Chapter 6, Problem 4P

 (a)

To determine

The production process without capital input.

Explanation of Solution

A production function of a firm shows the relationship between the output and two inputs, capital and labor. In this case, the firm follows the Cobb‑Douglas production function.

$\text{Q}=4{\text{K}}^{.5}{\text{L}}^{.5}$

Here, this production function clearly shows that the firm produces the output using the inputs, capital, and labor at an equal proportion. Therefore, producer cannot build any coffee tables without tools (capital). If did so, the output would be zero.

 (b)

To determine

The production process without labor input.

Explanation of Solution

As described in part (a), the firm follows the Cobb‑Douglas production function where the firm uses the inputs, capital, and labor at an equal proportion to produce the given level of output. Therefore, the producer cannot build any coffee tables without labor inputs. If did so, the output would be zero.

 (c)

To determine

The short run production function.

Explanation of Solution

The production economies state that the short-run is the period of time during which one or more inputs into production cannot be changed, which mean that the short run production function of a firm holds variable inputs as well as fixed inputs. Therefore, the short run production function shows the output as a function of variable input and the fixed input.

In this case, the fixed input, capital, is given as 16. Now substitute the respective value into the given Cobb‑Douglas production function to get the short run production function of the firm.

$\begin{array}{c}\text{Q}=4{\left(16\right)}^{0.5}{\text{L}}^{0.5}\\ \text{Q}=4\left(4\right){\text{L}}^{0.5}\\ \text{Q}=16{\text{L}}^{0.5}\end{array}$

Therefore, the short run production function is $\text{Q}=16{\text{L}}^{0.5}$.

Economics Concept Introduction

Fixed inputs: The fixed inputs are inputs that cannot be changed in the short run

Variable inputs: The variable inputs are inputs that can be changed in the short run.

 (d)

To determine

The graphical representation of the production function.

Explanation of Solution

Short run production function is given below (Derived in subpart (c)).

$\text{Q}=16{\text{L}}^{0.5}$ (1)

Substitute the labor of 1 unit in Equation (1) to calculate the output at that particular level.

$\begin{array}{c}\text{Q}=16{\left(1\right)}^{0.5}\\ =16\left(1\right)\\ =16\end{array}$

In the short run, output with one unit of labor is 16 units.

Table 1 shows the output of different levels of labor.

Table 2

Labor	Output
1	16
4	32
9	48
16	64
25	80

Figure 1 shows the short-run production function.

In Figure 1, the vertical axis shows the output and the horizontal axis shows the quantity of labor inputs. The production curve shows the possible level of output with different labor quantities.

 (e)

To determine

The average and marginal product of labor.

Explanation of Solution

The marginal product can be calculated using the following equation.

$\text{Marginal product}=\frac{{\text{Total product}}_{\text{Present}}-{\text{Total product}}_{\text{Previous}}}{{\text{Quantity}}_{\text{Present}}-{\text{Quantity}}_{\text{Previous}}}$ (2)

Now substitute the respective in Equation (1) to get the value of marginal product at 4 units of labor inputs.

$\begin{array}{c}\text{Marginal product}=\frac{32-16}{4-1}\\ =5.33\end{array}$

Therefore, the value of marginal product at 4 units of labor is 5.33.

The average product can be calculated using the following equation.

$\text{Average product}=\frac{\text{Total Product}}{\text{Quantity}}$ (3)

Now substitute the respective values in Equation (3) to get the value of average product at 4 units of labor inputs.

$\begin{array}{c}\text{Average product}=\frac{32}{4}\\ =8\end{array}$

Therefore, the value of the average product at 4 units of labor is 8.

Table 2 shows the values of total product, marginal product, and average product using Equations (1), (2), and (3).

Table 2

Number of workers	Total product	Marginal product	Average product
1	16	16	16
4	32	5.33	8
9	48	3.2	5.33
16	64	2.29	4
25	80	1.78	3.2

Economics Concept Introduction

Marginal product: A marginal product is the additional output that a firm can produce using an additional unit of an input, holding use of the other input constant.

Average product: An average product is the quantity of output produced per unit of input.

 (f)

To determine

The total product, average product, and marginal product.

Explanation of Solution

In this case, the capital was fixed at 16 units. If there is a fall of 7 machines, then the capital will be 9 $\left(16-7\right)$ machines. Therefore, the new shot run production function can be represented as follows:

$\begin{array}{c}\text{Q}=4{\left(9\right)}^{0.5}{\text{L}}^{0.5}\\ \text{Q}=4\left(3\right){\text{L}}^{0.5}\\ \text{Q}=12{\text{L}}^{0.5}\end{array}$

Therefore, the new short run production function is $\text{Q}=12{\text{L}}^{0.5}$.

Table 3 shows the values of total product, marginal product, and average product at each level of new output using equation (1), (2), and (3).

Table 3

Number of workers	Total product	Marginal product	Average product
1	12	12	12
4	24	4	6
9	36	2.4	4
16	48	1.71	3
25	60	1.33	2.4

Economics Concept Introduction

Total product: The total product is the total quantity of output produced by a firm at the given time period.

Average product: An average product is the quantity of output produced per unit of input.

Marginal product: A marginal product is the additional output that a firm can produce using an additional unit of an input.