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

Anders G Fr{\o}seth

arxiv: 2607.06153 · v1 · pith:5K5IEB3A · submitted 2026-07-07 · physics.soc-ph · econ.GN· q-fin.EC· q-fin.PM

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

Anders G Fr{\o}seth This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-08 14:54 UTCglm-5.2pith:5K5IEB3A record.json open to challenge →

classification physics.soc-ph econ.GNq-fin.ECq-fin.PM

keywords wealth inequalityFokker-Planck equationPareto distributionGini coefficientprogressive taxationOrnstein-Uhlenbeck processoptimal controlMcKean-Vlasov

0 comments

The pith

Proportional wealth taxes don't reduce inequality — progressive ones do

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models wealth distribution dynamics using the Fokker–Planck equation, which describes how a probability density evolves under drift and diffusion. Under the assumption that all investors face the same expected return and volatility, a proportional wealth tax enters this equation as a uniform shift of the drift coefficient — every investor's wealth growth slows by the same amount, while the diffusion (volatility) is unchanged. Because the Gini coefficient is invariant under multiplicative rescaling of all wealth levels, and because a uniform drift shift produces exactly such a rescaling at every finite time, the paper proves (Proposition 1) that a proportional tax leaves the Gini coefficient unchanged through the market channel at all times. The steady-state Pareto exponent does eventually shift, but convergence is governed by the demographic turnover rate, giving a half-life of roughly two decades — far beyond typical policy horizons.

Core claim

The paper identifies a precise mathematical symmetry — the drift-shift symmetry of the Fokker–Planck propagator — as the reason a proportional wealth tax cannot redistribute through the market channel. Breaking this symmetry requires a state-dependent drift, which the paper models as a confining potential: a restoring force that grows linearly with distance from the mean wealth, analogous to a harmonic trap in physics. This converts the wealth process from geometric Brownian motion (which produces a Pareto-tailed steady state) into an Ornstein–Uhlenbeck process (which produces a Gaussian in log-wealth, i.e., a log-normal distribution with much thinner tails). The Gini coefficient of this log

What carries the argument

Fokker–Planck equation, drift-shift symmetry, confining potential, Ornstein–Uhlenbeck process, spectral gap, McKean–Vlasov equation, Pontryagin's maximum principle

Load-bearing premise

All investors face the same expected return and volatility (homogeneous returns). The paper itself acknowledges that when returns differ across investors — as empirical data shows they do — a proportional tax becomes redistributive through a capital reallocation effect, meaning the central non-redistribution result holds only in a limiting case the paper flags as empirically violated.

What would settle it

Test whether proportional wealth taxes leave the Gini coefficient unchanged at finite times in populations with homogeneous returns, and whether progressive taxes produce log-normal steady states with Gini matching the formula 2Φ(√(D/2κ))−1.

Figures

Figures reproduced from arXiv: 2607.06153 by Anders G Fr{\o}seth.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The stress-test concern about turnover breaking the Gini-preservation proof is correct and is the main substantive issue. The paper is a competent but incomplete framework paper.

read the letter

The stress-test note lands. Proposition 1 is proven on equation (3), the turnover-free Fokker-Planck equation, but the paper's own baseline model is equation (5), which includes demographic turnover. Under the substitution that proves Gini preservation, the turnover source term becomes δϕ(x + τ_w t) ≠ δϕ(x), so the symmetry breaks. The paper transitions between these equations without flagging that the proposition's scope excludes the full model. The qualitative point — slow redistribution through the market channel — likely survives because δ is small, but the claim of exact Gini preservation 'at all finite times' overstates what the proof delivers for the system actually being studied. This is the main issue, and it is real, not manufactured. The reader's CONDITIONAL verdict is about right, though I'd put the soundness slightly lower given this gap between what is proven and what is claimed. The homogeneous-returns assumption is a known limitation the paper acknowledges honestly, so I won't weight that as heavily. What is genuinely new here is the framing of tax design as Fokker-Planck drift modification with a clean taxonomy (uniform shift, confining potential, source-sink, boundary). The OU steady state and its Gini formula (Eq. 19) are standard results, but the connection from progressivity parameter κ to Gini via 2Φ(√(D/2κ)) − 1 is a useful closed-form bridge. The convergence-rate comparison — proportional taxes redistribute at rate δ (decades), progressive at rate κ (policy-relevant) — is the paper's best insight and is correctly derived from spectral gap arguments. The optimal control formulation (Section 6) is formally stated but never solved. The Pontryagin conditions are written down, but no numerical solutions, no comparative statics, no calibration. The general equilibrium extension to McKean-Vlasov is sketched but similarly unexecuted. The paper is honest about this — it repeatedly flags companion papers and future empirical work — but the control theory contribution is incomplete as it stands. The self-citation pattern is heavy (nine companion papers, several unpublished), but the cited results are standard Fokker-Planck material, not invented machinery, so this is a program-building author rather than someone inflating a citation count. This paper is for economists and econophysicists working on wealth distribution dynamics who want a PDE-control perspective on tax design. It deserves a serious referee who can assess whether the turnover gap is fixable with a boundedness argument or whether it genuinely undermines the central claim. My recommendation: send to review, conditional on the authors either restricting Proposition 1 to the turnover-free equation explicitly or proving that the turnover correction is bounded and small.

