Thomas Piketty, Emmanuel Saez · Econometrica 81(5): 1851-1886

A Theory of Optimal Inheritance Taxation

14 problems found: 5 moderate · 9 minor

What the paper claims

Optimal linear (and two-bracket) inheritance-tax rates can be written in sufficient statistics (aggregate bequest and labor elasticities eB, eL; distributional ratios b_received, b_left, y_L; R or R/G; bequest-motive share nu). The formulas carry over to social discounting, dynamic efficiency (R -> 1), partly accidental bequests and the Barro-Becker dynastic model, and nest Atkinson-Stiglitz/Farhi-Werning (tauB = 0 or < 0) and Chamley-Judd (eB infinite).
For realistic parameters the optimal inheritance tax rate 'might be as large as 50%-60%' (optimum for zero-bequest receivers, the 'Meritocratic Rawlsian' criterion).
'...or even higher for top bequests, in line with historical experience': optimal top rates above an exemption of about 60%, matching Anglo-Saxon top estate-tax rates 'systematically above 60%' from the 1930s to the 1980s.
Inheritance taxation involves deeply conflicting interests: the bottom ~70% of receivers prefer ~50%, rates fall abruptly above P70 and become negative for the top 10-20%.
In the rentier society of Paris 1872-1937, bottom-80% receivers' b_left is 20-30%, implying zero-receiver optimal rates 'typically above 75%' for moderate eB ('above 80%' in Supplement S.3).
Sensitivity of the zero-receiver rate: eB = 0 -> ~70%, eB = 0.5 -> ~50%, eB = 1 -> ~35%; r-g = 3% -> ~70%; R/G = 1 (dynamic efficiency) -> ~40%; nu = 0 -> >80%; nu = 1 -> ~50%; eL matters little.

Every problem found

1moderateconfirmed · previously known · overclaiming

The 50-60% figure is the tax rate that would best serve people who inherited nothing, and the paper says so. Its supplement adds that a more conventional goal, counting everyone's well-being equally, would give almost the same answer; that shortcut is mathematically wrong, and a later study using the paper's own formula and data finds such a goal can even call for subsidizing inheritances. The headline therefore stands or falls with the value judgment behind it.

Technical detail and evidence

Why it matters, in detail. Bequests are extremely concentrated (brec/yL far above 1 in the top decile) and the donor's warm glow enters welfare, so with equal or mildly concave weights the b_received term dominates: with all ratios equal to 1, tauB = (1 - 1.2D - D*nu/1.822)/(1.2 - 1.2D) = -582% (nu = 1). Under GM's labor-income-based weights, concavity makes it more negative. Well-being proxies that include inheritance can bring a concave utilitarian optimum closer to the zero-receiver rate (auditor A's percentile CRRA exercise for France: 49% at rho = 1, 56% at rho = 2, nu = 0.7), so the utilitarian answer is undetermined in sign and size rather than 'very close' to 50-60%. Whether donors' warm glow should count socially is itself contested; dropping it pushes rates toward 1/(1+eB).

Evidence. Supplement S.3 p. 15 ('Utilitarian Optimum'); PS13 pp. 1858, 1875-1876, fn. 19. The FOC on p. 1857 rearranges to 0 = N - tauB*H with N = yL - D*brec*(1+eBhat) - D*nu*bleft/(R/G), H = yL*(1+eB) - D*brec*(1+eBhat), D = 1 - eL*tauL/(1-tauL), both linear in the weights (scripts/theory_check.py). Garcia-Miralles (2020), B.E. J. Econ. Anal. Policy 20(2), art. 20190274, Table 1 (nu = 1, eb = 0.2, R/G = 1.8, SCF 2010): utilitarian -582.3% (linear utility; reproduced analytically here with all ratios = 1), -5,040% (rho = 0.3), at or below the -10,000% bound (rho = 0.7, 1.4); responsibility-and-compensation 38.5-50.0%; Rawlsian 83.3%; Table 2: utilitarian -435.9% at nu = 0.7. GM fn. 13 proxies consumption by labor income. Farhi & Werning (2013, NBER WP 18792, fn. 1): altruism heterogeneity 'coupled with their choice of welfare functions may be key' to PS's positive and negative rates.

Why this severity. The abstract explicitly conditions the high rate on society caring mostly about those receiving little inheritance, so the headline is not overturned; but the supplement's assurance that a standard utilitarian objective would give nearly the same answer, which broadens the headline's appeal, is wrong as a method and contradicted by the only published implementation.

