We’ve just learned how to derive a firm’s total cost curve from its production function. Our next step is to take a deeper look at total cost by deriving two extremely useful measures: marginal cost and average cost. As we’ll see, these two measures of the cost of production have a somewhat surprising relationship to each other. Moreover, they will prove to be vitally important in Chapter 12, where we will use them to analyze the firm’s output decision and the market supply curve.

We defined marginal cost in Chapter 9: it is the change in total cost generated by producing one more unit of output. We’ve already seen that the marginal product of an input is easiest to calculate if data on output are available in increments of one unit of that input, all other things equal. Similarly, marginal cost is easiest to calculate if data on total cost are available in increments of one unit of output. When the data come in less convenient increments, it’s still possible to calculate marginal cost. But for the sake of simplicity, let’s work with an example in which the data come in convenient one-unit increments.

Selena’s Gourmet Salsas produces bottled salsa and Table 11-1 shows how its costs per day depend on the number of cases of salsa it produces per day. The firm has fixed cost of $108 per day, shown in the second column, which represents the daily cost of its food-preparation equipment. The third column shows the variable cost, and the fourth column shows the total cost. Panel (a) of Figure 11-8 plots the total cost curve. Like the total cost curve for Alec and Janet’s farm in Figure 11-6, this curve slopes upward, getting steeper as you move up it to the right.

The significance of the slope of the total cost curve is shown by the fifth column of Table 11-1, which calculates marginal cost: the additional cost of each additional unit, all other things equal. The general formula for marginal cost is:

or

As in the case of marginal product, marginal cost is equal to “rise” (the increase in total cost) divided by “run” (the increase in the quantity of output). So just as marginal product is equal to the slope of the total product curve, marginal cost is equal to the slope of the total cost curve.

Now we can understand why the total cost curve gets steeper as we move up it to the right: as you can see in Table 11-1, marginal cost at Selena’s Gourmet Salsas rises as output increases. Panel (b) of Figure 11-8 shows the marginal cost curve corresponding to the data in Table 11-1. Notice that, as in Figure 11-2, we plot the marginal cost for increasing output from 0 to 1 case of salsa halfway between 0 and 1, the marginal cost for increasing output from 1 to 2 cases of salsa halfway between 1 and 2, and so on.

Why does the marginal cost curve slope upward? Because there are diminishing returns to inputs in this example. All other things equal, as output increases, the marginal product of the variable input declines. This implies that more and more of the variable input must be used to produce each additional unit of output as the amount of output already produced rises. And since each unit of the variable input must be paid for, the additional cost per additional unit of output also rises.

In addition, recall that the flattening of the total product curve is also due to diminishing returns: the marginal product of an input falls as more of that input is used if the quantities of other inputs are fixed. The flattening of the total product curve as output increases and the steepening of the total cost curve as output increases are just flip-sides of the same phenomenon. That is, as output increases, the marginal cost of output also increases because the marginal product of the variable input decreases.

The inverse relationship between productivity and costs is discussed in greater detail in Appendix 11A. We will return to marginal cost in Chapter 12, when we consider the firm’s profit-maximizing output decision. Our next step is to introduce another measure of cost: average cost.

In addition to total cost and marginal cost, it’s useful to calculate another measure, average total cost, often simply called average cost. The average total cost is total cost divided by the quantity of output produced; that is, it is equal to total cost per unit of output. If we let ATC denote average total cost, the equation looks like this:

Average total cost is important because it tells the producer how much the average or typical unit of output costs to produce. Marginal cost, meanwhile, tells the producer how much one more unit of output costs to produce. Although they may look very similar, these two measures of cost typically differ. And confusion between them is a major source of error in economics, both in the classroom and in real life, as illustrated by the upcoming Economics in Action.

Table 11-2 uses data from Selena’s Gourmet Salsas to calculate average total cost. For example, the total cost of producing 4 cases of salsa is $300, consisting of $108 in fixed cost and $192 in variable cost (from Table 11-1). So the average total cost of producing 4 cases of salsa is $300/4 = $75. You can see from Table 11-2 that as quantity of output increases, average total cost first falls, then rises.

Figure 11-9 plots that data to yield the average total cost curve, which shows how average total cost depends on output. As before, cost in dollars is measured on the vertical axis and quantity of output is measured on the horizontal axis. The average total cost curve has a distinctive U shape that corresponds to how average total cost first falls and then rises as output increases. Economists believe that such U-shaped average total cost curves are the norm for producers in many industries.

To help our understanding of why the average total cost curve is U-shaped, Table 11-2 breaks average total cost into its two underlying components, average fixed cost and average variable cost. Average fixed cost, or AFC, is fixed cost divided by the quantity of output, also known as the fixed cost per unit of output. For example, if Selena’s Gourmet Salsas produces 4 cases of salsa, average fixed cost is $108/4 = $27 per case. Average variable cost, or AVC, is variable cost divided by the quantity of output, also known as variable cost per unit of output. At an output of 4 cases, average variable cost is $192/4 = $48 per case. Writing these in the form of equations:

Average total cost is the sum of average fixed cost and average variable cost. It has a U shape because these components move in opposite directions as output rises.

