Problem:

Suppose society consists of 5 households 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-household economy.

Answer:

1. Total income in the society is $160,000 = $8,000 + $12,000, $20,000 + $40,000 + $80,000.
2. The poorest quintile consists only of the lowest-income household, 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 household, 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 households. The values are below:

   Households	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 households are comprised of households whose shares are 5%, 7.5%, and 12.5%, for a cumulative total of 25%.

Problem:

Consider the three hypothetical public assistance plans:

1. What is the minimum monthly income for each plan?
2. What is the benefit-reduction amount for each plan?
3. What is the break-even income for each plan?
4. Which plan has the strongest incentive to work? Which the weakest?

 

Answer:

1. The minimum income is equal to the amount of the transfer payment received at a zero income. Plans 1 and 3 have a minimum income of $6,000 while plan two provides a minimum of $8,000.
2. The benefit-reduction rate is the rate at which the transfer payment is reduced for every additional dollar of earned income. For plans 1 and 2, the transfer is reduced by $2,000 for each $4,000 addition to earned income. The benefit-reduction rate is therefore 50% = $2,000/$4,000. The benefit-reduction rate for plan 3 is 25% = $1,000/$4,000.
3. The break-even income is the amount of earned income at which the benefit falls to zero. In plan 1, whose benefit-reduction rate is 50%, the break-even income is $12,000—an additional $4,000 beyond the last entry in the table would reduce the transfer payment from $2,000 to zero. The break-even incomes for plans 2 and 3 are found similarly to be $16,000 and $24,000, respectively.
4. Plan 3 has the lowest benefit-reduction rate, so it has the strongest work incentives. Plans 1 and 2 have the same effective "tax" on earnings, but plan 2 offers a greater guaranteed minimum income. It therefore has the weakest work incentives.