A given Lorenz curve is used to represent the income distribution of a country. The coefficient of inequality for this Lorenz curve needs to be determined.

The Lorenz Curve

The coefficient of inequality is defined as

where L(x) is the equation of the Lorenz curve, which is given as

      L(x) = 5x³/(x² + x +3)

Therefore, the coefficient of inequality equals the following integration.

Preliminary Step of
Dividing Q(x) by P(x)

The second term of the above expression is an integration of an improper rational function.The first step is to simplify it to a proper ration function by the dividing (x² + x + 3) by 10x³ (steps are shown on the left).

Then the COI becomes,

                           (1)

The last term of the above expression is a proper rational function. Since (x² + x + 3) is an irreducible quadratic factor, it can be integrated by completing the square and making a tangent substitution.

Make the substitution u = x +1/2, then du = dx, x = u - 1/2

Substitute the value into equation (1) gives the COI

The coefficient of inequality equals 0.435.