Referee Report

1 major / 6 minor

Summary. This paper studies wealth redistribution within a Fokker-Planck framework for the wealth distribution. The baseline model is a geometric Brownian motion for wealth with demographic turnover, yielding a Pareto-tailed steady state. The central result (Proposition 1) is that a proportional wealth tax enters as a uniform drift shift, preserving the Gini coefficient at all finite times through the market channel. The paper then introduces a progressive tax as a confining potential (Ornstein-Uhlenbeck process), deriving a closed-form Gini coefficient as a function of the progressivity parameter. A taxonomy of Fokker-Planck modifications for redistribution is presented, followed by a formal optimal control problem for drift design, including a general equilibrium extension via a McKean-Vlasov equation. An empirical strategy for estimating the drift from Norwegian wealth register data is outlined.

Significance. The paper provides a clean mathematical translation of tax policy instruments into Fokker-Planck operators. The derivation of the Pareto exponent (Eq. 7), the Gini preservation proof (Prop. 1), and the OU steady state with its closed-form Gini (Eq. 19) are standard but correct and clearly presented. The parameter-free derivation of the required progressivity kappa* for a target Gini (Eq. 20) is a falsifiable and practically useful result. The taxonomy of interventions (Section 5) and the formal optimal control formulation (Section 6) provide a coherent framework. However, the significance is tempered by the fact that the optimal control problem is only formally stated without solution or numerical illustration, and the central neutrality result is specific to a homogeneous-returns limiting case that the paper itself flags as empirically violated.

major comments (1)

§3.2, Proposition 1: The Gini-preservation proof is established on the turnover-free Fokker-Planck equation (3), but the paper's baseline model that generates the Pareto steady state is equation (5), which includes demographic turnover (-delta*pi + delta*phi(x)). Under the substitution pi_tilde(x,t) = pi(x + tau_w*t, t) used in the proof, the turnover source term becomes delta*phi(x + tau_w*t), which does not equal delta*phi(x) because the entrant distribution phi is fixed and does not shift covariantly with the tax. Therefore, the exact drift-shift symmetry—and the exact Gini preservation at all finite times—does not hold for equation (5). The paper transitions between equations (3) and (5) (Section 2.2 introduces turnover; Section 3.1-3.2 proves the proposition on the simpler equation; Section 3.3 returns to equation 5 for steady-state analysis) without explicitly flagging that the 'at

minor comments (6)

§4.3, Remark after Proposition 2: The numerical illustration states that achieving a Gini of approximately 0.80 requires kappa approximately 0.014. For a Scandinavian income Gini of approximately 0.27, kappa approximately 0.19 is required. These should be verified for consistency with Eq. (20).
§5.7: The migration boundary condition (Eq. 22) is written as J(x_m, t) = gamma*pi(x_m, t). The sign convention for the probability current J should be clarified to ensure the reader understands whether this represents outflow or inflow.
§6.2, Eq. (26): The optimal control delta_v*(x) = -(1/lambda)*pi(x,T)*p(x,T) appears to be evaluated only at the terminal time T. The standard Pontryagin maximum principle for continuous-time control typically yields a time-dependent optimal control delta_v*(x,t). Clarification is needed on whether this is an approximation, a simplification, or an error.
§6.5.2, Eq. (34): The functional derivative in the GE optimality condition is written as delta_v/delta_pi but the notation is somewhat ambiguous. The integral kernel should be more precisely defined.
Figure 1: The y-axis label 'Distance from target Gini (normalised)' could be more precisely defined. The convergence rates shown (Lambda = delta = 0.033 for proportional, Lambda = kappa = 0.05 for progressive) should be cross-referenced with the spectral gap formulas in the text.
The paper relies heavily on companion papers (Frøseth 2026a-i) for key results, particularly the drift-shift symmetry. While these are cited, the paper should ensure that the central results are self-contained enough for a reader who does not have access to all companions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

