M_{AT}, T_{AT}, \ldots)$ . The control variables are $(\mu, C)$ . The current-value Hamiltonian:

\mathcal{H} = L\frac{(C/L)^{1-\sigma}-1}{1-\sigma} + q_K\dot{K} + q_{AT}\dot{M}_{AT} + q_{T}\dot{T}_{AT} + \ldots

(6)

where $q_K$ is the shadow price of capital (Tobin’s $q$ ), $q_{AT}$ is the shadow price of atmospheric carbon (the SCC), and $q_T$ is the shadow price of temperature.

Definition 35.1 (Social Cost of Carbon). The social cost of carbon is the negative of the costate variable associated with atmospheric CO₂:

SCC(t) \equiv -\frac{\partial\mathcal{H}}{\partial E}\bigg/\frac{\partial\mathcal{H}}{\partial C} = -\frac{q_{AT}(t)}{q_K(t)}.

(7)

This is the marginal cost of emitting one additional ton of CO₂, measured in units of current consumption.

Pontryagin conditions for $\mu$ :

\frac{\partial\mathcal{H}}{\partial\mu} = 0 \implies \text{optimal abatement: MAC} = SCC,

(8)

where MAC is the marginal abatement cost ( $\partial\Lambda/\partial\mu \cdot Y$ ). The optimal abatement fraction equates marginal abatement cost to the SCC.

The SCC dynamics (costate equation for $q_{AT}$ ):

\dot{q}_{AT} = \rho q_{AT} - \frac{\partial\mathcal{H}}{\partial M_{AT}} = \rho q_{AT} + q_T \cdot \frac{\partial F}{\partial M_{AT}}\cdot\frac{1}{C_{AT}}.

(9)

The SCC grows (in absolute value) over time at a rate related to the discount rate $\rho$ and the marginal impact of carbon on temperature.

Theorem 35.1 (Ramsey Formula for the SCC). In the DICE model, the steady-state growth rate of the SCC satisfies:

\frac{\dot{SCC}}{SCC} \approx \rho + \sigma g_c - g_{damages},

(10)

where $g_c$ is consumption growth, $g_{damages}$ is the growth rate of climate damages, and $\rho + \sigma g_c$ is the Ramsey discount rate [P:Ch.37.2].

Derivation. The costate equation gives $\dot{q}_{AT}/q_{AT} = \rho - \partial\ln\mathcal{H}/\partial M_{AT}$ . The Ramsey discount rate $\rho + \sigma g_c$ adjusts for the declining marginal utility of consumption as the economy grows. $\square$

35.4 Sensitivity Analysis: Stern vs. Nordhaus¶

The Stern Review (2006) and Nordhaus (2008) reached dramatically different conclusions about optimal climate policy. The difference is almost entirely driven by the discount rate parameters $\rho$ (pure time preference) and $\sigma$ (CRRA curvature).

Table 35.1: Stern vs. Nordhaus parameters.

Parameter	Stern	Nordhaus	Effect on SCC
$\rho$ (pure time preference)	0.001	0.015	Lower $\rho$ → higher SCC
$\sigma$ (CRRA)	1.0	2.0	Lower $\sigma$ → higher SCC
Ramsey rate ( $\rho+\sigma g$ )	≈1.4%	≈5.5%	Much lower for Stern
SCC (2020 estimate)	~$300/t	~$40/t	7.5× difference

The Stern–Nordhaus debate is fundamentally a disagreement about the appropriate intergenerational discount rate — how much we should discount welfare of future generations relative to the present [P:Ch.37.3].

35.5 Numerical Solution: Discretizing DICE¶

The continuous-time DICE problem is discretized to 5-year time steps, converting the optimal control problem to a nonlinear programming (NLP) problem.

Discrete DICE system (each period $t = 0, 1, \ldots, T$ ):

State transition: $K_{t+1} = (1-\delta_K)K_t + Y_t - C_t$ , $M_{AT,t+1} = \sum_j\phi_{1j}M_{j,t} + E_t$ , $T_{t+1}$ from climate equations.

Objective: $\max_{\mu_t, C_t}\sum_t R_t L_t u(C_t/L_t)$ , where $R_t = e^{-\rho\Delta t}/(1+R_t^{market})$ (discount factor).

This is a $2T$ -dimensional NLP solved by scipy.optimize.minimize (Python) or Ipopt (Julia). The Hamiltonian provides analytical gradients, accelerating convergence.

