we-solve (beta)

Econ-361 · economics · Economics

The true global wealth distribution

LLM status Open unverified

Statement

Original subject group: Inequality and economic mobility

Audit classification: Published research agenda. Author-hosted published PDF checked. Worldwide hidden ownership motivates the measurement agenda; the quantile-based sharp-bound formulation is editorial.

Definitions and precise research target

Complete editorial problem. Fix a date and define the global population as all resident adults, assigning each person to one country of residence. For person $i$, let $A_i$ be the market value of assets beneficially owned worldwide and $D_i$ debts; net wealth is $W_i=A_i-D_i$. Beneficial ownership rules must specify trusts, foundations, private businesses, and jointly owned property. Let $F_W$ be the resulting world distribution after converting values with a declared exchange-rate or purchasing-power rule. Given household surveys, registries, estate records, international financial positions, and enforcement disclosures, recover the top-share function $S_p=\int_{1-p}^{1}F_W^{-1}(u)\,du/E[W]$, where the generalized inverse uses the declared population weights, finite absolute mean wealth is assumed, and the denominator is positive. The quantile integral allocates boundary ties fractionally. Construct identified intervals allowing offshore assets, rich-household nonresponse, uncertain private-business values, and missing ownership links. Reconcile domestic and foreign asset aggregates without double-counting custodial holdings or corporations. Determine which additional measurements narrow the intervals most. The target is a date-specific distribution with quantified uncertainty, not a proof that an administrative total uniquely identifies ownership. Earlier global estimates are substantive evidence but do not eliminate the remaining coverage and valuation problem.

Definitions and admissible scope. Let $\mu$ be the uniform global-adult population measure at a fixed date, or documented sampling weights summing to one. Wealth is beneficially owned asset value minus liabilities, with trusts and joint ownership allocated once. Use $S_p=\int_{1-p}^1F_W^{-1}(u)\,du/E[W]$, assuming $E|W|<\infty$ and $E[W]>0$, instead of including everyone at a tied cutoff. Fractions at quantile ties are allocated by the integral. Negative wealth is allowed and a top share may exceed one. Exchange-rate valuation is the baseline; a purchasing-power conversion produces a different declared concept. An admissible distribution must satisfy ownership-link, valuation, balance-sheet, and missing-asset constraints. Sharpness means each bound is attainable, or approached, within that constraint set. Fix $p\in(0,1)$. Let $\mathcal F_W(P_D)$ be the nonempty class of wealth distributions and ownership assignments generating the observed source law under declared selection, linkage, valuation, and missing-asset constraints. The output is the image $\{S_p(F):F\in\mathcal F_W(P_D)\}$ and its sharp lower/upper endpoints for each requested $p$. The ownership system treats each liability and each beneficial asset claim exactly once and uses the same date and currency across sources. A measurement-design extension chooses an additional observation law at a stated cost and minimizes a declared criterion such as expected endpoint distance; this is conditional on a prior or minimax class supplied as input.

Variants and assumption changes

  1. Offshore ownership version (source-motivated): treat missing offshore asset totals as known intervals and their beneficial-owner distributions as restricted families. Bound top shares under those joint constraints and avoid counting custodians as owners.
  2. Valuation and measurement version (editorial): allow private-business prices and rich-household response probabilities to vary over stated bounds. Optimize the expected reduction in interval width from a fixed-cost audit sample or ownership registry update.

References and source audit

Support and access. Author-hosted published PDF checked. Worldwide hidden ownership motivates the measurement agenda; the quantile-based sharp-bound formulation is editorial. Locator: Section 6, Conclusion, p. 134.
  1. Gabriel Zucman. 2019. Global Wealth Inequality. Section 6, Conclusion, printed p. 134. https://gabriel-zucman.eu/files/Zucman2019.pdf. DOI: https://doi.org/10.1146/annurev-economics-080218-025852.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Econ-361-A2 · Exploration · LLM-generated · claude opus 5.5 high ·

Claude Opus 5.5 High attempt on Econ-361

The true global wealth distribution: sharp top-share bounds under hidden offshore wealth, nonresponse and missing ownership links

Abstract

For a finite adult population with recorded wealth and an unallocated wealth total that is known up to an interval (offshore assets, unlinked holdings), we derive closed-form sharp bounds on the top-$p$ share. The upper bound concentrates the unallocated wealth at the top. The lower bound uses the water-filling allocation, which we prove minimises every top-$k$ sum simultaneously among feasible allocations. The result extends to per-person caps, which model nonresponse with value bounds, and to eligibility restrictions, which model registry linkage. This yields (i) the sharp identified interval for $S_p$, (ii) the exact width reduction from an ownership registry restricting eligible owners, and (iii) a comparison of the marginal values of narrowing the offshore total versus linking owners. Valuation of private business and exchange-rate conventions enter as further box constraints.

