Question

Household Group Share of Aggregate Income

One-fifth with lowest income 4.7%

Next lowest one-fifth 8.5%

Middle one-fifth 19.4%

Next highest one-fifth 27.6%

One-fifth with highest income 39.8%

What is the Gini coefficient?

Answer 1

Solution:

Clearly, the population and been divided in 5 groups, increasing income wise, so required percentage (cumulative) for groups are: 20%, 40%, 60%, 80%, and 100%. Their corresponding percentage of income share (cumulative) is: 4.7%, (4.7% + 8.5% =) 13.2%, (13.2 + 19.4 =) 32.6%, (32.6 + 27.6 =) 60.2%, and (60.2 + 39.8 =) 100%.

Gini coefficient can be calculated as: (0.5 - X)/0.5; where X is sum of all 5 individual parts for each of the above divisions; these divisions include one triangle, and for reasoning four as trapeziums, with height as population percentage (which in this case is 20%), and base as self and old (cumulative) income share.

Area of triangle = (1/2)*base*height = (1/2)*0.047*0.20 = 0.0047

Area of trapezium = (base 1 + base 2)*height/2; then

Area of trapezium 1 = (0.047 + 0.132)*0.20/2 = 0.0179

Area of trapezium 2 = (0.132 + 0.326)*0.20/2 = 0.0458

Area of trapezium 3 = (0.326 + 0.602)*0.20/2 = 0.0928

Area of trapezium 4 = (0.602 + 1)*0.20/2 = 0.1602

So, X = 0.0047 + 0.0179 + 0.0458 + 0.0928 + 0.1602 = 0.3214

Gini coefficient = (0.5 - 0.3214)/0.5 = 0.3572