Errors in the research of Emmanuel Saez and Gabriel Zucman

The savings limit studied in the paper governs two benefits at once: the yearly cash payment and a health subsidy for medicine, dental care and glasses. The paper credits the whole savings response to the cash payment, and the 2003 and 2010 rule changes it uses as tests also changed the health subsidy. The response is real, but how much of it is due to the cash payment alone is not known.

The paper counts every dollar of corporate tax as paid by the companies' shareholders. Government agencies instead assume part of it is paid by workers and by other investors. Under the agencies' rules the top 1%'s tax rate still falls a lot after 1950, but by about 8-10 points instead of 13. So roughly a fifth to a third of the measured fall comes from this accounting choice.

When the authors actually published 'real-time' numbers during the pandemic, those numbers showed income inequality before taxes and benefits barely changing. Once tax data arrived, it turned out that the top 1% had gained a lot and the bottom half had lost ground. The paper reports its method as accurate in recessions without showing this record, so it overstates how much one can trust its real-time numbers on who gains from growth, although its real-time numbers on the effect of government aid were accurate.

The paper says Swedish holding companies explain about half of why top incomes seem to have grown much less in Sweden than in the US, where tax rules discourage such companies. But the US figure it relies on uses a narrower legal definition than the one applied to Sweden, so the US side was assumed to need no correction. The best-known US correction would probably strengthen the paper's point, so its direction holds, but 'half' is a rough number.

The tax only charges the gap between 2% of a household's wealth and the income tax it already pays. The paper assumes everyone worth over GBP100m already pays very little (0.2% of their wealth). But the study it cites assumes that people worth GBP100m-1bn pay six times as much (1.2%), and the paper itself says they pay 'considerably' more. Using the cited study's own figures, the tax would raise about GBP8bn, not GBP10bn.

The paper's first finding is that, after 2013, getting rich stopped being linked to sending money to tax havens. But the paper never checks whether the before/after gap is bigger than chance alone could produce, and its own numbers show it is not. The pattern points the same way in every version of the analysis, but the evidence is too noisy to say how large the change was, or even to be sure there was one.

The paper measures how much faster declared wealth grew in tax havens than in ordinary countries. It then applies that extra growth to the money in the ordinary countries too, even though they were the comparison group. Done consistently, the paper's own method gives about a quarter of the increase rather than 38%.

The paper argues Forbes misses hundreds of smaller billionaires, which would push the tax take up to $115-128 billion. That estimate comes from a curve-fitting exercise that gives wildly different answers depending on the year and on whether the very richest people are included. A reconstruction from state tax data finds far fewer missing billionaires, adding about $5 billion rather than $24 billion.

The paper says US companies moved 1-5 percentage points of their profits back home after the 2017 reform, but it measures this only against 2017, an unusually good year for profits abroad. Compared with the average of the years just before the reform, the share booked abroad in 2018-2019 was about the same or higher. After allowing for ups and downs in the economy, the fall is about 1-2 points and could be chance. The paper's broader point, that the reform changed little, is if anything strengthened.

The USD 24bn assumes that every very rich family pays in full and that nobody moves abroad, reorganises their holdings or gets their companies valued low. The authors' own earlier work and the studies they cite expect a fifth or more of such revenue to be lost, so on this count alone a realistic figure is nearer USD 19-20bn.

Billionaires pay very little personal income tax, but the companies they own pay corporate tax, which the author himself counts when he says billionaires pay less than everyone else. Counted that way, they already pay roughly 2% of their wealth in tax each year, not 'almost nothing'. The proposed 2% would come on top of that, which the article does not explain. The numbers are right, but the 'almost untaxed' message overstates the case.

The paper argues that trade raised taxes on capital by moving activity from small family businesses into corporations. But for most developing countries that split is not measured; it is estimated with a formula based on average income, so the data cannot show that trade, rather than ordinary economic growth, caused the shift. The separate finding that corporations paid higher effective tax rates holds up better.

The note says it makes no sense to count lost income taxes forever while counting the new tax only once. That logic works against the critics' guess about extra unannounced departures, which assumed the tax would be charged every year. It does not settle the case of people who genuinely moved to dodge the one-time tax. Moving back is costly, so they may stay away and their California income tax may be lost for good, and the note gives no evidence that they will return. This does not overturn its conclusion, because the people reported to have left pay relatively little California income tax.

The memo says anyone who thinks the default formula overvalues their stake can hand in an appraisal and pay tax on that number. The law is stricter. The owner must prove with 'clear and convincing evidence' that the formula 'substantially' overstates the value, and the tax agency can use the same route to push values up. Small overvaluations would stick, and private-business owners could find the escape hatch harder to use than the memo suggests.

The brief says a French study found that a one-point tax rise drives out only 0.003-0.03% of rich taxpayers a year, and puts this next to a Scandinavian figure of about 2%. The two numbers measure different things (a yearly flow per point of income tax versus a long-run total per point of wealth tax), and the French study itself says French reactions are about as strong as the Scandinavian ones. The conclusion that emigration was not the main problem survives, but the French evidence is made to look far more reassuring than it is.

Two people, Google's founders, hold about a quarter of all the wealth the tax would hit. Both bought homes and moved business entities out of California in the weeks before the January 1, 2026 cutoff. One of the report's own authors wrote in a companion paper that Page might be able to show he had already left. If they don't owe the tax, $100 billion becomes about $78 billion, but the report dismisses the risk instead of showing it.

The book says billionaires pay half the overall tax rate of the average French person, but the two figures do not count the same taxes: the billionaires' figure leaves out several taxes that are counted for everyone else, such as VAT on spending, some business and property taxes, and inheritance taxes. Measured the same way, billionaires in 2016 paid roughly half to two-thirds of the average rate, so the gap is real and large but probably smaller than 'half'. Cuts to company taxes since 2016 would lower billionaires' rate today, which partly offsets this.

