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A firm is an organization that 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.

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. For example, in the long run, George and Martha can vary the amount of land they farm by buying or selling land. 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 in this chapter, 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 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 a variable input, is shown in the first two columns of the table in Figure 6-1; the diagram there shows the same information graphically. The curve in Figure 6-1 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 6-1 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 why the slope changes, look at the third column of the table in Figure 6-1, which shows the change in the quantity of output that is generated by adding one more worker. This is called the marginal product of labor, or MPL: the additional quantity of output from using one more unit of labor (where one unit of labor is equal to one worker). In general, the marginal product of an input is the additional quantity of output that is produced by using one more unit of that input.

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

or

In this equation, 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” (see the appendix to Chapter 2). This implies that the slope of the total product curve is the change in the quantity of output (the “rise”, ΔQ) divided by the change in the quantity of labor (the “run”, ΔL). And this, as we can see from Equation 6-1, is simply the marginal product of labor. So in Figure 6-1, 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.

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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 6-2 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 6-1 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 6-2. 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.

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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 of land. As the number of workers increases, the land is farmed more intensively and the number of bushels produced 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 6-3. 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 6-1). TP₂₀ is the total product curve when the farm 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 6-2), and MPL₂₀ is the marginal product of labor curve given 20 acres.

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Both curves slope downward because, in each case, the amount of land is fixed, albeit 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 6-3 demonstrates a general result: the position of the total product curve of a given input depends on the quantities of other inputs. If you change the quantity of the other inputs, both the total product curve and the marginal product curve of the remaining input will shift.

Once George and Martha know their production function, they know the relationship between inputs of labor and land and output of wheat. But if they want to maximize their profits, they need to translate this knowledge into information about the relationship between the quantity of output and cost. Let’s see how they can do this.

To translate information about a firm’s production function into information about its costs, we need to know how much the firm must pay for its inputs. We will assume that George and Martha face a cost of $400 for the use of the land. It is irrelevant whether George and Martha must rent 10 acres of the land for $400 from someone else or whether they own the land themselves and forgo earning $400 from renting it to someone else. Either way, they pay an opportunity cost of $400 by using the land to grow wheat. Moreover, since the land is a fixed input, the $400 George and Martha pay for it is a fixed cost, denoted by FC—a cost that does not depend on the quantity of output produced (in the short run). In business, fixed cost is often referred to as “overhead cost.”

We also assume that George and Martha must pay each worker $200. Using their production function, George and Martha know that the number of workers they must hire depends on the amount of wheat they intend to produce. So the cost of labor, which is equal to the number of workers multiplied by $200, is a variable cost, denoted by VC—a cost that depends on the quantity of output produced. It is variable because in order to produce more they have to employ more units of input. Adding the fixed cost and the variable cost of a given quantity of output gives the total cost, or TC, of that quantity of output. We can express the relationship among fixed cost, variable cost, and total cost as an equation:

(6-2) Total cost = Fixed cost + Variable cost

or

TC = FC + VC

The table in Figure 6-4 shows how total cost is calculated for George and Martha’s farm. The second column shows the number of workers employed, L. The third column shows the corresponding level of output, Q, taken from the table in Figure 6-1. The fourth column shows the variable cost, VC, equal to the number of workers multiplied by $200, the cost per worker. The fifth column shows the fixed cost, FC, which is $400 regardless of how many workers are employed. The sixth column shows the total cost of output, TC, which is the variable cost plus the fixed cost.

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The first column labels each row of the table with a letter, from A to I. These labels will be helpful in understanding our next step: drawing the total cost curve, a curve that shows how total cost depends on the quantity of output.

George and Martha’s total cost curve is shown in the diagram in Figure 6-4, where the horizontal axis measures the quantity of output in bushels of wheat and the vertical axis measures total cost in dollars. Each point on the curve corresponds to one row of the table in Figure 6-4. For example, point A shows the situation when 0 workers are employed: output is 0, and total cost is equal to fixed cost, $400. Similarly, point B shows the situation when 1 worker is employed: output is 19 bushels, and total cost is $600, equal to the sum of $400 in fixed cost and $200 in variable cost.

Like the total product curve, the total cost curve slopes upward: due to the variable cost, the more output produced, the higher the farm’s total cost. But unlike the total product curve, which gets flatter as employment rises, the total cost curve gets steeper. That is, the slope of the total cost curve is greater as the amount of output produced increases. As we will soon see, the steepening of the total cost curve is also due to diminishing returns to the variable input. Before we can understand this, we must first look at the relationships among several useful measures of cost.

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