Chapter 13, Problem 13.4P

a)

To determine

To analyze the equilibrium level of wage rate and number of labors employed.

Explanation of Solution

Given:

Demand for labor: $\text{L = -50 w + 450}$

Supply of labor: $\text{L = 100w}$

At equilibrium,

  $\begin{array}{l}\text{Supply of labor = Demand for labor}\hfill \\ \text{ -50 w + 450 = 100w}\hfill \\ \text{ 450= 100w+50w}\hfill \\ \text{ 450 = 150w}\hfill \\ \text{ w = \$3}\hfill \end{array}$

Substituting the value of w in supply equation we get,

  $\begin{array}{l}\text{L =100×3}\hfill \\ \text{L= 300}\hfill \end{array}$

Thus, at equilibrium $\text{w = \$3}$ and $\text{L= 300}$ .

b)

To determine

To analyze the amount of subsidy and the new equilibrium level of employment after subsidy.

Explanation of Solution

New equilibrium wage that government wishes to rise is $4.

  $\text{Current wage = \$3}$

Hence,

  $\begin{array}{l}\text{Subsidy= New equilibrium wage – existing wage}\hfill \\ \text{ = \$4-\$3}\hfill \\ \text{ = \$1}\text{.}\hfill \end{array}$

Now,

To compute new equilibrium wage,

  $\begin{array}{l}\text{New demand: L = -50}\left(\text{w-1}\right)\text{ + 450}\hfill \\ \text{ = -50w + 50 + 450}\hfill \\ \text{ = -50w + 500}\hfill \end{array}$

At new equilibrium,

  $\begin{array}{l}\text{New demand = Supply}\hfill \\ \text{ -50w+500 = 100w}\hfill \\ \text{ 150w = 500}\hfill \\ \text{ w = \$3}\text{.333}\hfill \end{array}$

Hence,

  $\begin{array}{l}\text{employment }\left(\text{L}\right)\text{=100×3}\text{.333}\hfill \\ \text{ = 334}\hfill \end{array}$

Similarly,

  $\begin{array}{l}\text{total subsidy = \$1×334}\hfill \\ \text{ = \$334}\hfill \end{array}$

c)

To determine

To analyze the number of labors demanded at the wage $4 and unemployment level at this wage rate.

Explanation of Solution

Given:

  $\begin{array}{l}\text{Wage= \$4}\hfill \\ \text{Demand for labor: L= -50 w + 450}\hfill \\ \text{Supply of labor: L = 100w}\hfill \end{array}$

To compute the number of labors demanded at $4 we will substitute the value of w as $4 in demand equation,

  $\begin{array}{l}\text{L= -50 w + 450}\hfill \\ \text{ = -50 ×\$4 + 450}\hfill \\ \text{ = -200 + 450}\hfill \\ \text{ = 250}\hfill \end{array}$

Similarly, the number of labors supplied at wage $4 can be computed by substituting the value of $4 in supply equation,

  $\begin{array}{l}\text{Supply of labor = 100w}\hfill \\ \text{ = 400}\hfill \end{array}$

At wage rate of $4, 250 labors are demanded, and 400 labors are available. Therefore, unemployment can be computed as:

  $\begin{array}{l}\text{Unemployment = Labor supplied – labor demanded}\hfill \\ \text{ = 400 – 250}\hfill \\ \text{ = 150}\hfill \end{array}$

d)

To determine

To draw the graph of the above situations.

Explanation of Solution

When the government gives subsidy, the demand curve for the labor will shift to the right because the firms will be more willing to hire labors. The diagram below shows the effect of subsidy on the demand curve for labor.

  

In the diagram initial equilibrium is at point E₁ where, the wage rate is $3 and number of labors employed is 300. When the government gives subsidy, the demand for labor shifts to the right due to which the equilibrium shifts to E₂ and the number of labors employed rises to 334.