The paper's statistical case that equality itself raises productivity, beyond education and health spending, comes from comparing whole world regions: rich, equal Europe against poorer, unequal regions. Comparing countries within the same region, or the same country over time, the link vanishes or reverses. The data show that equality and prosperity can coexist, but they cannot show that equality caused the prosperity, and they cannot 'rule out' that equality had some cost.

The 30% average includes Social Security and Medicare payroll taxes, a large part of what ordinary workers pay, which fund benefits that come back to them; those benefits are not counted as their income, and billionaires pay almost none of these taxes. Counting the benefits as income, as the Congressional Budget Office does, lowers the average to about 27%, so billionaires' 24% is still lower but the gap halves; with other common choices the two rates are within a point or two.

The paper's headline average tax rate of 42.5% includes some payments that are not taxes, such as contributions to private pension funds and bookkeeping entries for civil servants' pensions. On the official figures the paper itself cites, the average is about 38-39%. The rich still pay far less than average, but the gap is smaller than the headline suggests.

To claim that a 2% levy brings billionaires up to, but not above, what ordinary people pay, the author has to assume how much income billionaires earn each year relative to their fortunes. The lecture does not say what it assumed (9%), and the author's own tax-record studies point to 5.6-7.4%. At those levels a 2% levy would take 43-50% of billionaires' income, more than any other group pays. The tax would raise the same money, but it would not be the 'minimal' fix the lecture describes.

The note says a 2% tax on fortunes above EUR 100 million would end the situation in which Europe's richest pay lower tax rates than ordinary people. Its own source says this is true only on a rough average dominated by the United States. It also warns that in the Netherlands and Italy people far below EUR 100 million already pay less than the middle class, which this tax would not change. The money estimates are not affected, but this fairness claim is overstated.

The paper credits all of the 1996-1997 rise in reported wealth taxpayers to pre-filled returns, assuming nothing else would have changed. But the number of taxpayers was rising by roughly 5-10% a year anyway, so the true effect is probably about 12-16% more taxpayers rather than 'almost 20%'. The effect is still large; it is overstated by roughly a quarter.

The '$1 trillion' assumes the gains would be taxed at about 40%. Yet the paper says they would be taxed like ordinary capital gains, whose top US rate is about 24%. At that rate the same calculation gives roughly half a trillion dollars. That is still a lot of money, but the headline depends on a rate rise that was proposed and never passed, and the paper does not say so.

The 70% comes from dividing by a number that is the difference between two large, uncertain estimates. Using the paper's own uncertainty, the true share could plausibly be anything from about 40% to over 100%, far wider than the 56-95% range it reports, and a 10% change in the assumed interest rate alone moves it between 64% and 96%.

The report says the wealth of the richest one-in-a-million households grew about 7% a year after inflation, which it reads as a 7.5% annual return on their money. But that 7% is how fast the group's combined wealth grew while the group itself kept getting bigger as the world's population grew. Per household the growth was about 5.3%, implying a return of roughly 5.8%, no higher than the average return the report cites for everyone. The revenue estimates are unaffected, but the claim that billionaires earn unusually high returns, and the reassurance built on it, are much weaker.

Most people who live in Dubai are foreigners, so many 'foreign-owned' homes are simply homes that expatriates live in. The note compares its figure with other cities where only owners living abroad are counted. Counted the same way in Dubai, as the authors later did, the share is about 30% instead of 43%. That is still far higher than anywhere else, but about a third less 'offshore' wealth than the note implies.

The essay quotes a 2021 draft saying the richest 1% fail to report over 20% of their income, against 7% for the bottom half. Before the essay was published the same authors revised this to roughly 16% against 10%. The rich still under-report more, but the gap is about half as large as the essay says.

The paper's single biggest item is a large, fast-growing block of untaxed partnership income. The paper assigns it to the rich by assuming it is spread like the partnership income people report on their tax returns. The rival authors later traced who actually owns the partnerships that claim this depreciation, and found it much less concentrated at the top. The paper does not discuss that evidence. If the evidence is right, this item explains a small part of the disagreement rather than its largest part.

The paper's eye-catching claim that Danish tax audits actually lower multinationals' total worldwide tax bills depends on valuing Denmark's extra revenue at a 22% tax rate, but Denmark's rate in the years studied was 23.5-25%. With the correct rate the effect is roughly zero or slightly positive, so the claim that enforcement lowers global taxes is not supported, although the broader point that most of what Denmark gains is simply taken from other high-tax countries still holds.

The report says profit shifting costs governments nearly 10% of the world's corporate tax revenue and has not declined. For 2021-22 these numbers were largely assumed rather than measured, and the same team's later, more complete estimates put the cost at about 7% and show a modest fall after 2018. The central fact, that roughly $1 trillion of profits is booked in tax havens each year, still stands.

The chart says it compares tax burdens measured the same way in all three countries, but the French line measures something different: only working-age people with jobs, with pension contributions counted as taxes. That choice makes French taxes look flat. Measured the way the chart's note describes, the same French study finds that tax rates rise clearly with income, from about 27% to about 50%, and fall only slightly at the very top.

The essay says household wealth in the US, Western Europe and China grew from about two times the nation's yearly income in the 1950s to six times today. For the US, the essay's own sources put the 1950s level at about three and a half times, so the US increase was only about 60% as large as stated, and those sources give no 1950s figure for China at all. The tripling is right for France and Germany, and the broader point that wealth has grown much faster than incomes still stands.

When France raised the wealth-tax entry point to EUR 1.3 million in 2011, anyone just above it suddenly owed EUR 1,250-2,750 while anyone just below owed nothing. That jump alone is a strong reason to report wealth just under the line, so the evidence at this threshold cannot show that paperwork rather than tax drove the response. The paper's main evidence, at EUR 2.57 million, is not affected.

The paper checks what happens if one trusts tax havens' own (incomplete) statistics instead of the authors' corrections, and reports that the estimate would fall by only 18%. Because of a units mistake in the spreadsheet, the correct fall is about half, and the paper's stated minimum of 22% becomes roughly 10-12%. The main 36% figure is unchanged, but it leans on the authors' data corrections far more than readers are told.

