PART 1
CHAPTER 1
People often make bad decisions because they fail to compare the relevant costs and benefits.©Nick Dolding/Cultura/Corbis
LEARNING OBJECTIVES
After reading this chapter, you should be able to:
1. LO1Explain and apply the Scarcity Principle, which says that having more of any good thing necessarily requires having less of something else.
2. LO2Explain and apply the Cost-Benefit Principle, which says that an action should be taken if, but only if, its benefit is at least as great as its cost.
3. LO3Discuss three important pitfalls that occur when applying the Cost-Benefit Principle inconsistently.
4. LO4Explain and apply the Incentive Principle, which says that if you want to predict people’s behavior, a good place to start is by examining their incentives.
How many students are in your introductory economics class? Some classes have just 20 or so. Others average 35, 100, or 200 students. At some schools, introductory economics classes may have as many as 2,000 students. What size is best?
If cost were no object, the best size might be a single student. Think about it: the whole course, all term long, with just you and your professor! Everything could be custom-tailored to your own background and ability. You could cover the material at just the right pace. The tutorial format also would promote close communication and personal trust between you and your professor. And your grade would depend more heavily on what you actually learned than on your luck when taking multiple-choice exams. Let’s suppose, for the sake of discussion, that students have been shown to learn best in the tutorial format.
Why, then, do so many introductory classes still have hundreds of students? The simple reason is that costs do matter. They matter not just to the university administrators who must build classrooms and pay faculty salaries, but also to you. The direct cost of providing you with your own personal introductory economics course might easily top $50,000. Someone has to pay these costs. In private universities, a large share of the cost would be recovered directly from higher tuition payments. In state universities, the burden would be split between higher tuition payments and higher tax payments. But, in either case, the course would be unaffordable for most students.
With larger classes, of course, the cost per student goes down. For example, an introductory economics course with 300 students might cost as little as $200 per student. But a class that large would surely compromise the quality of the learning environment. Compared to the custom tutorial format, however, it would be dramatically more affordable.
In choosing what size introductory economics course to offer, then, university administrators confront a classic economic trade-off. In making the class larger, they lower the quality of instruction—a bad thing. At the same time, they reduce costs and hence the tuition students must pay—a good thing.
Are small classes “better” than large ones?
In this chapter, we’ll introduce three simple principles that will help you understand and explain patterns of behavior you observe in the world around you. These principles also will help you avoid three pitfalls that plague decision makers in everyday life.
ECONOMICS: STUDYING CHOICE IN A WORLD OF SCARCITY
Even in rich societies like the United States, scarcity is a fundamental fact of life. There is never enough time, money, or energy to do everything we want to do or have everything we’d like to have. Economics is the study of how people make choices under conditions of scarcity and of the results of those choices for society.
In the class-size example just discussed, a motivated economics student might definitely prefer to be in a class of 20 rather than a class of 100, everything else being equal. But other things, of course, are not equal. Students can enjoy the benefits of having smaller classes, but only at the price of having less money for other activities. The student’s choice inevitably will come down to the relative importance of competing activities.
That such trade-offs are widespread and important is one of the core principles of economics. We call it the Scarcity Principle because the simple fact of scarcity makes trade-offs necessary. Another name for the scarcity principle is the No-Free-Lunch Principle (which comes from the observation that even lunches that are given to you are never really free—somebody, somehow, always has to pay for them).
Scarcity
The Scarcity Principle (also called the No-Free-Lunch Principle): Although we have boundless needs and wants, the resources available to us are limited. So having more of one good thing usually means having less of another.
Inherent in the idea of a trade-off is the fact that choice involves compromise between competing interests. Economists resolve such trade-offs by using cost-benefit analysis, which is based on the disarmingly simple principle that an action should be taken if, and only if, its benefits exceed its costs. We call this statement the Cost-Benefit Principle, and it, too, is one of the core principles of economics:
Cost-Benefit
The Cost-Benefit Principle: An individual (or a firm or a society) should take an action if, and only if, the extra benefits from taking the action are at least as great as the extra costs.
With the Cost-Benefit Principle in mind, let’s think about our class-size question again. Imagine that classrooms come in only two sizes—100-seat lecture halls and 20-seat classrooms—and that your university currently offers introductory economics courses to classes of 100 students. Question: Should administrators reduce the class size to 20 students? Answer: Reduce if, and only if, the value of the improvement in instruction outweighs its additional cost.
