Brazil's measured inequality depends more on the data source than on the index

There is a long-running argument in development economics about which single number should stand for a country's inequality. In 2013 Alex Cobham and Andy Sumner proposed that the Gini coefficient give way to what they named the Palma ratio: the income share of the richest tenth divided by that of the poorest four tenths.¹ The Gini, they argued, is opaque to non-specialists and insensitive to the parts of a distribution people actually argue about.¹ The quarrel matters, because the measure a government reports is the measure it is judged by. But it is worth settling in the right order, and the order is this paper's subject.

The Gini begins with a picture. Rank every household from poorest to richest, plot the cumulative share of income they hold against the cumulative share of households, and you have the Lorenz curve, first drawn in 1905. Under perfect equality it is the diagonal; otherwise it sags below, and the Gini is twice the area of the sag, running from nought for complete equality to one for complete concentration. Corrado Gini set it out in 1914, calling it a concentration ratio.² Building it on ranks has a consequence easily passed over: the index responds to a transfer according to how many people stand between giver and receiver in the queue, not to how poor either of them is. A sum moved from the sixtieth percentile to the fortieth shifts it appreciably; the same sum moved between two households three percentiles apart in the bottom decile barely registers.

Anthony Atkinson noticed that in 1970: the Gini, he wrote, attaches more weight to transfers affecting middle income classes. His response was not to discard it but to make its buried judgement explicit.³ He built a measure from equally distributed equivalent income: the income per head which, shared equally, would leave a society exactly as well off as the unequal distribution does.³ The gap between that and actual mean income is the share of national income a society would surrender to buy perfect equality, and computing it means choosing a parameter that states how much extra weight the poorest carry.³ Every index of inequality holds a view about where a change matters most. Atkinson's puts the view in a dial the reader can see.

The sensitivity complaint has since been tested and partly defeated. Joseph Gastwirth asked in 2017 whether the Gini really is overly sensitive to the middle, and found on the comparisons he ran that a transfer to a middle-income household moved it less than the same transfer to a low-income one.⁴ The structural property survives him, since the movement still depends on the ranks of donor and recipient, but the strong charge that the Gini is nearly blind to the poor does not.⁴ The case for the tails deserves full strength too: Cobham and Sumner reported that the Gini is almost entirely accounted for by two points of a distribution, and that between 1979 and 2010 the Palma ratio for American post-tax income rose by 43 per cent against 21 per cent for the Gini.¹

Now measure the other term. Brazil is the place to do it, because more than one serious institution publishes a number for it. The World Bank's poverty and equity brief reported that Brazilian inequality fell sharply in 2020, to a Gini of 48.9 on the hundred-point scale from 53.4 the year before, and credited Auxilio Emergencial, the pandemic cash transfer.⁵ The Bank's technical note on how it reached the 2021 figure is the more interesting document. Auxilio Emergencial reached 36 million people that year; the survey the estimate rests on observed 8 million of them, so the team adjusted its estimates using the programme's eligibility rules rather than what respondents had reported.⁶ A measure of the distribution of income was computed partly from a model of who should have received a transfer.

Then take the same country and year from FGV Social in Rio de Janeiro, whose Wealth Map study links the household survey to personal income tax records, so that income declared to the tax authority, including financial gains that cluster at the top, is counted. On the survey alone the study's Gini fell from 0.6117 in 2019 to 0.6013 in 2020, agreeing in direction with the World Bank. On the survey linked to the tax files it moved from 0.7066 to 0.7068: flat, or fractionally the wrong way.⁷ Its own explanation: the richest one per cent lost about 1.5 per cent of their income that year, the middle 4.2 per cent.⁷ A measured fall in inequality produced by the middle losing ground is a different event from one produced by the bottom gaining, and a Gini reports the two identically.

