Imagine a social planner who wants to make society as well-off as possible over the next two centuries. She controls how much the economy consumes today versus invests for the future—but there's a catch: production generates CO₂, which warms the planet, which damages the economy. More output today means more emissions, more warming, and less output tomorrow.
The Social Cost of Carbon (SCC) answers a simple question: if we emit one extra ton of CO₂ right now, how much total economic damage does that cause (in today's dollars)? It's the price tag on a ton of pollution. If a carbon tax were set equal to the SCC, it would give the economy exactly the right incentive to pollute less—it's the optimal Pigouvian tax.
Three things matter most for the SCC:
1. The discount rate \(\rho\) — How much do we value future generations' welfare compared to our own? A low discount rate says "future people matter almost as much as us," which makes distant climate damages count heavily and drives the SCC up. This is the single most important parameter and the source of the biggest disagreements (Stern vs. Nordhaus).
2. The damage function \(D(T) = \pi_1 T + \pi_2 T^2\) — How badly does each degree of warming hurt the economy? The linear term \(\pi_1 T\) captures damages proportional to warming; the quadratic term \(\pi_2 T^2\) captures damages that accelerate as it gets hotter. Bigger coefficients mean more damage per degree, raising the SCC.
3. Climate sensitivity \(\eta\) — How much does temperature rise when CO₂ doubles? Higher sensitivity means each ton of CO₂ causes more warming, which again raises the SCC.
Play with the sliders to see how these assumptions change the answer. Try setting \(\rho\) very low (Stern's argument) or \(\pi_1 > 0\) to see how even small linear damages at low temperatures can matter.
The social planner maximizes the sum of discounted utility from per-capita consumption \(c_t = C_t / L_t\):
Output \(Q_t\) is produced from capital and labor. A fraction \(D(T_t)\) is destroyed by climate damages, so net output is:
Temperature rises as emissions accumulate in the atmosphere. The SCC is the shadow price on emissions — it answers "how much would welfare improve if we could magically remove one ton of CO₂?"
Reading right to left: \(\partial T_s / \partial E_t\) is how much today's extra ton warms the planet at future time \(s\). Multiplied by \((\pi_1 + 2\pi_2 T_s) Q_s\), that gives the dollar damage. The ratio \(c_t / c_s\) adjusts for the fact that future people are richer. The discount factor weighs near-term damages more. Sum over all future periods and you get the SCC.