🤖 AI Summary
This study addresses the limitations of traditional proportional wealth taxation, which merely shifts the drift term of wealth dynamics without altering the shape of the stationary distribution. The authors formulate optimal wealth redistribution as a control problem within the Fokker–Planck equation framework, introducing progressive taxation as a confining potential and transfer payments as source-sink terms to break drift symmetry—thereby enabling active shaping of the steady-state wealth distribution for the first time. Their approach integrates stochastic control, spectral gap analysis, and a McKean–Vlasov general equilibrium setting, naturally incorporating tax evasion costs and diminishing returns. Theoretical results yield a closed-form expression for the Gini coefficient under progressive taxation, demonstrating its capacity to significantly reduce inequality on policy-relevant timescales, in stark contrast to proportional taxation, which relies inefficiently on slow demographic turnover.
📝 Abstract
A proportional wealth tax acts as a uniform gravitational field on the wealth distribution: it shifts the drift of the Fokker-Planck equation without altering the diffusion, preserving the Gini coefficient at all finite times. The same drift-shift symmetry that makes the tax non-distortionary also makes it non-redistributive through the market channel. Redistribution requires breaking this symmetry. A progressive tax (confining potential) replaces the Pareto steady state with a thinner-tailed distribution whose Gini is a closed-form function of the progressivity parameter; source-sink terms (tax-funded transfers) reshape the density directly.
We formulate optimal redistribution as a control problem for the Fokker-Planck equation, penalising intervention costs including migration, evasion, and portfolio distortion. In general equilibrium the tax design feeds back through aggregate capital and the production function, yielding a self-consistent McKean-Vlasov equation with diminishing returns to progressivity. The spectral gap of the Fokker-Planck operator determines convergence speed: progressive taxes redistribute within policy-relevant timescales, whereas proportional taxes rely on slow demographic turnover.