The authors' response. No direct reply to Garcia-Miralles located. PS13 itself notes that weight on bequest receivers can make the optimal rate negative (pp. 1852, 1858), and Piketty, Saez & Zucman (2023, Oxford Review of Economic Policy 39(3): 575-591, p. 582) repeat that 'if the normative objective puts weight on bequest receivers, then lower or even negative bequest tax rates can be optimal' while keeping the Meritocratic Rawlsian benchmark.

2moderateconfirmed · newly identified here · speculative assumption

The 50-60% rests on particular guesses about how fast inherited wealth grows compared with the economy, why people leave bequests, and how much they cut back when taxed. Changing two of these guesses to values the paper itself uses elsewhere gives about 20-30% instead. The rate stays clearly positive in every case, but 50-60% is one possibility among many rather than a firm estimate.

Technical detail and evidence

Why it matters, in detail. Holding the other inputs at the benchmark (France/US): R/G = 1 -> 46/38%; nu = 1 -> 54/48%; eB = 0.5 -> 50/47%; r - g = 1% (R/G = 1.35, workbook variant i) -> 56/50%. Jointly, R/G = 1 and nu = 1 give 31/19% at eB = 0.2 and 24/15% at eB = 0.5. The headline is linear in D*nu*b_left/((R/G)*yL) (0.24 France, 0.30 US): a 10% change in that term moves tauB by 2-2.5 pp. None of R/G, nu or eB is estimated in the paper (nu is mis-sourced, F6; eB is borrowed, F7). The benchmark is interior (eB = 0 or r - g = 3% give 65-82%), so the issue is the width of the defensible range, not a one-sided bias.

Evidence. Table I panels 0-3 (p. 1874); pp. 1864, 1873, 1876-1877; Table A3; Supplement S.3 pp. 13-17. Recomputation from Table1!B39:D40 (scripts/joint_sensitivity.py, derived/joint_sensitivity.json): grid eB in {0, 0.2, 0.5} x nu in {0.7, 1} x R/G in {1, 1.35, 1.82, 2.46} gives 24.4-82.0% (France) and 15.4-78.1% (US); 6 of 24 cells per country lie in [50, 60]%. tauB >= 50% requires eB <= 0.51 (France) / 0.41 (US) at nu = 0.7, R/G = 1.82, but <= 0.31 / 0.16 at nu = 1; at R/G = 1, nu = 1 no eB >= 0 reaches 50%. Alvaredo, Garbinti & Piketty (2017), Economica 84: 239-260, Table 1: SCF 2010 self-reported inheritance flow = 22% of the economic flow.

Why this severity. The qualitative result (positive, substantial rates for zero receivers) holds across the whole grid (minimum about 15%), and each one-way sensitivity is disclosed; but combining the paper's own alternative values moves the headline far outside 50-60%, so the specific magnitude is one illustrative point rather than a robust estimate.

The authors' response. One-way sensitivities are reported (Table I panels 2-3; Figures S.1-S.6) and S.3 (p. 14) calls the calibrations 'exploratory... [they] should not be used to make fine policy recommendations'. No discussion of joint variation located.

3moderateconfirmed · newly identified here · overclaiming

The abstract says the best tax rate is even higher for the largest inheritances. The paper's own table shows this for France (65-66% versus 63%) but not for the US (57-58% versus 59%), and under the version of the formula the supplement actually derives, the US figure would be about 40%.

Technical detail and evidence

Why it matters, in detail. Top rates are tauB = [1 - D*nu*Bleft/((R/G)*yL)]/(1 + a*e_b), with Pareto a = 1.80/1.97 (France) and 1.32/1.40 (US) and Bleft = 0.31/0.22 (France) vs 0.72/0.72 (US). US zero receivers are measured as nearly as likely as average to leave large taxable estates (Bleft about 0.72), possibly because of SCF under-reporting of inheritances, which removes the gain from an exemption. The cross-sectional analogue with R/G follows from the same derivation as (17) and keeps panel 6 comparable with the linear benchmark, so the R/G mismatch is a documentation inconsistency. It does mean the top rates inherit the full R/G sensitivity (R/G = 1 gives about 40% for the US), which Table I does not show.

