Simplified Distributional National Accounts

Why it matters, in detail. The simplified share is AD*T + AT*(W+Y) + AR*Z: it assumes the fiscal top 1% receives AT of untaxed labor/pension and pension-fund income and AR of other untaxed capital, with no re-ranking. In PSZ's micro-files the fiscal top 1% actually receives 8.4%->12.4% (1962->2016) of the former (AT: 5.9%->15.4%) and a flat ~32% of the latter (AR: 38%->48%). The resulting overstatement (+0.7 pp in 1962, +2.7 pp in 2016) is offset by the omitted re-ranking (+0.6, +2.7 pp), which contributes about 2.1 of PSZ's 8.2 pp 1962-2016 tax-unit rise. Realistic shares for Social Security and employer health insurance alone would cut the 2016 simplified share by 1.2 pp and its 1960-2016 rise by about 1.1 pp. Ranking on gains-inclusive fiscal income captures only 0.35-1.03 pp of re-ranking, so the offset is partly structural but not by design.

Evidence. Adjudicator re-implementation (scripts/adj_micro_test.py; outputs/adj_micro_test.json) reproduces Claude's results to within about 0.01 pp. Fiscal top 1% (ranked by PS-matching fiscal income, PS tax-unit count) share of other untaxed capital: 32.0% (1962), 25.6% (1979), 24.9% (1986), 26.3% (1988), 31.1% (2000), 40.2% (2007), 32.4% (2016), vs the workbook key AR 38.0, 32.1, 47.3, 37.7, 49.8, 54.4, 47.7%. Share of untaxed labor + pension-fund income: 8.4% (1962) -> 12.4% (2016) vs the key AT 5.9% -> 15.4%; employer health insurance 2.1-2.5%, Social Security old-age benefits 0.7-1.1%. Overstatement implied by the workbook keys: +0.74 (1962), +0.41 (1979), +1.83 (1986), +0.85 (1988), +1.65 (2000), +1.71 (2007), +2.70 pp (2016). Re-ranking (PSZ-ranked minus fiscal-ranked top-1% share of factor-price pretax income): +0.56, +1.35, +1.66, +1.27, +1.86, +1.96, +2.68 pp (adult weights: +0.56 ... +2.71 pp); only 63-79% of the fiscal top 1% are in PSZ's top 1%. Codex's sensitivity (reproduced by adjudicator): giving employer health insurance a 2% and Social Security a 4% top-1% share lowers the 2016 simplified share from 21.13% to 19.93% (1960: -0.08 pp). Adjudicator extension (scripts/adj_rerank_mechanism.py): ranking on gains-inclusive fiscal income captures only 0.35-1.03 pp of the re-ranking effect; units entering PSZ's top 1% derive just 4-32% of their pretax income from fiscal income, realize gains worth 14-50% of it, and 8-38% report business losses. The workbook ReadMe (A8) itself notes assumption (i) 'might be a bit excessive as about 1/3 = SS benefits, equally distributed'. Annual gaps of 0.9-1.65 pp occur not only in 1976-85 (the period the paper discusses) but also in 1987, 1989-91 and 2001-02.

Why this severity. It leaves Fig. 3's outputs intact, so C1 survives as a statement about outputs. But it contradicts C2 (the claim to reveal what PSZ 'amounts to'), means the 'PSZ-equivalent' parameter paths used as the yardstick in Section II are not PSZ's (the other-capital key is about 15 pp too high in 2016), and shows the fit depends on an omitted re-ranking term that grows by about 2.1 pp over 1962-2016, roughly a quarter of PSZ's tax-unit trend.

The authors' response. The paper concedes that the assumptions are 'clearly too coarse' and that 'the sophisticated PSZ approach is required to deliver more accurate results' (pp.292-293). No later response to the micro-level point was located; the authors' later work (PSZ 2024 comment, fn 12) switched to a component-by-component comparison.

