We start by analyzing production in the short run because it is the simplest case. In our earlier discussion of the assumptions of this model, we defined the short run as the period during which a firm cannot change the amount of capital. While the capital stock is fixed, the firm can choose how much labor to hire to minimize its cost of making the output quantity. Table 6.1 shows some values of labor inputs and output quantities from a short-run production function. Here, we use the Cobb–Douglas production function of our earlier example and fix capital ( ) at 4 units, so the numbers in Table 6.1 correspond to the production function

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Figure 6.1 plots the short-run production function from Table 6.1. The plot contains the numbers in Table 6.1, but also shows output levels for all amounts of labor between 0 and 5 hours per week, as well as amounts of labor greater than 5 hours per week.

Even though capital is fixed, when the firm increases its labor inputs, its output rises. This reflects Assumption 6 above: More inputs mean more output. Notice, however, that the rate at which output rises slows as the firm hires more and more labor. This phenomenon of additional units of labor yielding less and less additional output reflects Assumption 7: The production function exhibits diminishing marginal returns to inputs. To see why there are diminishing marginal returns, we need to understand just what happens when one input increases while the other input remains fixed. This is what is meant by an input’s marginal product.

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The incremental output that a firm can produce by using an additional unit of an input (holding use of the other input constant) is called the marginal product. In the short run, the marginal product that is most relevant is the marginal product of labor because we are assuming that capital is fixed at 4. The marginal product of labor (MP_L) is the change in quantity (ΔQ) resulting from a 1-unit change in labor inputs (ΔL):

MP_L = ΔQ/ΔL

The marginal product of labor for our short-run production function Q = 4^0.5L^0.5 is shown in the fourth column of Table 6.1. If a firm with this production function (in which capital is fixed at 4 units) uses 0 units of labor, it produces 0 units of output. (Perhaps the firm can’t make any output without anyone to run the machines.) If it hires 1 unit of labor to combine with the 4 units of capital, it can produce 2 units of output. Therefore, the marginal product of that first unit of labor is 2.00. With 2 units of labor (and the same 4 units of capital), the firm can produce 2.83 units of output, making the marginal product of the second unit of labor the change in output or 2.83 – 2.00 = 0.83. With 3 units of labor, output rises by 0.63 to 3.46, so the marginal product of labor is 0.63, and so on.

This diminishing marginal product of labor is the reduction in the extra output obtained from adding more and more labor; it is embodied in our Assumption 7 and is a common feature of production functions. This is why the production function curve in Figure 6.1 flattens at higher quantities of labor. A diminishing marginal product makes intuitive sense. With a fixed amount of capital, then every time you add a worker, each worker has less capital to use. If a coffee shop has one espresso machine and one worker, she has a machine at her disposal during her entire shift. If the coffee shop hires a second worker to work the same hours as the first, the two workers must share the machine and will get in each other’s way. With three workers per shift and still only one machine, the situation gets worse. A fourth worker produces a tiny bit more output, but this fourth worker’s marginal product will be smaller than the first’s. The coffee shop’s solution to this problem is to buy more espresso machines—that is, add more capital. But we’ve assumed it can’t do so in the short run. Eventually, in the long run, it can. We look at what happens when the firm can change its capital level later in this chapter.

Keep in mind, however, that diminishing marginal returns do not have to occur all the time; they just need to occur eventually. A production function could have increasing marginal returns at low levels of labor before running into the problem of diminishing marginal product.

A Graphical Analysis of Marginal Product We can plot the marginal product on a graph of the production function. Recall that the marginal product is the change in output quantity that comes from adding one additional unit of input: MP_L = ΔQ/ΔL. ΔQ/ΔL is the slope of the short-run production function in Figure 6.1. Thus, the marginal product of labor at a given level of labor input is the slope of the production function at that point. Panel a of Figure 6.2 shows how the MP_L can be derived from our production function. At L = 1, the slope of the production function (i.e., the slope of the line tangent to the production function at L = 1) is relatively steep. Adding additional labor at this point will increase output by a substantial amount. At L = 4, the slope is considerably flatter, and adding additional labor will boost output by a smaller increment than when L = 1. The marginal product of labor falls between L = 1 and L = 4, as we saw in Table 6.1. Panel b of Figure 6.2 shows the corresponding marginal-product-of-labor curve. Because this production function exhibits diminishing marginal returns at all levels of labor, the marginal product curve is downward-sloping.

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A Mathematical Representation of Marginal Product To find MP_L, we need to calculate the additional output obtained by adding an incremental unit of labor, holding capital constant. So let’s compute the firm’s increase in output when it uses L + ΔL units of labor instead of L units (holding capital constant). ΔL is the incremental unit of labor. Mathematically, MP_L is

Applying this to our short-run production function gives

To consider what happens as we let ΔL get really tiny involves some calculus. If you don’t know calculus, however, here is the marginal product formula for our Cobb–Douglas production function: The MP_L values calculated using this formula are shown in Figure 6.2. (These values are slightly different from those in Table 6.1 because while the table sets ΔL = 1, the formula allows the incremental unit to be much smaller. The economic idea behind both calculations is the same, however.)

It’s important to see that the marginal product is not the same as the average product. Average product is the total quantity of output divided by the number of units of input used to produce it. The average product of labor (AP_L), for example, is the quantity produced Q divided by the amount of labor L used to produce it:

AP_L = Q/L

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The average product of labor for our short-run production function is shown in the last column of Table 6.1. Notice that the average product of labor falls as labor inputs increase. This decline occurs because the marginal product of labor is less than the average product at each level of labor in the table, so each unit of labor added on the margin brings down labor’s average product. The easiest way to see this is to consider your grades in a course. Suppose that your mid-semester average is 80%, then you take an exam and get a 90%. Your average will rise because the score on the last exam you took (the marginal exam) is greater than your previous average. If your exam score is 65% instead of 90%, however, your course average will fall, because the marginal score is lower than the average score.

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