1.1 Module 21: The Production Function

The Production Function

A firm produces goods or services for sale. To do this, it must transform inputs into output. The quantity of output a firm produces depends on the quantity of inputs; this relationship is known as the firm’s production function. As we’ll see, a firm’s production function underlies its cost curves. As a first step, let’s look at the characteristics of a hypothetical production function.

Inputs and Output

To understand the concept of a production function, let’s consider a farm that we assume, for the sake of simplicity, produces only one output, wheat, and uses only two inputs, land and labor. This particular farm is owned by a couple named George and Martha. They hire workers to do the actual physical labor on the farm. Moreover, we will assume that all potential workers are of the same quality—they are all equally knowledgeable and capable of performing farmwork.

George and Martha’s farm sits on 10 acres of land; no more acres are available to them, and they are currently unable to either increase or decrease the size of their farm by selling, buying, or leasing acreage. Land here is what economists call a fixed input—an input whose quantity is fixed for a period of time and cannot be varied. George and Martha are, however, free to decide how many workers to hire. The labor provided by these workers is called a variable input—an input whose quantity the firm can vary at any time.

In reality, whether or not the quantity of an input is really fixed depends on the time horizon. In the long run—that is, given that a long enough period of time has elapsed—firms can adjust the quantity of any input. So there are no fixed inputs in the long run. In contrast, the short run is defined as the time period during which at least one input is fixed. Later, we’ll look more carefully at the distinction between the short run and the long run. But for now, we will restrict our attention to the short run and assume that at least one input (land) is fixed.

George and Martha know that the quantity of wheat they produce depends on the number of workers they hire. Using modern farming techniques, one worker can cultivate the 10-acre farm, albeit not very intensively. When an additional worker is added, the land is divided equally among all the workers: each worker has 5 acres to cultivate when 2 workers are employed, each cultivates 3⅓ acres when 3 are employed, and so on. So as additional workers are employed, the 10 acres of land are cultivated more intensively and more bushels of wheat are produced.

The relationship between the quantity of labor and the quantity of output, for a given amount of the fixed input, constitutes the farm’s production function. The production function for George and Martha’s farm, where land is the fixed input and labor is the variable input, is shown in the first two columns of the table in Figure 21-1; the diagram there depicts the same information graphically. The curve in Figure 21-1 shows how the quantity of output depends on the quantity of the variable input for a given quantity of the fixed input; it is called the farm’s total product curve. The physical quantity of output, bushels of wheat, is measured on the vertical axis; the quantity of the variable input, labor (that is, the number of workers employed), is measured on the horizontal axis. The total product curve here slopes upward, reflecting the fact that more bushels of wheat are produced as more workers are employed.

Although the total product curve in Figure 21-1 slopes upward along its entire length, the slope isn’t constant: as you move up the curve to the right, it flattens out. To understand this changing slope, look at the third column of the table in Figure 21-1, which shows the change in the quantity of output generated by adding one more worker. That is, it shows the marginal product of labor, or MPL: the additional quantity of output from using one more unit of labor (one more worker).

In this example, we have data at intervals of 1 worker—that is, we have information on the quantity of output when there are 3 workers, 4 workers, and so on. Sometimes data aren’t available in increments of 1 unit—for example, you might have information on the quantity of output only when there are 40 workers and when there are 50 workers. In this case, you can use the following equation to calculate the marginal product of labor:

or

Note that Δ, the Greek uppercase delta, represents the change in a variable.

Now we can explain the significance of the slope of the total product curve: it is equal to the marginal product of labor. The slope of a line is equal to “rise” over “run.” This implies that the slope of the total product curve is the change in the quantity of output (the “rise”) divided by the change in the quantity of labor (the “run”). And this, as we can see from Equation 21-1, is simply the marginal product of labor. So in Figure 21-1, the fact that the marginal product of the first worker is 19 also means that the slope of the total product curve in going from 0 to 1 worker is 19. Similarly, the slope of the total product curve in going from 1 to 2 workers is the same as the marginal product of the second worker, 17, and so on.

In this example, the marginal product of labor steadily declines as more workers are hired—that is, each successive worker adds less to output than the previous worker. So as employment increases, the total product curve gets flatter.

Figure 21-2 shows how the marginal product of labor depends on the number of workers employed on the farm. The marginal product of labor, MPL, is measured on the vertical axis in units of physical output—bushels of wheat—produced per additional worker, and the number of workers employed is measured on the horizontal axis. You can see from the table in Figure 21-1 that if 5 workers are employed instead of 4, output rises from 64 to 75 bushels; in this case the marginal product of labor is 11 bushels—the same number found in Figure 21-2. To indicate that 11 bushels is the marginal product when employment rises from 4 to 5, we place the point corresponding to that information halfway between 4 and 5 workers.

In this example the marginal product of labor falls as the number of workers increases. That is, there are diminishing returns to labor on George and Martha’s farm. In general, there are diminishing returns to an input when an increase in the quantity of that input, holding the quantity of all other inputs fixed, reduces that input’s marginal product. Due to diminishing returns to labor, the MPL curve is negatively sloped.

To grasp why diminishing returns can occur, think about what happens as George and Martha add more and more workers without increasing the number of acres. As the number of workers increases, the land is farmed more intensively and the number of bushels increases. But each additional worker is working with a smaller share of the 10 acres—the fixed input—than the previous worker. As a result, the additional worker cannot produce as much output as the previous worker. So it’s not surprising that the marginal product of the additional worker falls.

The crucial point to emphasize about diminishing returns is that, like many propositions in economics, it is an “other things equal” proposition: each successive unit of an input will raise production by less than the last if the quantity of all other inputs is held fixed.

What would happen if the levels of other inputs were allowed to change? You can see the answer illustrated in Figure 21-3. Panel (a) shows two total product curves, TP₁₀ and TP₂₀. TP₁₀ is the farm’s total product curve when its total area is 10 acres (the same curve as in Figure 21-1). TP₂₀ is the total product curve when the farm’s area has increased to 20 acres. Except when 0 workers are employed, TP₂₀ lies everywhere above TP₁₀ because with more acres available, any given number of workers produces more output. Panel (b) shows the corresponding marginal product of labor curves. MPL₁₀ is the marginal product of labor curve given 10 acres to cultivate (the same curve as in Figure 21-2), and MPL₂₀ is the marginal product of labor curve given 20 acres. Both curves slope downward because, in each case, the amount of land is fixed, although at different levels. But MPL₂₀ lies everywhere above MPL₁₀, reflecting the fact that the marginal product of the same worker is higher when he or she has more of the fixed input to work with.

Figure 21-3 demonstrates a general result: the position of the total product curve depends on the quantities of other inputs. If you change the quantities of the other inputs, both the total product curve and the marginal product curve of the remaining input will shift.

Module 21 Review

Check Your Understanding

Use the data in the table below to graph the production function and the marginal product of labor. Do the data illustrate diminishing returns to labor? Explain.