Working Paper: NBER ID: w25821
Authors: Stefan Steinerberger; Aleh Tsyvinski
Abstract: We demonstrate how a static optimal income taxation problem can be analyzed using dynamical methods. We show that the taxation problem is intimately connected to the heat equation and derive a new property of the optimal tax which we call the fairness principle. The optimal tax at a given income is equal to the weighted by the heat kernels average of optimal taxes at other incomes and income densities. The fairness principle arises not due to equality considerations but represents an efficient way to smooth the burden of taxes and generated revenues across incomes. Just as nature distributes heat evenly, the optimal way for a government to raise revenues is to distribute the tax burden and raised revenues evenly among individuals. We then construct a gradient flow of taxes – a dynamic process changing the existing tax system in the direction of the increase in tax revenues – and show that it takes the form of a heat equation. The fairness principle holds also for the short-term asymptotics of the gradient flow. The gradient flow is a continuous process of a reform of the nonlinear tax and thus unifies the variational approach to taxation and optimal taxation
Keywords: optimal income taxation; heat equation; fairness principle; gradient flow
JEL Codes: E62; H21
Edges that are evidenced by causal inference methods are in orange, and the rest are in light blue.
| Cause | Effect |
|---|---|
| Optimal tax at a given income (H21) | Fairness principle (D63) |
| Fairness principle (D63) | Weighted average of optimal taxes at other income levels (H21) |
| Gradient flow of taxes (H29) | Fairness principle (D63) |
| Gradient flow of taxes (H29) | Optimal tax (H21) |
| Optimal tax (H21) | Tax revenue (H29) |
| Tax changes (H29) | Behavioral responses of individuals (D91) |