1.1 Module 21: The Production Function

© Terrance Klassen/AgeFotostock

WHAT YOU WILL LEARN

  • The importance of the firm’s production function, the relationship between the quantity of inputs and the quantity of output
  • Why production is often subject to diminishing returns to inputs

The Production Function

A production function is the relationship between the quantity of inputs a firm uses and the quantity of output it produces.

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.

A fixed input is an input whose quantity is fixed for a period of time and cannot be varied.

A variable input is an input whose quantity the firm can vary at any time.

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.

The long run is the time period in which all inputs can be varied.

The short run is the time period in which at least one input is fixed.

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 total product curve shows how the quantity of output depends on the quantity of the variable input, for a given quantity of the fixed input.

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.

The table shows the production function, the relationship between the quantity of the variable input (labor, measured in number of workers) and the quantity of output (wheat, measured in bushels) for a given quantity of the fixed input. It also shows the marginal product of labor on George and Martha’s farm. The total product curve shows the production function graphically. It slopes upward because more wheat is produced as more workers are employed. It also becomes flatter because the marginal product of labor declines as more and more workers are employed.

The marginal product of an input is the additional quantity of output produced by using one more unit of that input.

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.

The marginal product of labor curve plots each worker’s marginal product, the increase in the quantity of output generated by each additional worker. The change in the quantity of output is measured on the vertical axis and the number of workers employed on the horizontal axis. The first worker employed generates an increase in output of 19 bushels, the second worker generates an increase of 17 bushels, and so on. The curve slopes downward due to diminishing returns to labor.

There are diminishing returns to an input when an increase in the quantity of that input, holding the levels of all other inputs fixed, leads to a decline in the marginal product of that input.

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.

As more workers are added to a fixed amount of land, each worker adds less to total output than the previous worker.
Cultura Limited/SuperstockReview

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.

This figure shows how the quantity of output—illustrated by the total product curve—and marginal product depend on the level of the fixed input. Panel (a) shows two total product curves for George and Martha’s farm, TP10 when their farm is 10 acres and TP20 when it is 20 acres. With more land, each worker can produce more wheat. So an increase in the fixed input shifts the total product curve up from TP10 to TP20. This also implies that the marginal product of each worker is higher when the farm is 20 acres than when it is 10 acres. As a result, an increase in acreage also shifts the marginal product of labor curve up from MPL10 to MPL20. Panel (b) shows the marginal product of labor curves. Note that both marginal product of labor curves still slope downward due to diminishing returns to labor.

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.

THE MYTHICAL MAN-MONTH

The concept of diminishing returns to an input was first formulated by economists during the late eighteenth century. These economists, notably including Thomas Malthus, drew their inspiration from agricultural examples. Although still valid, examples drawn from agriculture can seem somewhat musty and old-fashioned in our modern economy.

However, the idea of diminishing returns to an input applies with equal force to the most modern of economic activities—such as, say, the design of software. In 1975 Frederick P. Brooks Jr., a project manager at IBM during the days when it dominated the computer business, published a book titled The Mythical Man-Month that soon became a classic.

The chapter that gave its title to the book is basically about diminishing returns to labor in the writing of software. Brooks observed that multiplying the number of programmers assigned to a project did not produce a proportionate reduction in the time it took to get the program written. A project that could be done by one programmer in 12 months could not be done by 12 programmers in one month. The “mythical man-month” is the false notion that the number of lines of programming code produced is proportional to the number of code writers employed. In fact, above a certain number, adding another programmer on a project actually increased the time to completion.

The argument of The Mythical Man-Month is summarized in Figure 21-4. The upper part of the figure shows how the quantity of the project’s output, as measured by the number of lines of code produced per month, varies with the number of programmers. Each additional programmer accomplishes less than the previous one, and beyond a certain point an additional programmer is actually counterproductive. The lower part of the figure shows the marginal product of each successive programmer, which falls as more programmers are employed and eventually becomes negative.

In other words, programming is subject to diminishing returns so severe that at some point more programmers actually have negative marginal product. The source of the diminishing returns lies in the nature of the production function for a programming project: each programmer must coordinate his or her work with that of all the other programmers on the project, leading to each person spending more time communicating with others—exchanging e-mails, devising project plans, attending meetings, and so on. In other words, other things equal, there are diminishing returns to labor. It is likely, however, that if fixed inputs devoted to programming projects are increased—say, installing a faster Wiki system—the problem of diminishing returns for additional programmers can be mitigated.

Module 21 Review

Solutions appear at the back of the book.

Check Your Understanding

1. Bernie’s ice-making company produces ice cubes using a 10-ton machine and electricity (along with water, which we will ignore as an input for simplicity). The quantity of output, measured in pounds of ice, is given in the accompanying table.

  • a. What is the fixed input? What is the variable input?

  • b. Construct a table showing the marginal product of the variable input. Does it show diminishing returns?

  • c. Suppose a 50% increase in the size of the fixed input increases output by 100% for any given amount of the variable input. What is the fixed input now? Construct a table showing the quantity of output and the marginal product in this case.

Quantity of electricity (kilowatts) Quantity of ice (pounds)
0        0
1 1,000
2 1,800
3 2,400
4 2,800

Multiple-Choice Questions

Question

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Question

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Question

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Question

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Question

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Critical-Thinking Questions

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.

Quantity of labor Quantity of output
L Q
0  0
1 19
2 36
3 51
4 64
5 75
6 84
7 91
8 96

PITFALLS: WHAT ARE THE RIGHT UNITS TO USE?

WHAT ARE THE RIGHT UNITS TO USE?

The marginal product of labor (or any other input) is defined as the increase in the quantity of output when you increase the quantity of that input by one unit. But when we say a “unit” of labor, do we mean an additional hour of labor, an additional week, or a person-year?

It doesn’t matter, as long as you are consistent. whatever units you use, always be careful that you use the same units throughout your analysis of any problem. One common source of error in economics is getting units confused—say, comparing the output added by an additional hour of labor with the cost of employing a worker for a week.

To learn more, review the definition of marginal product of labor.