Any activity that decision makers might wish to undertake will generate both benefits and costs. Consequently, decision makers will want to choose the level of activity to obtain the maximum possible net benefit from the activity, where the net benefit (NB) associated with a specific amount of activity (A) is the difference between total benefit (TB) and total cost (TC) for the activity:
NB = TB - TC
Net benefit, then, serves as the objective function to be maximized, and the amount of activity, A, represents the choice variable. Furthermore, decision makers can choose any level of activity they wish, fromzero to infinity, in either discrete or continuous units. Thus we are studying unconstrained maximization in this section.
The Optimal Level of Activity (A*)
We begin the analysis of unconstrained maximization with a rather typical set of total benefit and total cost curves for some activity, A, as shown in Panel A of Figure 3.1. Total benefit increases with higher levels of activity up to 1,000 units of activity (point G); then total benefit falls beyond this point. Total cost begins at a value of zero and rises continuously as activity increases. These “typical” curves allow us to derive general rules for finding the best solution to all such unconstrained problems, even though specific problems Encounter sometimes involve benefit and cost curves with shapes that differ somewhat from those shown in Panel A. For example, total benefit curves can be linear. Total cost curves can be linear or even S-shaped. And, as you will see , total cost curves can include fixed costs when they take positive values at zero units of activity. In all of these variations, however, the rules for making the best decisions do not change. By learning how to solve the optimization problem as set forth in Figure 3.1, you will be prepared to solve all variations of these problems.
The level of activity that maximizes net benefit is called the optimal level of activity, which we distinguish from other levels of activity with an asterisk: A*. In Panel A of Figure 3.1, net benefit at any particular level of activity is measured by the vertical distance between the total benefit and total cost curves. At 200 units of activity, for example, net benefit equals the length of line segment CC', which happens to be $1,000 as shown in Panel B at point c”. Panel B of Figure 3.1 shows the net benefit curve associated with the TB and TC curves in Panel A. As you can see from examining the net benefit curve in Panel B, the optimal level of activity, A*, is 350 units, where NB reaches its maximum value. At 350 units in Panel A, the vertical distance between TB and TC is maximized, and this maximum distance is $1,225 (= NB*).
Two important observations can now be made about A* in unconstrained maximization problems. First, the optimal level of activity does not generally result in maximization of total benefits. In Panel Aof Figure 3.1, you can see that total benefit is still rising at the optimal point B. As we will demonstrate later in this book, for one of the most important applications of this technique, profit maximization, the optimal level of production occurs at a point where revenues are not maximized. This outcome can confuse managers, especially ones who believe any decision that increases revenue should be undertaken. We will have much more to say about this later in the text. Second, the optimal level of activity in an unconstrained maximization problem does not result in minimization of total cost. In Panel A, you can easily verify that total cost isn’t minimized at A* but rather at zero units of activity.
Finding A* in Figure 3.1 seems easy enough. A decision maker starts with the total benefit and total cost curves in Panel A and subtracts the total cost curve from the total benefit curve to construct the net benefit curve in Panel B. Then, the decision maker chooses the value of A corresponding to the peak of the net benefit curve. You might reasonably wonder why we are going to develop an alternative method, marginal analysis, for making optimal decisions. Perhaps the most important reason for learning how to use marginal analysis is that economists regard marginal analysis as “the central organizing principle of economic theory.” The graphical derivation of net benefit shown in Figure 3.1 serves only to define and describe the optimal level of activity; it does not explain why net benefit rises, falls, or reaches its peak. Marginal analysis, by focusing only on the changes in total benefits and total costs, provides a simple and complete explanation of the underlying forces causing net benefit to change. Understanding precisely what causes net benefit to improve makes it possible to develop simple rules for deciding when an activity needs to be increased, decreased, or left at its current level.
We are also going to show that using marginal analysis to make optimal decisions ensures that you will not consider irrelevant information about such things as fixed costs, sunk costs, or average costs in the decision-making process. As you will see shortly, decision makers using marginal analysis can reach the optimal activity level using only information about the benefits and costs at the margin. For this reason, marginal analysis requires less information than would be needed to construct TB, TC, and NB curves for all possible activity levels, as in Figure 3.1. There is no need to gather and process information for levels of activity that will never be chosen on the way to reaching A*. For example, if the decision maker is currently at 199 units of activity in Figure 3.1, information about benefits and costs is only needed for activity levels from 200 to 351 units. The optimal level of activity can be found without any information about benefits or costs below 200 units or above 351 units.
