Wealth Tax Neutrality as Drift-Shift Symmetry: A Statistical Physics Formulation

Abstract

We reformulate the neutral wealth tax framework of Froeseth (2026) in the language of stochastic dynamics and statistical physics. Individual wealth under geometric Brownian motion satisfies a Langevin equation with multiplicative noise; the probability density of wealth across a population then evolves according to a Fokker-Planck equation. A proportional wealth tax at market value enters as a uniform reduction of the drift coefficient, preserving the diffusion structure and all relative probability currents. This drift-shift symmetry is the physical content of tax neutrality. Each channel through which neutrality breaks down in practice, book-value assessment, liquidity frictions, forced dividend extraction, migration, and market impact, corresponds to a specific violation of this symmetry: a state-dependent, asset-dependent, or flow-dependent modification of the Fokker-Planck equation. The framework clarifies when wealth taxation is a benign rescaling of the dynamics and when it introduces genuinely new physics.

Figures (4)

• Figure 1: The separability result visualised. A population of investors starts at log-wealth $x_0 = 3$ (dashed). After $t = 20$ years with $\mu = 10\%$ and $\sigma = 20\%$, the untaxed distribution (blue) and the distribution under a $\tau_w = 3\%$ proportional wealth tax (red) have identical shape and width ($\sqrt{2Dt} = 0.89$) but different centres. The tax shifts the propagator to the left by $\tau_w t = 0.6$ in log-wealth without deforming it. This is the visual content of neutrality: the tax is a pure translation in log-wealth space.
• Figure 2: How each distortion channel deforms the wealth distribution relative to the neutral case. Blue: untaxed propagator. Red: taxed propagator. (a) Neutral tax: pure translation, no deformation. (b) Book-value assessment: different assets shift by different amounts, splitting the distribution into modes. (c) Liquidity frictions: the distribution broadens (increased effective $D$) as forced selling at unfavourable prices adds noise. (d) Dividend extraction: the drift becomes time-dependent as firm capital erodes, introducing asymmetry. (e) Migration: high-wealth investors exit the system above a threshold $x^*$, truncating the right tail. (f) Market impact: the collective selling pressure shifts the distribution further left than the tax alone would predict, with additional compression from price feedback.
• Figure 3: Relaxation toward the new steady-state wealth distribution after a tax change, for three tax rates. Parameters: $\sigma = 0.30$, $\mu = 0.08$, $\delta = 1/30$ (generational turnover). The vertical axis measures the distance $\|p(\cdot,t) - p_\infty\|$ normalised to unity at $t = 0$. At $\tau_w = 2\%$ the time-average growth rate remains positive ($v_\tau > 0$), so convergence depends entirely on demographic turnover ($\lambda = \delta$). At $\tau_w = 5\%$, $v_\tau$ turns negative but only marginally, adding less than $4\%$ to the convergence rate. Even at the extreme rate $\tau_w = 10\%$, the half-life is still over a decade. The shaded region marks a typical electoral cycle ($\sim$4--8 years), during which the distribution has moved less than a quarter of the way to its new steady state in all three scenarios.
• Figure 4: Geometric interpretation of neutrality and its breakdown. (a) Under CRRA preferences, the optimal portfolio $\mathbf{w}^*$ is a horizontal section of the fiber bundle: it does not vary with log-wealth $x$. The proportional wealth tax (red arrows) translates the base leftward without rotating the fibers, leaving the section invariant. (b) When a distortion channel is active (e.g. book-value assessment, liquidity frictions), the section becomes wealth-dependent: the connection acquires curvature, and the portfolio varies with $x$. The dashed line shows the flat (neutral) section for comparison.

Theorems & Definitions (22)

• Remark : Analogy with statistical mechanics
• Proposition 1: Fundamental theorem of asset pricing
• Remark : Risk is not variance
• Remark : When does the SDF change?
• Remark : No restoring force
• Remark : Correlation structure
• Remark : Terminology
• Remark : Propagator
• Definition 1: Drift-shift transformation
• Proposition 2: Neutrality as invariance