
(a)
Calculate the marginal tax rate, average tax rate, average household income, pay roll bill, total tax bill, percentage of total income tax, and total pay roll tax by four types of households.
(a)

Explanation of Solution
The marginal tax rate for each household can be calculated as follows:
Table 1
Earnings of households | Percentage of tax | Pay roll tax | Marginal tax rate |
40,000 | 15% | 7.5% | 22.5% |
70,000 | 25% | 7.5% | 32.5% |
120,000 | 25% | 0 | 25% |
500,000 | 35% | 0 | 35% |
The income tax bill for each household can be calculated as follows:
Table 2
Earnings | Percentage of marginal tax | Tax on excess earnings | Total income tax |
40,000 | 15 of $10,000 | 0 | 1,500 |
70,000 | 25% of 10,000 | 4,500 | 7,000 |
120,000 | 25% of 60,000 | 4,500 | 19,500 |
500,000 | 35% of 350,000 | 27,000 | 149,500 |
Since the pay roll tax is imposed up to $80,000, the pay roll bill for each household can be calculated as follows:
Table 3
Earnings | Pay roll tax rate | Pay roll tax up to | Total income tax |
40,000 | 7.5 | 40,000 | 3,000 |
70,000 | 7.5 | 70,000 | 5,250 |
120,000 | 0 | 80,000 | 6,000 |
500,000 | 0 | 80,000 | 6,000 |
The total tax bill and average tax rate can be calculated as follows:
Table 4
Earnings | Total tax (income tax + payroll tax) | Average total tax |
40,000 | 4,500 | 11.25% |
70,000 | 12,250 | 17.50% |
120,000 | 25,500 | 21.25% |
500,000 | 155,500 | 31.10% |
The average tax bills of total households can be calculated as follows:
Thus, the average total tax is $17,525.
The percentage of total income tax that is paid by each household can be calculated as follows:
The total income tax paid by household with $40,000 is calculated as follows:
The total income tax paid by household with $70,000 is calculated as follows:
The total income tax paid by household with $120,000 is calculated as follows:
The total income tax paid by household with $500,000 is calculated as follows:
The percentage of total payroll tax that is paid by each household can be calculated as follows:
The total pay roll tax paid by household with $40,000 is calculated as follows:
The total pay roll tax paid by household with $70,000 is calculated as follows:
The total pay roll tax paid by household with $120,000 is calculated as follows:
The total pay roll tax paid by household with $500,000 is calculated as follows:
(b)
Calculate Gini coefficient, while comparing the after-tax income of all households.
(b)

Explanation of Solution
Assume that there are 1,000 householders so that the total income after tax can be calculated as follows:
The share of cumulative gross income by each household can be calculated as follows:
From this, the area under Lorenz curve can be calculated as follows:
Therefore, the value of Gini coefficient can be calculated as follows:
Thus, the Gini coefficient is 0.3506.
When there is 20 percent flat tax on income, the income after tax can be calculated as follows:
Thus, the income after tax is $67,200,000.
The share of cumulative gross income by each household can be calculated as follows:
From this, the area under Lorenz curve can be calculated as follows:
Therefore, the value of Gini coefficient can be calculated as follows:
Thus, the Gini coefficient is 0.3996.
(c)
The amount of payroll tax reduces when keeping the total tax collected at the same.
(c)

Explanation of Solution
It is assumed that there are 1,000 households, where 500 households pay $3,000, 300 households pay $5,250, and 200 households pay $6,000 so that when the total income is $84 million, the total payroll tax will be $4,275,000. Therefore, the tax rate of 5.089 percent is needed to produce $4.275 million from the total income.
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