Why it matters, in detail. Fig. 4 is a residual: with AD and T held at PS tax-unit values, any shortfall of AS relative to PSZ must be absorbed by the untaxed keys, and the split between the labor/pension and other-capital keys is a free choice (alternative splits require negative shares). AS's fiscal/unit/sample corrections alone (-2.3 pp in 2015) roughly double the required 2015 other-capital key (11.1% -> 21.8%), and AS's own 2014 decomposition places 57% of their gap with PSZ in underreporting and retirement income (categories (i)/(ii) here) against the 69% Fig. 4 puts in other capital. Rising wealth concentration does not identify the top-1% share of this residual component: in PSZ's own micro-files it is flat at ~32% for the fiscal top 1% while wealth concentration rises 13 pp. The defensible conclusion is only that AS's untaxed allocations are much less top-heavy than PSZ's, as the authors' own later component-by-component work (PSZ 2024) shows, locating the gap mainly in legally exempt pass-through income and evasion.

Evidence. Identification (adjudicator, scripts/adj_recompute.py; reproduces Claude and Codex): with the other-capital key at its PSZ-style value (AR), the labor/pension key needed to hit AS is -9.6% (1960) and -17.4% (2015); with the labor key at AT, the other-capital key needed is negative from 1990; with the frozen key it is 28.1% (1960) and 11.1% (2015). Fiscal base: AS (2018, Table 1) corrected fiscal top-1% share 7.9% (1960) -> 16.0% (2015) vs PS 8.3% -> 18.6%, i.e. 2.2 pp less rise before any untaxed income is added (grouping by individuals with size-adjusted ranking alone: -0.8/-1.9 pp); AS's AEA workbook shows -2.30 pp in 2015 from grouping, ranking and sample corrections; lowering the fiscal input by 2.3 pp raises the required 2015 other-capital key from 11.1% to 21.8% (Codex). Locus: Fig. 4 assigns 4.91 of the 7.14 pp 2016 gap (69%) to other untaxed capital (labor 1.21, pension capital 1.02 pp); AS (2018, Table 4) put 2.2 of 5.8 pp (38%) of their 2014 gap with PSZ in underreported income, 1.1 pp (19%) in retirement income, about 1.4 pp (24%) in fiscal-definition, sample, unit and sales-tax items, and only about 1.0-1.7 pp (17-29%) in items mapping onto other untaxed capital (corporate income 0.6, imputed rent 0.3, Fed 0.1, non-profits 0.1, inflation -0.1, plus ~30% of underreporting). AS (2019, p.310 and appendix Sec. IV.A) report that their top 1% gets ~40% of non-retirement untaxed capital excluding imputed rent and ~9% of imputed rent (AS-reported, not verifiable; AS's bridge equation in which these appear sums to $2.329Tr, not the stated $2.23Tr); weighting these gives roughly 29% for that bundle, not 10%. Wealth: in PSZ's micro-files the wealth-ranked top-1% wealth share rises from 27.1% (1962) to 40.0% (2016) while the fiscal top 1%'s share of other untaxed capital stays near 32%, so rising wealth concentration does not imply the paper's rising 40%->48% key. Adjudicator addition: the 'equalization must have occurred' logic (p.293) also ignores that income moving across the tax base (e.g. C-to-S conversions after TRA86) changes the composition of the untaxed pool. What survives: holding PS fiscal shares fixed, AS imply that the top 1%'s average share of all untaxed income falls from 18.6% (1960) to 7.7% (2015), versus 24.6% -> 27.7% for PSZ.

Why this severity. C3's diagnosis -- that AS effectively assume capital income and wealth became extremely equal, refuted by wealth data -- is not established, because the exercise cannot identify where AS differ and loads AS's fiscal-base and unit corrections onto untaxed capital. The qualitative kernel (AS imply a far less top-concentrated untaxed component than PSZ) survives, so the problem is major rather than critical.

The authors' response. Saez & Zucman (2020, NBER WP 27921, Sec. 3) restate the argument ('you need to assume that income not reported on tax returns has in reverse become much less concentrated to replicate the AS results') and say the PSZ (2019) computations show AS's implicit wealth distribution is 'too equal'. PSZ (2024 comment, fn 12) describe their later component-by-component comparison as fleshing out this paper's argument; that analysis locates the gap mainly in legally exempt pass-through income (62% vs 16% to the top 1%; 1.6 pp in 2019) and evasion, not in 'other untaxed capital'. PSZ (2026) restate the general point.