Evidence. Table I panels 0 and 6 (p. 1874); p. 1877; Figure 2 (publisher fig2R.pdf matches TableA6g/A7g column O: P1-50 means 63.6% France, 47.4% US); Supplement S.4 pp. 17-19; AppendixTablesFigures.xls Table1!A52:J57 (I53:I57 = 1.8221). Recomputation (scripts/recompute_table1.py): with R/G = 1 as in (S.10), eB = 0.2, nu = 0.7: France 57.8%/61.2%, US 39.8%/39.1% (500k/1m); with nu = 1 as well, US 23.0%/22.3%. NBER WP 17989 p. 28.

Why this severity. A secondary claim in the abstract is unsupported for one of the two countries. The gap at the benchmark is small (1-2 pp below rather than above the linear rate), but the claim is stated as a general result and the S.4-consistent variant is far lower for the US.

The authors' response. Section 4 (p. 1877) describes the Figure 2 top rate as 'roughly comparable' to the linear rate and 'slightly higher in France', attributing the gap partly to reporting biases; no correction of the abstract located.

4moderateconfirmed · newly identified here · internal inconsistency

The paper's numbers for people who inherited middling or large amounts depend on spreadsheet choices that are undocumented and differ between its main table and its figure. Depending on the choice, the preferred rate of the French upper-middle group ranges from about 13% to 33% instead of the printed 31%. The main 50-60% figure for people who inherited nothing is not affected.

Technical detail and evidence

Why it matters, in detail. Because b_received rises steeply in the upper tail, brec/yL approaches 1/D (about 1.09), where the denominator of (17) vanishes; small construction choices then move tauB(p) by tens of points and flip signs (P90-95). The correct same-group construction is not unique (smoothing windows; own-sample vs Table A3 denominators; capitalization rate), so no single corrected value exists. The defensible statement is that the P70-90 rate lies somewhere in about 13-33% (France) and 38-52% (US).

Evidence. AppendixTablesFigures.xls: TableA4/A5 vs FRr/USr sheets (scripts/find_windows.py: wealth and labor from the r = 0 sheets in 100/100 rows; r3/r5 receipts from the r3/r5 sheets); Table1!B43:D50 equal group means of TableA6a/A7a with b_received from the r = 5% column (scripts/trace_inputs.py); fig1R.pdf vs TableA6g/A7g columns G/H/I; scripts/rank_consistency.py. France P70-90 at eB = 0.2, nu = 0.7: 30.6% (printed 31%), 24.8% (same-rank), 33.4% (raw, unsmoothed, own-sample denominators), 12.8% (r = 3% as in Figure 1). US: 45.7% (46%), 44.3%, 51.9%, 38.2%. P90-95: printed -330%/-84%; same-rank -1114%/-175%; at r = 3% the formula's denominator approaches or crosses zero.

Why this severity. The headline bottom-50% rates are unchanged under every construction (63.06%/58.71%), and the qualitative pattern (rates fall sharply and turn negative for heavy inheritors) survives; but reported secondary numbers move by up to 18 pp (France P70-90) depending on undocumented choices.

5moderatelikely · newly identified here · overclaiming

The paper's 'inheritance tax' is really the total tax on inherited wealth plus everything it earns over a lifetime. Because that income is already taxed today, the extra inheritance tax the calculation implies is lower, roughly 30-55% depending on how the model is read. Comparing 60% directly with the 60-77% estate-tax rates of the mid-20th century, which came on top of very high income taxes, compares different things.

Technical detail and evidence

Why it matters, in detail. Formula (17) is derived with tauB as the sole tax on capitalized bequests. If existing capital taxes impose a wedge tau0 and tauB is the total, the add-on rate is 1 - (1 - tauB)/(1 - tau0). If instead R is after-tax, a budget-neutral rise in tauB also erodes existing capital-tax revenue, replacing eB*tauB/(1-tauB) with eB*(tauB + tau0/(1-tau0))/(1-tauB); this lowers the add-on rate by 7-11 pp at tau0 = 30-40%. The add-on rate returns to about 55-64% only if one also uses a pre-tax r - g of about 5%, well above the paper's 2%. Historical statutory top estate-tax rates (US 77% in 1941-76; UK 65-85% in 1940-83) coexisted with top income-tax rates of 70% or more on dividends and interest, so historical total wedges on large estates were far above 60%; the 'in line with history' comparison is not like-for-like.

