Chapter 9, Problem 4P

(a)

To determine

Determine the number of workers of each race that a non-discriminating firm would hire and the amount of profit earned if there are no other costs.

Explanation of Solution

If a firm is non-discriminating, then it will hire all black workers as labor enters the function of the marginal product as a sum and black wage is lower than the white wage. Therefore, the firm will prefer to hire black workers Therefore, the number of worker can be calculated as follows:

${\text{VMP}}_{\text{E}}={\text{W}}_{\text{B}}$        (1)

Rearrange Equation 1 as follows:

$\text{P}\times {\text{MP}}_{\text{E}}={\text{W}}_{\text{B}}$        (2)

Use Equation 2, to get the value of ${\text{MP}}_{\text{E}}$, since Price is $20 and wage for black is 50 as follows:

$\begin{array}{c}{\text{MP}}_{\text{E}}=\frac{{\text{W}}_{\text{B}}}{\text{P}}\\ {\text{MP}}_{\text{E}}=\frac{50}{20}\\ =2.5\end{array}$

Thus, ${\text{MP}}_{\text{E}}$ is 2.5.

Substitute the value of ${\text{MP}}_{\text{E}}$, into the marginal product of labor, given as ${\text{MP}}_{\text{E}}=\frac{40}{{\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1}$

$\begin{array}{c}{\text{MP}}_{\text{E}}=\frac{40}{{\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1}\\ 2.5=\frac{40}{\left({\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1\right)}\\ \left({\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1\right)=\frac{40}{2.5}\\ \left({\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1\right)=16\end{array}$

Since firm prefer black workers, it is assumed that ${\text{E}}_{\text{W}}$ is zero and $\left({\text{E}}_{\text{B}}+1\right)=16$.

$\begin{array}{c}{\text{E}}_{\text{B}}=16-1\\ =15\end{array}$

Therefore, the non-discriminate firm hires 15 black workers and 0 white workers. Substitute the value into the production Function (given in the question).

$\begin{array}{c}\text{Q}=40\ln \left({\text{E}}_{\text{W}}+{\text{E}}_{\text{W}}+1\right)\\ \text{Q}=40\ln \left(16\right)\\ \approx 111.\end{array}$

Thus, the output is approximately 111.

Therefore, the profit can calculate as follows:

$\begin{array}{c}\text{Profit}\left(\pi \right)=\text{Total revenue}-\text{Total cost}\\ =\left(\text{Price}\times \text{output}\right)-\left(\text{Number of employees}\times \text{wage}\right)\\ =\left(\$20\times 111\right)-\left(15\times \$50\right)\\ =\$2,220-\$750\\ =\$1,470\end{array}$

Thus, the profit earned is $1,470.

(b)

To determine

Determine the number of workers of each race that a firm with discrimination coefficient of 0.6 against blacks would hire and the amount of profit earned if there are no other costs. 

Explanation of Solution

Here, the discriminating firm is associated with a discrimination coefficient of 0.6. This coefficient will compare the white wage of $100 to the adjusted black wage of $80 $\left(\left(1+0.6\right)\times 50\right)$. Still the adjusted wage is lower than the white wage, the firm will prefer to hire black workers. However, in the calculation the adjusted wage rate will include.

Using the similar method explained in part “a”, the number of workers can be calculated as follows:

${\text{VMP}}_{\text{E}}=\left(1+\text{d}\right)\times {\text{W}}_{\text{B}}$ (3)                               

Rearrange Equation 3 as follows:

$\text{P}\times {\text{MP}}_{\text{E}}=\left(1+\text{d}\right)\times {\text{W}}_{\text{B}}$ (4)                                   

Use Equation 4, to get the value of ${\text{MP}}_{\text{E}}$, since Price is $20 and wage for black is 50 as follows:

$\begin{array}{c}{\text{MP}}_{\text{E}}=\frac{\left(\text{1}+\text{d}\right)\times {\text{W}}_{\text{B}}}{\text{P}}\\ {\text{MP}}_{\text{E}}=\frac{\left(\text{1}+0.6\right)50}{20}\\ =4\end{array}$

Thus, ${\text{MP}}_{\text{E}}$ is 4.