Marginal Benefit and Marginal Cost
In order to understand and use marginal analysis, you must understand the two key components of this methodology: marginal benefit and marginal cost. Marginal benefit is the change in total benefit caused by an incremental change in the level of an activity. Similarly, marginal cost is the change in total cost caused by an incremental change in activity. Dictionaries typically define “incremental” to mean “a small positive or negative change in a variable.” You can think of “small” or “incremental” changes in activity to be any change that is small relative to the total level of activity. In most applications it is convenient to interpret an incremental change as a one-unit change. In some decisions, however, it may be impractical or even impossible to make changes as small as one-unit. This causes no problem for applying marginal analysis as long as the activity can be adjusted in relatively small increments. We should also mention that “small” refers only to the change in activity level; “small” doesn’t apply to the resulting changes in total benefit or total cost, which can be any size.
Marginal benefit and marginal cost can be expressed mathematically as
where the symbol “Δ” means “the change in” and A denotes the level of an activity. Since “marginal” variables measure rates of change in corresponding “total” variables, marginal benefit and marginal cost are also slopes of total benefit and total cost curves, respectively.
The two panels in Figure 3.2 show how the total curves in Figure 3.1 are related to their respective marginal curves. Panel A in Figure 3.2 illustrates the procedure for measuring slopes of total curves at various points or levels of activity. Recall from your high school math classes or a pre-calculus course in college that the slope of a curve at any particular point can be measured by first constructing a line tangent to the curve at the point of measure and then computing the slope of this tangent line by dividing the “rise” by the “run” of the tangent line.4 Consider, for example, the slope of TB at point C in Panel A. The tangent line at point C rises by 640 units (dollars) over a 100-unit run, and total benefit’s slope at point C is $6.40 (= $640/100). Thus the marginal benefit of the 200th unit of activity is $6.40, which means adding the 200th unit of activity (going from 199 to 200 units) causes total benefit to rise by $6.40.
You should understand that the value of marginal benefit also tells you that subtracting the 200th unit (going from 200 to 199 units) causes total benefit to fall by $6.40. Since the slope of TB at point C is $6.40 per unit change in activity, marginal benefit at point c in Panel B is $6.40. You can verify that the same relation holds for the rest of the points shown on total benefit (B, D, and G), as well as for the points shown on total cost (C', B', and D'). We summarize this important discussion in a principle:
Principle Marginal benefit (marginal cost) is the change in total benefit (total cost) per unit change in the level of activity. The marginal benefit (marginal cost) of a particular unit of activity can be measured by the slope of the line tangent to the total benefit (total cost) curve at that point of activity.
At this point, you might be concerned that constructing tangent lines and measuring slopes of the tangent lines presents a tedious and imprecise method of finding marginal benefit and marginal cost curves. It is quite useful, nonetheless, for you to be able to visualize a series of tangent lines along total benefit and total cost curves in order to see why marginal benefit and marginal cost curves, respectively, are rising, falling, or even flat. Even if you don’t know the numerical values of the slopes at points C, B, D, and F in Figure 3.2, you can still determine that marginal benefit in Panel B must slope downward because, as you can tell by looking, the tangent lines along TB get flatter (slopes get smaller) as the activity increases. Marginal cost, on the other hand, must be increasing in Panel B because, as you can tell by looking, its tangent lines get steeper (slope is getting larger) as the activity increases.
Finding Optimal Activity Levels with Marginal Analysis
As we stated earlier, the method of marginal analysis involves comparing marginal benefit and marginal cost to see if net benefit can be increased by making an incremental change in activity level. We can now demonstrate exactly how this works using the marginal benefit and marginal cost curves in Panel B of Figure 3.2. Let’s suppose the decision maker is currently undertaking 199 units of activity in Panel B and wants to decide whether an incremental change in activity can cause net benefit to rise. Adding the 200th unit of activity will cause both total benefit and total cost to rise. As you can tell from points c and c in Panel B, TB increases by more than TC increases ($6.40 is a larger increase than $3.40). Consequently, increasing activity from 199 to 200 units will cause net benefit to rise by $3 (= $6.40 - $3.40). Notice in Figure 3.3 that, at 200 units of activity (point c”), net benefit is rising at a rate of $3 (= $300/100) per unit increase in activity, as it must since MB equals $6.40 and MC equals $3.40.
