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

From Gravity to Confinement: Designing Optimal Wealth Redistribution

Why do some wealth taxes fail to reduce inequality even when they generate massive revenue? Many governments implement proportional taxes to capture capital. However, these policies often struggle to move the needle on social mobility.

A new study by Anders G. Frøseth uses mathematical frameworks from physics to explain this failure. The paper suggests that simple wealth taxes act like a uniform gravitational field. They pull everyone down equally without changing the relative gap between the rich and the poor. To truly compress inequality, the author argues, taxes must act as a "confining potential." This is a mathematical trap that pulls the extremely wealthy back toward the center of the distribution.

The Redistribution Paradox

The research addresses how different tax structures alter the shape of wealth distribution. Most economists focus on revenue. Instead, the author focuses on "drift" and "diffusion."

In this framework, "drift" is the average direction in which wealth moves (such as through investment returns). "Diffusion" represents the randomness or volatility of those returns. Think of a crowd in a plaza. Drift is the steady walking pace toward an exit. Diffusion is the erratic, stumbling movement of individuals.

The paper identifies a "redistribution paradox." A proportional wealth tax—charging the same percentage from everyone—merely shifts the drift downward. Because it scales all wealth by the same factor, relative inequality remains unchanged through the market channel. The author notes that while such a tax might eventually affect the steady-state distribution, the transition takes decades or even centuries. This timeline far exceeds typical policy horizons.

The Fokker–Planck Framework

To model these dynamics, the study employs the Fokker–Planck equation. This is a partial differential equation used in statistical mechanics to describe how the probability density (the likelihood of finding someone at a specific wealth level) of particles evolves over time. Here, the "particles" are individual investors.

The author builds upon the work of Gabaix et al. (2016). They analyzed how random multiplicative growth generates heavy-tailed Pareto distributions. These are "fat tails" where a tiny percentage of the population holds most of the wealth. The paper establishes that a baseline wealth system, driven by random growth and demographic turnover (the rate at which people enter and leave the economy), naturally settles into this unequal Pareto state.

The research categorizes tax designs into five mathematical modifications:

• Class 1 (Proportional Tax): Acts as a uniform drift shift. It is analogous to a uniform gravitational field. It pulls all wealth levels down but preserves the shape of the distribution.
• Class 2 (Progressive Tax): Introduces a state-dependent drift, or a "confining potential." This acts like a harmonic trap in physics. It exerts a stronger pull on those with higher wealth.
• Class 3 (Transfers): Uses "source-sink" terms to inject wealth at the bottom and extract it at the top. This is similar to pumping water between two reservoirs.
• Class 4 (Wealth Cap): Creates an absorbing boundary. This truncates the distribution entirely at a certain threshold.
• Class 5 (Volatility Policy): Modifies the diffusion coefficient. This reduces fluctuations (randomness) at high wealth levels.

Breaking Symmetry with Progressivity

Redistribution requires "breaking the symmetry" of the proportional tax. Moving from a Class 1 proportional tax to a Class 2 progressive tax changes the mathematical nature of the steady state.

Instead of a Pareto distribution with a heavy tail, a progressive tax replaces the tail with a "thinner" log-normal distribution. The author derives a relationship between the degree of progressivity ($\kappa$) and the resulting Gini coefficient (a measure of inequality). Crucially, the paper shows that progressive taxes possess their own internal convergence mechanism.

As shown in, the speed at which a society reaches its new wealth distribution depends on the tax structure.

A proportional tax relies on slow demographic turnover. Conversely, a progressive tax actively drives the distribution toward the target. The study finds that progressive taxes can achieve meaningful redistribution within typical electoral cycles of 4 to 8 years. In contrast, proportional taxes remain largely stagnant during those same periods.

Optimal Control and General Equilibrium

The paper formulates redistribution as an "optimal drift design" problem. Using Pontryagin’s maximum principle, the author seeks a tax structure that minimizes the distance to a target distribution. It also minimizes the "cost" of the intervention.

These costs include the economic and political penalties of distortion. A highly aggressive tax might lead to "leakage," such as migration or tax evasion. Furthermore, the paper extends the model into a "general equilibrium" setting. In this view, tax design feeds back into the economy. Taxing wealth reduces aggregate capital. This, in turn, raises the expected return on the remaining capital.

This creates a self-consistent loop, modeled as a McKean–Vlasov equation. Here, the tax policy and economic returns adjust to one another. The authors report that this feedback leads to "diminishing returns to progressivity." Eventually, the economic pushback from reduced capital accumulation begins to offset the gains from the tax.

Limits of the Model

The framework has clear boundaries. The "redistribution paradox" is specific to the "homogeneous-returns" limit. This is a scenario where all investors face the same expected returns. If investors have different inherent abilities, a proportional tax might redistribute capital toward higher-ability investors.

Additionally, the model treats "leakage" channels as exogenous (external) factors. In reality, behaviors like capital flight and evasion are likely endogenous (internal to the system). As a tax becomes more progressive, the incentive to evade or migrate increases. The paper also notes that the Ornstein–Uhlenbeck model used for progressive taxation is a simplification. Real-world tax brackets are rarely perfectly linear.