Chapter 3, Problem 4PA

(a)

To determine

Fraction of income that the capital and labor receive.

Explanation of Solution

Given information:

Production function is $\text{F}\left(\text{K,L}\right)={\text{AK}}^{\text{α}}{\text{L}}^{\text{1}-\text{α}}$.

Value of “$\alpha$” is 0.3.

Calculation:

Under Cobb-Douglas production function, explain the input-output relationship in a production process. The general form of Cobb-Douglas production function is given below:

$\text{F}\left(\text{K,L}\right)={\text{AK}}^{\text{α}}{\text{L}}^{\text{1}-\text{α}}$ (1)

The marginal product labor can be derived as follows:

$\begin{array}{c}\text{MPL}=\frac{\partial \left(\text{Total production}\right)}{\partial \left(\text{Total labor}\right)}\\ =\frac{\partial \left({\text{AK}}^{\text{α}}{\text{L}}^{\text{1}-\text{α}}\right)}{\partial \text{L}}\\ =\left(\text{1}-\text{α}\right)\frac{\text{Y}}{\text{L}}\end{array}$

Thus, the marginal product of capital is $\left(\text{1}-\text{α}\right)\frac{\text{Y}}{\text{L}}$.

The marginal product of capital can be derived as follows:

$\begin{array}{c}\text{MPK}=\frac{\partial \left(\text{Total production}\right)}{\partial \left(\text{Total labor}\right)}\\ =\frac{\partial \left({\text{AK}}^{\text{α}}{\text{L}}^{\text{1}-\text{α}}\right)}{\partial \text{L}}\\ =\left(\text{α}\right)\frac{\text{Y}}{\text{K}}\end{array}$

Thus, the marginal product of capital is $\left(\text{α}\right)\frac{\text{Y}}{\text{L}}$.

In a competitive, the profit maximizing firm hires labor when the MPL is equal to real wage rate. When MPK is equal to real rental rate, they will hire capital. Thus, by using this information, the marginal products for Cobb-Douglas production function can be written as follows:

$\begin{array}{c}\frac{\text{W}}{\text{P}}=\text{MPL}\\ \frac{\text{W}}{\text{P}}=\left(\text{1}-\text{α}\right)\frac{\text{Y}}{\text{L}}\end{array}$ (2)

$\begin{array}{l}\frac{\text{R}}{\text{P}}=\text{MPK}\\ \frac{\text{R}}{\text{P}}=\left(\text{α}\right)\frac{\text{Y}}{\text{K}}\end{array}$ (3)

Rearrange the MPL equation (Equation (2)) as follows:

$\begin{array}{c}\left(\frac{\text{W}}{\text{P}}\right)\times \text{L}=\text{MPL}\times \text{L}\\ =\left(\text{1}-\text{α}\right)\frac{\text{Y}}{\text{L}}\times \text{L}\\ =\left(\text{1}-\text{α}\right)\text{Y}\end{array}$

Rearrange the MPK equation (Equation (3)) as follows:

$\begin{array}{c}\left(\frac{\text{R}}{\text{P}}\right)\times \text{K}=\text{MPK}\times \text{K}\\ \text{=}\left(\text{α}\right)\frac{\text{Y}}{\text{K}}\times \text{K}\\ =\left(\text{α}\right)\text{Y}\end{array}$

From the rearranged form, the term $\left(\frac{\text{W}}{\text{P}}\right)\times \text{L}$ is the wage bill and the term $\left(\frac{\text{R}}{\text{P}}\right)\times \text{K}$ is the total returns to capital.

When the value of “$\alpha$” is 0.3, the labor receives 70 percent of total output (income) $\left(\left(\text{1}-\text{α}\right)\text{Y}=1-0.3=0.7\right)$. The capital receives 30 percent of total output (income) $\left(\left(\text{α}\right)\text{Y}=0.3\right)$.

Economics Concept Introduction

Cobb-Douglas production function: Cobb-Douglas production function is a well-known production function; it shows the relationship between inputs used and output produced in a production process.

(b)

To determine

Impact of increase in labor force on total output, rental price of capital, and real wage rate.

Explanation of Solution

Given information:

Production function is $\text{F}\left(\text{K,L}\right)={\text{AK}}^{\text{α}}{\text{L}}^{\text{1}-\text{α}}$.

Value of “$\alpha$” is 0.3.

Labor increases by 10%.

Calculation:

Separate the Cobb-Douglas production function as Y1 and Y2 to explain the initial quantity of output and final quantity of output, respectively.

${\text{Y}}_{\text{1}}={\text{AK}}^{\text{0}\text{.3}}{\text{L}}^{\text{0}\text{.7}}$ (4)

${\text{Y}}_{\text{2}}={\text{AK}}^{\text{0}\text{.3}}{\left(\text{1}\text{.1L}\right)}^{\text{0}\text{.7}}$ (5)

In Y₂ equation, L is multiplied with 1.1 to reflect the 10% increase in the labor force.