After increasing the activity to 200 units, the decision maker then reevaluates benefits and costs at the margin to see whether another incremental increase in activity is warranted. In this situation, for the 201st unit of activity, the decision maker once again discovers that MB is greater than MC, which indicates the activity should be further increased. This incremental adjustment process continues until marginal benefit and marginal cost are exactly equal at point M(A* = 350). As a practical matter, the decision maker can make a single adjustment to reach equilibrium, jumping from 199 units to 350 units in one adjustment of A, or make a series of smaller adjustments until MB equals MC at 350 units of activity. In any case, the number of adjustments made to reach A* does not, of course, alter the optimal decision or the value of net benefit at its maximum point.
Now let’s start from a position of too much activity instead of beginning with too little activity. Suppose the decision maker begins at 600 units of activity, which you can tell is too much activity by looking at the NB curve (in either Figure 3.1 or 3.3). Subtracting the 600th unit of activity will cause both total benefit and total cost to fall. As you can tell from points d and d' in Panel B of Figure 3.2, TC decreases by more than TB decreases ($8.20 is a larger decrease than $3.20).
Consequently, reducing activity from 600 to 599 units will cause net benefit to rise by $5 (= $8.20 - $3.20). You can now verify in Figure 3.3 that at 600 units of activity (point d”) net benefit is rising at a rate of $5 per unit decrease in activity. Since MC is still greater than MB at 599 units, the decision maker would continue reducing activity until MB exactly equals MC at 350 units (point M).
Table 3.1 summarizes the logic of marginal analysis by presenting the relation between marginal benefit, marginal cost, and net benefit set forth in the previous discussion and shown in Figure 3.3. We now summarize in the following principle the logic of marginal analysis for unconstrained maximization problems in which the choice variable is continuous:
Principle If, at a given level of activity, a small increase or decrease in activity causes net benefit to increase, then this level of the activity is not optimal. The activity must then be increased (if marginal benefit exceeds marginal cost) or decreased (if marginal cost exceeds marginal benefit) to reach the highest net benefit. The optimal level of the activity—the level that maximizes net benefit—is attained when no further increases in net benefit are possible for any changes in the activity, which occurs at the activity level for which marginal benefit equals marginal cost: MB = MC.
While the preceding discussion of unconstrained optimization has allowed only one activity or choice variable to influence net benefit, sometimes managers will need to choose the levels of two or more variables. As it turns out, when decision makers wish to maximize the net benefit from several activities, precisely the same principle applies: The firm maximizes net benefit when the marginal benefit from each activity equals the marginal cost of that activity. The problem is somewhat more complicated mathematically because the manager will have to equate marginal benefits and marginal costs for all of the activities simultaneously. For example, if the decision maker chooses the levels of two activities A and B to maximize net benefit, then the values for A and B must satisfy two conditions at once: MBA = MCA and MBB = MCB. As it happens in this text, all the unconstrained maximization problems involve just one choice variable or activity.
Maximization with Discrete Choice Variables
In the preceding analysis, the choice variable or activity level was a continuous variable. When a choice variable can vary only discretely, the logic of marginal analysis applies in exactly the same manner as when the choice variable is continuous. However, when choice variables are discrete, decision makers will not usually be able to adjust the level of activity to the point where marginal benefit exactly equals marginal cost. To make optimal decisions for discrete choice variables, decision makers must increase activity until the last level of activity is reached for which marginal benefit exceeds marginal cost. We can explain this rule for discrete choice variables by referring to Table 3.2, which shows a schedule of total benefits and total costs for various levels of some activity, A, expressed in integers between 0 and 8.