To fill a gap in official statistics, the authors assume that three-quarters of the stock-market value of all companies registered in the Cayman Islands (mostly large Chinese tech firms) is owned from abroad in a way that ends up counted as hidden household wealth. This guess is not tested, yet it explains most of the increase in offshore wealth since about 2016 and roughly $1.7-1.9 trillion of the $13 trillion figure for 2023. Without it the series is flatter, which still fits the paper's 'stable' story but makes the 2023 total smaller.

The report assigns the emissions from building factories, machines and buildings to the people who own the wealth that pays for them, not to the people who buy what is produced. That is a defensible choice but not the usual one, and the report never shows what happens if it is changed. Counted the more usual way, the richest 1% produce about 8-11% of emissions rather than 17%. The claim that they added more to emissions growth than the poorer half of humanity then no longer holds, although emissions remain very unequally spread.

The paper says profit shifting peaked just before the 2017 US tax reform and fell afterwards. Its own numbers show the share of profits sent to tax havens was about the same in 2023 as in 2018, and higher in 2019-2020 than in 2017-2018. Shifting did dip, but only in 2021-2022, and it partly bounced back in 2023.

About two-fifths of the EU's EUR 55 billion comes from supposedly low-taxed profits that big European companies record in their home countries. Those profits look low-taxed mainly because they include dividends passed between companies of the same group, and similar income the new rules ignore. Correcting for this alone brings the EU figure down to about EUR 42 billion or less.

The study shows convincingly that people did not simply swap illegal hiding for legal tax tricks one-for-one: their taxes went up and stayed up. But the paper goes further and says it can rule out any significant shift to legal tax avoidance. Its tests are not sharp enough for that. Some avoidance routes are measured imprecisely, moving abroad is barely measured at all, and the one clear response, setting up holding companies, is played down.

To work out how much of a non-European company's gain the EU could tax, the paper assumes that almost every such company makes 10% of its sales in the EU. Many of them, such as Indian and American electricity companies, sell almost nothing in Europe. Using realistic figures removes roughly EUR 6-17 billion from the EUR 65 billion estimate.

The essay says the richest 1% get 'almost 20%' of income and own 'almost 40%' of wealth. The wealth number counts people in a way (tax-filing units) that gives a higher figure than the way the income number counts them. On the same basis the authors' own figure is about 35%. Other researchers estimate roughly 31-34% for wealth and about 14% for income, though those estimates are disputed. The top 1% clearly hold a very large share, but readers are not told that these exact numbers are contested.

Governments, especially China's and Saudi Arabia's, own a big slice of the companies that would pay this tax. When a state-owned company pays the tax to its own government, no new money is raised: the government is paying itself. Taking this into account cuts the real yield by about one-seventh, although it stays roughly the same size.

The study finds no clear rise in multinationals' tax payments, but its measurement is too imprecise to rule out a sizeable rise, perhaps enough to recover about half of the tax the paper says Chile was losing. 'We could not detect an effect' is not the same as 'there was no effect'.

The paper's main table compares what each country's wealth tax raised with what a simple 1% tax on the richest 1% would have raised. Using the data source the paper names, the 'would have raised' numbers come out much lower than printed for Norway, Spain and Switzerland; only France matches. Corrected, Norway's tax raised about as much as the simple tax would, so the table no longer shows Norway's or Switzerland's taxes leaking, although other evidence suggests they did.

To decide whether a foreign owner lives abroad, the paper mostly relies on a rule of thumb: an owner whose address is unknown but who rents out all their Dubai flats is assumed to live abroad, and that pattern is then extended to everyone else. The rule is sensible but untested, so the $68 billion could plausibly be somewhat lower or higher (roughly $50-75 billion). Dubai would still stand far above the other cities compared.

The note says about half of all Russian household wealth is held abroad. The study it cites says about half of Russians' financial savings (bank deposits, shares, bonds) are abroad. Once homes, land and businesses are counted, the share held abroad is about a quarter. Russians' offshore wealth is still very large, but the note doubles its share.

The paper says that after its fixes the two studies agree, but it never re-runs the numbers with all the fixes together. Adding up the fixes that have actually been calculated closes about half of the gap. The rest closes only if every uncertain assumption is set the way that favours the authors.

The authors said their estimate of how much hidden income raises the top 1%'s share was, if anything, too low. Their own later work cut it by about a third, from 1.5 to about 1 percentage point, and their new range runs down to zero. The main reason is that redoing the total amount of hidden income with the IRS's newer method gave about a quarter less, the kind of downward revision the reply had dismissed.

For 13 EU countries without detailed multinational data, the note uses the profits of all domestically owned companies, including small, purely domestic firms that the minimum tax would never cover. This inflates the figures for countries such as Poland and Germany and slightly enlarges the estimated carve-out loss. Even removing this component entirely still leaves a loss of 13-17%.

For 13 EU countries, including Germany, Spain and Poland, the 'home-country' part of the estimate is calculated from the profits of all domestic companies, not just multinationals. For Poland this makes the figure at least four times too big; for Germany it may even be too small, so the EU total is probably somewhat too high, but by an uncertain amount.

The estimate assumes the government could tax billionaires on the full value Forbes magazine puts on their fortunes, losing nothing to disputes over what private companies are worth or to tax dodging. In 2019 and early 2021 the same authors assumed about 15% would be lost. Using their own assumption brings the yield down to roughly $0.77-0.87 trillion, so '$1 trillion' sits near the top of the plausible range rather than in the middle.

The chapter's key number, that the 400 richest Americans paid 23% of their income in taxes in 2018, less than anyone else, was an early estimate made before full data existed. When the authors later updated their own figures (in 2020, before the chapter appeared, and again in 2022 and 2026), the same group came out at 27-29%, about what the average American pays. A separate 2025 study by the authors using actual tax records for the 400 wealthiest finds about 24%, so the claim may still hold, but the chapter gives no warning that its number was this uncertain.