Set the figures side by side and the gap is about twenty-two points for one country in one year, from 48.9 to 70.68 on a common scale.⁸ No disagreement between the Gini and the Palma approaches that, because rival indices are applied to the same numbers. The comparison is of course unfair, and its unfairness is the finding. These are not two estimates of one quantity: one ranks households by income per head, the other ranks people by a broader income including gains the survey never sees. But a reader meeting the phrase Brazil's Gini coefficient in a newspaper or a ministry's strategy document has no way to know which chain of choices produced it, and those choices are made upstream of the index and disclosed, if at all, downstream of the headline.

The stronger objection is that this proves too much: within one consistent series the concept is fixed and the index is the only choice left, so the 43 against 21 result stands undisturbed. That is right, and it concedes the ordering. A country that adopted the Palma ratio tomorrow and went on computing it from an uncorrected household survey would have improved the smaller term and left the larger alone. It would also have moved its headline onto the income share of the top decile, the part of a survey least reliably observed, since the richest households are likeliest to refuse the interview or understate what they hold.⁹ A tail-sensitive index is the better instrument only where the tail is well seen.

So the conclusion is an order of operations rather than a verdict on the Gini. State the concept out loud: whose income, which income, before or after the state. Then find the missing top, from tax records where a country keeps them, which is the step FGV took. Only then is it worth asking whether the Gini, the Palma ratio or an Atkinson index with a declared aversion parameter best summarises what remains. The ordering is testable: compare, across countries, the spread attributable to the index with the data held fixed against the spread attributable to concept and source with the index held fixed. If the first is larger, this paper is mistaken. Brazil in 2020 suggests the contest will not be close.

Notes

1Alex Cobham and Andy Sumner, "Is It All About the Tails? The Palma Measure of Income Inequality," Center for Global Development Working Paper 343, September 2013, for the definition of the Palma ratio, the claim that the Gini is almost entirely accounted for by two points of a distribution, the 43 against 21 per cent comparison for American post-tax income, and the authors' own preference for several measures rather than one. CGD working paper; SSRN record

2Simone Pellegrino, "The Gini Coefficient: Its Origins," working paper, University of Turin, on Gini's 1914 memoir, the 1905 Lorenz curve, and the confusion between the 1914 concentration ratio and the mean difference introduced in Gini's 1912 book. University of Turin repository copy

3A. B. Atkinson, "On the Measurement of Inequality," Journal of Economic Theory 2 (1970), 244-263, for equally distributed equivalent income, the inequality aversion parameter, and the statement that the Gini coefficient attaches more weight to transfers affecting middle income classes while the standard deviation weights transfers at the lower end more heavily. Full text hosted at Stanford

4Joseph L. Gastwirth, "Is the Gini Index of Inequality Overly Sensitive to Changes in the Middle of the Income Distribution?," Statistics and Public Policy (2017), on the rank dependence of the index and on the finding that a transfer to a low-income household moved it more than the same transfer to a middle-income household. Publisher's open-access text

5World Bank, Brazil poverty and equity brief, for the 2020 Gini of 48.9 against 53.4 in 2019, the rebound to 52.9 in 2021, and the poverty figures at the 6.85 dollar a day line. World Bank Brazil poverty and equity brief

6World Bank, Brazil 2021 data update, on Auxilio Emergencial covering 36 million people in 2021 against 8 million observed in the household survey, and on the resulting adjustment using the programme's eligibility rules. World Bank technical note

7FGV Social, Mapa da Riqueza no Brasil, coordinated by Marcelo Neri, for the survey-only Gini of 0.6117 in 2019 and 0.6013 in 2020, the tax-linked Gini of 0.7066 and 0.7068, and the income changes of the richest one per cent and of the middle. FGV Social project page; study text

8The twenty-two point gap is this paper's own subtraction, 70.68 less 48.9, performed on the two cited figures. The two are not measures of the same income concept, which is the point the paragraph goes on to make rather than a defect in the arithmetic. Sources as at notes 5 and 7: World Bank brief; FGV Social

9Antonio David and others, "Inequality in Brazil: A Regional Perspective," IMF Working Paper WP/17/225, on the under-reporting of top incomes in the Brazilian household survey and the use of income tax returns to correct it. IMF working paper