A model for the Lorentz curve for family income in a country has (x)=x^p. The Gini index for family income in this country is thought to be 2/4. How can I solve for p with this information?

Answer 1

First off, it's Lorenz Curve, not Lorentz Curve.

Recall that the formula for the Gini Index is G=A/(A+B) where A represents the area above the Lorenz Curve and B represents the area below the Lorenz Curve.

As an integral, the formula for the Gini Index is G=2∫[0,1][x-L(x)]dx where L(x) is the Lorenz Curve. If G=0, then all families share equal income. If G=1, then all families share unequal income.

Because we are given that G=2/4=1/2 and our Lorenz Curve is L(x)=x^p, this means that our integral is 1/2=∫[0,1][x-x^p]dx. Plugging in our lower and upper bounds, we can solve for p:

1/2 = 2∫[0,1][x-x^p]dx
1/2 = 2[(x^2)/2 - (x^(p+1))/(p+1)] [0,1]
1/2 = x^2 - 2(x^(p+1)/(p+1)) [0,1]
1/2 = [1^2 - 2(1^(p+1))/(p+1)] - [0^2 - 2(0^(p+1))/(p+1)]
1/2 = 1 - 2/(p+1)
-1/2 = -2/(p+1)
-1/2p - 1/2 = -2
-1/2p = -3/2
p = 3

Therefore, the value of p is 3