Figure 1. Lorenz Curve and Gini Index

To compute the Gini index, we first measure the area between the Lorenz Curve and the y=x equality line. This area is divided by the entire area below the y=x line (which is always exactly one half). The quotient is the Gini index, a measure of inequality. In other words, the Gini index is the area shaded in pink divided by the total of the areas shaded in pink and light blue-green. In some examples we have a formula, y = f(x), for the Lorenz curve. The areas are then computed using integrals. The computation of the Gini index for that case is summarized by a single formula

For a perfectly equal distribution, there would be no area between the y=x line and the Lorenz curve -- a Gini index of zero. For complete inequality, in which only one person has any income (if that were possible) the Lorenz curve would coincide with the straight lines at the lower and right boundaries of the curve, so the Gini index would be one. Real economies have some, but not complete inequality, so the Gini indices for real economic systems are between zero and one.

Share of Aggregate Income, US Households

This data is from Money Income in the United States: 2001, U.S. Census Bureau. The year 1968 is the year in recent history during which the income distribution in the U.S. was most equal.

Sample computations:

Finding the Lorenz curve: We will use the 1968 data. The first thing we need to do is compute the cumulative percentages. By adding, starting at the leftmost, column we find:

Cumulative Fraction of Aggregate Income, US Households, 1968

This gives us six points on the Lorenz curve; (0, 0), (0.2, 0.042), (0.4, 0.153), (0.6, 0.328), (0.8, 0.572), (1,1). The coordinates of the first and last points are forced by the definitions; the other points summarize our data. The points are plotted in the graph below and are connected by a smooth curve. That curve is the Lorenz curve. (Actually it is the approximation to the curve built from 6 points.) The diagonal line is where the curve would be if there were absolute equality in the distribution of income.

Finding the Gini Index: The first step in computing the Gini index is to find the area under the Lorenz curve between x = 0 and x = 1. The area under the curve is given by an integral but we do not have a formula for the curve, we just have a few data points. Hence we will do numerical integration ( = approximate integration). The first way we do the approximation is to compute the right sum. In each of the five vertical strips we select the rightmost point of the curve. We draw a horizontal from that point to the left edge of that vertical strip. This gives us the horizontal lines for the tops of our rectangles. The area under those rectangles is our estimate of the area under the curve. The area enclosed by the rectangles is

(0.2 × 0.042) + (0.2 × 0.153) + (0.2 × 0.328) + (0.2 × 0.572) + (0.2 × 1.00) = 0.419

This estimate is the right sum; see the following figure:

The estimate for the area under the Lorenz curve using left endpoints, the left sum, is

(0.2 × 0) + (0.2 × 0.042) + (0.2 × 0.153) + (0.2 × 0.328) + (0.2 × 0.572) = 0.219

Clearly from the pictures the right sum overestimates the area and the left sum underestimates the area. Hence we get a better estimate by averaging the two:

(0.419 + 0.219)/2 = 0.319.

The Gini index is defined as the area between the diagonal line y=x and the Lorenz curve y=f(x) (the shaded area in the figure below). The triangular area under the diagonal y=x has size 0.5 (it is ½ of a 1 × 1 square). The area under the Lorenz curve that we just estimated is 0.319. So our estimate of the shaded area is 2(0.5 - 0.319) = 0.362 which is, to the accuracy our data allows us, the Gini Index.

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