## Tuesday, November 1, 2011

### 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."