Question

"Society consists of fifteen single individuals. Their incomes are as follows. Use the data to construct a table showing the percentage of total income received by each quintile. Then compute the cumulative distribution and use it to construct a Lorenz curve and Gini coefficient using the method described in Chapter 1 Appendix A from the textbook The Economics of Inequality, Discrimination, Poverty, and Mobility.

$100,00         $75.000            $25,000           $35,000              $150,000

$80,000         $15,000            $45,000           $85,000              $90,000

$110,000       $135,000           $95,000           $70,000              $60,000

Answer 1

Solution:

We are given total of 15 individuals, hence, to form quintiles (division of population in 5 equal groups), for each quintile, we will have 3 individuals in their order of income. Income of all 15 individuals in ascending order:

$15,000, $25,000, $35,000, $45,000, $60,000, $70,000, $75,000, $80,000, $85,000, $90,000, $95,000, $100,000, $110,000, $135,000, $150,000

Then, total income in the

First quintile = 15000 + 25000 + 35000 = $75,000

Second quintile = 45000 + 60000+ 70000 = $175,000

Third quintile = 75000+ 80000+ 85000 = $240,000

Fourth quintile = 90000+ 95000 + 100000 = $285,000

Fifth quintile = 110000+ 135000+ 150000 = $395,000

Economy = sum of all quintiles' income = 75000 + 175000 + 240000 + 285000 + 395000 = $1,170,000

Thus, we have the following table:

Quintile	%age of total income	Cumulative frequency (in %)
First	(75000/1170000)*100 = 6.41%	6.41%
Second	(175000/117000)*100 = 14.96%	6.41 + 14.96 = 21.37%
Third	(240000/1170000)*100 = 20.51%	21.37 + 20.51 = 41.88%
Fourth	(285000/1170000)*100 = 24.36%	41.88 + 24.36 = 66.24%
Fifth	(395000/1170000)*100 = 33.76%	66.24 + 33.76 = 100%

Then, the Lorenz curve can be constructed as follows: The red curve represents the Lorenz curve.

100 90 80 (100, 100) Line of complete equality Cumulative 70 share of ncome earned (in %) 0, 66.24) 60 50 40 0:41:88) Lorenz Curve 30 20 10 0, 21.87) 20, 6.41) 20 40 60 80 100 Cumulative share of people (lowest to highest quintile share)(in %)

There are two ways to calculate the Gini coefficient: one is using the Lorenz curve, and the other is formula based. Since, I was unable to find the mentioned reference book online, so please provide in the comment section below, from Chapter 1, Appendix A which is the mentioned method in the question; I'll update the solution accordingly.