This rule sounds simple. But to apply it we need some way to measure the rele-vant costs and benefits, a task that’s often difficult in practice. If we make a few simplifying assumptions, however, we can see how the analysis might work. On the cost side, the primary expense of reducing class size from 100 to 20 is that we’ll now need five professors instead of just one. We’ll also need five smaller classrooms rather than a single big one, and this too may add slightly to the expense of the move. Let’s suppose that classes with 20 cost $1,000 per student more than those with 100. Should administrators switch to the smaller class size? If they apply the Cost-Benefit Principle, they will realize that doing so makes sense only if the value of attending the smaller class is at least $1,000 per student greater than the value of attending the larger class.
Cost-Benefit
Would you (or your family) be willing to pay an extra $1,000 for a smaller class? If not, and if other students feel the same way, then sticking with the larger class size makes sense. But if you and others would be willing to pay the extra tuition, then reducing the class size makes good economic sense.
Notice that the “best” class size, from an economic point of view, will generally not be the same as the “best” size from the point of view of an educational psychologist. That’s because the economic definition of “best” takes into account both the benefits and the costs of different class sizes. The psychologist ignores costs and looks only at the learning benefits of different class sizes.
In practice, of course, different people feel differently about the value of smaller classes. People with high incomes, for example, tend to be willing to pay more for the advantage. That helps to explain why average class size is smaller, and tuition higher, at private schools whose students come predominantly from high-income families.
If Mark Zuckerberg saw a $100 bill lying on the sidewalk, would it be worth his time to pick it up?©catwalker/Shutterstock
The cost-benefit framework for thinking about the class-size problem also suggests a possible reason for the gradual increase in average class size that has been taking place in American colleges and universities. During the last 30 years, professors’ salaries have risen sharply, making smaller classes more costly. During the same period, median family income—and hence the willingness to pay for smaller classes—has remained roughly constant. When the cost of offering smaller classes goes up but willingness to pay for smaller classes does not, universities shift to larger class sizes.
Scarcity and the trade-offs that result also apply to resources other than money. Mark Zuckerberg is one of the richest men on Earth. His wealth is estimated at more than $60 billion. That’s more than the combined wealth of the poorest 40 percent of Americans. Zuckerberg could buy more houses, cars, vacations, and other consumer goods than he could possibly use. Yet he, like the rest of us, has only 24 hours each day and a limited amount of energy. So even he confronts trade-offs. Any activity he pursues—whether it be building his business empire or redecorating his mansion—uses up time and energy that he could otherwise spend on other things. Indeed, someone once calculated that the value of Zuckerberg’s time is so great that pausing to pick up a $100 bill from the sidewalk simply wouldn’t be worth his while.
APPLYING THE COST-BENEFIT PRINCIPLE
In studying choice under scarcity, we’ll usually begin with the premise that people are rational, which means they have well-defined goals and try to fulfill them as best they can. The Cost-Benefit Principle is a fundamental tool for the study of how rational people make choices.
As in the class-size example, often the only real difficulty in applying the cost-benefit rule is to come up with reasonable measures of the relevant benefits and costs. Only in rare instances will exact dollar measures be conveniently available. But the cost-benefit framework can lend structure to your thinking even when no relevant market data are available.
To illustrate how we proceed in such cases, the following example asks you to decide whether to perform an action whose cost is described only in vague, qualitative terms.
EXAMPLE 1.1Comparing Costs and Benefits
Should you walk downtown to save $10 on a $25 video game?
Imagine you are about to buy a $25 video game at the nearby campus store when a friend tells you that the same game is on sale at a downtown store for only $15. If the downtown store is a 30-minute walk away, where should you buy the game?
Cost-Benefit
The Cost-Benefit Principle tells us that you should buy it downtown if the benefit of doing so exceeds the cost. The benefit of taking any action is the dollar value of everything you gain by taking it. Here, the benefit of buying downtown is exactly $10, because that’s the amount you’ll save on the price of the game. The cost of taking any action is the dollar value of everything you give up by taking it. Here, the cost of buying downtown is the dollar value you assign to the time and trouble it takes to make the trip. But how do we estimate that value?