Why it matters, in detail. The paper's 40%->48% key is not the top-1% share of taxable capital income (32%->38% without gains) but a gains-inclusive key whose gains term mixes the Table A3 and Table A1 rankings, overstating the no-gains group's gains by a selection effect. Applied to other untaxed capital (13% of factor-price NI in 2016), the consistent fixed-rank (A2) key lowers the simplified share by 1.42 pp (1960) and 1.14 pp (2016), leaving it 1.2-1.4 pp below PSZ at both endpoints and raising the mean absolute gap from 0.68 to 1.45 pp. Trends are essentially unchanged (8.0 vs 7.7 pp), so the issue concerns the claimed closeness in levels, not the rise.

Evidence. Adjudicator re-implementation (scripts/adj_recompute.py), matching both auditors: key AR 39.6% (1960) -> 47.7% (2016); consistent A2-based key 31.3% -> 38.9%; no-gains key AP 32.3% -> 38.3%. Simplified share: published 13.40% (1960) / 21.13% (2016) vs PSZ 13.22/21.40%; with the A2 key 11.98/19.98% (-1.24/-1.42 pp vs PSZ); with the no-gains key 12.14/19.90% (-1.08/-1.50 pp). Mean absolute gap vs PSZ: 0.68 pp (published), 1.45 pp (A2), 1.74 pp (no gains); maximum 1.65, 2.60, 3.33 pp. Rise 1960-2016: 7.72 (published), 8.00 (A2), 7.76 pp (no gains) vs PSZ 8.18 pp. The gains term also imports volatility: AR 37.9% (1985), 47.3% (1986), 32.0% (1987).

Why this severity. It shifts the simplified level by 1.1-1.5 pp (5-11%) and doubles the mean absolute fit error, so the 'almost perfect match in 1960 and 2016' depends on an undescribed construction. The trend, and hence the reproduction of PSZ's rise, is unaffected.

The authors' response. Disclosed only in the workbook ReadMe (A10). A defensible rationale exists -- PSZ allocate retained earnings by equity wealth capitalized from dividends and gains -- but it is not given in the paper; no response on the mixed ranking was located.

Why it matters, in detail. J&S's Table 5 result (true and reported top-1% AGI shares both 17.8%) is the benchmark that an AGI-proportional or audit-based allocation reproduces (AS 2001: +0.015 pp), whereas PSZ's source-proportional allocation puts 42-47% of underreporting in the top 1% and raises the top share by ~1.2 pp (PSZ's own 2013 estimate). J&S Table 3 (27% by true income) and Table 4 (18-19% misreporting for partnership/S-corp income, vs 57% for Schedule C) do not support allocating total underreported business income in proportion to positive business income, which is mostly partnership/S-corp income at the top. PSZ's real argument -- that random audits miss sophisticated top-end evasion (supported by Guyton et al. 2021; Gale, Sabelhaus & Thorpe 2023; Iselin & Reck 2025 put PSZ at the high end of plausible values) -- is a reason to depart from J&S, not evidence from J&S.

Evidence. J&S (2010) Table 5: top-1% share of estimated true AGI 3.8+14.0 = 17.8% vs reported 3.7+14.1 = 17.8%; Table 4: Schedule C 55% (top 0.5%) vs 57% (all), but partnership/S-corp/estate 19% vs 18%; Table 3: the true-income top 1% account for 7+20 = 27% of AGI underreporting (5% by reported AGI). PSZ (2017) Data Appendix (9 Nov 2017), p.35 and fn 30: PSZ allocate underreported business income by reported business income (47% to the top 1%) and underreported wages by wages (11%), $262bn of $618bn in 2013, which they estimate raises the top-1% share by 1.17 pp relative to an AGI-proportional allocation; PSZ there describe the AS allocation as the one 'consistent with the distribution of tax evasion estimated by IRS random audit studies (see Johns and Slemrod, 2010)' and argue that audits 'severely under-estimate tax evasion at the top-end'. AS AEA workbook (sheet 'TRA86 & online app', rows 24-26): AS's 2001 filer underreporting allocation changes their top-1% share by +0.015 pp (-0.153 pp more with non-filers).