The chapter's most quoted claim, that in 2018 the 400 richest Americans paid a lower tax rate than any other group, depends on a gap of half a percentage point. That gap disappears or reverses with the authors' own updated figures, released before the chapter came out, or with the standard way of counting tax credits for low-income workers. What does hold is that the very richest pay noticeably less than the merely rich just below them.

About a fifth of the '$2 trillion from withholding' in the abstract comes from another part of the package: taxing gains when people die or give assets to charity. Some of that is paid by people well below the $50 million line. In a billionaires-only version, withholding itself raises about $440 billion, not $925 billion.

The chart says it compares the same tax measure in both countries, but the French line comes from a different calculation (working-age people only, measured against a broader income concept). When the same source's comparable numbers are used, richer people pay a clearly larger share of income in tax in both countries, so the claim that the tax systems are 'approximately proportional' does not hold as stated. The lecture's wider point that public spending does much of the equalising is not overturned, but this piece of support for it is weaker than presented.

The article's striking claim that the 400 richest Americans paid a lower tax rate in 2018 than the average American comes from an early version of the authors' data. In the updated version that the article uses for its income and wealth charts, the difference almost vanishes (27.2% against 28.3%), and in a 2022 update the richest 400 pay slightly more than average. A later study that matched the richest Americans to their actual tax records does support the claim, so it is not disproved, but the article's own evidence for it is fragile.

The paper says one group of critics mishandles income that never appears on tax returns, and that this explains about half of why their numbers differ. Checked against the government's accounts and the IRS audit study it relies on, the effect more plausibly explains a quarter to a third of the difference. The critics' method does still seem to understate the top 1%'s income somewhat, so the rebuttal points the right way but is roughly twice as strong as the evidence allows.

The memo says the IRS gets no balance sheets from foreign companies owned by Americans. In fact, Americans who control a foreign company or partnership must already send the IRS its balance sheet. The real gap is foreign firms that Americans own without controlling them, so the problem is smaller than the memo says, though it is real.

The note says its revised numbers now match the Federal Reserve's own wealth statistics almost exactly. The authors' own spreadsheet shows they still differ: in 2018 the richest 1% hold about 25% of their wealth in bonds and deposits in the authors' series versus 22% in the Fed's, and the two estimates of the top 1%'s share of all wealth differ by about 2 percentage points in most years before 2017. The series are close, but 'almost identical' and 'fully consistent' overstate the match.

The bill limits how much of a billionaire's gains can be taxed in one year to a quarter of their wealth above $1 billion, but the report read the limit as applying to the tax bill instead. As written, the bill would have raised about $9-10 billion for 2020, not $23 billion. Most of the rest would come due in 2021-22, and only from billionaires still living in New York, which undercuts the report's claim that the tax gives them no new reason to leave.

The authors say they knocked 20% off their estimate to allow for tax dodging. In their table, though, that cut is not applied to California's 99 billionaires, who hold more than a third of the taxable wealth. Correcting this trims the headline only a little, to about $6.9 billion. It does make the figures for a billionaires-only version of the tax roughly a fifth to a quarter too high.

The paper says the US could tax foreign tax-haven companies in proportion to their US sales, using reports that already exist. But those reports show where a company's subsidiaries are based, not where its customers are, and the agreements for sharing them between governments forbid using them to set tax bills. New reporting would be needed, so this part of the plan is harder to carry out than the paper suggests.

The 'about 25%' figure is not measured; it is simulated by assuming that 80% of the extra earnings of graduates of better colleges is caused by the colleges rather than by the kind of students they admit. Other research finds anything from almost no such effect to a large one. The simulated gain moves almost one-for-one with that assumption, and more than half of it comes from assuming that richer students who lose their places would earn less, so the true figure could easily be 10-20% rather than 25%.

The paper's headline says world inequality went up after 1980. In the authors' own numbers that is true only if you look at the share going to the very richest. By the most common overall yardstick (the Gini coefficient) world inequality was flat, and the poorer half of the world increased its share of income by about a fifth. So 'the top pulled away' is supported, but 'inequality rose' as a general statement is not.

Economists agree that US income inequality has risen since about 1980 but disagree about how much. A leading alternative estimate finds a rise only about half as large as the author's, partly because a 1986 tax reform changed how income was reported. The essay doesn't mention this debate. Most data sets, including the critics' own, still show today's inequality back near its level of the early 1900s, so the essay's basic message stands.

The essay blames rising inequality on a list of free-market policies, as if that were settled, without giving evidence. Research, including a study printed in the same journal issue, finds that changes in technology and in the demand for educated workers explain a large part of the growing wage gaps. The policies on the list did play a role, so the claim is incomplete rather than wrong, and the essay's main argument does not rest on it.

The column says its revenue figure already allows for wealthy people dodging the tax. In fact the calculation multiplies the tax rates by today's wealth and assumes nobody changes behaviour. Suppose the rich shield only 15% of their wealth, the figure the same authors assumed for a US proposal a year earlier. The tax then raises about 0.9% of GDP instead of 1.05%, and repaying the debt takes about 11 years rather than 10. With larger but plausible responses it takes 13-16 years. The tax would still raise a lot of money, but the precise promise is not backed by the calculation.

To project 30 years ahead, the model needs to know how much people value leaving wealth at death, and eight years of data cannot tell. The paper fills the gap by assuming that the tax cut does not change how fast people's wealth grows in their 80s. The evidence it cites for that assumption fits any value equally well, so the long-run numbers could easily be half as large, or somewhat larger.

When the paper's own most cautious assumptions are used together, counting hidden income slightly lowers the top 1%'s share of income (about 19.7% instead of 20.3%) rather than raising it. The paper shows its cautious assumptions only one at a time, so its flat statement that evasion makes measured inequality higher goes beyond its evidence. Its central estimate (about +1 point) is still a reasonable best guess.