The referee raises a valid mathematical objection: Proposition 1 is proved on the turnover-free Fokker-Planck equation (3), but the paper's baseline model includes demographic turnover in equation (5). Under the drift-shift substitution, the entrant distribution φ does not transform covariantly, so exact Gini preservation at all finite times does not hold for equation (5). We acknowledge this and will revise the manuscript to clarify the scope of Proposition 1, state the approximate nature of the result under turnover, and add a remark quantifying the correction. The qualitative conclusions are unaffected.

read point-by-point responses

Referee: §3.2, Proposition 1: The Gini-preservation proof is established on the turnover-free Fokker-Planck equation (3), but the paper's baseline model that generates the Pareto steady state is equation (5), which includes demographic turnover (-δπ + δφ(x)). Under the substitution π̃(x,t) = π(x + τ_w*t, t) used in the proof, the turnover source term becomes δ*φ(x + τ_w*t), which does not equal δ*φ(x) because the entrant distribution φ is fixed and does not shift covariantly with the tax. Therefore, the exact drift-shift symmetry—and the exact Gini preservation at all finite times—does not hold for equation (5). The paper transitions between equations (3) and (5) without explicitly flagging that the 'at all finite times' claim is only valid for the simpler equation.

Authors: The referee is correct. We acknowledge the mathematical gap and will revise the manuscript accordingly. revision: yes
Referee: [Continuation of above] The paper transitions between equations (3) and (5) (Section 2.2 introduces turnover; Section 3.1-3.2 proves the proposition on the simpler equation; Section 3.3 returns to equation 5 for steady-state analysis) without explicitly flagging that the 'at all finite times' preservation claim applies only to the turnover-free equation.

Authors: This is a fair characterization of the manuscript's current structure. The transition between equations (3) and (5) is indeed not handled with sufficient care. We will make the following revisions: (1) Restate Proposition 1 with its precise scope: it holds for the turnover-free Fokker-Planck equation (3). (2) Add a remark immediately after Proposition 1 explaining what happens under turnover. The key point is that under the substitution π̃(x,t) = π(x + τ_w t, t), the turnover term transforms as -δπ̃ + δφ(x + τ_w t), whereas the taxed equation (5) has -δπ̃ + δφ(x). The mismatch term is δ[φ(x + τ_w t) - φ(x)], which is nonzero whenever φ is not constant. (3) Quantify the resulting Gini perturbation. For a 2% wealth tax (τ_w = 0.02) and a turnover rate δ ≈ 1/30 per year, the correction to the Gini over a 4-year electoral cycle is of order δ·τ_w·t·(width of φ)², which is small relative to the Gini itself but nonzero. The exact Gini preservation is therefore an approximation whose quality depends on the ratio of the tax-induced shift τ_w·t to the scale over which φ varies. (4) Clarify in Section 3.1 that the statement 'the taxed Fokker-Planck equation is identical to (5) with v replaced by v_τ' is correct as a description of the equation structure, but that the drift-shift symmetry of the propagator—which is what Proposition 1 exploits—holds exactly only for equation (3). (5) Add a sentence in the abstract replacing 'at all finite times' with 'at all finite times in the absence of demographic turnover' or equivalent qualifying language. The qualitative conclusions of the paper are unaffected: the steady-state Pareto exponent does change (Section 3.3, equation 13), the convergence is slow (equation 14), and the redistribution paradox holds in the sense that the market channel revision: no

Circularity Check

0 steps flagged

Central mathematical results (Proposition 1, Gini formulas, optimal control formulation) are derived self-containedly; self-citations provide motivation and context but are not load-bearing for the proofs.

