Chapter 6, Problem 9P

 (a)

To determine

The total cost function.

Explanation of Solution

In this case, the labor (L) costs $12 per hour and the capital (K) costs rent is $7 per hour. Now the total cost (TC) function can be represented as follows:

$\text{TC}=\left(\text{Wage}\times \text{Labor}\right)+\left(\text{Rent}\times \text{Capital}\right)$

$\text{TC}=\left(\$12\times \text{L}\right)+\left(\$7\times \text{K}\right)$ (1)

Thus, cost function is $\text{TC}=\left(\$12\times \text{L}\right)+\left(\$7\times \text{K}\right)$.

Economics Concept Introduction

Total cost: The total cost is the sum of firm’s fixed and variable costs.

 (b)

To determine

The total cost function and the isocost line.

Explanation of Solution

If the producers wish to keep their total cost to exactly $100 then, the total cost function, given in part (a) can be used to find the equation for the isocost line of the firm.

$\begin{array}{c}100=\left(12\times \text{L}\right)+\left(7\times \text{K}\right)\\ 7\text{K}=100-12\text{L}\\ \text{K}=\frac{100}{7}-\frac{12}{7}\text{L}\end{array}$

Therefore, this is the equation for an isocost line corresponding to exactly $100 of costs. Is $\text{K}=\frac{100}{7}-\frac{12}{7}\text{L}$

Economics Concept Introduction

Isocost line: An isocost line is a curve that shows all of the input combinations that yield the same cost.

 (c)

To determine

The graphical representation of the isocost line.

Explanation of Solution

Maximum quantity of capital (When labor quantity is 0 units) can be calculated by substituting the respective values in Equation (1).

$\begin{array}{c}\text{100}=\left(\$12\times \text{0}\right)+\left(\$7\times \text{K}\right)\\ 7\text{K}=100\\ \text{K}=\frac{100}{7}\end{array}$

Thus, maximum quantity of capital is $\frac{100}{7}$.

Maximum quantity of labor (When capital quantity is 0 units) can be calculated by substituting the respective values in Equation (1).

$\begin{array}{c}\text{100}=\left(\$12\times \text{L}\right)+\left(\$7\times \text{0}\right)\\ 12\text{L}=100\\ \text{L}=\frac{100}{12}\end{array}$

Thus, maximum quantity of labor is $\frac{100}{12}$.

The Figure 1 illustrates the isocost line.

In Figure 1, the vertical axis shows the capital and the horizontal axis shows the labor where each point lies on the isocost line that shows all the possible input combinations that yield the same cost.

Economics Concept Introduction

Isocost line: An isocost line is a curve that shows all of the input combinations that yield the same cost.

 (d)

To determine

The vertical intercept and the horizontal intercept of the isocost line.

Explanation of Solution

In Figure 1, the vertical axis shows the different quantity of capital input used in the production process where the vertical intercept indicates the maximum quantity of capital that can be rented with the total cost of $100, without hiring labor.

The horizontal axis shows the different quantity of labor input used in the production process where the horizontal intercept indicates the maximum quantity of labor that can be hired with the total cost, here its $100, without renting for capital.

 (e)

To determine

The slope of the isocost line.

Explanation of Solution

In Figure 1, each point lies on the isocost line that shows all the possible input combinations that yield the same cost where the slope of the isocost line indicates a negative sign $\left(-\frac{12}{7}\right)$ because the slope of the isocost line indicates the ratio of two input prices. This price ratio of inputs provide the information about the cost consequences of trading off one unit of capital for another unit of labor.

 (f)

To determine

The new isocost line.

Explanation of Solution

In this case, the labor cost increases from $12 per hour to $14 per hour and the capital remains the same, $7 per hour. If the producers wish to keep their total cost to exactly $100 then, the new total cost function can be represented as follows:

Maximum quantity of capital (When labor quantity is 0 units) can be calculated by substituting the respective values in Equation (1).

$\begin{array}{c}\text{100}=\left(\$14\times \text{0}\right)+\left(\$7\times \text{K}\right)\\ 7\text{K}=100\\ \text{K}=\frac{100}{7}\end{array}$

Thus, maximum quantity of capital is $\frac{100}{7}$.

Maximum quantity of labor (When capital quantity is 0 units) can be calculated by substituting the respective values in Equation (1).

$\begin{array}{c}\text{100}=\left(\$14\times \text{L}\right)+\left(\$7\times \text{0}\right)\\ 14\text{L}=100\\ \text{L}=\frac{100}{14}\end{array}$

Thus, maximum quantity of labor is $\frac{100}{14}$.

The Figure 2 illustrates the new isocost line.

In Figure 2, the vertical axis shows the capital and the horizontal axis shows the labor when the relative prices of input changes, the isocost line rotates. In this case, the wage rises from $12 to $14. As a result, the initial isoquant line ${\text{C}}_1$ rotates and the new isoquant line is ${\text{C}}_2$ which is steeper than the initial one.

Economics Concept Introduction

Isocost line: An isocost line is a curve that shows all of the input combinations that yield the same cost.