How is it possible to measure the distribution of income among the inhabitants of a given country? One such measure is the Gini index, named after the Italian economist Corrado Gini who first devised it in 1912.
We first rank all households in a country by income and then we compute the percentage of households whose income is at most a given percentage of the country's total income. We define a Lorenz curve y = L(x) on the interval [0, 1] by plotting the point (a/100, b/100) on the curve if the bottom a% of households receive at most b% of the total income. For instance, in Figure 1 the point (0.4, 0.12) is on the Lorenz Clint for the United States in 2008 because the poorest 40% of the population received just 12% of the total income. Likewise the bottom 80% of the population received 50% of the total income, so the point (0.8.0.5) lies on the Lorenz curve. (The Lorenz curve is named after the American economist Max Lorenz.)
Figure 2 shows some typical Lorenz curves. They all pass through the points (0, 0) and (1, 1) and are concave upward. In the extreme case L (x) = x. society is perfectly egalitarian: The poorest a% of the population receives a% of the total income and so everybody receives the same income. The area between a Lorenz curve y = L (x) and the line y = x measures how much the income distribution differs from absolute equality. The Gini Index (sometimes called the Gini coefficient or the coefficient of Inequality) is the area between the Lorenz curve and the line y = x (shaded in Figure 3) divided by the area under y = x.
1. (a) Show that the Gini index C is twice the area between the Lorenz curve and the line y = x, that is G = 2 0∫1 [x - L(x)] dx
(b) What is the value of G for a perfectly egalitarian society (everybody has the same income)? What is the value of G for a perfectly totalitarian society (a single person receives all the income?)
2. The following table ((keyed from data supplied by the US Census Bureau) shows values of the Lorenz function for income distribution in the United States for the year 2008.
|
x
|
0.0
|
0.2
|
0.4
|
0.6
|
0.8
|
1.0
|
|
L (x)
|
0.000
|
0.034
|
0.120
|
0.267
|
0.500
|
1.00
|
(a) What percentage of the total US income was received by the richest 20% of the population in 2008?
(b) Use a calculator or computer to fit a quadratic function to the data in the table. Graph the data points and the quadratic function. Is the quadratic model a reasonable fit?
(c) Use the quadratic model for the Lorenz function to estimate the Gini index for the United States in 2008.
3. The following table gives values for the Lorenz function in the years 1970, 1980, 1990, and 2000. Use the method of Problem 2 to estimate the Gini index for the United States for those years and compare with your answer to Problem 2 c). Do you notice a trend?
|
x
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0.0
|
0.2
|
0.4
|
0.6
|
0.8
|
1.0
|
|
1970
|
0.000
|
0.041
|
0.149
|
0.323
|
0.568
|
1.000
|
|
1980
|
0.000
|
0.042
|
0.144
|
0.312
|
0.559
|
1.000
|
|
1990
|
0.000
|
0.038
|
0.134
|
0.293
|
0.530
|
1.000
|
|
2000
|
0.000
|
0.036
|
0.125
|
0273
|
0.503
|
1.000
|
4. A power model often provides a more &carat fit than a quadratic model for a Lorenz function. If you have a computer with Maple or Mathematics fit a power function(y = ax') to the data in Problem 2 and use it o estimate the Gini index for the United States in 2008. Compare with your answer to pans (b) and (c) of Problem 2.