The headline, that reshuffling students could cut the rich-child advantage by about a quarter, holds only if 80% of the earnings edge of graduates of better colleges comes from the college itself rather than from whom it admits. The paper’s test of that assumption is indirect. Later studies disagree: some find colleges matter about that much, others (in Texas) find far less. With more cautious assumptions the effect would be roughly 10–20% instead of 25%: still positive, but smaller.

The interview says the richest 1% of Americans went from owning 20% of all private wealth in the early 1980s to 40% today, a doubling. The authors' own published figures say it was about 24% then and about 39% recently: a big increase, but about 60% rather than 100%. The direction is right, but the size of the change is exaggerated by roughly a quarter or more.

The authors say a 6% tax on all income would by itself pay for health insurance for everyone. Their own spreadsheets show it raises roughly two-thirds to four-fifths of the cost they assign to it. Their overall plan still balances, but only because other new taxes chip in, or because of extra income tax on higher wages that the authors never count.

The famous 'billionaires pay less than the working class' result rests on a gap of about one percentage point. Count the tax credits that low-income workers receive through the tax system as reducing their taxes, as government agencies do and as the authors themselves did in a later version of their online calculator, and the working class pays less than billionaires. Other reasonable choices push the other way, so the honest conclusion is that the two groups pay roughly similar rates, not that billionaires clearly pay less.

One of the paper's two main sources, leaked client records from HSBC's Swiss private bank, suggests that more than half of all hidden wealth belongs to the richest 0.01% of households, but that result rests largely on a single Norwegian family with about $288 million at the bank, which the paper never mentions. Leaving that one family out lowers the paper's headline estimate of the share of taxes the very richest evade from about 24% to 21%, and undercuts the paper's claims that its different sources confirm one another. The main conclusion survives because the other main source, records of people who came forward in tax amnesties, does not involve this family and on its own gives 23%.

The note says the tax would completely close the gap between how fast billionaires' fortunes grow and how fast ordinary families' wealth grows. The authors' own simulation of the tax - the one drawn in the note's figure - shows the richest 400 still pulling ahead, only more slowly: about a third of the gap remains, and more if some evasion is allowed. The tax would shrink the gap a lot, but not eliminate it.

The $212 billion figure depends on a guess that the very rich would hide or shelter only 15% of their wealth. That guess came from averaging four foreign studies in a way that leaned toward small responses; for example, it used the small Danish estimate for the moderately rich rather than the much larger one the same study (co-written by Zucman) found for the very rich. Reasonable alternative assumptions give about a quarter less money in the first year, so the estimate is better described as optimistic than as middle-of-the-road.

The paper's summary says multinationals pay lower tax rates than local firms, but the paper never measures tax as a share of profit. Its own chart implies multinationals pay roughly the same tax per peso of reported profit (about 11% versus 10%). They pay less relative to their wage bill only because they report less profit relative to it. So this paper does not show the 'lower tax rate' claim, though it might hold in other data.

The paper says the richest tenth's share of Chinese wealth jumped from 40% to 67% in twenty years. But the early years come from one survey and the later years from a different one, and the correction applied to the surveys was quietly changed from year to year, which made the rise look smooth. Wealth very likely did become more concentrated; how much and when is far less certain than the paper suggests.

The paper's model says billionaires' fortunes could be taxed at about 6% a year before the tax starts bringing in less money, but the model assumes billionaires do nothing to shrink or hide their taxable wealth. Using the authors' own estimate of how much the rich hide, the best rate for revenue falls to about 4%, and lower if they also save less. So the claim that an 8-10% billionaire tax is about the best for revenue does not hold up, although taxes of a few percent would still raise substantial money.

The paper's 'world' wealth figures treat hundreds of millions of people in Eastern Europe and the former Soviet Union as if they were as rich as people in Britain, France and Spain, and convert currencies in a way the paper never mentions. With reasonable alternatives, the rise in the richest 1%'s share shrinks from about 5 points to between about 1 and 4 points, and the poorer three-quarters' share rises instead of staying flat. The richest 1%'s share still goes up in every version.

The authors' own method says people hide more wealth when the tax rate is higher, but they applied the same 15% to billionaires facing 3% or 6% rates. Applied consistently, most of the extra $0.9 trillion promised from the higher billionaire rate disappears; the effect on the main estimates is only a few percent.

The shortcut gets about the right total for the top 1% for partly the wrong reasons. It gives the richest group too large a share of the income that never shows up on tax returns, and it ignores that the people at the top of the full national-income ranking are not the same people as those at the top of the tax-return ranking. In the US data these two mistakes roughly cancel, so the match does not show that the authors' complex method boils down to two simple rules.

The authors' computer code quietly replaced the Middle East's average income for 1980-84 with a much lower number, and that number is contradicted by the data they were using. This makes the world's richest 1% look as if they started from a lower point in 1980, so the rise in their share of world income looks about a quarter larger than the data support. The headline 'twice as much' comparison barely changes.

About two-fifths of national income never appears on anyone's tax return, so the authors had to decide who receives it, and their choices tend to give more of it to the richest. With other reasonable choices, the rise in the top 1%'s share since 1980 ranges from about half as large to nearly as large as reported. Under the main critics' choices, the top 1% never overtakes the bottom half. The paper gives single numbers with no ranges, so figures like '81 times' are one estimate among several, even though the overall direction is not in doubt.

To estimate who gets the income that never shows up on tax forms, the authors had to make assumptions, and other researchers using different assumptions find that the rise in the share going to the richest 1% was only about half as big. Who is right is still debated; either way, the gap between the top and the bottom in the US widened.

The paper assumes that in every country the richest 0.01% of households own about half of all hidden offshore wealth, because that is what Scandinavian data showed. It says it checked that other assumptions make little difference, but those checks are not in its published appendix or data. With a plausible lower figure, the extra inequality it reports for France, Britain and Spain shrinks by half or more, and France no longer overtakes Scandinavia.