One way is to perform the following hypothetical auction. Imagine that a stranger has offered to pay you to do an errand that involves the same walk downtown (perhaps to drop off a letter for her at the post office). If she offered you a payment of, say, $1,000, would you accept? If so, we know that your cost of walking downtown and back must be less than $1,000. Now imagine her offer being reduced in small increments until you finally refuse the last offer. For example, if you’d agree to walk downtown and back for $9 but not for $8.99, then your cost of making the trip is $9. In this case, you should buy the game downtown because the $10 you’ll save (your benefit) is greater than your $9 cost of making the trip.
But suppose your cost of making the trip had been greater than $10. In that case, your best bet would have been to buy the game from the nearby campus store. Confronted with this choice, different people may choose differently, depending on how costly they think it is to make the trip downtown. But although there is no uniquely correct choice, most people who are asked what they would do in this situation say they would buy the game downtown.
ECONOMIC SURPLUS
Suppose that in Example 1.1 your “cost” of making the trip downtown was $9. Compared to the alternative of buying the game at the campus store, buying it downtown resulted in an economic surplus of $1, the difference between the benefit of making the trip and its cost. In general, your goal as an economic decision maker is to choose those actions that generate the largest possible economic surplus. This means taking all actions that yield a positive total economic surplus, which is just another way of restating the Cost-Benefit Principle.
Cost-Benefit
Note that the fact that your best choice was to buy the game downtown doesn’t imply that you enjoy making the trip, any more than choosing a large class means that you prefer large classes to small ones. It simply means that the trip is less unpleasant than the prospect of paying $10 extra for the game. Once again, you’ve faced a trade-off. In this case, the choice was between a cheaper game and the free time gained by avoiding the trip.
OPPORTUNITY COST
Of course, your mental auction could have produced a different outcome. Suppose, for example, that the time required for the trip is the only time you have left to study for a difficult test the next day. Or suppose you are watching one of your favorite movies on cable, or that you are tired and would love a short nap. In such cases, we say that the opportunity cost of making the trip—that is, the value of what you must sacrifice to walk downtown and back—is high and you are more likely to decide against making the trip.
Strictly speaking, your opportunity cost of engaging in an activity is the value of everything you must sacrifice to engage in it. For instance, if seeing a movie requires not only that you buy a $10 ticket, but also that you give up a $20 babysitting job that you would have been willing to do for free, then the opportunity cost of seeing the film is $30.
Under this definition, all costs—both implicit and explicit—are opportunity costs. Unless otherwise stated, we will adhere to this strict definition.
We must warn you, however, that some economists use the term opportunity cost to refer only to the implicit value of opportunities forgone. Thus, in the example just discussed, these economists wouldn’t include the $10 ticket price when calculating the opportunity cost of seeing the film. But virtually all economists would agree that your opportunity cost of not doing the babysitting job is $20.
In the previous example, if watching the last hour of the cable TV movie is the most valuable opportunity that conflicts with the trip downtown, the opportunity cost of making the trip is the dollar value you place on pursuing that opportunity. It is the largest amount you’d be willing to pay to avoid missing the end of the movie. Note that the opportunity cost of making the trip is not the combined value of all possible activities you could have pursued, but only the value of your best alternative—the one you would have chosen had you not made the trip.
Cost-Benefit
Throughout the text we’ll pose concept checks like the one that follows. You’ll find that pausing to answer them will help you to master key concepts in economics. Because doing these concept checks isn’t very costly (indeed, many students report that they’re actually fun), the Cost-Benefit Principle indicates that it’s well worth your while to do them.
CONCEPT CHECK 1.1
You would again save $10 by buying the game downtown rather than at the campus store, but your cost of making the trip is now $12, not $9. By how much would your economic surplus be smaller if you bought the game downtown rather than at the campus store?
THE ROLE OF ECONOMIC MODELS
Economists use the Cost-Benefit Principle as an abstract model of how an idealized rational individual would choose among competing alternatives. (By “abstract model” we mean a simplified description that captures the essential elements of a situation and allows us to analyze them in a logical way.) A computer model of a complex phenomenon like climate change, which must ignore many details and includes only the major forces at work, is an example of an abstract model.
Noneconomists are sometimes harshly critical of the economist’s cost-benefit model on the grounds that people in the real world never conduct hypothetical mental auctions before deciding whether to make trips downtown. But this criticism betrays a fundamental misunderstanding of how abstract models can help to explain and predict human behavior. Economists know perfectly well that people don’t conduct hypothetical mental auctions when they make simple decisions. All the Cost-Benefit Principle really says is that a rational decision is one that is explicitly or implicitly based on a weighing of costs and benefits.