Average fixed cost falls as more output is produced because the numerator (the fixed cost) is a fixed number but the denominator (the quantity of output) increases as more is produced. Another way to think about this relationship is that, as more output is produced, the fixed cost is spread over more units of output; the end result is that the fixed cost per unit of output—the average fixed cost—falls. You can see this effect in the fourth column of Table 11-2: average fixed cost drops continuously as output increases.

Average variable cost, however, rises as output increases. As we’ve seen, this reflects diminishing returns to the variable input: each additional unit of output incurs more variable cost to produce than the previous unit. So variable cost rises at a faster rate than the quantity of output increases.

So increasing output has two opposing effects on average total cost—the “spreading effect” and the “diminishing returns effect”:

At low levels of output, the spreading effect is very powerful because even small increases in output cause large reductions in average fixed cost. So at low levels of output, the spreading effect dominates the diminishing returns effect and causes the average total cost curve to slope downward. But when output is large, average fixed cost is already quite small, so increasing output further has only a very small spreading effect. Diminishing returns, however, usually grow increasingly important as output rises. As a result, when output is large, the diminishing returns effect dominates the spreading effect, causing the average total cost curve to slope upward. At the bottom of the U-shaped average total cost curve, point M in Figure 11-9, the two effects exactly balance each other. At this point average total cost is at its minimum level, the minimum average total cost.

Figure 11-10 brings together in a single picture four members of the family of cost curves that we have derived from the total cost curve for Selena’s Gourmet Salsas: the marginal cost curve (MC), the average total cost curve (ATC), the average variable cost curve (AVC), and the average fixed cost curve (AFC). All are based on the information in Tables 11-1 and 11-2. As before, cost is measured on the vertical axis and the quantity of output is measured on the horizontal axis.

Let’s take a moment to note some features of the various cost curves. First of all, marginal cost slopes upward—the result of diminishing returns that make an additional unit of output more costly to produce than the one before. Average variable cost also slopes upward—again, due to diminishing returns—but is flatter than the marginal cost curve. This is because the higher cost of an additional unit of output is averaged across all units, not just the additional units, in the average variable cost measure. Meanwhile, average fixed cost slopes downward because of the spreading effect.

Finally, notice that the marginal cost curve intersects the average total cost curve from below, crossing it at its lowest point, point M in Figure 11-10. This last feature is our next subject of study.

For a U-shaped average total cost curve, average total cost is at its minimum level at the bottom of the U. Economists call the quantity of output that corresponds to the minimum average total cost the minimum-cost output. In the case of Selena’s Gourmet Salsas, the minimum-cost output is three cases of salsa per day.

In Figure 11-10, the bottom of the U is at the level of output at which the marginal cost curve crosses the average total cost curve from below. Is this an accident? No—it reflects general principles that are always true about a firm’s marginal cost and average total cost curves:

To understand these principles, think about how your grade in one course—say, a 3.0 in physics—affects your overall grade point average. If your GPA before receiving that grade was more than 3.0, the new grade lowers your average.

Similarly, if marginal cost—the cost of producing one more unit—is less than average total cost, producing that extra unit lowers average total cost. This is shown in Figure 11-11 by the movement from A₁ to A₂. In this case, the marginal cost of producing an additional unit of output is low, as indicated by the point MC_L on the marginal cost curve. When the cost of producing the next unit of output is less than average total cost, increasing production reduces average total cost. So any quantity of output at which marginal cost is less than average total cost must be on the downward-sloping segment of the U.

But if your grade in physics is more than the average of your previous grades, this new grade raises your GPA. Similarly, if marginal cost is greater than average total cost, producing that extra unit raises average total cost. This is illustrated by the movement from B₁ to B₂ in Figure 11-11, where the marginal cost, MC_H, is higher than average total cost. So any quantity of output at which marginal cost is greater than average total cost must be on the upward-sloping segment of the U.

Finally, if a new grade is exactly equal to your previous GPA, the additional grade neither raises nor lowers that average—it stays the same. This corresponds to point M in Figure 11-11: when marginal cost equals average total cost, we must be at the bottom of the U, because only at that point is average total cost neither falling nor rising.

Up to this point, we have emphasized the importance of diminishing returns, which lead to a marginal product curve that always slopes downward and a marginal cost curve that always slopes upward. In practice, however, economists believe that marginal cost curves often slope downward as a firm increases its production from zero up to some low level, sloping upward only at higher levels of production: they look like the curve MC in Figure 11-12.

This initial downward slope occurs because a firm often finds that, when it starts with only a very small number of workers, employing more workers and expanding output allows specialization to take place. This specialization leads to increasing returns to the hiring of additional workers (i.e., the marginal product of labour curve begins by sloping upward) and results in a marginal cost curve that initially slopes downward. But once the benefits of specialization have been exhausted, diminishing returns to labour set in (i.e., the marginal product of labour curve begins sloping downward) and the marginal cost curve changes direction and slopes upward. So typical marginal cost curves actually have the “swoosh” shape shown by MC in Figure 11-12. For the same reason, average variable cost curves typically look like AVC in Figure 11-12: they are U-shaped rather than strictly upward sloping. The appendix of this chapter uses mathematics to investigate the relationship between marginal product and marginal cost, and between average product and average cost.

However, as Figure 11-12 also shows, the key features we saw from the example of Selena’s Gourmet Salsas remain true: the average total cost curve is U-shaped, and the marginal cost curve passes through the point of minimum average total cost.