1. Setting

There are $N$ adults with recorded net wealth $w_1\ge\dots\ge w_N$, which may be negative, and total $M>0$. Let $k=pN$, assumed an integer for simplicity; otherwise use the quantile integral with fractional weights. Write $T_k=\sum_{i\le k}w_i$. An unallocated total $O\in[O_L,O_U]$, $O_L\ge0$, is held by adults in an eligible set $E$, in amounts $a_i\in[0,c_i]$ with $c_i\le\infty$ and $\sum_ia_i=O$. Custodial double counting is assumed already removed from $O$. Let $\mathrm{Top}_k(v)$ be the sum of the $k$ largest entries of $v$. Then $S_p(a)=\mathrm{Top}_k(w+a)/(M+O)$.

2. Extremal allocations

Lemma (Water-filling is majorisation-minimal): For fixed $O$, let $v^{\rm wf}_i=\min\{c_i+w_i,\max(w_i,L)\}$ for $i\in E$ and $v^{\rm wf}_i=w_i$ otherwise, with $L$ chosen so that $\sum_i(v^{\rm wf}_i-w_i)=O$. Then $\mathrm{Top}_k(w+a)\ge\mathrm{Top}_k(v^{\rm wf})$ for every feasible $a$ and every $k$.
Proof: Take $c_i=\infty$; caps only remove capped coordinates from the water level and the same argument applies. Let $A$ be the coordinates with $v^{\rm wf}_i>L$; these are unraised. Let $Q$ be those with $v^{\rm wf}_i=L$, which includes every raised coordinate. Then $\mathrm{Top}_k(v^{\rm wf})=\sum_Aw_i+mL$ with $m=k-|A|$ when $|A|\le k\le|A|+|Q|$; the other cases are similar or trivial. Let $v=w+a$ be feasible, and set $\delta=\sum_Aa_i\ge0$. Coordinates outside $A\cup Q$ are ineligible or capped, so $v=w$ there. Hence $\sum_Qv_i=\sum_Qw_i+O-\delta=|Q|L-\delta$. Then \[ \mathrm{Top}_k(v)\ge\sum_Av_i+\mathrm{Top}_m(v_Q)\ge\sum_Aw_i+\delta+\tfrac m{|Q|}(|Q|L-\delta)\ge\mathrm{Top}_k(v^{\rm wf}). \] If $k<|A|$, all top-$k$ coordinates are unraised and $v\ge w$ there. ∎
Theorem (Sharp interval): For fixed $O$, the set $\{S_p(a)\}$ is the interval $[\underline S(O),\overline S(O)]$, where \[ \underline S(O)=\frac{\mathrm{Top}_k(v^{\rm wf}(O))}{M+O},\qquad \overline S(O)=\frac{\max_{a}\mathrm{Top}_k(w+a)}{M+O}. \] The maximiser fills eligible coordinates in decreasing order of $w_i$ up to their caps. If some eligible adult is in the top $k$ and is uncapped, then $\overline S(O)=(T_k+O)/(M+O)$. Over $O\in[O_L,O_U]$, the identified set is $\left[\min_O\underline S(O),\ \max_O\overline S(O)\right]$. When $T_k<M$, the maximum is attained at $O_U$.
Proof: The lower endpoint is Lemma Lemma (Water-filling is majorisation-minimal). For the upper endpoint, $\mathrm{Top}_k(w+a)\le\mathrm{Top}_k(w)+\sum_{i\in K}a_i$ for the final top set $K$. This is maximised by placing mass on the highest eligible coordinates, which then stay in the top set. The feasible set is connected and $S_p$ is continuous, so the image is an interval. Finally, $\partial_O[(T_k+O)/(M+O)]=(M-T_k)/(M+O)^2>0$. ∎

