Chapter 10, Problem 4.1P

(a)

To determine

Which process will be employed.

Explanation of Solution

The total variable cost incurred for each process can be calculated using the following equation:

$\text{Total variable cost}=\left(\begin{array}{c}\left(\text{Price of capital}\times \text{Quantity of capital}\right)\\ +\left(\text{Price of labor}\times \text{Quantity of labor}\right)\end{array}\right)$ (1)

Given the cost of capital per unit is $5, and the cost of labor per unit is $3. Process 1 requires 5 units of labor and 2 units of capital. Process 2 requires 3 units of labor and 3 units of capital. Process 3 requires 1 unit of labor and 6 units of capital.

Substitute the respective values in Equation 1 to calculate the total variable cost incurred in process 1, which is calculated as follows:

$\begin{array}{c}\text{Total variable cost}=\left(5\times 2\right)+\left(3\times 5\right)\\ =10+15\\ =25\end{array}$

The total variable cost of process 1 is $25.

Table 1 shows the total variable cost of the three processes calculated using Equation 1.

Table 1

	Quantity of capital	Quantity of labor	Total variable cost
Process 1	2	5	$                25.00
Process 2	3	3	$                24.00
Process 3	6	1	$                33.00

Since the cheapest cost of production is for process 2, the process to be employed is process 2.

Economics Concept Introduction

Total Variable Cost:  Total variable cost is defined as the cost that depends on the level of production or output of a firm.

(b)

To determine

The total variable cost curve.

Explanation of Solution

Figure 1 shows the total variable cost curve.

The horizontal axis of Figure 1 measures the quantity of output, and the vertical axis measures the cost. The total variable cost curve is an upward sloping curve as the cost increases the output increases. TVC shows the total variable cost when the cost of capital is $5, and cost of labor is $3 per unit. TVC shows the total variable cost when the cost of capital is $7, and the cost of labor is $3 per unit.

Economics Concept Introduction

Total Variable Cost:  Total variable cost is defined as the cost that depends on the level of production or output of a firm.

(c)

To determine

The demand for capital and labor.

Explanation of Solution

Since 1 unit of output requires 3 units of capital and 3 units of labor using process 1, the amount of capital and labor required to produce any quantity of output can be calculated using Equations 2 and 3.

$\text{Amount of capital required}=3\times \text{Quantity}$ (2)

$\text{Amount of labor required}=3\times \text{Quantity}$ (3)

Substitute the respective values in Equation 2 to calculate the quantity of capital required to produce 25 units of output, which is calculated as follows:

$\begin{array}{c}\text{Amount of capital required}=3\times 25\\ =75\end{array}$

The amount of capital required to produce 25 units of output is 75.

Similarly, the amount of labor required to produce 25 units of output can be calculated using Equation 3 as follows:

$\begin{array}{c}\text{Amount of labor required}=3\times 25\\ =75\end{array}$

The amount of labor required to produce 25 units of output is 75.

The total variable cost incurred to produce 25 units of output using process 2 can be calculated using Equation 1 as follows:

$\begin{array}{c}\text{Total variable cost}=\left(5\times 75\right)+\left(3\times 75\right)\\ =375+225\\ =600\end{array}$

Thus, the total variable cost to produce 25 units of output is $600.

Table 1 shows the quantity of capital and labor required to produce various levels of output through process 1, which is calculated using Equations (1), (2), and (3).

Table 1

Quantity of output	Amount of capital required	Amount of labor required	Total variable cost
25	75	75	$              600.00
50	150	150	$           1,200.00
100	300	300	$           2,400.00

Economics Concept Introduction

Total Variable Cost:  Total variable cost is defined as the cost that depends on the level of production or output of a firm.

(d)

To determine

The demand for capital and labor.

Explanation of Solution

When the cost of capital increases to $7, the total variable cost incurred in process 1 can be calculated by substituting the values in Equation 1 as follows:

Substitute the respective values in Equation 1 to calculate the total variable cost incurred in process 1, which is calculated as follows:

$\begin{array}{c}\text{Total variable cost}=\left(7\times 2\right)+\left(3\times 5\right)\\ =14+15\\ =29\end{array}$

The total variable cost of process 1 is $29.

Table 3 shows the total variable cost of the three processes, which are calculated using Equation (1).

Table 3

	Quantity of capital	Quantity of labor	Total variable cost
Process 1	2	5	$                29.00
Process 2	3	3	$                30.00
Process 3	6	1	$                45.00

When the cost of capital increases to $7, the process to be chosen is process 1, as it is the cheapest.

Since 1 unit of output requires 2 units of capital and 5 units of labor using process 1, the amount of capital and labor required to produce any quantity of output can be calculated using Equations 4 and 5.

$\text{Amount of capital required}=2\times \text{Quantity}$ (4)

$\text{Amount of labor required}=5\times \text{Quantity}$ (5)

Substitute the respective values in Equation (4) to calculate the quantity of capital required to produce 25 units of output, which is calculated as follows:

$\begin{array}{c}\text{Amount of capital required}=2\times 25\\ =50\end{array}$

The amount of capital required to produce 25 units of output is 50.

Similarly, the amount of labor required to produce 25 units of output can be calculated using Equation (5) as follows:

$\begin{array}{c}\text{Amount of labor required}=5\times 25\\ =125\end{array}$

The amount of labor required to produce 25 units of output is 125.

The total variable cost incurred to produce 25 units of output using process 1 can be calculated using Equation 1 as follows:

$\begin{array}{c}\text{Total variable cost}=\left(7\times 50\right)+\left(3\times 125\right)\\ =350+375\\ =725\end{array}$

Thus, the total variable cost to produce 25 units of output is $725.

Table 4 shows the quantity of capital and labor required to produce various levels of output through process 1, which is calculated using Equations (1), (4), and (5).

Table 4

Quantity of output	Amount of capital required	Amount of labor required	Total variable cost
25	50	125	$              725.00
50	100	250	$           1,450.00
100	200	500	$           2,900.00

Economics Concept Introduction

Total Variable Cost:  Total variable cost is defined as the cost that depends on the level of production or output of a firm.