Evidence. PS13 fn. 4 p. 1854; pp. 1878-1880; Fig. 3; Supplement S.3 p. 16; NBER WP 17989 Section 3 ('If tauB = 60% then tauK = 86%'); Piketty (2011, QJE 126(3), Table II p. 1122): g = growth of national income, after-tax rd - g = 3.6% and pre-tax r - g = 5.0% for 1820-2009. Recomputation of the add-on statutory inheritance tax when existing capital taxes impose a wedge tau0 = 30-40% (scripts/theory_check.py, misc_checks.py): (a) (17) read as the total wedge: France 38-47%, US 31-41%; (b) R read as after-tax, with the fiscal externality on existing capital-tax revenue added to the elasticity term: France 52-56%, US 48-52%; (c) total wedge recomputed with a pre-tax r - g of 5% (R/G = 4.48): France 58-64%, US 55-62%. Piketty, Saez & Zucman (2023, p. 581) present the inheritance tax as 'in addition to progressive income and wealth taxes' while using the same single-capital-tax formula.

Why this severity. The computed optimum is not wrong, but its natural reading as a statutory inheritance-tax rate, and the claim that it matches historical top estate-tax rates, rest on an unstated assumption. Under the paper's own Section 5 and S.3 logic the implied add-on inheritance tax is roughly 30-55%, a noticeable but not transformative change to the policy reading.

The authors' response. Partly anticipated in fn. 4 and Section 5 (the equivalence with a capital-income tax and the instruction to lower tauB if capital income is taxed), but not applied in the abstract, Section 4 or Figure 3. Piketty, Saez & Zucman (2023, p. 584) restate that US top inheritance-tax rates until the 1970s-1980s 'were close to optimal from the viewpoint of the bottom two-thirds of the population'.

6minorconfirmed · newly identified here · misrepresentation of source

The paper cites a study as finding that one-half to two-thirds of people have a bequest motive; the study actually reports about three-quarters, and that about half of what people leave is due to that motive. Correcting the citation barely changes the result, and the authors' own later reading would raise the rate, but this key input is not pinned down by the evidence cited.

Technical detail and evidence

Why it matters, in detail. Mapping KL's prevalence (about 0.75) to nu lowers the benchmark by 1.5-1.8 pp; mapping KL's share of bequests due to the motive (0.53) raises it by 5-7 pp. Neither is the g*b-weighted marginal ratio R(1-tauB)V_b/V_c among zero receivers that (17) requires, so nu is effectively unidentified; the larger nu uncertainty (nu = 1: -9 to -11 pp) is part of F2.

Evidence. PS13 p. 1877, fn. 16 p. 1873, p. 1866. Kopczuk & Lupton (2007), Review of Economic Studies 74(1): 207-235: abstract; introduction ('roughly 75%'); Table 3 (predicted probabilities 0.735-0.794 across estimated columns); p. 229 ('53% is accounted for by a bequest motive'); conclusion ('about 75%'). Recomputation (France/US): nu = 0.75 -> 61.6%/56.9%; nu = 0.53 -> 68.0%/64.7%; nu = 0.5 -> 68.8%/65.7%; nu = 1 -> 54.4%/48.2%.

Why this severity. The miscitation removes the stated empirical anchor for nu, but KL-consistent values move the headline by only about -2 pp (prevalence reading) or +5-7 pp (wealth-share reading, which the authors later adopt), so it does not bias the headline upward.

The authors' response. Piketty, Saez & Zucman (2023, p. 583) later write that 'an average value around nu = 0.5 might be realistic (see, for example, Kopczuk and Lupton (2007))', i.e. the wealth-share reading, which implies higher rates (about 66-69%).

7minorconfirmed · newly identified here · overclaiming

The Paris records are complete for what people left at death, but they show what people had inherited only for a subset of married people; for the majority who left nothing, inheritances were assumed to be zero. The resulting high preferred rate (about 75%) also holds under the authors' alternative assumption, so it is the description, not the number, that is overstated.

Technical detail and evidence

Why it matters, in detail. Because most decedents leave nothing and are assigned zero inheritance, the bottom-80% receiver group is dominated by zero-estate decedents, which builds part of the low b_left into the construction; version 2 (giving bottom wealth-holders the inheritance distribution of wealth ranks 70-90) raises b_left only to 0.14-0.31, so the high-rate result is robust within the supplied variants.