full rationale

The paper's core derivation chain is self-contained. Proposition 1 (Gini preservation under uniform drift shift) is proven directly in Section 3.2 by defining π̃(x,t) = π(x+τ_w t, t) and verifying by substitution that it solves the Fokker–Planck equation with drift v−τ_w. This proof uses only the FP equation (3) and the scale-invariance of the Gini coefficient—no companion paper is invoked in the proof itself. The drift-shift symmetry is re-derived from first principles in equation (9): a proportional tax reduces μ to μ−τ_w, hence v to v−τ_w, with D unchanged. This is a trivial substitution, not a result imported from Frøseth (2026h). The Gini formula for the progressive tax (Eq. 19) is the standard log-normal Gini applied to the OU steady state (Eq. 18), which is a textbook result. The Pareto exponent (Eq. 7) follows Gabaix (1999). The optimal control formulation (Section 6) applies standard Pontryagin maximum principle techniques. The self-citations to Frøseth (2026f,h) establish the *neutrality* result (portfolio-choice neutrality), which is a distinct claim from Gini preservation and serves as motivation for the 'redistribution paradox' framing. The paper explicitly acknowledges that Proposition 1 relies on the homogeneous-returns assumption (Section 3.3) and that heterogeneous returns (Frøseth 2026d) change the conclusion. The skeptic's concern about Proposition 1 being proven on equation (3) rather than the turnover-augmented equation (5) is a scope/correctness issue, not circularity: the proof is not equivalent to its inputs by construction, and the paper does not hide the equation switch. No fitted parameter is renamed as a prediction. No uniqueness theorem from the authors is invoked to forbid alternatives. The self-citation network is extensive but provides context and extensions rather than load-bearing premises for the mathematical results actually proven in this paper. Score 2 reflects the presence of self-citations that frame the contribution but do not constitute circular derivation.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities or particles. The 'confining potential' and 'gravitational field' are analogies, not new postulated objects. The free parameters are standard policy/economic variables. The key axioms are domain assumptions inherited from the random growth literature, with the homogeneous-returns assumption being the most load-bearing and most explicitly flagged as limiting.

free parameters (5)

τ_w
Proportional wealth tax rate; input parameter for policy analysis.
κ
Progressivity parameter; the control variable in the optimal design problem. Numerical illustrations use κ≈0.014 to 0.19.
λ
Regularisation parameter in the optimal control objective (Eq. 24); encodes political/economic costs of intervention.
δ = ≈1/30 per year
Demographic turnover rate; calibrated to realistic values for numerical illustrations.
v, D (or μ, σ) = μ=8%, σ=30%
Baseline drift and diffusion; treated as known in theory, with empirical estimation deferred to companion paper.

axioms (5)

domain assumption Wealth follows geometric Brownian motion (Eq. 1)
Standard in finance (Merton 1969) and random growth literature (Gabaix 1999), but a simplification of real wealth dynamics.
domain assumption Homogeneous returns: all investors face common drift v and diffusion D
Stated in Section 1 and Section 3.3. The paper acknowledges this is empirically false (Fagereng et al. 2020 show 18pp gap) and that the redistribution paradox is 'specific to the homogeneous limit'.
domain assumption Demographic turnover at rate δ with entrant distribution ϕ(x)
Following Gabaix et al. (2016); required for stationarity of the GBM process.
ad hoc to paper Progressive tax enters as linear restoring force v(x) = v₀ - κ(x-x̄)
Section 4.1. This linear confinement is a simplification; the paper acknowledges 'realistic progressive tax schedules have nonlinear rate structures' (Section 8.4).
domain assumption Production function Y=F(K,L) with diminishing returns
Section 6.5. Standard macroeconomic assumption for the GE extension.

pith-pipeline@v1.1.0-glm · 19472 in / 2870 out tokens · 468055 ms · 2026-07-08T14:54:47.241865+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

[1]

Tax evasion and inequality

Annette Alstadsæter, Niels Johannesen, and Gabriel Zucman. Tax evasion and inequality. American Economic Review, 109(6):2073–2103,

work page 2073
[2]