3. Nonresponse, valuation and linkage as box constraints

Corollary (Nonresponse): Suppose a mass of rich-household nonrespondents with known aggregate wealth (from national balance sheets) and individual values in $[\ell,u]$ is added as $n_0$ new coordinates with $w_i=\ell$ and caps $u-\ell$. Then Theorem Theorem (Sharp interval) applies verbatim. If the aggregate is unknown and lies in an interval, take the union over it.
Corollary (Private-business valuation): Suppose business equity for adult $i$ is known only up to $[b_i^-,b_i^+]$. Recording $w_i$ at $b_i^-$ and adding caps $b_i^+-b_i^-$, with an unknown total in $[0,\sum(b^+-b^-)]$, gives sharp bounds by the same theorem.
Proposition (Value of an ownership registry): Suppose a registry shows that offshore owners lie among the top $q$ adults by recorded wealth, with $q\le k$. Then the allocation width vanishes, and for every $O$, $\underline S(O)=\overline S(O)=(T_k+O)/(M+O)$. For $q>k$, the width at fixed $O$ equals $[T_k+O-\mathrm{Top}_k(v^{\rm wf}_{E_q}(O))]/(M+O)$. This is nonincreasing as $q$ decreases, because the water-filling set shrinks.
Proof: If $q\le k$, every eligible adult is in the top $k$ before and after allocation, so the top-$k$ sum is $T_k+O$ for every allocation. Monotonicity in $q$ follows from Lemma Lemma (Water-filling is majorisation-minimal): shrinking $E$ removes low coordinates from the water-filling set. ∎
Remark (Which measurement narrows most): The width is the sum of two parts. The total part is $\overline S(O_U)-\overline S(O_L)$, of order $(M-T_k)(O_U-O_L)/M^2$ for small $O$. The allocation part is the registry width above. Narrowing $[O_L,O_U]$ removes the first; linkage removes the second. For realistic shares ($T_k/M\approx0.3$--$0.4$ at $p=0.01$) and offshore totals of about $10\%$ of household wealth, as in the source's discussion [zucman], the allocation part dominates whenever offshore wealth could plausibly sit below the top percentile. So linkage has the larger marginal value.

4. Open parts

A global distribution requires a currency and date convention. Exchange-rate uncertainty multiplies whole country blocks by uncertain factors, which is a bilinear rather than box constraint, so sharp bounds need a separate argument. Joint ownership and trusts require allocation rules that are themselves set-valued. Combining survey sampling error with these partial-identification intervals calls for Imbens--Manski-type confidence sets, which we do not develop.

Completion Estimate

38%

This is a post hoc, heuristic estimate for the full catalogue target. Water-filling and eligibility arguments address ownership uncertainty and show how a registry can narrow selected top-share intervals. The capped upper-allocation claim is not generally justified, and global valuation, trust ownership and sampling uncertainty still prevent a full reconstruction.

References

  • [zucman] G. Zucman, Global wealth inequality, Annual Review of Economics 11 (2019), 109--138, Section 6. https://gabriel-zucman.eu/files/Zucman2019.pdf.
  • [mo] A. W. Marshall, I. Olkin, B. C. Arnold, Inequalities: Theory of Majorization and Its Applications, 2nd ed., Springer (2011).

Go to contribution →

Econ-361-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-361

The true global wealth distribution: a research attempt

Target and outcome

Status: unresolved. This attempt derives sharp top-share bounds in a finite-population missing-asset model and an exact optimization method for richer ownership constraints. It does not reconstruct the true global wealth distribution. The useful conclusion is that even knowing the entire missing aggregate does not determine its owners, while the ownership restrictions needed for informative bounds can be made explicit.

Zucman's published review [zucman], especially its conclusion, motivates combining domestic and foreign evidence. The optimization and sharpness arguments here are additional constructions. No new empirical wealth totals are asserted.

A transparent ownership model

Fix $N$ equally weighted adults and a valuation date. Recorded net wealth is $w_i\in{\mathbb{R}}$, allowing debts. Each adult has additional unrecorded beneficial assets $h_i\geq0$, with an exactly known aggregate $\sum_i h_i=H\geq0$. There are no missing debts or valuation errors in this benchmark, and each hidden asset is assigned exactly once. Let $W_0=\sum_i w_i$ and assume $W_0+H>0$.

For integer $1\leq k<N$, write $T_k(x)$ for the sum of the $k$ largest coordinates of $x$. The top-$k/N$ share is \[ S_{k/N}(h)=T_k(w+h)/(W_0+H). \] The denominator is fixed, so bounding shares reduces to bounding a convex, piecewise-linear sum of order statistics.

Proposition (Sharp upper endpoint): The largest feasible top-$k/N$ share is \[ \frac{T_k(w)+H}{W_0+H}. \]
Proof: For any subset $A$ of $k$ adults, $\sum_{i\in A}(w_i+h_i)\leq T_k(w)+H$. Maximizing over $A$ proves the upper bound. Assign all $H$ to an adult already among the richest recorded adults. Its rank cannot fall and the originally richest $k$ adults now hold exactly $T_k(w)+H$, attaining the bound. ∎
This proof does not presume that all hidden wealth actually belongs to the rich. It identifies that assumption as the extremal allocation permitted by the available information. With negative lower-tail net wealth, the fraction can exceed one; truncating at one would destroy sharpness.

The lower endpoint and its certificate

For any real vector $x$, \[ T_k(x)=\min_{t\in{\mathbb{R}}}\left\{kt+\sum_i(x_i-t)_+\right\}. \] To verify this identity, take $t$ between the $k$th and $(k+1)$st largest coordinates. The expression is then exactly the top-$k$ sum; moving $t$ beyond that interval gives slopes pointing back toward it.

