The Lorenz curve is a graphical representation for the distribution of the cumulative share of income across successive income intervals. The Gini coefficient is a measure of inequality in the income distribution.
Data collection:
(1) Separate population by income and sort by ascending income.
(2) Divide population into tenths (deciles) of the population.
(3) Graph the cumulative percentage of income (y-axis) vs the cumulative percentage of households (x-axis).
|
Decile |
Percentage of Total Income |
Cumulative Sum of Total Income |
|
0 |
0 |
0 |
|
1 (10%) |
x1 |
x1 |
|
2 (20%) |
x2 |
x1 + x2 |
|
3 (30%) |
x3 |
x1 + x2 + x3 |
|
4 (40%) |
x4 |
x1 + …+ x4 |
|
5 (50%) |
x5 |
x1 + …+ x5 |
|
6 (60%) |
x6 |
x1 + …+ x6 |
|
7 (70%) |
x7 |
x1 + …+ x7 |
|
8 (80%) |
x8 |
x1 + …+ x8 |
|
9 (90%) |
x9 |
x1 + …+ x9 |
|
10 (100%) |
x10 |
x1 + …+ x10 |
In a population with an equal distribution of wealth, the Lorenz curve is a 45 degree line running from (0,0) to (100,100) or (1,1) depending on whether percentages or decimal fractions are used.
In most populations, the Lorenz curve is a bow shaped exponential curve running below the equal distribution line.
Gini index =
= (area between the Lorenz curve and the 45 degree line) / (area below the 45 degree line) = ((area below the 45 degree line) – (area below curve)) / (area below the 45 degree line)
Steps:
(1) Enter the decile and cumulative sum into curve approximation software.
(2) Derive the exponential equation to best describe the Lorenz curve.
(3) Use integration to determine the area below the curve.
area below the Lorenz curve =
= INTEGRATION ((curve) * dx) from 0 to 100
For example, plugging the data in the Appendix of Kennedy et al (1996) into JPL:
line approximation for the Lorenz curve =
= (0.0000971 * ((cumulative percent)^3)) - (0.003097 * ((cumulative percent)^2)) + (0.3250925 * (cumulative percent)) – 1.163916
area under the curve =
= (0.0000971 * ((cumulative percent)^4) / 4) - (0.003097 * ((cumulative percent)^3) / 3) + (0.3250925 * ((cumulative percent)^2) / 2) – (1.163916 * (cumulative percent))
Over the interval of 0 to 100, the area under the curve is 2,907.
The area under the 45 degree curve is 5,000.
Gini coefficient =
= 2093 / 5000 = 0.42
Interpretation:
• The smaller the Gini coefficient, the more equal the income distribution.
• The larger the Gini coefficient, the more unequal the income distribution.