Jess Benhabib, Alberto Bisin, and Shenghao Zhu

doi: 10.1257/aer.20172043. Jess Benhabib, Alberto Bisin, and Shenghao Zhu. The distribution of wealth and fiscal policy in economies with finitely lived agents.Econometrica, 79(1):123–157,

work page doi:10.1257/aer.20172043
[3]

24 Marie Bjørneby, Simen Markussen, and Knut Røed

arXiv:2503.23189v3. 24 Marie Bjørneby, Simen Markussen, and Knut Røed. An imperfect wealth tax and employ- ment in closely held firms.Economica, 90(358):557–583,

work page arXiv
[4]

Fischer Black and Myron Scholes

doi: 10.1111/ecca.12456. Fischer Black and Myron Scholes. The pricing of options and corporate liabilities.Journal of Political Economy, 81(3):637–654,

work page doi:10.1111/ecca.12456
[5]

David Burgherr

doi: 10.1257/pol.20200258. David Burgherr. The costs of net wealth taxation: Evidence from a tax reform in Zurich. ETH Zurich, Public Finance Discussion Paper,

work page doi:10.1257/pol.20200258
[6]

Anders G. Frøseth. Minimum-distortion wealth taxation, i: Information-theoretic versus transport-geometric optimality on the proportional class. Working paper, 2026a. Anders G. Frøseth. Extensions to the wealth tax neutrality framework. 2026b. arXiv:2603.05277 [physics.soc-ph]. Anders G. Frøseth. Flow taxes, stock taxes, and portfolio choice: A generalise...

work page internal anchor Pith review Pith/arXiv arXiv
[7]

Xavier Gabaix, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll

doi: 10.1146/annurev.economics.050708.142940. Xavier Gabaix, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll. The dynamics of inequality.Econometrica, 84(6):2071–2111,

work page doi:10.1146/annurev.economics.050708.142940 2071
[8]

Katrine Jakobsen, Kristian Jakobsen, Henrik Kleven, and Gabriel Zucman

doi: 10.1093/qje/qjac047. Katrine Jakobsen, Kristian Jakobsen, Henrik Kleven, and Gabriel Zucman. Wealth tax- ation and wealth accumulation: Theory and evidence from Denmark.The Quarterly Journal of Economics, 135(1):329–388,

work page doi:10.1093/qje/qjac047
[9]

Katrine Jakobsen, Henrik Kleven, Jonas Kolsrud, Camille Landais, and Mathilde Mu˜ noz

doi: 10.1093/qje/qjz032. Katrine Jakobsen, Henrik Kleven, Jonas Kolsrud, Camille Landais, and Mathilde Mu˜ noz. Taxing top wealth: Migration responses and their aggregate economic implications. NBER Working Paper No. 32153, February 2024, revised January 2026; forthcoming, American Economic Review,

work page doi:10.1093/qje/qjz032 2024
[10]

Eric Pichet

doi: 10.2307/1926560. Eric Pichet. The economic consequences of the French wealth tax.La Revue de Droit Fiscal, 14:1–25,

work page doi:10.2307/1926560
[11]

Progressive wealth taxation.Brookings Papers on Economic Activity, 2019(Fall):437–533,

Emmanuel Saez and Gabriel Zucman. Progressive wealth taxation.Brookings Papers on Economic Activity, 2019(Fall):437–533,

work page 2019
[12]

Yang Song, Jascha Sohl-Dickstein, Diederik P

doi: 10.1353/eca.2019.0017. Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Er- mon, and Ben Poole. Score-based generative modeling through stochastic differential equations. InInternational Conference on Learning Representations (ICLR),

work page doi:10.1353/eca.2019.0017 2019
[13]

A Gini coefficient for standard distributions For reference, we collect the Gini coefficient formulas for the distributions appearing in the main text. Pareto distributionwith tail exponentα >1: GiniPareto = 1 2α−1 .(39) Log-normal distributionwith log-varianceσ 2 x: GiniLN = 2Φ σx√ 2 −1 = erf σx 2 .(40) Exponential distribution(Boltzmann–Gibbs, following...

work page 2000

[1] [1]

Tax evasion and inequality

Annette Alstadsæter, Niels Johannesen, and Gabriel Zucman. Tax evasion and inequality. American Economic Review, 109(6):2073–2103,

work page 2073

[2] [2]