The paper says almost half of the fall in the foreign taxes paid by US multinationals came from moving more profits into tax havens. Redoing the calculation the paper describes, with the authors' own data, gives roughly one-sixth to one-quarter. Lower tax rates in ordinary countries did most of the work. The paper's numbers appear to measure how much profit was sitting in havens in 2015, not how much more was moved there after 2000.

The brief's best-known number, that 40% of multinationals' foreign profits are moved to tax havens, rounds up an estimate its source puts at 36%. About half of that estimate comes from the authors' own top-ups to tax havens' incomplete statistics. Other reasonable methods give figures nearer a quarter to a third; some company-level studies find even less, and one finds more. Profit shifting is clearly very large, but the 40% is probably too high and much less certain than the brief suggests.

The paper claims that if a tax on wealth were announced far in advance, people would react so strongly that the best tax would be zero. It uses this to argue for its own new method. In the paper's own model the reaction stays limited, and the best tax stays positive. The paper's main tax formulas are unaffected; one of its arguments for preferring its new method falls away.

Russian tax statistics group people by their gross takings: for a trader, total sales; for someone selling shares or a flat, the full sale price. To turn those takings into income, the authors assume everyone deducts 70% as costs. But most people in the top brackets are business owners and asset sellers who typically deduct 80-95%. With more realistic deductions, the share of income going to the top 1% falls from about 20% to somewhere between 14% and 19%. That is no longer clearly as high as in the US, though still far above what surveys show.

The authors' computer code quietly replaced the Middle East's average income for 1980-84 with a figure almost half the size of the official statistics they had loaded, without any explanation. Because the rich of the Gulf oil states were among the world's top earners in 1980, making them poorer in 1980 makes the world's richest look as if their incomes grew much faster afterwards. Undoing the change cuts the reported growth of the world's richest by about a fifth, although the richest 1% still gained about twice the poorer half's share of growth.

Russian tax records list only people's gross takings, such as total proceeds from selling shares or business turnover, so the authors had to guess how much of that is actual income. They assumed that 30% of the takings at the top are income, more than the tax records themselves imply, even though top filers are largely business owners and share sellers with large costs. With more realistic assumptions the richest 1% of Russians get roughly 16-19% of national income rather than 20%. Russia's lead over China shrinks by a quarter to two-thirds, but Russia stays ahead.

The chapter says it measures wages plus benefits plus self-employment pay. Its charts actually use a slightly different measure, which subtracts payroll contributions and gives each wife half of her husband's pension and Social Security. On the chapter's own description, women would earn about 55-56 cents per man's dollar instead of 57, and the typical man's earnings would have grown about 10% over 50 years rather than not at all. The main message (a large gap, slow growth for men) is unchanged.

The paper's key medium-run number relied on an early projection of 2015 incomes. Once the IRS's complete 2015 data arrived, top incomes turned out about 3% lower than projected. Rerun with those data, the paper's own calculation says about 31% of the expected extra revenue was lost to behavioral responses, not 'at most 20%'. Most of the money was still raised, so the broad message stands, but the headline figure does not.

The lecture argues that the rich cannot have been hiding much income in the 1960s, because their charitable donations, which lowered their taxes, were not unusually high then. But donations also depend on tax incentives and on changing habits of giving, and the data cover only the US after 1962. They cannot settle how much tax avoidance mattered, let alone over a whole century or in other countries. The finding that inequality rose is not affected; the claim that avoidance has been ruled out is.

Taken together, the poorer half of American families owns only about 1% of all wealth, so in that sense the comment is right. But it is not true that half of all families individually have nothing: the typical family had about $81,000 in 2013, and only about one family in eight had zero or negative wealth. The wording is loose, but it does not change the comment's point that wealth is extremely concentrated.

The chapter says that since 1980 the bottom 90% of Americans got only about a third of the growth in average income. That figure is computed correctly from the authors' data, but other credible ways of counting put it between about 36% and 50%. Every method agrees the bottom 90% fell behind after 1980; how far behind is less certain than the chapter suggests.

The studies show clearly that, near the ceilings and in the short run, employers do not push their share of these contributions onto the individual workers concerned. But for several countries the data cannot tell 'exactly the legal split' apart from 'employers pay somewhat more or less than their legal share', and one German period shows employers paying noticeably more. The paper presents the exact legal split as an established result, which overstates how precise the evidence is.

Another team using the same US tax records, but different assumptions about income that never shows up on tax forms, finds that the top 1%'s share grew only about half as much as this paper says and that the poorer half's pre-tax incomes grew by about 40% rather than not at all. Experts still disagree about who is right, so this is a serious but unsettled doubt about the size of the US numbers, not about whether inequality rose.

To count public services like schools, police and defence in people's incomes, the guidelines share them out in proportion to income and call this 'neutral'. In fact it is the choice that shows the least redistribution. Giving everyone the same amount would cut the top 1%'s US share from about 15% to 12.5% and raise the poorer half's from 20% to 25%, and the 2016 text did not show this alternative.

The method infers how much each family holds in bank accounts and bonds from the interest it reports, assuming everyone earns the same interest rate. In reality richer families earn noticeably higher rates, because more of their savings are in bonds, which pay more than bank deposits, and the gap widened as deposit rates fell to almost nothing by 2012; as a result the method credits the rich with too much of this wealth and the middle class with too little. Fixing this still leaves a large rise since the late 1970s in the share owned by the richest 0.1%, but puts that share at roughly 16-20% in 2012 rather than 22%, removes most of the increase after 2000, and leaves it clearly below the 1929 peak of about 25%.

The paper's main selling point is that fairness ideas such as 'people who already pay a lot of tax deserve more' or 'focus on the poor' can be plugged straight into the standard tax formulas. A later published proof shows that when people's earnings respond to taxes, plugging in these particular ideas can produce circular judgments: policy A beats B, B beats C, and C beats A. The 'best' tax it recommends then reflects no consistent set of values. The method still works for more traditional fairness views, and newer work suggests a fix, but the paper's broadest claim is overstated.

