Question

The following table shows income distribution data for an economy in a particular year.

Household Group                                      Share of Aggregate Income   

One-fifth with lowest income                                   8.1%                     

Next lowest one-fifth 11.5%                   

Middle one-fifth                                                20.0%                          

Next highest one-fifth                                    22.8%                          

One-fifth with highest income                               37.6%                         

What is the Gini coefficient for this economy?

		.440
		.328
		.281
		1.453
		.459

Answer 1

see the lorenz curve here

% of population	cumulative % of population	% of income	cumulative % of income
0	0	0	0
20	20	8.1	8.1
20	40	11.5	19.6
20	60	20	39.6
20	80	22.8	62.4
20	100	37.6	100

i have calculated cumulative % of population and income to draw the lorenz curve.

further, using as fractions in the table below ( 20% is same as 0.2)

then, the % of Population that is Richer is the sum of all the “Fraction of Population” terms below. so 0.8 of poluation is richer than the first 0.2 (1/5th) of population. so on.

then we calculate score for each fraction

score = Fraction of Income * (Fraction of Population + 2 * % of Population Richer)

for first 1/5th, score = 0.081*(0.20 + 2 * 0.80) = 0.1458

% of population	cumulative % of population	% of income	cumulative % of income	% population that is richer	2*% population that is richer	% of population + 2*% population that is richer	Score = % of income*(% of population + 2*% population that is richer)
0.2	0.2	0.081	0.081	0.8	1.6	1.8	0.1458
0.2	0.4	0.115	0.196	0.6	1.2	1.4	0.161
0.2	0.6	0.2	0.396	0.4	0.8	1	0.2
0.2	0.8	0.228	0.624	0.2	0.4	0.6	0.1368
0.2	1	0.376	1	0	0	0.2	.0752

aggreggate score = 0.1458 + 0.161 + 0.2 + 0.1368 + 0.0752 = 0.7188

ginni coefficient = 1 - aggregate score = 1 - 0.7188 = 0.2812