Evidence. doEstates1872-1937(optimaltaxcalibration).txt (sample definition; zero-estate sampling); ResultsOptimalTaxCalibrationsNov2012.xls Results1/Results2 notes; workingfile.dta (scripts/paris_check.py): 99.5-100% of zero-estate decedents have zero inheritance, and 70-86% of zero-inheritance decedents are zero-estate. Bottom-80% b_left: 0.118-0.223 (version 1) and 0.137-0.311 (version 2), giving tauB of 76-79% and 73-79% at eB = 0.2, nu = 0.7, tauL = 15% (printed 75%). PS13 p. 1877; S.3 p. 15; Table I panel 5.

Why this severity. The 75% illustration survives both of the authors' imputation variants (73-79%), so the conclusion is unaffected; the problem is an overstated description of data coverage.

The authors' response. The supplied results workbook (Results1/Results2) documents the imputation and a robustness variant; no correction of the paper's description located.

8minorconfirmed · newly identified here · documentation gap

The authors share their final spreadsheet but not the code that built it from the survey data, and the French figures mix two slightly different samples. Using the public US survey we could rebuild the US result (about 59%), but the French result cannot be independently checked.

Technical detail and evidence

Why it matters, in detail. Without the processing code, sample selection (age, w >= 0), the income definition, receipt handling (additional receipts, negative values), inflation and capitalization conventions, the random tie-breaking seed and the smoothing windows (see F5) must be reverse-engineered. For France, where the Enquete Patrimoine is restricted-access, none of the inputs can be independently rebuilt.

Evidence. Inventories of ES_10712_programs_and_data_0.zip and doresults(optimaltaxcalibration).zip (no SCF/EP code); TableA3!B7 = 4,811 vs FRr070+!D3 = 4,076; FRr070+ mean wealth 189,961 vs Table A4 'All' 194,493. A single-sample France P0-50 rate is 62.0% vs 63.1% (scripts/rank_consistency.py). Independent SCF builds reproduce counts exactly and the headline closely (this adjudication 59.4%; auditor A 58.0-58.7%; auditor B 58.7%), but give 70+ labor-income means of $24,150 (wages + self-employment + Social Security/pensions) or $26,743 (auditor B's broader definition) vs $29,491.

Why this severity. The US headline reproduces independently from public data and the French sample mismatch moves the headline by about 1 pp; the gap limits verification of France and of construction choices rather than changing conclusions.

9minorconfirmed · previously known · methodology

The chart shows almost the same preferred rate for the bottom 70% of Americans, but only because the authors randomly sorted the many people who reported no inheritance. Among them, people with little wealth would want very high inheritance taxes and those with substantial self-made wealth would want low or even negative ones. The average for non-inheritors is unchanged, but their interests are less uniform than the chart suggests.

Technical detail and evidence

Why it matters, in detail. Under random ordering, b_left and yL are flat by construction up to the last zero-receiver percentile (about P68 in the US). Within-group dispersion of wealth and labor income implies that preferred rates among zero receivers range from the revenue-maximizing 1/(1+eB) = 83% (no wealth) to negative (self-made wealth holders), weakening the political-economy reading on p. 1878 ('shift in political power away from the bottom 80%').

Evidence. Supplement S.3 p. 14 and fn. 27; PS13 pp. 1875-1876; SCF 2010 (scripts/scf_zero_receivers.py): 69.8% of individuals aged 70+ report no inheritance; Garcia-Miralles (2020) Section 5 and Figure 2; auditor B's SCF wealth-decile sorting of non-receivers (nu = 1): 83.2% (poorest decile) to -10% (richest).

Why this severity. The headline uses only the zero-receiver group average, which is unaffected (59.4% in the SCF rebuild using all zero receivers); the issue concerns the interpretation that the bottom 70% share a common interest in rates around 50%.

The authors' response. The random assignment is disclosed (S.3 fn. 27: 'we tried several specifications, and this made little difference'). No reply to Garcia-Miralles located.

10minorconfirmed · newly identified here · arithmetic or transcription

Five numbers in the main table are off by up to 2 percentage points, and one spreadsheet cell is copied from its neighbour. None of them is a headline number.

Technical detail and evidence

Why it matters, in detail. Correct integer entries are 73%, 60%, 58%, 31% and 30% (cells C12, E12, F12, I33, J33). None enters a statement in the text except the P50-70 row, whose qualitative message (close to the P0-50 rate) is unchanged.