To make the state figures match the official national figures, every state's numbers were multiplied by the same correction each year. The note calls this correction minor, but for some groups and years it is large: up to 32% for the top 10% in the 1930s, and it more than doubles the top 0.01% in the late 1980s. In those cases much of what looks like each state's own history is really the national trend copied into every state. For the most-used figure, the top 1%, the correction is small.

The chapter's best-known inheritance number, that roughly two-thirds to three-quarters of French wealth in 2010 was inherited, comes from a series built by multiplying another estimate by round factors typed into a spreadsheet, not from the micro data the text describes. The authors' own reproducible estimate is closer to one half. The broad pattern (high, then low, then rising) still holds, but the 2010 level is much less certain than presented.

The chapter says the richest 1% went from owning under a quarter of US household wealth in the late 1970s to 42% in 2012. That figure came from a then-new method that is very sensitive to how much interest rich people earn on their savings; when this is measured better, including by the authors themselves, the rise is roughly a fifth to two-fifths smaller. Wealth did become much more concentrated, but by less than the chapter's figure implies.

The chapter says poor children's chances of moving up stayed the same, or even improved slightly, for children born between 1971 and 1993. Newer and much larger data, from a team led by the chapter's own lead author, show that for children born between 1978 and 1992 the link between parents' and children's incomes actually got about a tenth stronger. That is a modest fall in mobility, not stability. The forecasts the chapter relied on for the youngest children pointed the wrong way, although the change is still small compared with the differences between places.

The summary box says top earners went from earning 1.6 times as much as a middle earner in 1978 to three times as much in 2013. The briefing's own chart and the official data show the ratio went from about 1.65 to about 2. The gap did widen, but by about a fifth rather than nearly doubling, so the summary overstates the change about four-fold.

The authors tried to show that distrust of government makes people oppose redistribution, by showing some respondents negative material about government. But that material also changed other opinions, for example whether the rich deserve their income. So the experiment measures the effect of the whole package, not of trust alone. A later study that raised and lowered trust found almost no effect on support for redistribution.

The paper's famous 83% top tax rate depends on one number: how much top earners' extra income reflects real extra work and output rather than grabbing a bigger slice. The paper's own growth data are far too noisy to pin that number down. They are equally consistent with the conventional answer of about 57%, so the 83% is an illustration of what follows if the authors' guess is right, not a measured result.

The method turns interest income into wealth by assuming everyone earns the same interest rate. After 2008 ordinary savers earned almost nothing, while the very rich still earned real interest on their bonds. The method therefore credits the rich with far more savings than they probably had. Using the authors' own estate records and their own later revisions, the 2007-2012 jump in the richest families' share of wealth was roughly a quarter to well over half smaller than reported, though it was still a rise.

The Reporter says tax breaks for foreign footballers pull in stars but push out weaker foreign players and some local players. That result depends on assuming weaker and stronger players are equally attached to their home country, which the data clearly reject, and on the Dutch league. With a more realistic assumption weaker foreign players are attracted too, and without the Netherlands there is no measurable push-out of local players; only the finding that stars respond more holds up.

The paper says new saving explains about 60% of the rise in wealth relative to income since 1970, and rising prices of existing assets about 40%. That split only works if one credits saving with a large amount of research spending (which official statistics did not then count as saving) - several times what U.S. official figures support - and also counts statistical discrepancies and reclassifications as saving. With realistic figures the split reverses, so most of the rise came from higher prices, mainly for homes and land; this is the most serious problem found, weakening the paper's explanation of why wealth grew but not the finding that it grew.

The paper's main story is that evaders moved money from havens that signed treaties to havens that did not. That pattern largely disappears when the authors' own model is made stricter (a version printed in their drafts but cut from the published table). It is also absent in the eight havens whose data are now public. It may hold for a few small havens, but it is not firmly demonstrated.

The paper says about two-thirds of the fall in US companies' tax rate came from shifting profits to tax havens, but it never calculates this: it assumes the other causes explain only a small part. Using the paper's own numbers, the growth of profits booked in tax havens explains only about a quarter of the fall, and a study of company accounts found that purely domestic firms' tax rates fell about as much. Profit shifting mattered, but it was not the main cause the abstract describes.

The paper says the richest tenth's share of European wealth had already fallen from about 90% to under 60% by the 1950s-60s. Its own chart shows the share was still about 75% in the 1950s and 68% in the 1960s, and only dropped below 60% around 1980. Europe's wealth equalization was real, but slower and more of it happened in peacetime than the text says.

The paper's headline is that each 1% rise in the share of pay that foreigners keep after tax raised their number by 1.5-2%. That comes from one way of choosing comparison groups. The paper's own alternative check, built to rule out a general widening of top pay, gives about 1-1.3% and shows the effect fading over time, yet the text describes it as nearly the same. The tax break clearly attracted many foreigners, but probably by less than the headline says.

The paper says that places with more upward mobility have less segregation, less inequality, better schools, stronger community ties and more two-parent families, and that this holds however the data are sliced. That is true when every area counts equally, including many tiny rural ones. When areas are weighted by how many people live there, the links are roughly half as strong and several disappear, and the paper's own appendix shows this. The broad pattern still holds, but it is much weaker in the big cities where most children grow up than the text suggests.

For children born after 1982 the paper could not yet see adult incomes. For the youngest (born 1987-93) it predicted their future mobility from whether they enrolled in college at 19, assuming the link between college and later income never changes. The headline of 'no change, and at most a tiny one' depends on that prediction: the incomes actually measured are too noisy to rule out even the 20% worsening that some had predicted.

The paper's most careful statistical test of whether taxes attract foreign footballers essentially relies on one country, the Netherlands, and on an unstated assumption about which foreign players there received a tax break. Remove the Netherlands, or change that assumption, and this test finds little or no effect. Other evidence in the paper still points to a real effect, but its size is much less certain than the headline suggests.

The chapter suggests the best top income tax rate might be above 80%, because cutting top rates did not seem to boost economic growth across rich countries. But when the authors' own statistical analysis is re-run, the data turn out to be far too noisy to tell. They are about as consistent with a best rate in the mid-60s, close to the conventional view the chapter questions, as with one near 100%. The '80%-plus' figure is a hypothesis, not a finding.

