Afirm 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 and the state or generosity of the production technology; 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 labour. This particular farm is owned by a couple named Alec and Janet. They hire workers to do the actual physical labour 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.

Alec and Janet’s farm sits on 1 hectare of land; no more land is available to them, and they are currently unable to either increase or decrease the size of their farm by selling, buying, or leasing land. Land here is what economists call a fixed input—an input whose quantity is fixed for a period of time and cannot be varied. Alec and Janet are, however, free to decide how many workers to hire. The labour 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, Alec and Janet 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.

Alec and Janet 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 1 hectare farm, albeit not very intensively. When an additional worker is added, the land is divided equally among all the workers: each worker has ½ of a hectare to cultivate when 2 workers are employed, each cultivates 1/3 of a hectare when 3 are employed, and so on. So as additional workers are employed, the land is cultivated more intensively and more bushels of wheat are produced. The relationship between the quantity of labour and the quantity of output, for a given amount of the fixed input and the given production technology (i.e., farming techniques used in this example), constitutes the farm’s production function. The production function for Alec and Janet’s farm, where land is the fixed input and labour is a variable input, is shown in the first two columns of the table in Figure 11-1; the diagram there shows the same information graphically. The curve in Figure 11-1 shows how the quantity of output depends on the quantity of the variable input, for a given quantity of the fixed input and the given production technology; 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, labour (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 11-1 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 11-1, which shows the change in the quantity of output that is generated by adding one more worker. This is called the marginal product of labour, or MPL: the additional quantity of output from using one more unit of labour (where one unit of labour is equal to one worker), holding other inputs and the production technology fixed. 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.

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 labour:

or

In this equation, Δ, the Greek uppercase delta, represents the change in a variable. Here, the lowercase q represents the quantity of output made by an individual firm, as opposed to an uppercase Q, which represents the quantity of output in the market as a whole.

Now we can explain the significance of the slope of the total product curve: it is equal to the marginal product of labour. The slope of a line is equal to “rise” over “run” (see Appendix 2A). 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 labour (the “run”, ΔL). And this, as we can see from Equation 11-1, is simply the marginal product of labour. So in Figure 11-1, 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 labour 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 11-2 shows how the marginal product of labour depends on the number of workers employed on the farm. The marginal product of labour, 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 11-1 that if 5 workers are employed instead of 4, output rises from 64 to 75 bushels; in this case the marginal product of labour is 11 bushels—the same number found in Figure 11-2. 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 labour falls as the number of workers increases. That is, there are diminishing returns to labour on Alec and Janet’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 and the production technology fixed, reduces that input’s marginal product. Due to diminishing returns to labour, the MPL curve is negatively sloped.

To grasp why diminishing returns can occur, think about what happens as Alec and Janet add more and more workers without increasing the amount of land or improving their farming techniques. 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 1 hectare—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 and the production technology are held fixed.

What would happen if the levels of other inputs or production techniques were allowed to change? You can see the answer illustrated in Figure 11-3. Panel (a) shows two total product curves, TP₁ and TP₂. TP₁ is the farm’s total product curve when its total area is 1 hectare (the same curve as in Figure 11-1). TP₂ is the total product curve when the farm has increased to 2 hectares, holding farming techniques fixed. Except when 0 workers are employed, TP₂ lies above TP₁ everywhere because with more land available, any given number of workers produces more output. Panel (b) shows the corresponding marginal product of labour curves. MPL₁ is the marginal product of labour curve given 1 hectare to cultivate (the same curve as in Figure 11-2), and MPL₂ is the marginal product of labour curve given 2 hectares. Both curves slope downward because, in each case, the amount of land is fixed, albeit at different levels. But MPL₂ lies above MPL₁ everywhere, 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 11-3 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. Similarly, if there is an improvement in farming techniques (an example of an improvement in production technology), both the total product curve and the marginal product of labour curve would shift upward. With improved farming techniques, farmers can produce more bushels of wheat for any given amount of land.

Alec and Janet may also be interested in knowing how much wheat is harvested per worker. The average product of labour, or APL, is the amount of output produced per unit of labour. In general, the average product of an input is the amount of output produced per unit of that input employed in the production process:

or

The average product of labour is important because it tells the producer how many units of output the average or typical unit of labour can produce. Recall, the marginal product of labour shows the change in the quantity of output that is generated by adding one more worker. Although these quantities sound similar, they are typically not the same and confusion between them is a common source of errors in economics. Figure 11-4 shows the average product of labour curve for Alec and Janet’s farm compared to the marginal product of labour.

As you can see from the table in Figure 11-4, all other things equal, as the number of workers employed increases, output rises but the average product of labour falls. For example, when the number of workers employed increases from 4 to 5, output increases from 64 bushels to 75 bushels. However, the average product of labour falls from 16 bushels to 15 bushels. Why does this happen? It is because the marginal product is less than the average product; thus, the average product is decreasing. When the number of workers rises from 4 to 5, the marginal product of the fifth worker is 11 bushels (the table in Figure 11-1 shows the marginal product labour). The marginal contribution of that additional worker is below the average productivity of the first 4 workers, so the fifth worker lowers the average product of the whole group. On the farm, the marginal product of labour is less than the average product of labour for all quantities of labour, so the entire average product curve slopes downward.

Our example shows that both marginal product and average product are downward sloping. In practice, however, economists believe that both the marginal product and average product curves often slope upward as a firm increases its variable input from some low level (such as zero), sloping downward only at higher levels of this input: so they look like inverted U-shape curves as shown in Figure 11-5.

At a low level of the variable input, the marginal product is increasing because when the firm starts with a low number of workers. Adding more workers in the production process allows these workers to specialize in various tasks. Specialization can allow workers to become more productive, which raises their marginal product. For example, one individual producing salsa would have to perform all the tasks involved: selecting and preparing the ingredients, mixing the salsa, bottling and labelling it, packing it into cases, and so on. As more workers are employed, they can divide the tasks, with each worker specializing in one or a few aspects of salsa-making. This specialization leads to increasing returns and the hiring of additional workers, resulting in a marginal product curve that initially slopes upward. But once there are enough workers to have exhausted the benefits of specialization, diminishing returns to labour set in and the marginal product curve starts sloping downward. Thus, a typical marginal product curve has an inverted U-shape as shown in Figure 11-5.

By looking at Figure 11-5, you can also observe the following key facts about the relationship between marginal product and average product:

As a result, when the marginal product curve has an inverted U-shape, the average product of labour curve also has an inverted U-shape.

Once Alec and Janet know their production function, they know the relationship between inputs of labour and land and output of wheat for a given level of farming technique (or technology). 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 Alec and Janet face either an explicit or an implicit cost of $400 for the use of the land. As we learned in Chapter 9, it is irrelevant whether Alec and Janet must rent 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 Alec and Janet 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 Alec and Janet must pay each worker $200. Using their production function, Alec and Janet know that the number of workers they must hire depends on the amount of wheat they intend to produce. So the cost of labour, 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. 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:

or

TC = FC + VC

The table in Figure 11-6 shows how total cost is calculated for Alec and Janet’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 11-1. The fourth column shows the variable cost, VC, equal to the number of workers multiplied by $200. 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.

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.

Alec and Janet’s total cost curve is shown in the diagram in Figure 11-6, 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 11-6. 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.