# Marginal Analysis for Optimal Decisions

Edited by Paul Ducham

MARGINAL ANALYSIS

Optimizing Behavior on the part of a decision maker involves trying to maximize or minimize an objective function. For a manager of a firm, the objective function is usually profit, which is to be maximized. For a consumer, the objective function is the satisfaction derived from consumption of goods, which is to be maximized. For a city manager seeking to provide adequate law enforcement services, the objective function might be cost, which is to be minimized. For the manager of the marketing division of a large corporation, the objective function is usually sales, which are to be maximized. The objective function measures whatever it is that the particular decision maker wishes to either maximize or minimize.

If the decision maker seeks to maximize an objective function, the optimization problem is called a maximization problem. Alternatively, if the objective function is to be minimized, the optimization problem is called a minimization problem. As a general rule, when the objective function measures a benefit, the decision maker seeks to maximize this benefit and is solving a maximization problem. When the objective function measures a cost, the decision maker seeks to minimize this cost and is solving a minimization problem.

The value of the objective function is determined by the level of one or more activities or choice variables. For example, the value of profit depends on the number of units of output produced and sold. The production of units of the good is the activity that determines the value of the objective function, which in this case is profit. The decision maker controls the value of the objective function by choosing the levels of the activities or choice variables.

The choice variables in the optimization problems discussed in this text will at times vary discretely and at other times vary continuously. A discrete choice variable can take on only specified integer values, such as 1, 2, 3, . . . , or 10, 20, 30, . . . Examples of discrete choice variables arise when benefit and cost data are presented in a table, where each row represents one value of the choice variable. In this text, all examples of discrete choice variables will be presented in tables. A continuous choice variable can take on any value between two end points. For example, a continuous variable that can vary between 0 and 10 can take on the value 2, 2.345, 7.9, 8.999, or any one of the infinite number of values between the two limits. Examples of continuous choice variables are usually presented graphically but are sometimes shown by equations. As it turns out, the optimization rules differ only slightly in the discrete and continuous cases.

In addition to being categorized as either maximization or minimization problems, optimization problems are also categorized according to whether the decision maker can choose the values of the choice variables in the objective function from an unconstrained or constrained set of values. Unconstrained optimization problems occur when a decision maker can choose any level of activity he or she wishes in order to maximize the objective function. We show how to solve only unconstrained maximization problems since all the unconstrained decision problems we address in this text are maximization problems. Constrained optimization problems involve choosing the levels of two or more activities that maximize or minimize the objective function subject to an additional requirement or constraint that restricts the values of A and B that can be chosen. An example of such a constraint arises when the total cost of the chosen activity levels must equal a specified constraint on cost. In this text, we examine both constrained maximization and constrained minimization problems.

As we show later, the constrained maximization and the constrained minimization problems have one simple rule for the solution. Therefore, you will only have one rule to learn for all constrained optimization problems.

Even though there are a huge number of possible maximizing or minimizing decisions, you will see that all optimization problems can be solved using the single analytical technique: marginal analysis. Marginal analysis involves changing the value(s) of the choice variable(s) by a small amount to see if the objective function can be further increased (in the case of maximization problems) or further decreased (in the case of minimization problems). If so, the manager continues to make incremental adjustments in the choice variables until no further improvements are possible. Marginal analysis leads to two simple rules for solving optimization problems, one for unconstrained decisions and one for constrained decisions. We turn first to the unconstrained decision.

UNCONSTRAINED MAXIMIZATION

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.

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).

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.

CONSTRAINED OPTIMIZATION

On many occasions a manager will face situations in which the choice of activity levels is constrained by the circumstances surrounding the maximization or minimization problem. These constrained optimization problems can be solved, as in the case of unconstrained maximization, using the logic of marginal analysis. As noted in Section 3.1, even though constrained optimization problems can be either maximization or minimization problems, the optimization rule is the same for both types.

A crucial concept for solving constrained optimization problems is the concept of marginal benefit per dollar spent on an activity. Before you can understand how to solve constrained optimization problems, you must first understand how to interpret the ratio of the marginal benefit of an activity divided by the price of the activity.