Why this severity. Footnote 3 is the paper's only cited evidence for its benchmark on underreported income, the single largest AS-PSZ item (2.2 of 5.8 pp in 2014 per AS), and the cited source matches the critic's allocation in the year studied. This weakens the 'unrealistic' verdict on assumption 1 but does not settle the underlying dispute, which remains open.

The authors' response. PSZ (2017 Data Appendix p.35) and Saez & Zucman (2020, Sec. 3.1) argue that random audits miss sophisticated top-end evasion and that, ranked by true income, 27% of underreporting is in the top 1%. These are reasons to depart from J&S, not evidence that J&S support the PSZ allocation.

Why it matters, in detail. Multiplying a full-NI share (14.18%) by factor-price NI understates AS's top-1% dollar income by up to $108bn (5%). On AS's full-NI basis AS = $2.231Tr vs PS fiscal incl. gains $2.216Tr; on a factor-price basis that removes AS's own, less-than-proportional sales-tax allocation, AS is about $2.17-2.22Tr. The ordering is therefore indeterminate rather than reversed, and the comparison mixes non-nested concepts (realized gains vs national income) and units.

Evidence. AS AEA workbook, sheet 'AS': national income 2015 = $15,739.6bn; the same total is cached in the PSZ workbook's own external link (DINA(Aggreg) TA0!B111 = 15,739.598; sales taxes TSA12!F111 = 762.086; factor-price B61 = 14,977.5). AS top 1% = 14.176% x $15,739.6bn = $2,231bn (AS 2019, p.310: $2.23Tr), vs $2,123bn as computed in the paper and PS $2,216bn (21.592% x $10,265bn). Adjudicator refinement: on a like-for-like factor-price basis, AS's top 1% would lose only its own share of the $762bn of sales taxes (which AS allocate by disposable income less saving); at a 2-8% share that gives about $2.17-2.22Tr. The base enlargement to AS's denominator is 53%, not 46%. The comparison also sets realized gains (not part of national income) and tax units against national income and all individuals.

Why this severity. A rhetorical aside that enters no computation. Corrected, the two totals are within about 2% of each other, so the paper's 'more income' ordering is unsupported, while its broader message (AS's top 1% gets roughly as many dollars as the PS fiscal top 1% despite a ~50% larger base) survives.

Why it matters, in detail. The tax-unit/equal-split gap (0.65 -> 1.61 pp) and the AS unit (all individuals, size-adjusted ranking) are first-order relative to the fit errors being discussed (0.68 pp). Describing AS as an 'adult unit', 'pretax national income' series obscures that Fig. 1 and Fig. 4 compare different units and slightly different income concepts.

Evidence. Workbook AJ-AL (PSZ tax units minus equal-split adults): 0.65 pp (1960) rising to 1.61 pp (2016); rise 1960-2016 8.18 vs 7.22 pp (13% larger for tax units) -- larger than the simplified-vs-PSZ mean absolute gap of 0.68 pp. AS (2018) Sec. III.A.5: groups contain equal numbers of individuals including dependents, ranked by income divided by the square root of unit size. AS (2018) p.17: AS pretax income excludes social-insurance benefits and keeps social-insurance taxes, unlike PSZ (effect below 0.05 pp in 2014, Table 4). The AS replication nevertheless starts from PS tax-unit fiscal shares (BA=AD).

Why this severity. Labelling and characterization problems with effects of about 1 pp; they matter mainly as an input to F2 (the AS replication starts from tax-unit fiscal shares) and do not change any reported number.

The authors' response. The paper justifies tax units by consistency with the PS fiscal series (p.290); no response on the label was located.

Why it matters, in detail. Counting 70% of the top 1%'s S-corporation profits as labor in the numerator but none of aggregate S-corporation profits in the denominator raises AT by ~0.5 pp in 2016 relative to a consistent 70/30 split (and by 2.4 pp relative to PSZ-2018's S-corp-as-capital convention). Level effect on the simplified series is about +0.1 pp (0.5 pp x 23% of NI), but part of the '6% to 15%' rise in 'labor' concentration reflects S-corporation profits that PSZ (2018) classify as capital.

