The Optimal Carbon-Tax Trajectory: A Finite-Horizon HJB Approach
Closed Form, Numerical Algorithm, and a DICE-Calibrated 25-Year Path
Abstract
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 output cost 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 expressible as a Pigouvian steady-state term plus a finite-horizon correction that vanishes as the horizon T grows. We give an explicit finite-difference algorithm for the realistic-damage extension, calibrate the model to DICE-2018 parameters with T = 25 years, and show that the optimal initial carbon tax is materially lower than the infinite-horizon Pigouvian level. We close with contour plots of the value function and the optimal policy over the (t, E) plane.