Python¶

import numpy as np
from scipy.optimize import minimize

# Simplified DICE-like model (discrete time, 10-period simulation)
T_steps = 20      # 20 periods of 5 years each = 100 years
dt = 5.0          # years per period

# Parameters
rho   = 0.015     # Nordhaus: pure time preference
sigma_u = 2.0     # CRRA curvature (Nordhaus)
gamma = 0.30      # capital share
delta_K = 0.10    # depreciation (per 5-year period)
pi2   = 0.00236   # damage function coefficient
theta1, theta2 = 0.0318, 2.8  # abatement cost parameters

# Exogenous paths (simplified)
A_path = 5.0 * (1.04)**np.arange(T_steps)   # TFP growth
L_path = 7e9 * (1.005)**np.arange(T_steps)  # population
sigma_E = 0.6 * (0.98)**np.arange(T_steps)  # emission intensity

def dice_welfare(controls, rho=rho, sigma_u=sigma_u):
    """Evaluate DICE welfare given control sequence."""
    n = T_steps
    mu = controls[:n]   # abatement fraction [0,1]
    C  = controls[n:]   # consumption path (trillion $)
    mu = np.clip(mu, 0, 1); C = np.maximum(C, 1.0)
    
    # State trajectories
    K = np.zeros(n); T_AT = np.zeros(n); M_AT = np.zeros(n)
    K[0] = 200.0; T_AT[0] = 1.0; M_AT[0] = 800.0  # initial conditions
    
    welfare = 0.0
    for t in range(n):
        # Damage and abatement
        damage = 1.0 / (1 + pi2 * T_AT[t]**2)
        abate  = 1.0 - theta1 * mu[t]**theta2
        Y_t = damage * abate * A_path[t] * K[t]**gamma
        
        # Welfare contribution
        c_pc = C[t] / (L_path[t] / 1e9)  # per capita consumption
        welfare += L_path[t]/1e9 * np.exp(-rho*t*dt) * (c_pc**(1-sigma_u)-1)/(1-sigma_u) * dt
        
        # Update states
        E_t = sigma_E[t] * (1-mu[t]) * Y_t
        if t < n-1:
            K[t+1] = (1-delta_K)*K[t] + Y_t - C[t]
            K[t+1] = max(K[t+1], 1.0)
            # Simplified climate
            M_AT[t+1] = 0.88*M_AT[t] + E_t*dt*3.67  # mass balance
            T_AT[t+1] = T_AT[t] + 0.01*(np.log(M_AT[t]/590)/np.log(2)*3.8 - 1.2*T_AT[t])*dt
    
    return -welfare  # minimize negative welfare

# Optimize
mu0 = np.ones(T_steps) * 0.3  # initial abatement guess
C0  = np.ones(T_steps) * 50.0  # initial consumption guess
x0 = np.concatenate([mu0, C0])

result = minimize(dice_welfare, x0, method='SLSQP',
                  bounds=[(0,1)]*T_steps + [(1, 1e5)]*T_steps,
                  options={'maxiter': 500, 'ftol': 1e-8})

mu_opt = result.x[:T_steps]; C_opt = result.x[T_steps:]
print(f"Optimal abatement path (first 5 periods): {np.round(mu_opt[:5]*100,1)}%")
print(f"Optimal consumption (first 5 periods): {np.round(C_opt[:5],1)} T$")

# Compute SCC via numerical shadow price (finite difference)
def dice_welfare_delta(delta_E=0.01, t_shock=0):
    """Perturb emission at t_shock by delta_E, measure welfare change."""
    sigma_E_perturb = sigma_E.copy()
    sigma_E_perturb[t_shock] += delta_E / (A_path[t_shock] * 200**gamma * 0.7)
    # Re-run with perturbed path... (simplified)
    return dice_welfare(result.x)  # placeholder

# Stern comparison
result_stern = minimize(lambda x: dice_welfare(x, rho=0.001, sigma_u=1.0), x0,
                        method='SLSQP',
                        bounds=[(0,1)]*T_steps + [(1, 1e5)]*T_steps,
                        options={'maxiter': 300, 'ftol': 1e-6})
mu_stern = result_stern.x[:T_steps]
print(f"\nStern abatement (period 1): {mu_stern[0]*100:.1f}%  vs Nordhaus: {mu_opt[0]*100:.1f}%")
print("(Stern advocates near-100% abatement immediately due to low discount rate)")

Julia¶

using Optim

# DICE-like welfare calculation in Julia
function dice_welfare(controls; rho=0.015, sigma_u=2.0)
    T = 20; n = T; dt = 5.0
    mu = clamp.(controls[1:n], 0, 1)
    C  = max.(controls[n+1:2n], 1.0)
    
    # Exogenous paths
    A = 5.0 .* (1.04).^(0:n-1); L = 7e9 .* (1.005).^(0:n-1)
    sig_E = 0.6 .* (0.98).^(0:n-1)
    
    K = 200.0; T_AT = 1.0; M_AT = 800.0; welfare = 0.0
    for t in 1:n
        damage = 1/(1 + 0.00236*T_AT^2); abate = 1 - 0.0318*mu[t]^2.8
        Y = damage*abate*A[t]*K^0.30
        c_pc = C[t]/(L[t]/1e9)
        welfare += L[t]/1e9 * exp(-rho*(t-1)*dt) * (c_pc^(1-sigma_u)-1)/(1-sigma_u) * dt
        K = max((1-0.10)*K + Y - C[t], 1.0)
        M_AT = 0.88*M_AT + sig_E[t]*(1-mu[t])*Y*dt*3.67
        T_AT += 0.01*(log(M_AT/590)/log(2)*3.8 - 1.2*T_AT)*dt
    end
    return -welfare
end

x0 = vcat(0.3*ones(20), 50.0*ones(20))
res = optimize(dice_welfare, x0, LBFGS(), Optim.Options(iterations=500))
mu_opt = clamp.(Optim.minimizer(res)[1:20], 0, 1)
println("Nordhaus optimal abatement (period 1): $(round(mu_opt[1]*100,digits=1))%")

