Problem:

Suppose society consists of 5 families whose incomes are $8,000, $12,000, $20,000, $40,000, and $80,000.

1. What is the total income in this society?
2. What percentage of total income is earned by the poorest quintile?
3. What percentage of total income is earned by the richest quintile?
4. What percentages of total income are received by the second, third, and fourth quintiles?
5. Construct a Lorenz curve for this five-family economy.

Answer:

1. Total income in the society is $8,000 + $12,000, $20,000 + $40,000 + $80,000 = $160,000.
2. Since there are only 5 families, the poorest quintile consists only of the lowest-income family, that earning $8,000. As a percentage of total income, this is 5%: 8,000/160,000 = .05, or 5%.
3. The richest quintile consists only of the highest-income family, that earning $80,000. This is half, or 50%, of the total income of $160,000.
4. Second quintile: $12,000/$160,000 = 7.5%
   Third quintile: $20,000/$160,000 = 12.5%
   Fourth quintile: $40,000/$160,000 = 25%
5. The Lorenz curve plots the percentage of total income against the cumulative percentage of families. The values are below:
   
   

   Households	Percentage of Income
   0%	0%
   20%	5%
   40%	12.5%
   60%	25%
   80%	50%
   100%	100%

   
   Each value in the Income column is constructed by accumulating the quintile shares. For example, the poorest 60% of families are comprised of the bottom three families whose shares are 5%, 7.5%, and 12.5%, for a cumulative total of 25%. The Lorenz curve is shown below:


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