Evidence. scripts/recompute_table1.py -> derived/table1_recomputation.json; scripts/check_variant_sheets.py -> derived/variant_sheets_check.txt. The same values appear in the official PDF, the author PDF and the workbook (both auditors).

Why this severity. The largest discrepancy is 2.2 pp, all in secondary cells; the headline panel and all sensitivity panels reproduce.

11minorconfirmed · newly identified here · internal inconsistency

A handful of descriptive sentences are loose or contradict the paper's own tables, for example the share of inheritances going to the bottom half, the years when estate-tax rates exceeded 60%, and some figure references. They do not change the computed rates.

Technical detail and evidence

Why it matters, in detail. Only item (ii) has a quantitative counterpart: the calibration uses the survey's near-zero receipts for P0-50, which is appropriate for the zero-receiver criterion, so the sentence misdescribes the data rather than the calculation.

Evidence. Locations as listed; AppendixTablesFigures.xls sheets Table1 (B39:D40, C39), TableA1, TableA6a/A7a, DataFig3 (US and UK series by year); Supplement figure captions pp. 10-13; scripts/theory_check.py, misc_checks.py.

Why this severity. Descriptive slips and loose summaries that do not change any computed rate; item (ix) errs in the authors' disfavour.

12minorconfirmed · newly identified here · methodology

One step in an appendix argument identifies a quantity with the wrong statistic, though the conclusion the paper draws from it still holds. It does not affect any number in the paper.

Technical detail and evidence

Why it matters, in detail. e_B^anticip is proportional to (1-delta)/(1-kappa) with kappa the dominant eigenvalue of the transition restricted to positive-bequest states. With persistent bequest states kappa can be much closer to 1 than the cross-sectional share, so anticipatory elasticities can be much larger than the appendix suggests, but they remain finite.

Evidence. Two-state chain with transition matrix [[0.80, 0.20], [0.05, 0.95]] (zero, positive): stationary positive share 0.80, but uninterrupted survival from a positive state is 0.95^n, so the anticipatory sums are 20 vs 5 (auditor B, codex/scripts/theory_checks.py; verified by hand here).

Why this severity. The conclusion actually used (kappa < 1 whenever zero bequests recur, hence a finite anticipatory elasticity in the Aiyagari-type model) survives; only the stated identification of kappa is wrong, in an appendix the authors label as intuition.

The authors' response. Appendix A.3 is presented as 'intuitions rather than fully rigorous detailed proofs' (p. 1882).

13minorlikely · previously known · other

The paper says its formula contains the famous result that capital should not be taxed in the long run. Later research showed that famous result itself does not hold in general, so this link is weaker than stated; the paper's own formula is not affected.

Technical detail and evidence

Why it matters, in detail. PS13's own deterministic-model argument (e_B^anticip proportional to T, hence infinite) assumes convergence to an interior steady state from delta*R = 1; Straub-Werning's positive-tax paths violate the conditions under which the zero result was derived, so the nesting holds only under PS13's convergence assumptions.

Evidence. PS13 pp. 1852, 1869, 1882-1884; Straub & Werning (2020), American Economic Review 110(1): 86-119, DOI 10.1257/aer.20150210 (abstract: 'we prove that the long-run tax on capital is positive and significant, whenever the intertemporal elasticity of substitution is below one'); NBER WP 20441 (2014).

Why this severity. The sufficient-statistics formulas are unaffected (a positive long-run tax is consistent with a finite e_B^pdv); only the heuristic claim that the formulas nest the Chamley-Judd zero is weakened, and PS13 predates Straub-Werning.

The authors' response. Appendix A.3 is explicitly heuristic; PS13 predates Straub & Werning (first circulated as NBER WP 20441, 2014).

14minorplausible · previously known · speculative assumption

How much people save less or hide wealth when inheritances are taxed is the key unknown. The paper borrows one US estimate (about 0.16) and shows results for other values; later French evidence is similar, so the benchmark is reasonable. If the true response were twice as large, the 50-60% would fall to about 50-54%.

Technical detail and evidence

Why it matters, in detail. eB enters through the denominator (1 + eB): at the benchmark, eB = 0.4 gives 54/50% and eB = 0.35 gives 56/52%. A doubling of the benchmark therefore keeps the zero-receiver rate around 50%; only elasticities near or above the upper end of the wealth-tax evidence (eB of about 1) would take it to the mid-30s, which Table I already shows.