Marginal Benefit per Dollar Spent on an Activity

Retailers frequently advertise that their products give “more value for your money.” People don’t usually interpret this as meaning the best product in its class or the one with the highest value. Neither do they interpret it as meaning the cheapest. The advertiser wants to get across the message that customers will get more for their money or more value for each dollar spent on the product. When product rating services (such as Consumer Reports) rate a product a “best buy,” they don’t mean it is the best product or the cheapest; they mean that consumers will get more value per dollar spent on that product. When firms want to fill a position, they don’t necessarily hire the person who would be the most productive in the job—that person may cost too much. Neither do they necessarily hire the person who would work for the lowest wages—that person may not be very productive. They want the employee who can do the job and give the highest productivity for the wages paid.

In these examples, phrases such as “most value for your money,” “best buy,” and “greatest bang per buck” mean that a particular activity yields the highest marginal benefit per dollar spent. To illustrate this concept, suppose you are the office manager for an expanding law firm and you find that you need an extra copy machine in the office—the one copier you have is being overworked. You shop around and find three brands of office copy machines (brands A, B, and C) that have virtually identical features. The three brands do differ, however, in price and in the number of copies the machines will make before they wear out. Brand A’s copy machine costs \$2,500 and will produce about 500,000 copies before it wears out. The marginal benefit of this machine is 500,000 (MBA = 500,000) since the machine provides the law office with the ability to produce 500,000 additional copies.

To find the marginal benefit per dollar spent on copy machine A, marginal benefit is divided by price (PA = 2,500):

MBA/PA = 500,000 copies/2,500 dollars

= 200 copies/dollar

You get 200 copies for each of the dollars spent to purchase copy machine A.

Now compare machine A with machine B, which will produce 600,000 copies and costs \$4,000. The marginal benefit is greater, but so is the price. To determine how “good a deal” you get with machine B, compute the marginal benefit per dollar spent on machine B:

MBB/PB = 600,000 copies/4,000 dollars

= 150 copies/dollar

Even though machine B provides a higher marginal benefit, its marginal benefit per dollar spent is lower than that for machine A. Machine A is a better deal than machine B because it yields higher marginal benefit per dollar. The third copy machine produces 580,000 copies over its useful life and costs \$2,600. Machine C is neither the best machine (580,000 < 600,000 copies) nor is it the cheapest machine (\$2,600 > \$2,500), but of the three machines, machine C provides the greatest marginal benefit per dollar spent:

MBC/PC = 580,000 copies/2,600 dollars

= 223 copies/dollar

On a bang per buck basis, you would rank machine C first, machine A second, and machine B third.

When choosing among different activities, a decision maker compares the marginal benefits per dollar spent on each of the activities. Marginal benefit (the “bang”), by itself, does not provide sufficient information for decision-making purposes. Price (the “buck”), by itself, does not provide sufficient information for making decisions. It is marginal benefit per dollar spent (the “bang per buck”) that matters in decision making.

Constrained Maximization

In the general constrained maximization problem, a manager must choose the levels of two or more activities in order to maximize a total benefit (objective) function subject to a constraint in the form of a budget that restricts the amount that can be spent. Consider a situation in which there are two activities, A and B.

Each unit of activity A costs \$4 to undertake, and each unit of activity B costs \$2 to undertake. The manager faces a constraint that allows a total expenditure of only \$100 on activities A and B combined. The manager wishes to allocate \$100 between activities A and B so that the combined total benefit from both activities is maximized.

The manager is currently choosing 20 units of activityAand 10 units of activity B. The constraint is met for the combination 20A and 10B since (\$4 * 20) + (\$2 * 10) = \$100. For this combination of activities, suppose that the marginal benefit of the last unit of activity A is 40 units of additional benefit and the marginal benefit of the last unit of B is 10 units of additional benefit. In this situation, the marginal benefit per dollar spent on activity A exceeds the marginal benefit per dollar spent on activity B:

MBA/PA = 40/4 = 10 > 5 = 10/2 = MBB/PB

Spending an additional dollar on activity A increases total benefit by 10 units, while spending an additional dollar on activity B increases total benefit by 5 units. Since the marginal benefit per dollar spent is greater for activity A, it provides “more bang per buck” or is a better deal at this combination of activities.

To take advantage of this fact, the manager can increase activity A by one unit and decrease activity B by two units (now, A = 21 and B = 8). This combination of activities still costs \$100 [(\$4 * 21) + (\$2 * 8) = \$100]. Purchasing one more unit of activity A causes total benefit to rise by 40 units, while purchasing two less units of activity B causes total benefit to fall by 20 units. The combined total benefit from activities A and B rises by 20 units (= 40 - 20) and the new combination of activities (A = 21 and B = 8) costs the same amount, \$100, as the old combination. Total benefit riser without spending any more than \$100 on the activities.