Most of us make sensible decisions most of the time, without being consciously aware that we are weighing costs and benefits, just as most people ride a bike without being consciously aware of what keeps them from falling. Through trial and error, we gradually learn what kinds of choices tend to work best in different contexts, just as bicycle riders internalize the relevant laws of physics, usually without being conscious of them.
Even so, learning the explicit principles of cost-benefit analysis can help us make better decisions, just as knowing about physics can help in learning to ride a bicycle. For instance, when a young economist was teaching his oldest son to ride a bike, he followed the time-honored tradition of running alongside the bike and holding onto his son, then giving him a push and hoping for the best. After several hours and painfully skinned elbows and knees, his son finally got it. A year later, someone pointed out that the trick to riding a bike is to turn slightly in whichever direction the bike is leaning. Of course! The economist passed this information along to his second son, who learned to ride almost instantly. Just as knowing a little physics can help you learn to ride a bike, knowing a little economics can help you make better decisions.
RECAP
COST-BENEFIT ANALYSIS
Scarcity is a basic fact of economic life. Because of it, having more of one good thing almost always means having less of another (the scarcity principle). The Cost-Benefit Principle holds that an individual (or a firm or a society) should take an action if, and only if, the extra benefit from taking the action is at least as great as the extra cost. The benefit of taking any action minus the cost of taking the action is called the economic surplus from that action. Hence, the Cost-Benefit Principle suggests that we take only those actions that create additional economic surplus.
THREE IMPORTANT DECISION PITFALLS1
Rational people will apply the Cost-Benefit Principle most of the time, although probably in an intuitive and approximate way, rather than through explicit and precise calculation. Knowing that rational people tend to compare costs and benefits enables economists to predict their likely behavior. As noted earlier, for example, we can predict that students from wealthy families are more likely than others to attend colleges that offer small classes. (Again, while the cost of small classes is the same for all families, their benefit, as measured by what people are willing to pay for them, tends to be higher for wealthier families.)
Yet researchers have identified situations in which people tend to apply the Cost-Benefit Principle inconsistently. In these situations, the Cost-Benefit Principle may not predict behavior accurately. But it proves helpful in another way, by identifying specific strategies for avoiding bad decisions.
PITFALL 1: MEASURING COSTS AND BENEFITS AS PROPORTIONS RATHER THAN ABSOLUTE DOLLAR AMOUNTS
As the next example makes clear, even people who seem to know they should weigh the pros and cons of the actions they are contemplating sometimes don’t have a clear sense of how to measure the relevant costs and benefits.
EXAMPLE 1.2Comparing Costs and Benefits
Should you walk downtown to save $10 on a $2,020 laptop computer?
You are about to buy a $2,020 laptop computer at the nearby campus store when a friend tells you that the same computer is on sale at a downtown store for only $2,010. If the downtown store is half an hour’s walk away, where should you buy the computer?
Assuming that the laptop is light enough to carry without effort, the structure of this example is exactly the same as that of Example 1.1. The only difference is that the price of the laptop is dramatically higher than the price of the computer game. As before, the benefit of buying downtown is the dollar amount you’ll save, namely, $10. And because it’s exactly the same trip, its cost also must be the same as before. So if you are perfectly rational, you should make the same decision in both cases. Yet when people are asked what they would do in these situations, the overwhelming majority say they’d walk downtown to buy the game but would buy the laptop at the campus store. When asked to explain, most of them say something like, “The trip was worth it for the game because you save 40 percent, but not worth it for the laptop because you save only $10 out of $2,020.”
This is faulty reasoning. The benefit of the trip downtown is not the proportion you save on the original price. Rather, it is the absolute dollar amount you save. The benefit of walking downtown to buy the laptop is $10, exactly the same as for the computer game. And because the cost of the trip must also be the same in both cases, the economic surplus from making both trips must be exactly the same. That means that a rational decision maker would make the same decision in both cases. Yet, as noted, most people choose differently.
The pattern of faulty reasoning in the decision just discussed is one of several decision pitfalls to which people are often prone. In the discussion that follows, we will identify two additional decision pitfalls. In some cases, people ignore costs or benefits that they ought to take into account. On other occasions they are influenced by costs or benefits that are irrelevant.