# Stern comparison
dice_stern(x) = dice_welfare(x; rho=0.001, sigma_u=1.0)
res_stern = optimize(dice_stern, x0, LBFGS(), Optim.Options(iterations=500))
mu_stern = clamp.(Optim.minimizer(res_stern)[1:20], 0, 1)
println("Stern optimal abatement (period 1): $(round(mu_stern[1]*100,digits=1))%")

35.6 Programming Exercises¶

Exercise 35.1 (APL — DICE Scan Simulation)¶

Implement the DICE state update as a dfn dice_step ← {K T_AT M_AT mu C ← ⍵ ⋄ ...} and simulate the model forward using APL’s scan operator: states ← dice_step \ T ⍴ ⊂init_state. (a) Simulate 20 periods (100 years) with fixed $\mu = 0.3$ and $C = 50$ T $. (b) Plot temperature trajectory and cumulative welfare. (c) Use the outer product `∘.f` over a grid of$ (\rho, \sigma)$ values to compute the SCC sensitivity surface.

Exercise 35.2 (Python — SCC via Shadow Price)¶

Compute the SCC numerically: (a) solve DICE at the optimal $(\mu^*, C^*)$ ; (b) perturb $E_0$ by +1 ton of CO₂ (which changes $M_{AT,0}$ by the carbon cycle coefficient); (c) re-solve with the perturbed carbon path and compare welfare; (d) $SCC_0 = -\Delta V / \Delta E_0$ in dollars per ton. Compare to Nordhaus (2018) estimate of $\approx$ $40/t and Rennert et al. (2022) estimate of $\approx$ $185/t.

Exercise 35.3 (Julia — Regional RICE Model)¶

# Multi-region DICE (RICE): 5 regions
# Each region has own TFP, population, emission intensity
n_regions = 5
A_regional = [5.0, 8.0, 3.0, 1.5, 2.0]   # TFP by region
L_regional = [1.2, 4.5, 0.7, 0.4, 0.2].*1e9  # population
sigma_regional = [0.3, 0.8, 0.6, 0.5, 0.4]   # emission intensity
damage_regional = [0.00236, 0.00354, 0.00472, 0.00590, 0.00472] # higher in poor regions

println("Multi-region (RICE) SCC differs by region:")
println("Rich regions (low damage coeff): lower SCC from domestic perspective")
println("Poor regions (high damage coeff): higher SCC from domestic perspective")
println("Social planner sets global SCC = weighted average = Pareto optimum")
println("Nash bargaining leads to under-provision of abatement (prisoner's dilemma)")

Exercise 35.4 — Uncertainty and the SCC ( $\star$ )¶

Fat-tailed risks and Weitzman’s Dismal Theorem: Weitzman (2009) argues that if climate damages have a fat-tailed distribution (Pareto or similar), the expected SCC is infinite — because the marginal utility of a small probability of catastrophic outcomes dominates the calculation. (a) Modify the damage function to include a fat-tailed catastrophic scenario: with probability $p = 0.01$ , temperature rises by $10°C$ and damages are $90\%$ of GDP. (b) Compute the expected SCC with and without the catastrophic scenario. (c) Show that with CRRA utility ( $\sigma > 1$ ) and a fat upper tail on damages, the SCC grows without bound as the catastrophe probability $p \to 0$ but the tail becomes heavier.

35.7 Chapter Summary¶

Key results:

The DICE model integrates an economic growth model with a carbon-cycle and temperature model; the planner maximizes welfare over consumption and abatement paths, subject to climate dynamics.
The social cost of carbon is the costate variable of the atmospheric CO₂ stock: $SCC = -q_{AT}/q_K$ . The optimal abatement rule equates marginal abatement cost to the SCC (Pontryagin condition).
The Ramsey formula for SCC growth (Theorem 35.1): $\dot{SCC}/SCC \approx \rho + \sigma g_c - g_{damages}$ — the SCC grows at the Ramsey discount rate minus the growth of damages.
The Stern–Nordhaus gap ( $\approx 7.5\times$ ) arises almost entirely from the discount rate: Stern uses $\rho = 0.001$ (near-zero pure time preference); Nordhaus uses $\rho = 0.015$ (calibrated to observed market rates).
The discrete DICE NLP is solved by scipy.optimize.minimize (SLSQP) or Julia’s Optim.jl (L-BFGS); the APL implementation uses dice_step \ T ⍴ init for the state dynamics.
In APL: SCC_grid ← rho_grid ∘.{SCC_function ⍺ ⍵} sigma_grid — sensitivity surface via outer product.

Next: Chapter 36 — Agent-Based Computational Macroeconomics