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

1. Problem Statement

The paper addresses a fundamental tension in public finance theory regarding the equivalence of capital income taxes and wealth taxes.

• The Homogeneous Benchmark: In standard models where all investors face identical return processes (homogeneous returns), a proportional wealth tax is mathematically equivalent to a capital income tax (up to a rate adjustment). In this setting, a wealth tax acts as a uniform drift shift in the log-wealth dynamics, preserving the shape of the wealth distribution and portfolio allocations (the "neutrality" result).
• The Empirical Reality: Empirical evidence (e.g., Fagereng et al., 2020; Guvenen et al., 2023) demonstrates that returns are heterogeneous and persistent. High-ability investors (entrepreneurs, skilled managers) consistently outperform low-ability investors.
• The Core Question: Does the "neutrality" of a wealth tax hold when investors differ in a persistent, ability-driven parameter that governs their expected returns? Specifically, how does the interaction between heterogeneous returns and a uniform wealth tax alter the wealth distribution, asset prices, and portfolio allocations?

2. Methodology

The author extends the continuous-time Fokker–Planck framework previously developed by Frøseth (2026f) to a heterogeneous-agent setting.

• State Variables: The model introduces a joint state space $(x, z)$, where:
  • $x = \ln(W)$ is the log-wealth.
  • $z$ is a persistent, individual-specific ability parameter governing expected returns.
• Stochastic Dynamics:
  • Wealth Dynamics: Modeled via a Langevin equation where the drift $\mu(z)$ and volatility $\sigma(z)$ depend on ability $z$.
  • Ability Dynamics: Modeled as an Ornstein–Uhlenbeck process (mean-reverting diffusion) to capture the empirical finding that ability is persistent but not permanent.
• The Joint Fokker–Planck Equation: The evolution of the joint probability density function $f(x, z, t)$ is derived. This equation captures the advection (drift) and diffusion of wealth and ability simultaneously.
• Tax Implementation:
  • Wealth Tax ($\tau_w$): Enters as a uniform subtraction from the drift coefficient: $v(z) \to v(z) - \tau_w$.
  • Capital Income Tax ($\tau_k$): Enters as a multiplicative scaling of the excess return (flow tax).
• Analytical Approach: The paper analyzes the marginal wealth density $\pi(x, t) = \int f(x, z, t) dz$ to determine how the tax affects the aggregate distribution, focusing on the conditional expectation of ability given wealth, $\langle v(z) \rangle_x$.

3. Key Contributions

A. Breaking the Drift-Shift Symmetry

The paper proves that the drift-shift symmetry (which ensures neutrality in homogeneous models) breaks down when returns are heterogeneous.

• Mechanism: In the homogeneous case, a wealth tax shifts the entire distribution uniformly. In the heterogeneous case, high-ability investors accumulate wealth faster, creating a positive correlation between wealth ($x$) and ability ($z$).
• Result: The effective drift of the marginal wealth distribution becomes state-dependent: $\langle v(z) \rangle_x$. Since the rich are disproportionately high-ability, the tax burden relative to their growth capacity differs from that of the poor. The tax is no longer neutral; it alters the shape of the stationary wealth distribution.

B. The "Use-It-or-Lose-It" Mechanism in Continuous Time

The paper formalizes the "use-it-or-lose-it" mechanism (Guvenen et al., 2023) using Fokker–Planck dynamics:

• Wealth Tax: Preserves the difference in drift between high- and low-ability investors ($v(z_H) - v(z_L)$ remains constant). However, because the tax is a fixed subtraction, it imposes a larger relative burden on low-ability investors (whose net drift may turn negative). This causes wealth to migrate from low-ability to high-ability types, thinning the lower tail and altering the Pareto exponent.
• Income Tax: Compresses the drift difference by a factor $\lambda < 1$. It reduces the incentive for high-ability investors to accumulate relative to low-ability ones, effectively "compressing" the wealth gap.

C. Asymmetry of Flow vs. Stock Taxes

The paper derives Proposition 3, which highlights a structural asymmetry:

• Flow Taxes (Income Tax): Scale the excess return. They compress the drift gap between ability types.
• Stock Taxes (Wealth Tax): Shift the absolute drift. They preserve the drift gap.
• Implication: A revenue-neutral shift from income tax to wealth tax widens the drift gap between high and low ability, accelerating the reallocation of capital toward the skilled (or those with structural advantages).

D. Decomposition of "Ability"

The paper critically reinterprets the "ability" parameter $z$. It argues that persistent returns arise from three distinct sources, which have divergent welfare implications:

1. Productive Skill: Genuine alpha (efficient allocation).
2. Structural Advantage: Market frictions, scale economies, or incumbency (e.g., local monopolies).
3. Winner-Take-All Rents: Platform dynamics and network effects.
   The paper notes that a wealth tax preserves the drift gap regardless of the source. If the gap is driven by skill, the tax is efficient (reallocates to the skilled). If driven by structural rents, the tax reinforces market power and concentration.

4. Key Results

1. Neutrality Condition: A proportional wealth tax preserves the shape of the wealth distribution if and only if ability and wealth are statistically independent. Since ability is persistent and drives returns, this independence is impossible in the long run; thus, neutrality fails.
2. Pareto Exponent: The effective Pareto exponent of the wealth distribution is determined by the high-ability tail. A wealth tax reduces the effective drift of the tail, thinning the distribution (reducing the Pareto exponent), whereas an income tax compresses the tail differently.
3. Portfolio Allocations: In homogeneous models, wealth taxes do not change optimal portfolio weights. In heterogeneous models, the aggregate portfolio shifts because low-ability investors exit risky assets (due to negative net drift) while high-ability investors absorb the capacity.
4. Empirical Calibration: Using Norwegian data (Fagereng et al., 2020), the paper shows that return heterogeneity is substantial (standard deviation ~8.6%) and highly persistent. Crucially, the top of the wealth distribution is dominated by private business equity, where return heterogeneity is highest, implying that the aggregate deviation from neutrality is significant in modern economies.

5. Significance and Policy Implications

• Reframing the "Punishes Skill" Debate: The paper overturns the common argument that wealth taxes punish skill. It demonstrates that wealth taxes preserve the reward to skill (by maintaining the drift gap), whereas income taxes compress it. The "punishment" is actually a relative burden on low-ability investors, driving capital toward the skilled.
• Efficiency vs. Concentration: The welfare impact of a wealth tax depends on the source of return persistence.
  • If persistence is due to skill, wealth taxes improve allocative efficiency (capital moves to the best hands).
  • If persistence is due to structural rents (monopolies, platforms), wealth taxes exacerbate concentration and entrench incumbents.
• Tax Design: The paper suggests that a uniform wealth tax is a blunt instrument. Optimal policy requires a nuanced mix:
  • Wealth taxes may be preferred in sectors driven by genuine skill (e.g., early-stage venture capital) to accelerate reallocation.
  • Flow taxes (income taxes) may be preferred in sectors dominated by structural rents (e.g., digital platforms) to compress the drift gap and reduce concentration.
• Theoretical Bridge: The work connects the mathematical elegance of Fokker–Planck equations with the empirical reality of heterogeneous agent models, providing a rigorous continuous-time foundation for the "use-it-or-lose-it" mechanism.

In summary, the paper demonstrates that wealth tax neutrality is an artifact of the homogeneous assumption. Once persistent heterogeneity is introduced, wealth taxes become powerful redistributive tools that reshape the wealth distribution by altering the relative accumulation rates of different ability types, with profound implications for inequality, market structure, and optimal tax policy.