Naturally, the manager will continue to increase spending on activity A and reduce spending on activity B as long as MBA/PA exceeds MBB/PB. In most situations, the marginal benefit of an activity declines as the activity increases.7 Consequently, as activity A is increased, MBA gets smaller. As activity B is decreased, MBB gets larger. Thus as spending on A rises and spending on B falls, MBA/PA falls and MBB/PB rises. Apoint is eventually reached at which activity A is no longer a better deal than activity B; that is, MBA/PA equals MBB/PB. At this point, total benefit is maximized subject to the constraint that only \$100 is spent on the two activities.

If the original allocation of spending on activities A and B had been one where

MBA/PA < MBB/PB

the manager would recognize that activity B is the better deal. In this case, total benefit could be increased by spending more on activity B and less on activity A while maintaining the \$100 budget. Activity B would be increased by two units for every one-unit decrease in activity A (in order to satisfy the \$100 spending constraint) until the marginal benefit per dollar spent is equal for both activities:

MBA/PA = MBB/PB

If there are more than two activities in the objective function, the condition is expanded to require that the marginal benefit per dollar spent be equal for all activities.

Principle To maximize total benefits subject to a constraint on the levels of activities, choose the level of each activity so that the marginal benefit per dollar spent is equal for all activities

MBA/PA = MBB/PB = MBC/PC = ... = MBZ/PZ

and at the same time, the chosen level of activities must also satisfy the constraint.

Optimal Advertising Expenditures: An Example of Constrained Maximization

To illustrate how a firm can use the technique of constrained maximization to allocate its advertising budget, suppose a manager of a small retail firm wants to maximize the effectiveness (in total sales) of the firm’s weekly advertising budget of \$2,000. The manager has the option of advertising on the local television station or on the local AM radio station. As a class project, a marketing class at a nearby college estimated the impact on the retailer’s sales of varying levels of advertising in the two different media. The manager wants to maximize the number of units sold; thus the total benefit is measured by the total number of units sold. The estimates of the increases in weekly sales (the marginal benefits) from increasing the levels of advertising on television and radio are given in columns 2 and 4 below:

This indicates that sales rise by 1 unit per dollar spent on the first television ad and 1.2 units on the first radio ad. Therefore, when the manager is allocating the budget, the first ad she selects will be a radio ad—the activity with the larger marginal benefit per dollar spent. Following the same rule and using the MB/P values in columns 3 and 5 above, the \$2,000 advertising budget would be allocated as follows:

By selecting two television ads and four radio ads, the manager of the firm has maximized sales subject to the constraint that only \$2,000 can be spent on advertising activity. Note that for the optimal levels of television and radio ads (two TV and four radio):

The fact that the preceding application used artificially simplistic numbers shouldn’t make you think that the problem is artificial. If we add a few zeros to the prices of TV and radio ads, we have the real-world situation faced by advertisers.

Constrained Minimization

Constrained minimization problems involve minimizing a total cost function (the objective function) subject to a constraint that the levels of activities be chosen such that a given level of total benefit is achieved. Consider a manager who must minimize the total cost of two activities, A and B, subject to the constraint that 3,000 units of benefit are to be generated by those activities. The price of activity A is \$5 per unit and the price of activity B is \$20 per unit. Suppose the manager is currently using 100 units of activity A and 60 units of activity B and this combination of activity generates total benefit equal to 3,000. At this combination of activities, the marginal benefit of the last unit of activity A is 30 and the marginal benefit of the last unit of activity B is 60. In this situation, the marginal benefit per dollar spent on activity A exceeds the marginal benefit per dollar spent on activity B:

MBA/PA = 30/5 = 6 > 3 = 60/20 = MBB/PB

Since the marginal benefit per dollar spent is greater for activity A than for activity B, activity A gives “more for the money.”

To take advantage of activity A, the manager can reduce activity B by one unit, causing total benefit to fall by 60 units and reducing cost by \$20. To hold total benefit constant, the 60 units of lost benefit can be made up by increasing activity A by two units with a marginal benefit of 30 each. The two additional units of activity A cause total cost to rise by \$10. By reducing activity B by one unit and increasing activity A by two units, the manager reduces total cost by \$10 (= \$20 - \$10) without reducing total benefit.

As long as MBA/PA > MBB/PB, the manager will continue to increase activity A and decrease activity B at the rate that holds TB constant until

MBA/PA = MBB/PB

If there are more than two activities in the objective function, the condition is expanded to require that the marginal benefit per dollar spent be equal for all activities.