Evidence. PS13 p. 1876; Kopczuk & Slemrod, NBER WP 7960 (2000), pp. 28-33 ('0.16... would reduce the reported net worth of the richest half percent of the population by 10.5%'; 'suggestive rather than definitive'). Goupille-Lebret & Infante (2018), JPubE 168: 21-34: timing elasticity about 0.1, accumulation elasticities 0.23-0.35, described as upper bounds (PSZ 2023 fn. 24 cite 0.4 for the same study). PWBM (2019) brief on the Warren wealth tax, Table A1: difference-in-differences wealth-tax semi-elasticities of 11.6-40.5 over 2-8 years would map to eB of about 0.4-1.4 under the crude conversion eB = s/H (H = 30), but they measure reporting, avoidance and mobility responses to annual wealth taxes. Thresholds: tauB >= 50% needs eB <= 0.51/0.41 (France/US, nu = 0.7).

Why this severity. The uncertainty and its consequences (eB = 0.5 -> 50/47%; eB = 1 -> 38/35%) are disclosed, and later direct inheritance-tax evidence is consistent with the benchmark. Because the aggregate bequest flow is dominated by large estates, Kopczuk-Slemrod's top-estate population is the relevant one, leaving mainly a horizon and instrument mismatch.

The authors' response. PS13 p. 1876: 'There remains considerable uncertainty about eB. More empirical work on this key parameter would be valuable.' PSZ (2023, p. 583): 'one certainly cannot exclude the possibility of higher elasticities.'

Claims examined and rejected

Problems one auditor raised that did not survive checking.

I1: plugging the SCF cohort ratio of bequests left to capitalized bequests received into formula (17) puts the US zero-receiver rate at -13% (r = 3%) to 39% (r = 5%), so 'the US half of the headline moves from 59% to anything between about -13% and 50%'; France is roughly consistent with the steady state, the US is not. — The calculation takes survey-reported inheritance levels at face value. The paper (S.3 p. 13: survey flows below 50% of the fiscal flow) and Alvaredo, Garbinti & Piketty (2017, Economica 84, Table 1: SCF 2010 captures 22% of the economic flow; France 2009, 21%) document heavy under-reporting. At those capture rates the US cohort ratio is consistent with R/G = 1.82, while the French ratio would imply an even larger R/G, so the face-value ratio is uninformative for both countries. The qualitative point (R/G is assumed; the S.3 non-steady-state argument covers one channel only) is retained in F2.
I1: if g is per-capita growth, US population growth of about 1%/yr implies R/G = e^((r-g-n)H) of about 1.35 and a US rate of 50% (36% with nu = 1). — The paper benchmarks r - g against Piketty (2011, Table II), where g is the growth rate of national income, i.e. aggregate growth already including population growth. The NBER WP's g + n adjustment (App. B.3) applies when g denotes per-capita productivity growth, so subtracting n again double-counts. A lower US r - g because of faster aggregate growth is possible and is covered by the workbook's r - g = 1% variant (50%), included in F2.
I4: the panel-6 top rates 'do not follow from the formula as derived'; they use an underived cross-sectional analogue of (S.10). — Partly rejected. The cross-sectional-budget analogue (B_left divided by R/G) follows from the same derivation as (17), and auditor A verified it for zero receivers; using it keeps panel 6 comparable with the linear benchmark. The only inconsistency is that S.4 presents (S.10) as the dynamic-efficiency version, which is kept in F4 as a documentation point together with the R/G = 1 numbers.
I5: an optimal total wedge of 60% 'implies an additional statutory inheritance tax of about 33-43% ... not 60%'. — Partly rejected as a point estimate. 33-43% holds only if R/G = 1.82 is read as a pre-tax return; the paper's r - g citation is the after-tax gap. Under an after-tax reading with the fiscal externality the add-on rate is about 48-56%, and a consistent pre-tax total-wedge calculation (r - g = 5%) gives about 55-64%. F3 keeps the like-for-like point with the full 31-64% range.
I3: sensitivity is shown 'around the benchmark most favourable to high rates'. — The framing is rejected; the computation stands (F2). The benchmark is interior: eB = 0, r - g = 3% or nu = 0 give 65-100% in Table I; nu = 0.7 is below the authors' later reading of Kopczuk-Lupton (nu of about 0.5, which raises rates); and r - g = 2% is below the 3-4% historical gap the paper cites. The problem is the width of the joint range, not a one-sided choice.
I7: part of the eB mismatch is that Kopczuk-Slemrod cover 'estate-tax decedents, a small top fraction of all decedents'. — The eB in (17) is the elasticity of the aggregate bequest flow, i.e. a bequest-weighted average of individual responses, which is dominated by large estates; a top-estate population is therefore the relevant one. The horizon (single generation vs steady state) and instrument mismatches are retained in F7.
Checks passed: 'The distinction between a broad capitalized-inheritance wedge and an additional statutory estate tax is explicit at p.1854 note 4 and pp.1879-1880', so no issue arises. — The distinction is stated but not applied: the abstract and Section 4 compare tauB with historical statutory top estate-tax rates (Fig. 3), the paper never states whether r is pre- or after-tax, and S.3 itself maps existing capital taxes into 'tauB current = 40%'. Kept as F3 (moderate, likely).
I1: 'The workbook permits a direct repair without new microdata'; the same-rank recomputation (France P70-90 30.6% -> 24.8%) is the corrected value. — The mismatch is confirmed (F5), but the repair does not identify a corrected value: an equally defensible construction (raw unsmoothed percentile sheets with own-sample denominators) moves the same rate the other way (33.4%), and the capitalization rate used in Figure 1 (r = 3%) gives 12.8%. F5 reports the range instead.

