Tax Mechanisms and Gradient Flows

31 Pages Posted: 13 May 2019 Last revised: 22 Jul 2023

See all articles by Stefan Steinerberger

Stefan Steinerberger

University of Washington

Aleh Tsyvinski

Yale University - Cowles Foundation; Yale University

Date Written: May 2019


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

Suggested Citation

Steinerberger, Stefan and Tsyvinski, Aleh and Tsyvinski, Aleh, Tax Mechanisms and Gradient Flows (May 2019). NBER Working Paper No. w25821, Available at SSRN:

Stefan Steinerberger (Contact Author)

University of Washington ( email )

box 354350
Seattle, WA 98195-4350
United States

Aleh Tsyvinski

Yale University ( email )

493 College St
New Haven, CT CT 06520
United States

Yale University - Cowles Foundation ( email )

28 Hillhouse Ave
New Haven, CT 06520-8268
United States
203-432-9163 (Phone)

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