Let’s suppose the decision maker is currently doing none of the activity and wants to decide whether to undertake the first unit of activity. The marginal benefit of the first unit of the activity is $16, and the marginal cost is $2. Undertaking the first unit of activity adds $16 to total benefit and only $2 to total cost, so net benefit rises by $14 (from $0 to $14). The decision maker would choose to undertake the first unit of activity to gain a higher net benefit. Applying this reasoning to the second and third units of activity leads again to a decision to undertake more activity. Beyond the third unit, however, marginal cost exceeds marginal benefit for additional units of activity, so no further increase beyond three units of activity will add to net benefit. As you can see, the optimal level of the activity is three units because the net benefit associated with three units ($29) is higher than for any other level of activity. These results are summarized in the following principle:
Principle When a decision maker faces an unconstrained maximization problem and must choose among discrete levels of an activity, the activity should be increased if MB > MC and decreased if MB < MC. The optimal level of activity is reached—net benefit is maximized—when the level of activity is the last level for which marginal benefit exceeds marginal cost.
Before moving ahead, we would like to point out that this principle cannot be interpreted to mean “choose the activity level where MB and MC are as close to equal as possible.” To see why this interpretation can lead to the wrong decision, consider the fourth unit of activity in Table 3.2. At four units of activity, MB (= $8) is much closer to equality with MC (= $9) than at the optimal level of activity, where MB (= $10) is $5 larger than MC (= $5). Now you can see why the rule for discrete choice variables cannot be interpreted to mean “get MB as close to MC as possible.”
Sunk Costs, Fixed Costs, and Average Costs Are Irrelevant
In our discussion of optimization problems, we never mentioned sunk costs or fixed costs. Sunk costs are costs that have previously been paid and cannot be recovered. Fixed costs are costs that are constant and must be paid no matter what level of an activity is chosen. Such costs are totally irrelevant in decision making. They either have already been paid and cannot be recovered, as in the case of sunk costs, or must be paid no matter what a manager or any other decision maker decides to do, as in the case of fixed costs. In either case, the only relevant decision variables—marginal cost and marginal revenue—are in no way affected by the levels of either sunk or fixed costs.
Suppose you head your company’s advertising department and you have just paid $2 million to an advertising firm for developing and producing a 30-second television ad, which you plan to air next quarter on broadcast television networks nationwide. The $2 million one-time payment gives your company full ownership of the 30-second ad, and your company can run the ad as many times as it wishes without making any further payments to the advertising firm for its use. Under these circumstances, the $2 million payment is a sunk cost because it has already been paid and cannot be recovered, even if your firm decides not to use the ad after all.
To decide how many times to run the ad next quarter, you call a meeting of your company’s advertising department. At the meeting, the company’s media buyer informs you that 30-second television spots during American Idol will cost $250,000 per spot. The marketing research experts at the meeting predict that the 24th time the ad runs it will generate $270,000 of additional sales, while running it a 25th time will increase sales by $210,000. Using the logic of marginal analysis, the marketing team decides running the new ad 24 times next quarter is optimal because the 24th showing of the ad is the last showing for which the marginal benefit exceeds the marginal cost of showing the ad:
MB = $270,000 > 250,000 = MC
It would be a mistake to go beyond 24 showings, because the 25th showing would decrease net benefit; the change in net benefit would be -$40,000 (= $210,000 - $250,000).
Two days after this meeting, you learn about a serious accounting error: Your company actually paid $3 million to the advertising firm for developing and producing your company’s new television ad, not $2 million as originally reported. As you consider how to handle this new information, you realize that you don’t need to call another meeting of the marketing department to reconsider its decision about running the ad 24 times next quarter. Because the amount paid to the advertising firm is a sunk cost, it doesn’t affect either the marginal benefit or the marginal cost of running the ad one more time. The optimal number of times to run the ad is 24 times no matter how much the company paid in the past to obtain the ad.
Converting this example to a fixed cost, suppose that two days after your meeting you find out that, instead of making a sunk payment to buy the ad, your company instead decided to sign a 30-month contract leasing the rights to use the television ad for a monthly lease payment of $10,000. This amount is a fixed payment in each of the 30 months and must be paid no matter how many times your company decides to run the ad, even if it chooses never to run the ad. Do you need to call another meeting of the marketing department to recalculate the optimal number of times to run the ad during American Idol? As before, no new decision needs to be made. Since the fixed monthly loan payment does not change the predicted gain in sales (MB) or the extra cost of running the ad (MC), the optimal number of times to run the ad remains 24 times.