Divide the value of initial output and final value of output to calculate the total output after 10% increase in labor forces.

$\begin{array}{c}\frac{{\text{Y}}_{\text{2}}}{{\text{Y}}_1}=\left(\frac{{\text{AK}}^{\text{0}\text{.3}}{\left(\text{1}\text{.1L}\right)}^{\text{0}\text{.7}}}{{\text{AK}}^{\text{0}\text{.3}}{\text{L}}^{\text{0}\text{.7}}}\right)\\ ={\left(1.1\right)}^{0.7}\\ =1.069\end{array}$

Total output is increased by 1.069 (6.9%).

The rental price after 10% increase in labor force is calculated as follows:

$\left(\frac{\text{R}}{\text{P}}\right)=\text{MPK}={\text{αAK}}^{\text{α}-1}{\text{L}}^{1-\text{α}}$

Separate the equation as ${\left(\frac{\text{R}}{\text{P}}\right)}_1$ to show the initial value of rental price of capital, and ${\left(\frac{\text{R}}{\text{P}}\right)}_2$ to show the final rental price of capital after 10% increase in labor force.

${\left(\frac{\text{R}}{\text{P}}\right)}_1=\left(\text{0}\text{.3}\right){\text{AK}}^{-\text{0}\text{.7}}{\text{L}}^{\text{0}\text{.7}}$ (6)

${\left(\frac{\text{R}}{\text{P}}\right)}_2=\left(\text{0}\text{.3}\right){\text{AK}}^{-\text{0}\text{.7}}{\left(\text{1}\text{.1L}\right)}^{\text{0}\text{.7}}$ (7)

In ${\left(\frac{\text{R}}{\text{P}}\right)}_2$ equation, L is multiplied with 1.1 to reflect the 10% increase in the labor force.

Divide the value of initial rental price of capital by the final rental price of capital to calculate the rental price of capital after 10% increase in labor forces.

$\begin{array}{c}\frac{{\left(\frac{\text{R}}{\text{P}}\right)}_2}{{\left(\frac{\text{R}}{\text{P}}\right)}_1}=\left(\frac{\left(\text{0}\text{.3}\right){\text{AK}}^{-\text{0}\text{.7}}{\left(\text{1}\text{.1L}\right)}^{\text{0}\text{.7}}}{0.3{\text{AK}}^{-\text{0}\text{.7}}{\text{L}}^{0.7}}\right)\\ ={\left(1.1\right)}^{0.7}\\ =1.069\end{array}$

The rental price of capital is increased by 1.069 (6.9%).

The real wage after 10% increase in labor force is calculated as follows:

$\left(\frac{\text{W}}{\text{P}}\right)=\text{MPL}=\left(\text{1}-\text{α}\right){\text{AK}}^{\text{α}}{\text{L}}^{-\text{α}}$

Separate the equation as ${\left(\frac{\text{W}}{\text{P}}\right)}_1$ to show the initial value of real wage, and ${\left(\frac{\text{W}}{\text{P}}\right)}_2$ to show the final value of real wage after 10% increase in labor force.

${\left(\frac{\text{W}}{\text{P}}\right)}_1=\left(\text{1}-\text{ 0}\text{.3}\right){\text{AK}}^{\text{0}\text{.3}}{\text{L}}^{-\text{0}\text{.3}}$ (8)

${\left(\frac{\text{W}}{\text{P}}\right)}_2=\left(\text{1}-\text{ 0}\text{.3}\right){\text{AK}}^{\text{0}\text{.3}}{\left(\text{1}\text{.1L}\right)}^{-\text{0}\text{.3}}$ (9)

In ${\left(\frac{\text{W}}{\text{P}}\right)}_2$ equation, L is multiplied by 1.1 to reflect the 10% increase in the labor force.

Divide the initial value of real wage and final value of real wage to calculate the real wage rate after 10% increase in labor forces.

$\begin{array}{c}\frac{{\left(\frac{\text{W}}{\text{P}}\right)}_2}{{\left(\frac{\text{W}}{\text{P}}\right)}_1}=\left(\frac{\left(\text{1}-\text{ 0}\text{.3}\right){\text{AK}}^{\text{0}\text{.3}}{\left(\text{1}\text{.1L}\right)}^{-\text{0}\text{.3}}}{\left(\text{1}-\text{ 0}\text{.3}\right){\text{AK}}^{\text{0}\text{.3}}{\text{L}}^{{}^{-\text{0}\text{.3}}}}\right)\\ ={\left(1.1\right)}^{-0.3}\\ =0.972\end{array}$

The real wage is reduced by 0.972 (28%).

