conversableeconomist

Lorenz curves and Gini coefficients: CBO #3.

This is the third of three posts based on the the Congressional Budget Office report,  \”Trends in the Distribution of Household Income Between 1979 and 2007.\”  The first was Incomes of the Top 1%, and the second was about Federal Redistribution is Dropping.

This post focuses on explaining some basic tools for measuring inequality. The Lorenz curve offers an intuitively clear picture of inequality. The Gini coefficient, which is based on the curve, offers a way of measuring inequality across the income distribution as a single number–and thus is often used in graphs and figures about inequality. The CBO report has a nice clear explanation of these topics.

The Lorenz curve

The Lorenz curve was developed by an American statistician and economist named Max Lorenz when he was a graduate student at the University of Wisconsin. His article on the the topic

\”Methods of Measuring the Concentration of Wealth,\” appeared in Publications of the American Statistical Association , Vol. 9, No. 70 (Jun., 1905), pp. 209-219. The CBO report explains it this way:  

\”The cumulative percentage of income can be plotted against the cumulative percentage of the population, producing a so-called Lorenz curve (see the figure). The more even the income distribution is, the closer to a 45-degree line the Lorenz curve is. At one extreme, if each income group had the same income, then the cumulative income share would equal the cumulative population share, and the Lorenz curve would follow the 45-degree line, known as the line of equality. At the other extreme, if the highest income group earned all the income, the Lorenz curve would be flat across the vast majority of the income range,following the bottom edge of the figure, and then jump to the top of the figure at the very right-hand edge.

Lorenz curves for actual income distributions fall between those two hypothetical extremes. Typically, they intersect the diagonal line only at the very first and last points. Between those points, the curves are bow-shaped below the 45-degree line. The Lorenz curve of market income falls to the right and below the curve for after-tax income, reflecting its greater inequality. Both curves fall to the right and below the line of equality, reflecting the inequality in both market income and after-tax income.\”

The Gini coefficient

The Gini coefficient was developed by an Italian statistician (and noted fascist thinker) Corrado Gini in a 1912 paper written in Italian (and to my knowledge not freely available on the web). The intuition is straightforward (although the mathematical formula will look a little messier). On a Lorenz curve, greater equality means that the line based on actual data is closer to the 45-degree line that shows a perfectly equal distribution. Greater inequality means that the line based on actual data will be more \”bowed\” away from the 45-degree line. The Gini coefficient is based on the area between the 45-degree line and the actual data line. As the CBO writes:

\”The Gini index is equal to twice the area between the 45-degree line and the Lorenz curve. Once again, the
extreme cases of complete equality and complete inequality bound the measure. At one extreme, if
income was evenly distributed and the Lorenz curve followed the 45-degree line, there would be no area
between the curve and the line, so the Gini index would be zero. At the other extreme, if all income was
in the highest income group, the area between the line and the curve would be equal to the entire area
under the line, and the Gini index would equal one. The Gini index for [U.S.] after-tax income in 2007 was
0.489—about halfway between those two extremes.\”

conversableeconomist

Federal Redistribution is Dropping: CBO #2

This is the second of three posts on the recent Congressional Budget Office Report \”Trends in the Distribution of Household Income Between 1979 and 2007.\”  The first post on Incomes of the Top 1% is here, while the third explains the concepts of the Lorenz Curve and the Gini Coefficient.

The federal government can redistribute income in two ways: by taking those with high incomes relatively more, and by making transfer payments to those with lower income. Using the Gini index (explained in the third post in this grouping) as a measure of inequality, CBO reports: \”The dispersion of after-tax income in 2007 is about four-fifths as large as the dispersion of market income. Roughly 60 percent of the difference in dispersion between market income and after-tax income is attributable to transfers and roughly 40 percent is attributable to federal taxes.The redistributive effect of transfers and federal taxes was smaller in 2007 than in 1979 …\”

Redistribution through federal taxes

Here are three figures showing average federal tax rates paid by the top 1% of the income distribution in each year from 1979 to 2007, by the 81st to 99th percentiles, the 21st to 80th percentiles, and the lowest 20%. The first graph shows average payments as a share of income for individual income tax, the second shows payroll taxes, and the third shows all federal taxes combined. Here are a few patterns that jump out.

