The production engineers at Impact industries have derived the expansion path shown in the following figure. The price of labor is $100 per unit.
a. What prices does an impact industry pay for capital?
b. If the manager at impact decides to produce 180 units of output, how much labor and capital should be used in order to minimize total cost?
c. What is the total cost of producing 120,180, and 240 units of output in the long run?
d. Impact industries originally built the plant (i.e., purchased the amount of capital) designed to produce 180 units optimally. In the short run with capital fixed, if the manager decides to expand production to 240 units, what is the amount of labor and capital that will be used? (Hint: How must the firm expand output in the short run when capital is fixed?)
a. To find the price of capital, we need to look at the expansion path. The expansion path represents the combinations of labor and capital used to produce different levels of output while minimizing the cost. In this case, we know that the price of labor is $100 per unit.
b. In order to minimize total cost, we need to find the combination of labor and capital that corresponds to producing 180 units of output. To do this, we can follow these steps:
1. Look at the expansion path and find the point where the output level is closest to 180 units.
2. Once you've identified the point on the expansion path, read the corresponding values of labor and capital on the axes. This will give you the amount of labor and capital that should be used to minimize total cost.
c. To find the total cost of producing different units of output, we need to multiply the respective quantities of labor and capital used by their corresponding prices. For example, to find the total cost of producing 120 units of output, we can follow these steps:
1. Look at the expansion path and find the point where the output level is closest to 120 units.
2. Once you've identified the point on the expansion path, read the corresponding values of labor and capital on the axes.
3. Multiply the quantity of labor by the price of labor ($100 per unit) and the quantity of capital by the price of capital (which we need to determine in question a).
Repeat the same steps to find the total cost of producing 180 and 240 units of output.
d. In the short run, when capital is fixed, the firm can only adjust the quantity of labor to expand production. Since the manager wants to expand production from 180 units to 240 units, the firm needs to determine the additional quantity of labor needed.
1. Look at the expansion path and find the point where the output level is closest to 180 units.
2. Once you've identified the point on the expansion path, read the corresponding value of labor on the axis.
3. Find the difference between the quantity of labor used for 240 units and the quantity of labor used for 180 units. This will give you the additional quantity of labor needed.
Since the amount of capital is fixed, it will remain the same as in the optimal production level of 180 units.