Chapter 18, Problem 5PA

Subpart (a):

To determine

Wage rate.

Explanation of Solution

Daily wage can be derived as follows.

$\begin{array}{c}\text{Wage}=\text{VMP}\times {\text{Price}}_{\text{Apple}}\\ =\left(100-2\text{L}\right)\times 2\\ =200-4\text{L}\end{array}$

Wage is $200-4\text{L}$.

Rearrange the wage equation in terms of labor $\text{L}=50-0.25\text{W}$ . Market demand function can be derived as follows.

$\begin{array}{c}\text{Demand}=\text{L}\times \text{Number of firms}\\ =\left(50-0.25\text{W}\right)\times 20\\ =1,000-5\text{W}\end{array}$

Labor demand is $1,000-5\text{W}$.

Subpart (b):

To determine

Calculate wage rate, profit and income.

Explanation of Solution

Wage level with 200 labors can be calculated as follows.

$\begin{array}{c}\text{Supply of labor}=\text{<x-custom-btb-me data-me-id="2099" class="microExplainerHighlight">Demand</x-custom-btb-me> for labor}\\ 200=1,000-5\text{W}\\ \text{5W}=1,000-200\\ \text{W}=\frac{800}{5}\\ =160\end{array}$

Wage is 160.

Number of labor demanded by each firm can be calculated as follows.

$\begin{array}{c}\text{Demand for labor}=\frac{\text{Total labor}}{\text{Total nuber of firms}}\\ =\frac{200}{20}\\ =10\end{array}$

Each firm employs 10 labors.

Profit can be calculated as follows.

$\begin{array}{c}\text{Profit}=\left(\text{Q}\times \text{Price}\right)-\left(\text{Wage}\times \text{Labor}\right)\\ =\left(\left(100\text{L}-{\text{L}}^2\right)\times 2\right)-\left(160\times 10\right)\\ =\left(\left(100\left(10\right)-{\left(10\right)}^2\right)\times 2\right)-\left(160\times 10\right)\\ =1,800-1,600\\ =200\end{array}$

Total profit for each firm is 200.

Total income can be calculated as follows.

$\begin{array}{c}\text{Total income}=\left(\text{Profit}\times \text{Number of firm}\right)+\left(\text{Wage}\times \text{Number of labor}\right)\\ =\left(200\times 20\right)+\left(160\times 200\right)\\ =36,000\end{array}$

Total income is $36,000.

Subpart (c):

To determine

Calculate wage rate and income.

Explanation of Solution

New daily wage can be derived as follows.

$\begin{array}{c}\text{Wage}=\text{VMP}\times {\text{Price}}_{\text{Apple}}\\ =\left(100-2\text{L}\right)\times 4\\ =400-8\text{L}\end{array}$

Wage is $400-8\text{L}$.

Rearrange the wage equation in terms of labor $\text{L}=50-0.125\text{W}$ . Market demand function can be derived as follows.

$\begin{array}{c}\text{Demand}=\text{L}\times \text{Number of firms}\\ =\left(50-0.125\text{W}\right)\times 20\\ =1,000-2.5\text{W}\end{array}$

Labor demand is $1,000-2.5\text{W}$.

Wage level with 200 labors can be calculated as follows.

$\begin{array}{c}\text{Supply of labor}=\text{Demand for labor}\\ 200=1,000-2.5\text{W}\\ \text{2}\text{.5W}=1,000-200\\ \text{W}=\frac{800}{2.5}\\ =320\end{array}$

Wage is 320.

Profit can be calculated as follows.

$\begin{array}{c}\text{Profit}=\left(\text{Q}\times \text{Price}\right)-\left(\text{Wage}\times \text{Labor}\right)\\ =\left(\left(100\text{L}-{\text{L}}^2\right)\times 4\right)-\left(320\times 10\right)\\ =\left(\left(100\left(10\right)-{\left(10\right)}^2\right)\times 4\right)-\left(320\times 10\right)\\ =3,600-3,200\\ =400\end{array}$

Total profit for each firm is 400.

Total income can be calculated as follows.

$\begin{array}{c}\text{Total income}=\left(\text{Profit}\times \text{Number of firm}\right)+\left(\text{Wage}\times \text{Number of labor}\right)\\ =\left(400\times 20\right)+\left(320\times 200\right)\\ =72,000\end{array}$

Total income is $36,000.

Subpart (d):

To determine

Calculate profit and income.

Explanation of Solution

Market demand function after the destruction can be derived as follows.

$\begin{array}{c}\text{Demand}=\text{L}\times \text{Number of firms}\\ =\left(50-0.25\text{W}\right)\times 10\\ =500-2.5\text{W}\end{array}$

Labor demand is $500-2.5\text{W}$.

Wage level with 200 labors can be calculated as follows.

$\begin{array}{c}\text{Supply of labor}=\text{Demand for labor}\\ 200=500-2.5\text{W}\\ \text{2}\text{.5W}=500-200\\ \text{W}=\frac{300}{2.5}\\ =120\end{array}$

Wage is 120.

Number of labor demanded by each firm can be calculated as follows.

$\begin{array}{c}\text{Demand for labor}=\frac{\text{Total labor}}{\text{Total nuber of firms}}\\ =\frac{200}{10}\\ =20\end{array}$

Each firm employs 20 labors.

Profit can be calculated as follows.

$\begin{array}{c}\text{Profit}=\left(\text{Q}\times \text{Price}\right)-\left(\text{Wage}\times \text{Labor}\right)\\ =\left(\left(100\text{L}-{\text{L}}^2\right)\times 2\right)-\left(160\times 20\right)\\ =\left(\left(100\left(20\right)-{\left(20\right)}^2\right)\times 2\right)-\left(120\times 20\right)\\ =3,200-2,400\\ =800\end{array}$

Total profit for each firm is 800.

Total income can be calculated as follows.

$\begin{array}{c}\text{Total income}=\left(\text{Profit}\times \text{Number of firm}\right)+\left(\text{Wage}\times \text{Number of labor}\right)\\ =\left(800\times 10\right)+\left(120\times 200\right)\\ =32,000\end{array}$

Total income is $32,000. The total income decreased.