Chapter 3, Problem 6NP

a)

To determine

To find the equation of the labor demand curve, the equilibrium levels of the real wage and employment, the level of full-employment output, and the total after-tax wage income of workers.

Explanation of Solution

Given,

The production function:

  $Y=9K^{0.5}{N}^{0.5}$

  $MPN=4.5K^{0.5}{N}^{-0.5}$

The amount of capital is 25.

The labor supply curve:

  $NS=100{\left[\left(1-t\right)W\right]}^2$

Where, ‘W’ is the real wage rate and ‘t’ is the tax rate on labor income.

Since MP_N = W

  $\begin{array}{l}W=4.5K^{0.5}{N}^{-0.5}\\ N^{-0.5}=\frac{W}{4.5K^{0.5}}\\ N={\left[\frac{4.5K^{0.5}}{W}\right]}^2\\ N=\frac{20.25K}{W^2}\end{array}$

  $\begin{array}{c}N=\frac{20.25\times 25}{W^2}\\ =\frac{506.25}{W^2}\end{array}$

If t = 0, then labor supply is:

  $NS=100W^2$

At equilibrium, labor supply is equal to the labor demand. Setting labor supply equal to labor demand gives

  $\begin{array}{l}NS=ND\\ 100W^2=\frac{506.25}{W^2}\\ W^4=5.0625\\ W=1.5\end{array}$

If W = 1.5, then labor supply is:

  $\begin{array}{c}{N}_S=100\times {\left(1.5\right)}^2\\ \text{=}225\end{array}$

  $\begin{array}{c}N=\frac{506.25}{{1.5}^2}\\ =225\end{array}$

  $\begin{array}{c}Y=9K^{0.5}{N}^{0.5}\\ =9\times {25}^{0.5}\times {225}^{0.5}\\ =45\times 15\\ =675\end{array}$

The total after-tax wage income of workers is:

N_S × (1-t)W, which is equal to 337.5(225×1.5).

b)

To determine

To find the equation of the labor demand curve, the equilibrium levels of the real wage and employment, the level of full-employment output, and the total after-tax wage income of workers, when t = 0.6.

Explanation of Solution

From the previous part, marginal product of labor is:

  $\begin{array}{l}MPN=4.5K^{0.5}{N}^{-0.5}\\ W=4.5\times {25}^{0.5}{N}^{-0.5}\\ =\frac{22.5}{N^{0.5}}\end{array}$

  $\begin{array}{c}N=100{\left[\left(1-t\right)\frac{22.5}{N^{0.5}}\right]}^2\\ N=100{\left[\left(1-0.6\right)\frac{22.5}{N^{0.5}}\right]}^2\\ N=100\left(\frac{81}{N}\right)\\ N^2=8100\\ N=90\end{array}$

If t = 0.6, then labor supply is:

  $\begin{array}{c}NS=100{\left[\left(1-t\right)W\right]}^2\\ =100{\left[\left(1-0.6\right)W\right]}^2\\ =16W^2\end{array}$

  $\begin{array}{c}W=\frac{22.5}{N^{0.5}}\\ =\frac{22.5}{{90}^{0.5}}\\ =2.37\end{array}$

  $\begin{array}{c}Y=9K^{0.5}{N}^{0.5}\\ =9\times {25}^{0.5}\times {90}^{0.5}\\ =426.91\end{array}$

The total after-tax wage income of workers is:

N_S × (1-t)W, which is equal to 85.38 (90 × 0.4×2.37).

Here, output and income both decreases, although the wage is higher.

c)

To determine

The value of employment and real wage, when w = 2 and t = 0.

Explanation of Solution

The minimum wage w = 2 and tax rate t = 0.

  $\begin{array}{c}N=\frac{506.25}{W^2}\\ =\frac{506.25}{2^2}\\ =126.6\end{array}$

If t = 0, then labor supply is:

  $N_S=100W^2$

  $\begin{array}{c}NS=100\times 2^2\\ =400\end{array}$

Since labor supply is equal to 400 and labor demand is equal to 126.6, so labor supply exceeds labor demand.

  $\begin{array}{c}Unemployment=400–126.6\\ =273.4\end{array}$

Total income of the workers is:

wage × N = 2 × 126.6 = 253.2,which is lower than without a minimum wage (= 337.5), because employment has declined so much.

Therefore, introduction of minimum wage decreases the total income of workers.