Chapter 7, Problem 8P

(a)

To determine

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.

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:

$\begin{array}{c}\text{Total inxome tax}=0.\text{5}\left(\text{1,500}\right)+0.\text{3}\left(\text{7,000}\right)+0.\text{15}\left(\text{19,500}\right)+0.\text{05}\left(\text{149,500}\right)\\ =\text{13,250}\\ \text{Total pay roll tax}=\text{0}\text{.5}\left(\text{3,000}\right)+0.\text{3}\left(\text{5,250}\right)+\text{0}\text{.15}\left(\text{6,000}\right)+\text{0}\text{.05}\left(\text{6,000}\right)\\ =\text{4,275}\\ \text{Total tax}=\text{Total inxome tax}+\text{Total pay roll tax}\\ =\text{13,250}+\text{4,275}\\ =17,525\end{array}$

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:

$\begin{array}{c}\text{Total income tax}=\frac{0.\text{5}\left(\text{1,500}\right)}{13,250}\\ =\text{5}\text{.67\%}\end{array}$

The total income tax paid by household with $70,000 is calculated as follows:

$\begin{array}{c}\text{Total income tax}=\frac{0.\text{3}\left(\text{7,000}\right)}{13,250}\\ =1\text{5}\text{.85\%}\end{array}$

The total income tax paid by household with $120,000 is calculated as follows:

$\begin{array}{c}\text{Total income tax}=\frac{0.\text{15}\left(\text{19,500}\right)}{13,250}\\ =22.8\text{\%}\end{array}$

The total income tax paid by household with $500,000 is calculated as follows:

$\begin{array}{c}\text{Total income tax}=\frac{0.\text{5}\left(\text{149,500}\right)}{13,250}\\ =\text{56}\text{.42\%}\end{array}$

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:

$\begin{array}{c}\text{Total pay roll tax}=\frac{0.\text{5}\left(\text{3,000}\right)}{4,275}\\ =3\text{5}\text{.9\%}\end{array}$

The total pay roll tax paid by household with $70,000 is calculated as follows:

$\begin{array}{c}\text{Total income tax}=\frac{0.\text{3}\left(\text{5,250}\right)}{4,275}\\ =36.\text{84\%}\end{array}$

The total pay roll tax paid by household with $120,000 is calculated as follows:

$\begin{array}{c}\text{Total income tax}=\frac{0.\text{15}\left(\text{6,000}\right)}{4,275}\\ =21.05\text{\%}\end{array}$

The total pay roll tax paid by household with $500,000 is calculated as follows:

$\begin{array}{c}\text{Total income tax}=\frac{0.\text{5}\left(\text{6,000}\right)}{4,275}\\ =7.02\text{\%}\end{array}$

(b)

To determine

Calculate Gini coefficient, while comparing the after-tax income of all households.

Explanation of Solution

Assume that there are 1,000 householders so that the total income after tax can be calculated as follows:

$\begin{array}{c}\text{Total income after tax}=\text{500}\left(\text{35,500}\right)+\text{300}\left(\text{57,750}\right)+\text{150}\left(\text{94,500}\right)+\text{50}\left(\text{344,500}\right)\\ \text{=66,475,000}\end{array}$

The share of cumulative gross income by each household can be calculated as follows:

$\begin{array}{c}\left(\frac{\text{500×35,500}}{\text{66}\text{.47m}}\right)=\text{26}\text{.7 percent}\\ \text{26}\text{.7 percent}+\left(\frac{\text{300×57,750}}{\text{66}\text{.47m}}\right)\text{= 52}\text{.8 percent}\\ \text{52}\text{.8 percent}+\left(\frac{\text{150×94,500}}{\text{66}\text{.47m}}\right)=\text{74}\text{.1 percent}\\ \text{74}\text{.1 percent}+\left(\frac{\text{50}\times \text{344,500}}{\text{66}\text{.47m }}\right)=\text{100}\text{.0 percent}\end{array}$

From this, the area under Lorenz curve can be calculated as follows:

$\begin{array}{c}\text{Area}=\left(\text{0}\text{.5}\right)\left(\frac{\text{1}}{2}\right)\left(\text{0}\text{.267}\right)+\left(\text{0}\text{.3}\right)\left(\text{0}\text{.267}+\left(\frac{\text{1}}{2}\right)\left(\text{0}\text{.528}-\text{0}\text{.267 }\right)\right)+\left(\text{0}\text{.15}\right)\left(\text{0}\text{.528}+\left(\frac{\text{1}}{2}\right)\left(\text{0}\text{.741}-\text{0}\text{.528 }\right)\right)\\ +\left(\text{0}\text{.05}\right)\left(\text{0}\text{.741}+\left(\frac{1}{2}\right)\left(\text{1}-\text{0}\text{.741}\right)\right)\\ =\text{0}\text{.6675}+\text{0}\text{.11925}+\text{0}\text{.95175}+\text{0}\text{.043525 }\\ =\text{0}\text{.3247}\end{array}$

Therefore, the value of Gini coefficient can be calculated as follows:

$\begin{array}{c}\text{Gini coefficient}=\frac{\left(0.5-\text{0}.3247\right)}{0.5}\\ =\text{ 0}.3506\end{array}$

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:

$\begin{array}{c}\text{Total income after tax}=500\left(0.8\right)(40,000+300\left(0.8\right)\left(70,000\right)+150\left(0.8\right)\left(120,000\right)\\ +50\left(0.8\right)\left(500,000\right)\\ =67,200,000\end{array}$

Thus, the income after tax is $67,200,000.

The share of cumulative gross income by each household can be calculated as follows:

$\begin{array}{c}\left(\frac{\text{500×32,500}}{\text{67}\text{.2m}}\right)=23.8\text{ percent}\\ 23.8\text{ percent}+\left(\frac{\text{300×56,000}}{\text{67}\text{.2m}}\right)\text{= 48}\text{.8 percent}\\ \text{48}\text{.8 percent}+\left(\frac{\text{150×96,000}}{\text{67}\text{.2m}}\right)=\text{70}\text{.2 percent}\\ \text{70}\text{.2 percent}+\left(\frac{\text{50}\times 400\text{,000}}{\text{67}\text{.2m }}\right)=\text{100}\text{.0 percent}\end{array}$

From this, the area under Lorenz curve can be calculated as follows:

$\begin{array}{c}\text{Area}=\left(0.5\right)\left(\frac{1}{2}\right)\left(0.238\right)+\left(0.3\right)\left(0.238+\left(\frac{1}{2}\right)\left(0.488–.238\right)\right)+\left(0.15\right)\left(0.488+\left(\frac{1}{2}\right)\left(0.702-0.488\right)\right)\\ +\left(0.05\right)\left(0.702+\left(\frac{1}{2}\right)\left(1–0.702\right)\right)\\ =0.0595+\text{0}.1089+\text{0}.08925+0.04255\\ =\text{0}.3002\end{array}$

Therefore, the value of Gini coefficient can be calculated as follows:

$\begin{array}{c}\text{Gini coefficient}=\frac{\left(0.5-\text{0}.3002\right)}{0.5}\\ =\text{ 0}.3996\end{array}$

Thus, the Gini coefficient is 0.3996.

(c)

To determine

The amount of payroll tax reduces when keeping the total tax collected at the same.

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.