Substitute the value of ${\text{MP}}_{\text{E}}$, into the marginal product of labor, given as ${\text{MP}}_{\text{E}}=\frac{40}{{\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1}$ 

$\begin{array}{c}{\text{MP}}_{\text{E}}=\frac{40}{{\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1}\\ 4=\frac{40}{\left({\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1\right)}\\ \left({\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1\right)=\frac{40}{4}\\ \left({\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1\right)=10\end{array}$

For simplicity, it is assumed that ${\text{E}}_{\text{W}}$ is zero and $\left({\text{E}}_{\text{B}}+1\right)=10$.

$\begin{array}{c}{\text{E}}_{\text{B}}=10-1\\ =9\end{array}$

Therefore, the discriminate firm hires 9 black workers and 0 white workers. Substitute the value into the production Function (given in the question).

$\begin{array}{c}\text{Q}=40\ln \left({\text{E}}_{\text{W}}+{\text{E}}_{\text{W}}+1\right)\\ \text{Q}=40\ln \left(9\right)\\ \approx 88.\end{array}$

Thus, the output is approximately 88.

Therefore, the profit can be calculated as follows:

$\begin{array}{c}\text{Profit}\left(\pi \right)=\text{Total revenue}-\text{Total cost}\\ =\left(\text{Price}\times \text{output}\right)-\left(\text{Number of employees}\times \text{wage}\right)\\ =\left(\$20\times 88\right)-\left(9\times \$50\right)\\ =\$1,840-\$450\\ =\$1,310\end{array}$

Thus, the profit earned is $1,310. 

(c)

To determine

Determine the number of workers of each race that a firm with discrimination coefficient of 1.2 against blacks would hire and the amount of profit earned if there are no other costs. 

Explanation of Solution

Here, the discriminating firm is associated with a discrimination coefficient of 1.2. This coefficient will compare the white wage of $100 to the adjusted black wage of $100 $\left(\left(1+1.2\right)\times 50\right)$. Since the adjusted wage is greater than the white wage, the firm will prefer to hire white workers.

Using the similar method explained above, the number of workers can be calculated as follows:

${\text{VMP}}_{\text{E}}={\text{W}}_{\text{w}}$        (5)

Rearrange Equation 5 as follows:

$\text{P}\times {\text{MP}}_{\text{E}}={\text{W}}_{\text{w}}$        (6)

Use Equation 6, to get the value of ${\text{MP}}_{\text{E}}$, since Price is $20 and wage for white is $100 as follows:

$\begin{array}{c}{\text{MP}}_{\text{E}}=\frac{{\text{W}}_{\text{B}}}{\text{P}}\\ {\text{MP}}_{\text{E}}=\frac{100}{20}\\ =5\end{array}$

Thus, ${\text{MP}}_{\text{E}}$ is 5.

Substitute the value of ${\text{MP}}_{\text{E}}$, into the marginal product of labor, given as ${\text{MP}}_{\text{E}}=\frac{40}{{\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1}$ 

$\begin{array}{c}{\text{MP}}_{\text{E}}=\frac{40}{{\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1}\\ 5=\frac{40}{\left({\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1\right)}\\ \left({\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1\right)=\frac{40}{5}\\ \left({\text{E}}_{\text{W}}+{\text{E}}_{\text{B}}+1\right)=8\end{array}$

For simplicity, it is assumed that ${\text{E}}_{\text{b}}$ is zero and $\left({\text{E}}_{\text{w}}+1\right)=8$.

$\begin{array}{c}{\text{E}}_{\text{w}}=8-1\\ =7\end{array}$

Therefore, the discriminate firm hires 7 white workers and 0 black workers. Substitute the value into the production Function (given in the question):

$\begin{array}{c}\text{Q}=40\ln \left({\text{E}}_{\text{W}}+{\text{E}}_{\text{W}}+1\right)\\ \text{Q}=40\ln \left(7\right)\\ \approx 78.\end{array}$

Thus, the output is approximately 78.

Therefore, the profit can be calculated as follows:

$\begin{array}{c}\text{Profit}\left(\pi \right)=\text{Total revenue}-\text{Total cost}\\ =\left(\text{Price}\times \text{output}\right)-\left(\text{Number of employees}\times \text{wage}\right)\\ =\left(\$20\times 78\right)-\left(7\times \$100\right)\\ =\$1,560-\$350\\ =\$860\end{array}$

Thus, the profit earned is $860.