Chapter 9, Problem 6P

(a)

To determine

Determine the number of workers of each race that a non-discriminating firm would hire and the amount of profit earned.

Explanation of Solution

According to the trend, there are no complementarities between the types of labor as the quantity of labor enters the production function as a sum ${\text{E}}_{\text{w}}+{\text{E}}_{\text{b}}$. Moreover, the market wage for blacks is lower than the market wage for whites. Therefore, the firms are more interested to hire only black workers than white workers up to the point where the black wage equal the value of their marginal product. 

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

Rearrange Equation 1 as follows:

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

Substitute the respective values in Equation 2:

$\begin{array}{c}\$100\times \frac{5}{\sqrt{{\text{E}}_{\text{b}}}}={\text{W}}_{\text{B}}\\ {\text{W}}_{\text{B}}=\frac{100(5)}{\sqrt{{\text{E}}_{\text{b}}}}\end{array}$

Since the market wage for blacks is $10, the value of ${\text{E}}_{\text{b}}$ can calculate as follows:

$\begin{array}{c}\$100\times \frac{5}{\sqrt{{\text{E}}_{\text{b}}}}=\text{10}\\ 10=\frac{100(5)}{\sqrt{{\text{E}}_{\text{b}}}}\\ 10=\frac{500}{\sqrt{{\text{E}}_{\text{b}}}}\\ \sqrt{{\text{E}}_{\text{b}}}=\frac{500}{10}\\ \sqrt{{\text{E}}_{\text{b}}}=50\\ {\text{E}}_{\text{b}}=2,500.\end{array}$

Therefore, the firm prefer 2,500 black workers and zero white workers.

Substitute the value into the production Function (given in the question).

$\begin{array}{c}\text{Q}=10\sqrt{{\text{E}}_{\text{b}}}\\ \text{Q}=10\sqrt{\left(2500\right)}\\ =10\times 50\\ =500\end{array}$

Thus, the output is 500 units.

Therefore, the profit can calculate as follows:

$\begin{array}{c}\text{Profit}\left(\pi \right)=\text{Total revenue}-\text{Total cost}\\ =\left(\text{Price}\times \text{output}\right)-\left(\text{Number of employees}\times \text{wage}\right)\\ =\left(\$100\times 500\right)-\left(2,500\times \$10\right)\\ =\$50,000-\$25,000\\ =\$25,000\end{array}$

Thus, the profit earned is $25,000.

(b)

To determine

Determine the number of workers of each race that a firm would hire with discrimination coefficient of 0.25

Explanation of Solution

Here, the discriminating firm is associated with a discrimination coefficient of 0.25. This coefficient will compare the white wage of $20 to the adjusted black wage of $12.5$\left(\left(1+0.25\right)\times 10\right)$. Still the adjusted black wage is lower than the white wage, the firm will prefer only 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 $12.50:

$\begin{array}{c}\$100\times \frac{5}{\sqrt{{\text{E}}_{\text{b}}}}=\left(1+0.25\right)\times 10\\ 12.5=\frac{100(5)}{\sqrt{{\text{E}}_{\text{b}}}}\end{array}$

The number of employees can derived e as follows:

$\begin{array}{c}12.50=\frac{100(5)}{\sqrt{{\text{E}}_{\text{b}}}}\\ 12.50=\frac{100(5)}{\sqrt{{\text{E}}_{\text{b}}}}\\ 12.50=\frac{500}{\sqrt{{\text{E}}_{\text{b}}}}\\ \sqrt{{\text{E}}_{\text{b}}}=\frac{500}{12.50}\\ \sqrt{{\text{E}}_{\text{b}}}=40\\ {\text{E}}_{\text{b}}=1,600.\end{array}$

Therefore, the firm prefer 1,600 black workers, and zero white workers.

Substitute the value into the production Function (given in the question).

$\begin{array}{c}\text{Q}=10\sqrt{{\text{E}}_{\text{b}}}\\ \text{Q}=10\sqrt{\left(1,600\right)}\\ =10\times 40\\ =400\end{array}$

Thus, the output is 400 units. 

Therefore, the profit can calculate as follows:

$\begin{array}{c}\text{Profit}\left(\pi \right)=\text{Total revenue}-\text{Total cost}\\ =\left(\text{Price}\times \text{output}\right)-\left(\text{Number of employees}\times \text{wage}\right)\\ =\left(\$100\times 400\right)-\left(1,600\times \$10\right)\\ =\$40,000-\$16,000\\ =\$24,000\end{array}$

Thus, the profit earned is $24,000.

(c)

To determine

Determine the number of workers of each race that a firm would hire with discrimination coefficient of 1.25

Explanation of Solution

Here, the discriminating firm is associated with a discrimination coefficient of 1.25. This coefficient will compare the white wage of $20 to the adjusted black wage of $22.5$\left(\left(1+1.25\right)\times 10\right)$. Here, the adjusted black wage is higher than the white wage. Therefore, the firm will prefer only to hire white workers

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

$\begin{array}{c}\text{P}\times {\text{MP}}_{\text{E}}={\text{W}}_{\text{w}}\\ \$100\times \frac{5}{\sqrt{{\text{E}}_{\text{w}}}}=\text{20}\\ 20=\frac{100(5)}{\sqrt{{\text{E}}_{\text{w}}}}\end{array}$

The number of employees can be derived as follows:

$\begin{array}{c}20=\frac{100(5)}{\sqrt{{\text{E}}_{\text{w}}}}\\ 20=\frac{100(5)}{\sqrt{{\text{E}}_{\text{w}}}}\\ 20=\frac{500}{\sqrt{{\text{E}}_{\text{w}}}}\\ \sqrt{{\text{E}}_{\text{w}}}=\frac{500}{20}\\ \sqrt{{\text{E}}_{\text{w}}}=25\\ {\text{E}}_{\text{w}}=625.\end{array}$

Therefore, the firm prefer 625 white workers, and zero black workers.

Substitute the value into the production Function (given in the question).

$\begin{array}{c}\text{Q}=10\sqrt{{\text{E}}_{\text{W}}}\\ \text{Q}=10\sqrt{\left(625\right)}\\ =10\times 25\\ =250\end{array}$

Thus, the output is 250 units. 

Therefore, the profit can be calculated as follows:

$\begin{array}{c}\text{Profit}\left(\pi \right)=\text{Total revenue}-\text{Total cost}\\ =\left(\text{Price}\times \text{output}\right)-\left(\text{Number of employees}\times \text{wage}\right)\\ =\left(\$100\times 250\right)-\left(625\times \$20\right)\\ =\$25,000-\$12,500\\ =\$12,500\end{array}$

Thus, the profit earned is $12,500.