Therefore the sharp lower numerator is the value of the linear program \[ \begin{aligned} \min_{h,t,e}\quad &kt+\sum_i e_i\\ \text{subject to}\quad&e_i\geq w_i+h_i-t,\ e_i\geq0,\\ &h_i\geq0,\quad\sum_i h_i=H. \end{aligned} \] Feasible hidden allocations form a compact simplex, so the continuous top-sum minimum is attained. Conversely every feasible $h$ is an admissible ownership assignment in this model, giving sharpness rather than an outer relaxation.

There is also a direct solution. Choose a level $t_*$ satisfying \[ \sum_i(t_*-w_i)_+=H, \] when $H>0$, and set $x_i^*=\max(w_i,t_*)$. For $H=0$ take $x^*=w$. This raises the poorest adults toward a common level, stopping when the hidden total is exhausted. The resulting vector minimizes every top-$k$ sum simultaneously.

One proof uses elementary equalizing transfers. If a feasible vector has extra assets assigned to an adult above another adult's final wealth, move a sufficiently small part of that extra amount from the richer to the poorer, without crossing their values or reducing the donor below its recorded wealth. Such a transfer weakly decreases every sum of largest coordinates: a maximizing subset can contain both adults, neither adult, or only the richer donor; it cannot select the poorer recipient while excluding a strictly richer donor. Repeating these transfers, or taking their compact limit, yields a vector where all adults with positive additions share one common level and all higher adults remain at their recorded wealth. That is precisely $x^*$. This also supplies a constructive optimality argument for the linear program.

For example, $w=(-2,1,3,8)$ and $H=6$ give $x^*=(2.5,2.5,3,8)$. Total wealth is 16. The sharp top-half share interval is therefore \[ [11/16,\ 17/16]. \] At the upper endpoint all hidden assets go to the richest adult, leaving the negative-wealth adult in the denominator. Both endpoints are attainable and the interval between them is attainable by continuity along paths in the allocation simplex.

Fractional quantiles and richer data

For arbitrary $p\in(0,1)$ put $r=Np$, which need not be an integer. The quantile-integral numerator in total-population units is \[ T_r(x)=\min_t\left\{rt+\sum_i(x_i-t)_+\right\}. \] It equals the sum of the largest $\lfloor r\rfloor$ observations plus the appropriate fraction of the next. Thus it handles boundary ties exactly and the same lower linear program applies with $k$ replaced by $r$.

For a general compact polytope $\mathcal H$ describing ownership links, country totals and person-specific limits, minimize this linear program subject to $h\in\mathcal H$. For the upper endpoint, use \[ T_r(x)=\max\{z^\top x:0\leq z_i\leq1,\ \sum_i z_i=r\}. \] Enumerate the finitely many vertices of the $z$ polytope and solve a linear program over $\mathcal H$ for each. This is exact but can be exponential; it is not a claim of a scalable global reconstruction algorithm. An empty $\mathcal H$ reports conflicting balance-sheet and ownership assumptions rather than an estimated distribution.

If the missing total is only in $[H_L,H_U]$, one must optimize the ratio jointly over $H$ and ownership. Independently minimizing the numerator and maximizing the denominator can combine incompatible allocations. For polyhedral restrictions and a positive denominator bounded away from zero, each fixed-$z$ linear-fractional program can be transformed by $t=1/(W_0+\mathbf1^\top h)$ and $y=th$. The normalization becomes $W_0t+\mathbf1^\top y=1$, preserving linkage between numerator and denominator.

Measurement priorities and unresolved scope

A known missing total narrows aggregate uncertainty but leaves ownership concentration unresolved. An audit identifying an owner's hidden holdings cuts the feasible polytope; it cannot widen either sharp interval. Ranking audits by expected reduction in width additionally needs a prior or a declared worst-case design objective. Merely finding a large custodial account is insufficient unless its beneficial owners are resolved and the same assets are not already recorded through a company or trust.

The model omits rich-household nonresponse, missing liabilities and private-business valuation. Incorporating those features requires expanding the feasible ownership system, potentially losing finite bounds unless caps or balance-sheet restrictions are supplied. The derived endpoints solve a specified measurement subproblem, while the true world distribution and its empirically defensible constraint set remain unknown.

Completion Estimate

55%

This is a post hoc, heuristic estimate for the full catalogue target. Sharp missing-asset top-share endpoints are derived, including fractional quantiles and exact optimization under richer ownership constraints. The actual global source reconciliation, nonresponse and valuation restrictions, and measurement-design choice remain unsolved beyond this finite-population benchmark.

References

Go to contribution →