CONCEPT CHECK 1.2
Which is more valuable: saving $100 on a $2,000 plane ticket to Tokyo or saving $90 on a $200 plane ticket to Chicago?
PITFALL 2: IGNORING IMPLICIT COSTS
Sherlock Holmes, Arthur Conan Doyle’s legendary detective, was successful because he saw details that most others overlooked. In Silver Blaze, Holmes is called on to investigate the theft of an expensive racehorse from its stable. A Scotland Yard inspector assigned to the case asks Holmes whether some particular aspect of the crime requires further study. “Yes,” Holmes replies, and describes “the curious incident of the dog in the nighttime.” “The dog did nothing in the nighttime,”2 responds the puzzled inspector. But, as Holmes realized, that was precisely the problem! The watchdog’s failure to bark when Silver Blaze was stolen meant that the watchdog knew the thief. This clue ultimately proved the key to unraveling the mystery.
Just as we often don’t notice when a dog fails to bark, many of us tend to overlook the implicit value of activities that fail to happen. As discussed earlier, however, intelligent decisions require taking the value of forgone opportunities properly into account.
Implicit costs are like dogs that fail to bark in the night.
The opportunity cost of an activity, once again, is the value of all that must be forgone in order to engage in that activity. If buying a computer game downtown means not watching the last hour of a movie, then the value to you of watching the end of that movie is an implicit cost of the trip. Many people make bad decisions because they tend to ignore the value of such forgone opportunities. To avoid overlooking implicit costs, economists often translate questions like “Should I walk downtown?” into ones like “Should I walk downtown or watch the end of the movie?”
EXAMPLE 1.3Implicit Cost
Should you use your frequent-flyer coupon to fly to Cancun for spring break?
With spring break only a week away, you are still undecided about whether to go to Cancun with a group of classmates at the University of Iowa. The round-trip airfare from Cedar Rapids is $500, but you have a frequent-flyer coupon you could use for the trip. All other relevant costs for the vacation week at the beach total exactly $1,000. The most you would be willing to pay for the Cancun vacation is $1,350. That amount is your benefit of taking the vacation. Your only alternative use for your frequent-flyer coupon is for a trip to Boston the weekend after spring break to attend your brother’s wedding. (Your coupon expires shortly thereafter.) If the Cedar Rapids–Boston round-trip airfare is $400, should you use your frequent-flyer coupon to fly to Cancun for spring break?
Cost-Benefit
The Cost-Benefit Principle tells us that you should go to Cancun if the benefits of the trip exceed its costs. If not for the complication of the frequent-flyer coupon, solving this problem would be a straightforward matter of comparing your benefit from the week at the beach to the sum of all relevant costs. And because your airfare and other costs would add up to $1,500, or $150 more than your benefit from the trip, you would not go to Cancun.
Is your flight to Cancun “free” if you travel on a frequent-flyer coupon?©McGraw-Hill Education/Barry Barker, photographer
But what about the possibility of using your frequent-flyer coupon to make the trip? Using it for that purpose might make the flight to Cancun seem free, suggesting you’d reap an economic surplus of $350 by making the trip. But doing so also would mean you’d have to fork over $400 for your airfare to Boston. So the implicit cost of using your coupon to go to Cancun is really $400. If you use it for that purpose, the trip still ends up being a loser because the cost of the vacation, $1,400, exceeds the benefit by $50. In cases like these, you’re much more likely to decide sensibly if you ask yourself, “Should I use my frequent-flyer coupon for this trip or save it for an upcoming trip?”
We cannot emphasize strongly enough that the key to using the Cost-Benefit Principle correctly lies in recognizing precisely what taking a given action prevents us from doing. Concept Check 1.3 illustrates this point by modifying the details of Example 1.3 slightly.
CONCEPT CHECK 1.3
Refer to given information in Example 1.3, but this time your frequent-flyer coupon expires in a week, so your only chance to use it will be for the Cancun trip. Should you use your coupon?
PITFALL 3: FAILURE TO THINK AT THE MARGIN
When deciding whether to take an action, the only relevant costs and benefits are those that would occur as a result of taking the action. Sometimes people are influenced by costs they ought to ignore. Other times they compare the wrong costs and benefits. The only costs that should influence a decision about whether to take an action are those we can avoid by not taking the action. Similarly, the only benefits we should consider are those that would not occur unless the action were taken. As a practical matter, however, many decision makers appear to be influenced by costs or benefits that would have occurred no matter what. Thus, people are often influenced by sunk costs—costs that are beyond recovery at the moment a decision is made. For example, money spent on a nontransferable, nonrefundable airline ticket is a sunk cost.