Principle In order to minimize total costs subject to a constraint on the levels of activities, choose the level of each activity so that the marginal benefit per dollar spent is equal for all activities

MBA/PA = MBB/PB = MBC/PC = ... = MBZ/PZ

and at the same time, the chosen level of activities must also satisfy the constraint.

As you can see, this is the same condition that must be met in the case of constrained maximization.

I L L U S T R AT I O N 3 . 2

Seattle Seahawks Win on “Bang Per Buck” Defense

Behind every professional sports team, a team of business decision makers is constantly at work—there is no off-season for the business team—trying to figure out how to put together the most profitable team of players. In the NFL, the team-building process is a constrained optimization problem because the football league imposes restrictions on the amount each team can spend on players in a season, as well as the number of players the team can carry on its roster. Currently, NFL teams are limited to 53 players and a salary cap of \$85 million per season. While teams can, and do, structure cash bonuses to players in ways that allow them to exceed the salary caps in any single year, the NFL spending constraint nonetheless restricts the total amount a team can spend on its players. Based on what you have learned about constrained optimization, it should come as no surprise to you that, in the business of sports, finding and keeping players who can deliver the greatest bang for the buck may be the most important game a team must play. History has shown that most teams making it to the Super Bowl have played the “bang for the buck” game very well.

A recent article in The Wall Street Journala illustrates how much importance is attached to the “game behind the game”:

Every year, from the moment the last piece of Super Bowl confetti is bagged up, the NFL’s army of bean counters and personnel directors starts to deconstruct the strategy of the winning team to see what lessons they can learn. But they don’t start with the game films. The first order of business is to check the team’s salary-cap figures.

To see how personnel directors of NFL teams follow the principles of constrained maximization in choosing their teams’ rosters, we can look at the story of the Seattle Seahawks, who played (and lost to) the Pittsburgh Steelers in Super Bowl XL. According to the WSJ article, the Seahawks’ personnel director, Tim Ruskell, who enjoyed wide acclaim for building the highly regarded defenses at Tampa Bay and Atlanta, faced a particularly harsh salary-cap constraint in Seattle for the 2005 football season. The Seahawks’ salary cap in 2005 was penalized by \$18 million of “dead money”—the term used for money paid by previous contract to players no longer on the team—so Ruskell began with only \$67 million to spend on players. Making matters even worse for the team’s chief business decision maker was the fact that Seattle had signed giant contracts, even by NFL standards, to keep its biggest stars on offense. Obviously, this left Ruskell with very little money to spend on building the Seahawks’ defense. Compared with its Super Bowl rival, Seattle spent \$11 million less on defense than did the Steelers. Ruskell’s strategy for hiring defensive players, then, had to be extremely effective if Seattle was to have any chance of going to the Super Bowl in 2005.

The way that Ruskell built Seattle’s defense, subject to a very tight spending constraint, drew high praise from others in the league: “They did it (built a defense) without breaking the bank at Monte Carlo, and I think that’s extremely impressive,” remarked Gil Brandt, a former personnel director for the Dallas Cowboys.b As you know from our discussion, Ruskell must have been very successful at finding defensive players who could deliver the highest possible marginal benefits per dollar spent. To accomplish this, he recruited only inexpensive draft picks and young free agents, who were also likely to play with “exuberance” and perform defensive tasks well enough to get to the Super Bowl. We must stress that Ruskell’s strategy depended crucially on both the numerator and denominator in the MB/MC ratio. He understood that simply hiring the cheapest players would not produce a Super Bowl team. Team scouts had to find players who would also deliver high marginal benefits to the team’s defensive squad by making lots of tackles and intercepting lots of passes. Perhaps the best example of Ruskell’s success at getting the most “bang for the buck” in 2005 was Lofa Tatapu, a rookie linebacker. Tatapu, who was thought by many team scouts to be too small to be a great linebacker, became a star defensive player for Seattle and cost the team only \$230,000—one-tenth the amount paid on average for linebackers in the NFL.

As you can see from this Illustration, making optimal constrained maximization decisions in practice takes not only skill and experience, it sometimes involves a bit of luck! History shows, however, that NFL personnel directors who spend their salary caps to get either the very best players (the greatest bang) or the very cheapest players (the smallest buck), don’t usually make it to the Super Bowl. Thus, on Super Bowl game day, fans can generally expect to see the two NFL teams with the highest overall MB/MC ratios. Of course winning the Super Bowl is just a betting matter.