The Optimal Carbon-Tax Trajectory: A Finite-Horizon HJB Approach

We derive the optimal time-varying carbon-tax trajectory as the solution of a finite-horizon Hamilton–Jacobi–Bellman problem for a planner who balances the output cost of the tax against quadratic climate damage on the cumulative-emissions state. For the linear–quadratic specification the value function and the optimal tax admit a closed form: a stationary, Pigouvian term plus a finite-horizon correction that decays at an explicit rate. With physical anchors from the IPCC Sixth Assessment Report (cumulative emissions of $2\,390$ GtCO$_2$ by 2019, $0.45$ °C of warming per $1\,000$ GtCO$_2$) and an illustrative, DICE-style damage and cost calibration, the 25-year planner’s initial tax is $2.61/tCO$_2$ against a stationary benchmark of $5.80, 55 % lower. The dollar levels follow the illustrative anchors; the size of the gap is set mainly by how the terminal condition values the stock left at the horizon. Every headline number is computed twice, by the closed form and by an independent method: an ODE solve, a direct optimisation that does not use the HJB equation, Monte Carlo, or finite differences.