A simple model for the Lorentz curve for family income in a particular country is f(x)=0.5x+0.5x^6. The Gini index for family income in this country is?

Answer 1

Recall that the Gini index is just the area between the Lorenz curve and the line y=x, over the interval [0,1]. So, it is
∫[0,1] (x - 1/2 (x+x^6)) dx = 5/28

Answer 2

G = 2∫[0,1] (x - f(x))dx
G = 2∫[0,1] (x - (1/2)x - (1/2)x^6)dx
G = 2∫[0,1] ((1/2)x - (1/2)x^6)dx
G = 2[(1/4)x^2 - (1/14)x^7] [0,1]
G = (1/2)x^2 - (1/7)x^7 [0,1]
G = [(1/2)(1)^2 - (1/7)(1)^7] - [(1/2)(0)^2 - (1/7)(0)^7]
G = 1/2 - 1/7
G = 7/14 - 2/14
G = 5/14

Therefore, the Gini Index for family income in the country is G=5/14