While you should now understand that things over which you have no control should not affect decisions, some economic experiments do, surprisingly, find that many people fail to ignore fixed or sunk costs when making decisions. They say things such as, “I’ve already got so much invested in this project, I have to go on with it.” As you are aware, they should weigh the costs and benefits of going on before doing so. Then, if the benefits are greater than the additional costs, they should go on; if the additional costs are greater than the benefits, they should not go on. As Illustration 3.1 shows, failing to ignore fixed or sunk costs is a bad policy even in everyday decision making.
Another type of cost that should be ignored in finding the optimal level of an activity is the average or unit cost of the activity. Average (or unit) cost is the cost per unit of activity, computed by dividing total cost by the number of units of activity. In order to make optimal decisions, decision makers should not be concerned about whether their decision will push average costs up or down. The reason for ignoring average cost is quite simple: The impact on net benefit of making an incremental change in activity depends only on marginal benefit and marginal cost (∆NB = MB - MC), not on average benefit or average cost. In other words, optimal decisions are made at the margin, not “on the average.”
To illustrate this point, consider the decision in Table 3.2 once again. The average cost of two units of activity is $3 (= $6/2) and average cost for three units of activity is $3.67 (= $11/3). Recall from our earlier discussion, the decision to undertake the third unit of activity is made because the marginal benefit exceeds the marginal cost ($10 > $5), and net benefit rises. It is completely irrelevant that the average cost of three units of activity is higher than the average cost of two units of activity. Alternatively, a decision maker should not decrease activity from three units to two units just to achieve a reduction in average cost from $3.67 to $3 per unit of activity; such a decision would cause net benefit to fall from $29 to $24. The following principle summarizes the role of sunk, fixed, and average costs in making optimal decisions:
Principle Decision makers wishing to maximize the net benefit of an activity should ignore any sunk costs, any fixed costs, and the average costs associated with the activity because none of these costs affect the marginal cost of the activity and so are irrelevant for making optimal decisions.
I L L U S T R AT I O N 3 . 1
Is Cost–Benefit Analysis Really Useful?
We have extolled the usefulness of marginal analysis in optimal decision making—often referred to as cost– benefit analysis—in business decision making as well as decision making in everyday life. This process involves weighing the marginal benefits and marginal costs of an activity while ignoring all previously incurred or sunk costs. The principal rule is to increase the level of an activity if marginal benefits exceed marginal costs and decrease the level if marginal costs exceed marginal benefits. This simple rule, however, flies in the face of many honored traditional principles such as “Never give up” or “Anything worth doing is worth doing well” or “Waste not, want not.” So you might wonder if cost–benefit analysis is as useful as we have said it is.
It is, at least according to an article in The Wall Street Journal entitled “Economic Perspective Produces Steady Yields.” In this article, a University of Michigan research team concludes, “Cost–benefit analysis pays off in everyday living.” This team quizzed some of the university’s seniors and faculty members on such questions as how often they walk out on a bad movie, refuse to finish a bad novel, start over on a weak term paper, or abandon a research project that no longer looks promising. They believe that people who cut their losses this way are following sound economic rules: calculating the net benefits of alternative courses of action, writing off past costs that can’t be recovered, and weighing the opportunity to use future time and effort more profitably elsewhere.
The findings: Among faculty members, those who use cost–benefit reasoning in this fashion had higher salaries relative to their age and departments. Economists were more likely to apply the approach than professors of humanities or biology. Among students, those who have learned to use cost–benefit analysis frequently are apt to have far better grades than their SAT scores would have predicted. The more economics courses the students had taken, the more likely they were to apply cost–benefit analysis outside the classroom. The director of the University of Michigan study did concede that for many Americans cost–benefit rules often appear to conflict with traditional principles such as those we previously mentioned. Notwithstanding these probable conflicts, the study provides evidence that decision makers can indeed prosper by following the logic of marginal analysis and cost– benefit analysis.