Audit notes

What reproduced. Reproduced: (1) Formula (17) solves the p. 1857 first-order condition (max residual 2e-14 over 20,000 random draws); both auditors re-derived (7)-(10), (16) and (S.3). (2) Table I: 95/100 cells recompute from the simulator inputs (Table1!B39:I57) with formula (17); the five discrepancies (C12, E12, F12, I33, J33; at most 2.2 pp) are the same ones both auditors found. (3) All 10,000 percentile cells in Tables A6a-j/A7a-j recompute except TableA7e!H102, once the authors' -20% floor and singularity rule are applied. (4) Simulator inputs trace to group means of Tables A6a/A7a; panel-1 b_received uses r = 5% capitalization, while Figure 1 plots the r = 3% series (visual match at about 25 points); Figure 2 matches the workbook. (5) Independent SCF 2010 build (current Fed public files): individual counts 10,554 (18+) and 1,109 (70+) exact; 69.8% of 70+ report no inheritance; zero-receiver b_left 0.787 and yL 0.964 (paper P0-50: 0.779/0.926); tauB 59.4% vs 58.7% (bootstrap sd 1.2 pp); nu = 1 gives 49.2% (48%) and R/G = 1 gives 39.8% (38%); ages 60-69/70-79/80-89 give 59.9/58.7/59.7%, confirming fn. 17. Wealth levels differ from the paper by a constant price-basis factor (1.494), which cancels in ratios. (6) Paris: zero-estate shares 60.5-76.4%; bottom-80% b_left 0.12-0.22 (v1) and 0.14-0.31 (v2), giving 73-79% vs 75% printed. (7) Garcia-Miralles's utilitarian -582.3% reproduced analytically. Not reproduced: France from primary data (Enquete Patrimoine 2010 restricted; no code); the authors' US labor-income definition (70+ mean $29,491 vs $24,150-26,743 in independent builds); the S.3 claim that the endogenous-b_left correction takes 60% to about 55% (fixed points 60.1% France, 57.4% US, i.e. the paper's correction is conservative); the US Table A5 upper tail (smoothed percentiles average to 27-47% of the 70+ mean inheritance).

Limits of this audit. France could not be rebuilt from primary data (Enquete Patrimoine 2010 is restricted-access and no processing code is supplied); French inputs were checked only for internal consistency. The original Paris estate file is not distributed; checks use the authors' derived working files. The SCF rebuild uses the current Federal Reserve releases (full public file dated 2014; summary extract re-released in later-year dollars), not the 2012 vintage; its labor-income definition does not match the authors' absolute means, and the bootstrap ignores imputation variance and the dual-frame design. Figure 1/2 comparisons are visual matches of rendered publisher figures against workbook columns, not pixel digitization. The wealth-tax elasticity conversion in F7 is a deliberately crude back-of-envelope. Theory checks are numerical verifications of first-order conditions plus review of the auditors' derivations; no global-optimum, existence or general-equilibrium analysis was done. CEPR DP 8946/9241 full texts were not obtained (the version comparison uses NBER WP 17989 and the published article). No author contact was attempted; the absence of a located reply to Garcia-Miralles is not evidence that none exists.