Evidence. Workbook: G = 0.7 x (TC2!E - TC2!H + TC1!E) excludes S-corporation profits (TC2!H), which enter I at 100%; AS = AD x (0.7 x Entrep. + Wage) and AO = AD x (0.3 x Entrep. + Div. + Interest + Rents) apply 70/30 to PS 'Entrep.', which PS Table A7's notes define to include S-corporation profits. PSZ micro-files (scripts/adj_scorp_esi_check.py, matching Claude): AT in 2016 = 15.3% (workbook convention) vs 14.8% (consistent 70/30) vs 12.9% (S-corp fully capital); 2000: 14.5/14.3/13.0%; 1988: 11.0/10.9/10.0%. The top 1% receive 77% (2016) to 86% (2000) of S-corporation profits.

Why this severity. The effect on the simplified level is about 0.1 pp; it matters only for the rhetorical '6% -> 15%' rise in labor concentration used against AS's frozen 6%.

The authors' response. None located; PSZ Updates (Sept 2020, Sec. 2.1.5) later applied heterogeneous capital and labor shares to S-corporation profits in the full DINA.

Why it matters, in detail. Whether PSZ's untaxed income became more or less concentrated depends on the ranking: slightly less for the fiscal-ranked top 1% (21.5% -> 19.5%), more for the PSZ-ranked top 1% and under the paper's no-re-ranking accounting (24.6% -> 27.7%). The inconsistency is another symptom of the non-negligible re-ranking documented in F1.

Evidence. Adjudicator (scripts/adj_recompute.py): (AJ - AD x T)/(W+Y+Z) = 24.6% (1960) -> 27.7% (2016); simplified rule 25.2% -> 27.0%. PSZ micro-files, fiscal-ranked top 1%: 21.5% (1962) -> 19.5% (2016) (scripts/adj_micro_test.py). Claude also reports 24.3% -> 31.4% for PSZ's pretax-ranked top 1%.

Why this severity. The statement is measure-dependent and the paper does not say which measure it uses; correcting it would strengthen rather than weaken the paper's contrast with AS.

Why it matters, in detail. The benchmark's 2016 level fell by 0.76-0.86 pp and its 1960-2016 rise by ~1.0 pp across vintages, so the simplified series now overshoots the PSZ rise by ~0.5-0.6 pp instead of undershooting by 0.46 pp; part of the revision (-0.6 pp in 2016) implements AS's rollover point.

Evidence. PSZ appendix tables II, TB10 (scripts/adj_vintage_check.py): tax units 1960/2016 = 13.22/21.40% (workbook; identical to the Nov-2017 tables for overlapping years) -> 13.45/20.54% (Sep-2020) -> 13.47/20.64% (Feb-2022); equal-split adults 2016: 19.79 -> 18.61 -> 18.67%. PSZ Updates Sec. 2.2.1: the pension revision 'reduces the top 1% pre-tax income share by about 0.6 point in 2016'. Saez & Zucman (2020): 'We are grateful to AS for bringing this issue to our attention.' Against the current vintage the simplified series overshoots 2016 by about 0.5-0.6 pp, and the PSZ 1960-2016 rise falls from 8.18 to about 7.1-7.2 pp (simplified: 7.72 pp).

Why this severity. The revision was disclosed by PSZ and the fit remains within about 0.6 pp, so this is a durability caveat rather than an error; it modestly qualifies the paper's dismissal of AS's lower pension-income concentration.

The authors' response. Accepted in part: Saez & Zucman (2020) and PSZ Updates (Sept 2020, Sec. 2.2.1) reduced the weight on non-taxable pension distributions.

Why it matters, in detail. 0.2 pp level difference between the cited Bulletin (38.6%) and the paper (38.8%) for 2016; the 1989-2016 rise is unaffected.