The book's $190 billion yearly tax loss assumes hidden money earns 8% a year and that all of that gain is taxed every year at full rates, and it uses the highest inheritance-tax rates as if nobody qualified for exemptions. The author himself used 6%, 7% and 6.5% in other editions. With more realistic assumptions the loss is still large, but more like $110-170 billion than 'at least $200 billion'.

The 2002 working paper's table said, for example, that a fortune at the top-1% cutoff would be taxed down to the exempt level in about 9 years. The model's own equations make that impossible: it takes at least 43 years, and about 47 when the model is re-solved. Across the table the times are understated by 2 to 8 times. The 2013 journal version dropped the table, so its theorems are unaffected, but no corrected timeline was ever published.

The paper says more than 15% of America's national income moved from the bottom 90% to the top 10% over 30 years, but its figures measure only income reported on tax returns, which is about two-thirds of national income. Measured against all national income, including in the authors' own later estimates, the shift was roughly 8-11%: still very large, but a third to a half smaller than claimed, and about twice (not four times) the size of the trade deficit the paper compares it with.

The paper's figures are correct for the income people report on tax returns, but that measure leaves out a lot of income. When the same authors later counted all national income, the US increase was about a third smaller and France's increase was much bigger. The gap between the US and continental Europe is therefore far less dramatic than the paper's numbers suggest, although the US and other English-speaking countries still saw larger rises.

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.

The paper says that once hidden offshore money is counted, the eurozone stops being a net debtor and becomes a net lender to the rest of the world. That only works if eurozone residents own about half of the hidden money held outside Switzerland. The author's own later estimate is about a quarter. With that estimate the eurozone comes out roughly even, and on the paper's own assumptions it is still a net debtor in 2008. The hidden wealth clearly makes the eurozone look much less indebted, but the switch to net lender is not established.

The study shows clearly that low-paid workers were upset to learn what colleagues earn. It also says better-paid workers were not affected at all, and so people mind falling behind more than they enjoy being ahead. The data are too imprecise to show that: they are also consistent with better-paid workers becoming happier by about as much as low-paid workers became unhappy. 'No measurable effect' was treated as 'no effect'.

The reply says the tax advantage married one-earner couples get today is small except for the very rich. In fact it is worth about 5% of earnings for a modest earner (about EUR 100 a month at EUR 2,000 gross) and 6-8% higher up. What keeps most of these couples from losing much under the reform is its general tax cut, not a small advantage.

The note's figures are right, but the main public database that hosts them now shows a different set of numbers. Those numbers include one-off profits from selling land and shares, yet are labelled as excluding them, and in 2009 they switch to another team's series. Anyone downloading 'Japan's top 1% share' there gets figures up to about a fifth too high in the 2000s, far higher in the late-1980s bubble, and a fake drop in 2009.

The chapter's numbers depend heavily on one input: how much the rich cut the income they report when their tax rate goes up. The authors call their main figure 'midrange' and a higher figure a 'conservative upper bound'. But the studies they cite, including their own, allow noticeably larger responses. With those, the rate that raises the most money would be closer to 45-55% than 73%, and their 49% suggestion would be about right rather than 'far more likely too low'. A higher top rate would still raise revenue; how high it should go is much less certain than the chapter says.

The paper says that if top earners respond twice as strongly to taxes, the hidden economic cost of raising each extra dollar from them doubles, from about 34 cents to 68 cents. By the paper's own formula it roughly triples, to about $1.03. The paper's main figures are fine, but this one 'what if' check makes higher top tax rates look less costly than the paper's own method implies.

The notes say inequality fell abruptly during World War II and take that as evidence the war itself drove the change. A careful re-estimate from the same old tax records finds that pre-war inequality was lower, and that much of the fall came earlier, during the Great Depression. On that estimate the wartime drop is about a third smaller overall and far less sudden. The long-run 'U' pattern still holds, but the war-shock story is much less certain.

The reply says giving the RSA minimum income to each spouse individually would add only 'a few thousand' people. On the authors' own data and rules, between about half a million and three million people would become eligible, most of them women at home. The cost would be roughly EUR 1-8 billion a year, and a government estimate put it near EUR 9 billion. The cost side of the authors' argument is therefore wrong by a factor of hundreds.

The paper's best-known US numbers measure income as reported on tax returns. On the authors' own later and broader measure, which also counts income that never appears on tax returns, the fall in the richest 1%'s share of income between 1928 and 1949 and its rise between 1976 and 2007 are both roughly 40-45% smaller, and the US rise is about the same as Britain's and Canada's rather than larger. This is the most important problem found, but it scales back the US numbers rather than overturning the story: the US pattern of high, then low, then high again survives, and the big early-century falls in other countries are just as large on the broader measure.

To show its conclusion is safe, the paper takes the largest figure it reports for how strongly top earners react to tax rises and calls it a cautious upper limit. That figure comes from a single study and is very imprecise. The true reaction could be big enough that the 2011 US top rate was already close to the rate that raises the most money. Higher rates still look justified with typical figures, but calling the conclusion 'robust' goes further than the evidence.

The book counts pension and unemployment contributions as taxes. Yet they mostly buy workers their own future pensions and benefits, and the very rich pay almost none of them because they are capped and do not apply to investment income. Counted instead as deferred pay, as the authors' own unpublished tables do, they leave the lowest-paid full-time workers paying about 27% of their income, less than the richest (33%). The fall in rates at the very top is then about half as large.

Companies pay corporate tax, and the model has to decide which households ultimately bear it. It spreads the tax over everyone with savings, including holders of tax-free passbook accounts, rather than over shareholders, and it gives no reason for this choice. If shareholders bear it, as the authors themselves have described the tax, the richest group's rate rises from about 33% to 39% and the drop at the top shrinks by about a third. If all owners of capital bear it, homeowners included, the drop grows somewhat.