- Suppose you are told that the production function for widgets is given by the following equation where Q is the quantity of widgets, K is the number of units of capital used in producing widgets, and L is the number of units of labor used in producing widgets:
Q = 4K1/2L1/2
You are also told that total cost for this firm can be found by using the following equation where TC is total cost, Pk is the price per unit of capital, and Pl is the price per unit of labor:
TC = Pk K + Pl L
- Using this information, fill out the following table depicting the production function and the cost functions for this firm. Make sure you provide the units of measurement in your answers. If necessary, round your answers to the nearest hundredth.
| L | K | Q | VC | FC | TC | AVC | AFC | ATC | MC |
| 0 | 25 | — | — | — | — | ||||
| 1 | 25 | ||||||||
| 4 | 25 | $1.25 per unit of output | |||||||
| 9 | 25 | ||||||||
| 16 | 25 | $2.00 per unit of output | |||||||
| 25 | 25 |
- Given the above information, which input is the variable input? Explain why this is your answer.
- Given the above information, what is the price per unit of capital and what is the price per unit of labor?
- Does this firm experience diminishing returns to labor? To answer this question you will find it helpful to calculate the values for the marginal product of labor, MPl. Here’s a table to use in calculating these values. Make sure you provide the units of measurement in your answers. If necessary, round your answers to the nearest hundredth.
| L | K | Q | MPl |
| 0 | 25 | — | |
| 1 | 25 | ||
| 4 | 25 | ||
| 9 | 25 | ||
| 16 | 25 | ||
| 25 | 25 |
Answer:
| L | K | Q | VC | FC | TC | AVC | AFC | ATC | MC |
| 0 | 25 | 0 | $0 | $50 | $50 | — | — | — | — |
| 1 | 25 | 20 | $10 | $50 | $60 | $0.50 per unit of output | $2.50 per unit of output | $3.00 per unit of output | $0.50 per unit of output |
| 4 | 25 | 40 | $40 | $50 | $90 | $1 per unit of output | $1.25 per unit of output | $2.25 per unit of output | $1.50 per unit of output |
| 9 | 25 | 60 | $90 | $50 | $140 | $1.50 per unit of output | $0.83 per unit of output | $2.33 per unit of output | $2.50 per unit of output |
| 16 | 25 | 80 | $160 | $50 | $210 | $2.00 per unit of output | $0.63 per unit of output | $2.63 per unit of output | $3.50 per unit of output |
| 25 | 25 | 100 | $250 | $50 | $300 | $2.50 per unit of output | $0.50 per unit of output | $3.00 per unit of output | $4.50 per unit of output |
- Labor is the variable input: from the table we can see that the amount of labor varies as the level of output changes while the level of capital is constant and does not change as the level of output changes.
- To fill in the table you must figure out the price of labor and the price of capital.
From the table we know that the average variable cost is equal to $2.00 per unit of output and that this AVC occurs when the firm hires 16 units of labor and produces 80 units of output. Recall that AVC = VC/Q and in this case VC = Pl*L. Using this information we have $2.00 per unit of output = Pl*(16 units of labor)/(80 units of output) or Pl = ($2.00 per unit of output)(80 units of output)/(16 units of labor) = $10 per unit of labor. Review this calculation carefully and note in particular how the units of measurement work out to give you $ per unit of the variable input.
From the table we know that the average fixed cost is equal to $1.25 per unit of output and that this AFC occurs when the firm hires 25 units of capital and produces 40 units of output. Recall that AFC = FC/Q and in this case FC = Pk*K. Using this information we have $1.25 per unit of output = Pk*(25 units of capital)/(40 units of output) or Pk = ($1.25 per unit of output)(40 units of output)/(25 units of capital) = $2 per unit of capital. Review this calculation carefully and note in particular how the units of measurement work out to give you $ per unit of the fixed input.
- This firm does experience diminishing marginal returns to labor since as the amount of labor hired increases, output increases but at a diminishing rate. We can see this by examining the table below that shows the marginal product of labor. Notice that as we move down the table, the values of the marginal product of labor decrease: each additional unit of labor we hire results in output increasing, but at a decreasing rate.
| L | K | Q* | MPl |
| 0 | 25 | 0 | — |
| 1 | 25 | 20 | 20/1 = 20 units of output per unit of labor |
| 4 | 25 | 40 | 20/3 = 6.67 units of output per unit of labor |
| 9 | 25 | 60 | 20/5 = 4 units of output per unit of labor |
| 16 | 25 | 80 | 20/7 = 2.86 units of output per unit of labor |
| 25 | 25 | 100 | 20/9 = 2.22 units of output per unit of labor |
* Note that the values of Q were actually calculated in (a).