Jess Benhabib, Alberto Bisin, and Shenghao Zhu

doi: 10.1257/aer.20172043. Jess Benhabib, Alberto Bisin, and Shenghao Zhu. The distribution of wealth and fiscal policy in economies with finitely lived agents.Econometrica, 79(1):123–157,

work page doi:10.1257/aer.20172043

[3] [3]

24 Marie Bjørneby, Simen Markussen, and Knut Røed

arXiv:2503.23189v3. 24 Marie Bjørneby, Simen Markussen, and Knut Røed. An imperfect wealth tax and employ- ment in closely held firms.Economica, 90(358):557–583,

work page arXiv

[4] [4]

Fischer Black and Myron Scholes

doi: 10.1111/ecca.12456. Fischer Black and Myron Scholes. The pricing of options and corporate liabilities.Journal of Political Economy, 81(3):637–654,

work page doi:10.1111/ecca.12456

[5] [5]

David Burgherr

doi: 10.1257/pol.20200258. David Burgherr. The costs of net wealth taxation: Evidence from a tax reform in Zurich. ETH Zurich, Public Finance Discussion Paper,

work page doi:10.1257/pol.20200258

[6] [6]

Anders G. Frøseth. Minimum-distortion wealth taxation, i: Information-theoretic versus transport-geometric optimality on the proportional class. Working paper, 2026a. Anders G. Frøseth. Extensions to the wealth tax neutrality framework. 2026b. arXiv:2603.05277 [physics.soc-ph]. Anders G. Frøseth. Flow taxes, stock taxes, and portfolio choice: A generalise...

work page internal anchor Pith review Pith/arXiv arXiv

[7] [7]

Xavier Gabaix, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll

doi: 10.1146/annurev.economics.050708.142940. Xavier Gabaix, Jean-Michel Lasry, Pierre-Louis Lions, and Benjamin Moll. The dynamics of inequality.Econometrica, 84(6):2071–2111,

work page doi:10.1146/annurev.economics.050708.142940 2071

[8] [8]

Katrine Jakobsen, Kristian Jakobsen, Henrik Kleven, and Gabriel Zucman

doi: 10.1093/qje/qjac047. Katrine Jakobsen, Kristian Jakobsen, Henrik Kleven, and Gabriel Zucman. Wealth tax- ation and wealth accumulation: Theory and evidence from Denmark.The Quarterly Journal of Economics, 135(1):329–388,

work page doi:10.1093/qje/qjac047

[9] [9]

Katrine Jakobsen, Henrik Kleven, Jonas Kolsrud, Camille Landais, and Mathilde Mu˜ noz

doi: 10.1093/qje/qjz032. Katrine Jakobsen, Henrik Kleven, Jonas Kolsrud, Camille Landais, and Mathilde Mu˜ noz. Taxing top wealth: Migration responses and their aggregate economic implications. NBER Working Paper No. 32153, February 2024, revised January 2026; forthcoming, American Economic Review,

work page doi:10.1093/qje/qjz032 2024

[10] [10]

Eric Pichet

doi: 10.2307/1926560. Eric Pichet. The economic consequences of the French wealth tax.La Revue de Droit Fiscal, 14:1–25,

work page doi:10.2307/1926560

[11] [11]

Progressive wealth taxation.Brookings Papers on Economic Activity, 2019(Fall):437–533,

Emmanuel Saez and Gabriel Zucman. Progressive wealth taxation.Brookings Papers on Economic Activity, 2019(Fall):437–533,

work page 2019

[12] [12]

Yang Song, Jascha Sohl-Dickstein, Diederik P

doi: 10.1353/eca.2019.0017. Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Er- mon, and Ben Poole. Score-based generative modeling through stochastic differential equations. InInternational Conference on Learning Representations (ICLR),

work page doi:10.1353/eca.2019.0017 2019

[13] [13]

A Gini coefficient for standard distributions For reference, we collect the Gini coefficient formulas for the distributions appearing in the main text. Pareto distributionwith tail exponentα >1: GiniPareto = 1 2α−1 .(39) Log-normal distributionwith log-varianceσ 2 x: GiniLN = 2Φ σx√ 2 −1 = erf σx 2 .(40) Exponential distribution(Boltzmann–Gibbs, following...

work page 2000