Evidence. Bricker et al. (2017), Federal Reserve Bulletin 103(3), Box 3: 'The wealth share of the top 1 percent climbed from 36.3 percent in 2013 to 38.6 percent in 2016'. Zucman (2019) workbook, sheet scf_wealth: 29.70/38.79% (per Claude). Public SCF summary extracts: 29.9%/38.6% (Claude), 38.55% for 2016 (Codex). Codex also notes the Forbes-400 share is 1.02% in 1982 for the fixed 400 (0.83% population-normalized), not literally 'less than 1 percent', while the 'more than three' times rise is verified (x3.16).

Why this severity. A 0.2 pp attribution discrepancy with no effect on the trend or the argument.

Why it matters, in detail. Because AS's percentiles contain fixed numbers of people, ranking by income/sqrt(size) pushes large families out and lets more (smaller) tax units into the top group, raising the share by ~0.9 pp in 2015 relative to unit-income ranking. The footnote's 'not the highest top 1 percent incomes' argument applies only to a fixed-number-of-units or per-capita benchmark. The equivalence-scale choice is a legitimate conceptual dispute, but the stated sign is wrong for the comparison AS actually make.

Evidence. AS AEA workbook, sheet 'TRA86 & online app' (2015, fiscal income incl. gains): tax units 21.90%; groups by individuals with unit-income ranking 19.05% (-2.85 pp); size-adjusted ranking 19.96% (+0.91 pp); sample corrections 19.60% (-0.36 pp). AS (2018) fn 23 had already reported the ~1 pp positive ranking effect; AS (2019) appendix Sec. IV.C repeats it (+0.9 pp). AS (2018) Table 4 attributes 0.2 pp of the 5.8 pp 2014 gap to the whole unit/ranking difference.

Why this severity. The claimed mechanism is incorrect for AS's design, but the quantity involved is small (about 1 pp in AS's fiscal diagnostic, 0.2 pp of the 2014 AS-PSZ gap) and no computation in the paper depends on it.

The authors' response. Saez & Zucman (2020, NBER WP 27921, pp.31-32) maintain the conceptual objection: the size-adjustment argument 'reflects a confusion between consumption and income inequality' and produces a 'hybrid' measure.

Why it matters, in detail. Defining the PSZ top 1% on ~6% more tax units (2016) mechanically raises its share by 0.2-0.5 pp relative to the PS-count population behind the fiscal shares. Correcting it improves the level fit slightly and narrows the trend gap, so it is a documentation point rather than a threat to C1.

Evidence. PSZ April-2019 micro-files (scripts/adj_micro_test.py): 71.3m vs 73.7m units (1962) and 169.6m vs 180.0m (2016) under PS-count (dweghttaxu) vs adult (dweght) weights. PS-count weights reproduce PS fiscal shares (8.27% vs 8.27% in 1962; 17.85% vs 17.78% in 2016); adult weights reproduce AJ (factor-price 12.04% vs 11.95% in 1979, 21.54% vs 21.40% in 2016; full NI 21.38% vs 21.40% in 2016). PSZ's factor-price top-1% share on the PS count is lower by 0.18 (1962), 0.28 (1979), 0.36 (1986), 0.38 (1988), 0.52 (2000), 0.43 (2007) and 0.38 pp (2016).

Why this severity. The mismatch is undocumented but small (0.2-0.5 pp) and works in the paper's favour: on a like-for-like basis the simplified series' average undershoot of PSZ shrinks from -0.55 pp to roughly -0.2 pp, and PSZ's 1962-2016 rise is ~0.2 pp smaller.

Why it matters, in detail. Out-of-sample accuracy depends on whether the US offset between overstated untaxed concentration and omitted re-ranking recurs elsewhere; it need not where fiscal statistics exclude capital gains, the untaxed-income mix differs (e.g. more imputed rent or informal income), or re-ranking is larger or smaller.

Evidence. Abstract; p.290. The US fit relies on a gains-inclusive, mixed-ranking capital key (F3) and on an overstatement of untaxed-income concentration offsetting omitted re-ranking (F1). Both auditors note that no foreign-country validation is supplied (Codex records it as a conditional claim, not an issue).

Why this severity. The claim is hedged ('could be applied') and is not the paper's main result, but nothing in the paper shows the method's accuracy outside the US.