Chapter 7, Problem 14P

(a)

To determine

The total cost.

Explanation of Solution

The production function of the firm is given by Equation (1) as follows:

$\text{Q}=2\text{KL}$ (1)

The marginal product of capital MP_K can be calculated as follows:

$\begin{array}{c}{\text{MP}}_{\text{K}}=\frac{\partial \text{Total output}}{\partial \text{Capital}}\\ =\frac{\partial \left(2\text{KL}\right)}{\partial \text{K}}\\ =2\text{L}\end{array}$

The marginal product of capital is 2L.

The marginal product of labor MP_L can be calculated as follows:

$\begin{array}{c}{\text{MP}}_{\text{L}}=\frac{\partial \text{Total output}}{\partial \text{Capital}}\\ =\frac{\partial \left(2\text{KL}\right)}{\partial \text{L}}\\ =2\text{K}\end{array}$

The marginal product of labor is 2K.

The rent for capital is $100, and the cost of capital is $200. Since the firm is minimizing the costs, the ratio of marginal product of labor and marginal product of capital is equal to the ratio of cost of labor and capital.

$\begin{array}{l}\frac{{\text{MP}}_{\text{L}}}{{\text{MP}}_{\text{K}}}=\frac{\text{W}}{\text{R}}\\ \frac{2\text{K}}{2\text{L}}=\frac{200}{100}\\ \frac{\text{K}}{\text{L}}=2\\ \text{K}=2\text{L}\end{array}$

The total cost of producing 60 units can be calculated by substituting the value of K in the production function given by Equation (1).

$\begin{array}{c}\text{60}=2\left(\text{2L}\right)\text{L}\\ =4{\text{L}}^2\\ {\text{L}}^2=\frac{60}{4}\\ =15\\ \text{L}={15}^{\frac{1}{2}}\\ =3.87\end{array}$

The value of L is 3.87.

Therefore, the value of K can be calculated as follows:

$\begin{array}{c}\text{K}=2\times 3.87\\ =7.74\end{array}$

The value of K is 7.74.

The total cost can be calculated using Equation (2) as follows:

$\text{Total cost}=\text{Quantity of labor}\times \text{Cost of labor}+\text{Quantity of capital}\times \text{Cost of capital}$ (2)

$\begin{array}{c}\text{Total cost}=3.87\times 200+7.74\times 100\\ =774+774\\ =1,548\end{array}$

Thus, the total cost of producing 60 units is $1,548.

Economics Concept Introduction

Total cost: Total cost is defined as the total cost incurred for the production of a given level of output including the cost of labor and capital.

(b)

To determine

The total cost of production.

Explanation of Solution

The total cost of producing q units can be calculated by substituting the value of K in the production function given by Equation (1).

$\begin{array}{c}\text{q}=2\left(\text{2L}\right)\text{L}\\ =4{\text{L}}^2\\ {\text{L}}^2=\frac{\text{q}}{4}\\ \text{L}=\sqrt{\frac{\text{q}}{4}}\\ \text{L}=0.5\sqrt{\text{q}}\end{array}$

The value of L is $0.5\sqrt{\text{q}}$.

Therefore, the value of K can be calculated as follows:

$\begin{array}{c}\text{K}=2\times 0.5\sqrt{\text{q}}\\ =\sqrt{\text{q}}\end{array}$

The value of K is $\sqrt{\text{q}}$.

The total cost can be calculated using Equation (2) as follows:

$\text{Total cost}=\text{Quantity of labor}\times \text{Cost of labor}+\text{Quantity of capital}\times \text{Cost of capital}$ (2)

$\begin{array}{c}\text{Total cost}=0.5\sqrt{\text{q}}\times 200+\sqrt{\text{q}}\times 100\\ =100\sqrt{\text{q}}+100\sqrt{\text{q}}\\ =200\sqrt{\text{q}}\end{array}$

Thus, the total cost is $200\sqrt{\text{q}}$.

Economics Concept Introduction

Total cost: Total cost is defined as the total cost incurred for the production of a given level of output including the cost of labor and capital.

(c)

To determine

The average cost of production.

Explanation of Solution

The average cost of production can be calculated using Equation (3) as follows:

$\text{Average cost }=\frac{\text{Total cost}}{\text{Quantity of output}}$ (3)

Substituting the respective values in Equation (3), the average cost can be calculated as follows:

$\begin{array}{c}\text{Average cost }=\frac{\text{200}\sqrt{\text{q}}}{\text{q}}\\ =\frac{\text{200}}{\sqrt{\text{q}}}\end{array}$

The average cost is $\frac{\text{200}}{\sqrt{\text{q}}}$.

Economics Concept Introduction

Average total cost: Average total cost is defined as the total cost divided by the quantity of output.