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Heterogeneous Returns and Wealth Tax Neutrality: A Fokker--Planck Framework

Anders G Frøseth

Abstract

We extend the Fokker--Planck framework of Frøseth (2026, arXiv:2603.05283) to populations of investors with heterogeneous, persistent return-generating ability. When the drift coefficient in the Langevin equation for log-wealth varies across investors, the proportional wealth tax remains a uniform drift shift but ceases to be neutral in the economic sense: its real incidence differs across ability types, and the stationary wealth distribution changes shape. We derive the extended Fokker--Planck equation on the joint space of log-wealth and ability, characterise the conditions under which the drift-shift symmetry breaks, and identify the consequences for asset prices and portfolio allocations. The analysis connects the neutrality results of Frøseth (2026, arXiv:2603.05264) and the Fokker--Planck dynamics of Frøseth (2026, arXiv:2603.05283) to the heterogeneous-returns literature, notably the "use-it-or-lose-it" mechanism of Guvenen, Kambourov, Kuruscu, Ocampo-Diaz and Chen (2023).

Heterogeneous Returns and Wealth Tax Neutrality: A Fokker--Planck Framework

Abstract

We extend the Fokker--Planck framework of Frøseth (2026, arXiv:2603.05283) to populations of investors with heterogeneous, persistent return-generating ability. When the drift coefficient in the Langevin equation for log-wealth varies across investors, the proportional wealth tax remains a uniform drift shift but ceases to be neutral in the economic sense: its real incidence differs across ability types, and the stationary wealth distribution changes shape. We derive the extended Fokker--Planck equation on the joint space of log-wealth and ability, characterise the conditions under which the drift-shift symmetry breaks, and identify the consequences for asset prices and portfolio allocations. The analysis connects the neutrality results of Frøseth (2026, arXiv:2603.05264) and the Fokker--Planck dynamics of Frøseth (2026, arXiv:2603.05283) to the heterogeneous-returns literature, notably the "use-it-or-lose-it" mechanism of Guvenen, Kambourov, Kuruscu, Ocampo-Diaz and Chen (2023).
Paper Structure (27 sections, 3 theorems, 26 equations, 1 figure, 1 table)

This paper contains 27 sections, 3 theorems, 26 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The homogeneous framework: recap
  3. Individual dynamics
  4. Population dynamics
  5. The drift-shift symmetry
  6. Heterogeneous returns: the extended framework
  7. Ability as a second state variable
  8. The joint Fokker--Planck equation
  9. The taxed dynamics
  10. Where the symmetry breaks
  11. Consequences for neutrality, prices, and portfolios
  12. Neutrality breaking: the mechanism
  13. The effective Pareto exponent
  14. Asset prices
  15. Portfolio allocations
  16. ...and 12 more sections

Key Result

Proposition 1

The drift-shift $v(z) \mapsto v(z) - \tau_w$ preserves the marginal wealth distribution $\pi(x,t)$ if and only if the conditional average $\langle v(z) \rangle_x$ is independent of $x$---that is, ability and wealth are statistically independent in the joint distribution $f$.

Figures (1)

  • Figure 1: Stationary wealth distribution for a two-type economy (high-ability $\mu_H = 12\%$, low-ability $\mu_L = 2\%$) under three tax regimes. The wealth tax shifts both peaks leftward by the same amount, preserving the inter-type gap (Proposition \ref{['prop:differential']}). The income tax compresses the gap by the factor $\lambda = 1 - \tau_c$. Parameters: $\sigma = 0.15$, $\delta = 0.08$, $\tau_w = 2.5\%$, $\tau_c = 30\%$, equal population shares.

Theorems & Definitions (10)

  • Remark : Interpretation of $z$
  • Remark : The homogeneous limit
  • Proposition 1: Neutrality requires ability--wealth independence
  • proof
  • Proposition 2: Endogenous correlation
  • proof : Proof sketch
  • Proposition 3: Differential incidence of flow and stock taxes
  • proof
  • Remark : Guvenen's three channels, reinterpreted
  • Remark : Relationship to heterogeneous beliefs