From Gravity to Confinement: Wealth Redistribution as Optimal Drift Design in the Fokker-Planck Framework

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.