Question

3) A country with a population of 100,000,000 has the following income distribution: - Everyone in the first quintile earns exactly $200/month. - Everyone in the second quintile earns exactly $350/month. - Everyone in the third quintile earns $600/month - Everyone in the fourth quintile earns $1000/month - Everyone in the fifth quintile earns $2000/month Assume the poverty line is at $400/month a) Draw the Lorenz curve for this country b) Calculate the headcount ratio c) Calculate the average proportional income gap and the Poverty gap index (P1)

Answer 1

a) The first quintile earns (200/(200+350+600+1000+2000))*100 = 4.82% of total income

The second quintile earns (350/(200+350+600+1000+2000))*100 = 8.43% of total income

The third quintile earns (600/(200+350+600+1000+2000))*100 = 14.45% of total income

The fourth quintile earns (1000/(200+350+600+1000+2000))*100 = 24.10% of total income

The fifth quintile earns (2000/(200+350+600+1000+2000))*100 = 48.20% of total income

Lorenz curve is as follows

b) Headcount ratio = Number of people below povery line/ total population

Poverty line = $400/month. This shows that first and second quintile population is below poverty line.

Hence, Headcount ratio = (40 million/100 million)*100 = 40%

c) Income gap is the amount of income of an individual that falls short of the poverty line.

For first quintile i.e. 20 million people, income gap = (400-200)*20 million = $4000 million

For second quintile i.e. 20 million people, income gap = (400-350)*20 million = $1000 million

Average proportional income gap = Total income gap/number of people below poverty line

= (4000+1000)/40 = $125

Poverty gap index measures the intensity of poverty. It is the average poverty gap in the population as a proportion of the poverty line

Poverty gap index = (1/N)[{((400-200)/400)*20 million}+{((400-350)/400)*20 million}] where N is total population.

Poverty gap index = (1/100 million){(10 million+2.5 million)} = 0.125