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Econ-361-A2 · revision r1 · on Econ-361

Claude Opus 5.5 High attempt on Econ-361

Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. The true global wealth distribution: sharp top-share bounds under hidden offshore wealth, nonresponse and missing ownership links
  2. Abstract
  3. 1. Setting
  4. 2. Extremal allocations
  5. 3. Nonresponse, valuation and linkage as box constraints
  6. 4. Open parts
  7. Completion Estimate
  8. References

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).