  • The top 1% pays more of its income on average in income taxes, but much less in terms of payroll taxes. Of course, this is the income on which Social Security payroll taxes must be paid is  capped, so such taxes are a smaller share of income for those with very high incomes. 
  • With income taxes, the lowest quintile pays on average a negative tax rate: that is, with refundable tax credits, they receive more from the federal government through the tax code than they pay. 
  • Total taxes paid as a share of income have dropped off somewhat for all groups in the last decade or so; if one looks back to the mid-1990s, the drop in tax rates for the top 1% looks larger than for other groups.
  • The overall patterns here seem to be that the federal tax code as a whole became less progressive in the 1980s, more progressive in the 1990s, and since then has either not changed or become slightly less progressive, depending on what statistical measure one chooses to emphasize.

These sorts of graphs always turn my mind to Warren Buffett, and his claim that he pays less in taxes than his secretaries. For example, see Buffett\’s August 14 article, \”Stop Coddling the Super-Rich,\” in the New York Times, where he writes: \”Last year my federal tax bill — the income tax I paid, as well as payroll taxes paid by me and on my behalf — was $6,938,744. That sounds like a lot of money. But what I paid was only 17.4 percent of my taxable income — and that’s actually a lower percentage than was paid by any of the other 20 people in our office. Their tax burdens ranged from 33 percent to 41 percent and averaged 36 percent.\”
What Buffett pays as a share of income sounds plausible to me: he is well-known for taking a fairly small annual salary and then receiving most of his income in the form of gains from his investments. Because many of the investments have been held for long periods of time, they are subject to a lower capital gains tax rate. But the tax burdens that Buffett claims for his staff look unrealistically high. Let\’s say his office staff are in the 81st-99th percentiles of the income distribution. Average tax rates for that group are in the range of 22-23% in recent years. Buffett\’s staff might face a marginal tax rate might be 36% or 41%, depending on rules bout phase-outs of deductions and the like. But if Buffett\’s staff really are paying an average federal tax rate of 36% as a share of total income, they need access to better accountants or tax lawyer. 

 Redistribution through federal transfer payments

Federal government transfer payments are about 10-12% of household market income from 1979 to 2007. Spending in this category is heavily driven by Social Security and Medicare. In recent years, about half of federal transfer spending is Social Security. A third is health-related programs like Medicare and Medicaid. The rest is programs like unemployment insurance and welfare.

The share of federal transfer spending on the elderly is rising. In 1979 about 62% of all federal transfer payments went to elderly childless households, while about 19% went to nonelderly childless households and another 19% to households with children. By 2007, 69% of all federal transfer payments went to elderly childless households. The share going to nonelderly childless households stayed about the same, and the share going to households with children fell to about 11%.

Of course, Social Security and Medicare are not means-tested programs, so as they took a larger share of the federal transfer pie, the share going to the poor declined. Not coincidentally, back in 1979 about 54% of federal transfers went to households in the lowest quintile of income; by 2007, only about 36% of federal transfers went to households in the lowest quintile of income.

Summing up the redistribution by federal taxes and transfers
Here\’s a final figure showing how federal transfers and taxes affect income inequality, which is measured by the percent amount that these policies reduce the Gini index of inequality. The extent to which these policies reduce income inequality dropped in the 1980s, rose in the early 1990s, dropped in the late 1990s, rose in the early 2000s and has fallen since then. Interestingly, the diminished effect of federal redistribution since the mid-1990s is, by this measure, much more traceable to the changes in transfer payments than to the changes in progressivity of taxes.