(c)

To determine

Impact of increase in capital stock on total output, rental price of capital, and real wage rate.

Explanation of Solution

Given information:

Production function is $\text{F}\left(\text{K,L}\right)={\text{AK}}^{\text{α}}{\text{L}}^{\text{1}-\text{α}}$.

Value of “$\alpha$” is 0.3.

Capital stock increases by 10%.

Calculation:

Using the same logic used in sub-part (b), the total output after 10% increase in capital stock is calculated as follows:

$\begin{array}{c}\frac{{\text{Y}}_{\text{2}}}{{\text{Y}}_1}=\left(\frac{\text{A}{\left(\text{1}\text{.1K}\right)}^{\text{0}\text{.3}}{\text{L}}^{\text{0}\text{.7}}}{{\text{AK}}^{\text{0}\text{.3}}{\text{L}}^{\text{0}\text{.7}}}\right)\\ ={\left(1.1\right)}^{0.3}\\ =1.029\end{array}$

Total output is increased by 1.029 (2.9%).

The rental price of capital after 10% increase in capital stock is calculated as follows:

$\begin{array}{c}\frac{{\left(\frac{\text{R}}{\text{P}}\right)}_2}{{\left(\frac{\text{R}}{\text{P}}\right)}_1}=\left(\frac{\left(\text{0}\text{.3}\right)\text{A}{\left(\text{1}\text{.1K}\right)}^{-\text{0}\text{.7}}{\text{L}}^{\text{0}\text{.7}}}{0.3{\text{AK}}^{-\text{0}\text{.7}}{\text{L}}^{0.7}}\right)\\ ={\left(1.1\right)}^{-0.7}\\ =0.935\end{array}$

The rental price of capital is decreased by 0.935.

The real wage rate after 10% increase in capital stock is calculated as follows:

$\begin{array}{c}\frac{{\left(\frac{\text{W}}{\text{P}}\right)}_2}{{\left(\frac{\text{W}}{\text{P}}\right)}_1}=\left(\frac{\left(\text{1}-\text{ 0}\text{.7}\right)\text{A}{\left(\text{1}\text{.1K}\right)}^{\text{0}\text{.3}}{\text{L}}^{-\text{0}\text{.3}}}{\left(\text{1}-\text{ 0}\text{.7}\right){\text{AK}}^{\text{0}\text{.3}}{\text{L}}^{{}^{-\text{0}\text{.3}}}}\right)\\ ={\left(1.1\right)}^{0.3}\\ =1.029\end{array}$

The real wage is increased by 1.029 (2.9%).

(d)

To determine

New output, new rental price of capital, and real wage.

Explanation of Solution

Given information:

Production function is $\text{F}\left(\text{K,L}\right)={\text{AK}}^{\text{α}}{\text{L}}^{\text{1}-\text{α}}$.

Value of “$\alpha$” is 0.3.

The value of “A” increases by 10%.

Calculation:

Using the same logic used in sub-part (b) and in sub-part (c), the total output with 10% increase in the value of parameter is calculated as follows:

$\begin{array}{c}\frac{{\text{Y}}_{\text{2}}}{{\text{Y}}_1}=\left(\frac{\left(\text{1}\text{.1A}\right){\text{K}}^{\text{0}\text{.3}}{\text{L}}^{\text{0}\text{.7}}}{{\text{AK}}^{\text{0}\text{.3}}{\text{L}}^{\text{0}\text{.7}}}\right)\\ =1.1\end{array}$

Total output is increased by 1.1 (10%).

The rental price of capital is calculated as follows:

$\begin{array}{c}\frac{{\left(\frac{\text{R}}{\text{P}}\right)}_2}{{\left(\frac{\text{R}}{\text{P}}\right)}_1}=\left(\frac{\left(\text{0}\text{.3}\right)\left(\text{1}\text{.1A}\right){\text{K}}^{-\text{0}\text{.7}}{\text{L}}^{\text{0}\text{.7}}}{0.3{\text{AK}}^{-\text{0}\text{.7}}{\text{L}}^{0.7}}\right)\\ =1.1\end{array}$

The rental price of capital is 1.1.

The real wage is calculated as follows:

$\begin{array}{c}\frac{{\left(\frac{\text{W}}{\text{P}}\right)}_2}{{\left(\frac{\text{W}}{\text{P}}\right)}_1}=\left(\frac{\left(\text{1}-\text{ 0}\text{.7}\right)\left(\text{1}\text{.1A}\right){\text{K}}^{\text{0}\text{.3}}{\text{L}}^{-\text{0}\text{.3}}}{\left(\text{1}-\text{ 0}\text{.7}\right){\text{AK}}^{\text{0}\text{.3}}{\text{L}}^{{}^{-\text{0}\text{.3}}}}\right)\\ =1.1\end{array}$

The real wage is 1.1.