As the following example illustrates, sunk costs must be borne whether or not an action is taken, so they are irrelevant to the decision of whether to take the action.
EXAMPLE 1.4Sunk Cost
How much should you eat at an all-you-can-eat restaurant?
Sangam, an Indian restaurant in Philadelphia, offers an all-you-can-eat lunch buffet for $10. Customers pay $10 at the door, and no matter how many times they refill their plates, there is no additional charge. One day, as a goodwill gesture, the owner of the restaurant tells 20 randomly selected guests that their lunch is on the house. The remaining guests pay the usual price. If all diners are rational, will there be any difference in the average quantity of food consumed by people in these two groups?
Having eaten their first helping, diners in each group confront the following question: “Should I go back for another helping?” For rational diners, if the benefit of doing so exceeds the cost, the answer is yes; otherwise it is no. Note that at the moment of decision, the $10 charge for the lunch is a sunk cost. Those who paid it have no way to recover it. Thus, for both groups, the (extra) cost of another helping is exactly zero. And because the people who received the free lunch were chosen at random, there’s no reason their appetites or incomes should be any different from those of other diners. The benefit of another helping thus should be the same, on average, for people in both groups. And because their respective costs and benefits are the same, the two groups should eat the same number of helpings, on average.
Psychologists and economists have experimental evidence, however, that people in such groups do not eat similar amounts.3 In particular, those for whom the luncheon charge is not waived tend to eat substantially more than those for whom the charge is waived. People in the former group seem somehow determined to “get their money’s worth.” Their implicit goal is apparently to minimize the average cost per bite of the food they eat. Yet minimizing average cost is not a particularly sensible objective. It brings to mind the man who drove his car on the highway at night, even though he had nowhere to go, because he wanted to boost his average fuel economy. The irony is that diners who are determined to get their money’s worth usually end up eating too much.
The fact that the cost-benefit criterion failed the test of prediction in Example 1.4 does nothing to invalidate its advice about what people should do. If you are letting sunk costs influence your decisions, you can do better by changing your behavior.
In addition to paying attention to costs and benefits that should be ignored, people often use incorrect measures of the relevant costs and benefits. This error often occurs when we must choose the extent to which an activity should be pursued (as opposed to choosing whether to pursue it at all). We can apply the Cost-Benefit Principle in such situations by repeatedly asking the question, “Should I increase the level at which I am currently pursuing the activity?”
In attempting to answer this question, the focus should always be on the benefit and cost of an additional unit of activity. To emphasize this focus, economists refer to the cost of an additional unit of activity as its marginal cost. Similarly, the benefit of an additional unit of the activity is its marginal benefit.
When the problem is to discover the proper level for an activity, the cost-benefit rule is to keep increasing the level as long as the marginal benefit of the activity exceeds its marginal cost. As the following example illustrates, however, people often fail to apply this rule correctly.
EXAMPLE 1.5Focusing on Marginal Costs and Benefits
Should SpaceX expand its launch program from four launches per year to five?
SpaceX accountants have estimated that the gains from the company’s new jumbo rocket launch program are currently $24 billion a year (an average of $6 billion per launch) and that its costs are currently $20 billion a year (an average $5 billion per launch). On the basis of these estimates, they have recommended that the company should increase its number of launches. Should SpaceX CEO Elon Musk follow their advice?
To discover whether the advice makes economic sense, we must compare the marginal cost of a launch to its marginal benefit. The accountants’ estimates, however, tell us only the average cost and average benefit of the program. These are, respectively, the total cost of the program divided by the number of launches and the total benefit divided by the number of launches.
Knowing the average benefit and average cost per launch for all rockets launched thus far is simply not useful for deciding whether to expand the program. Of course, the average cost of the launches undertaken so far might be the same as the cost of adding another launch. But it also might be either higher or lower than the marginal cost of a launch. The same holds true regarding average and marginal benefits.
Suppose, for the sake of discussion, that the benefit of an additional launch is in fact the same as the average benefit per launch thus far, $6 billion. Should SpaceX add another launch? Not if the cost of adding the fifth launch would be more than $6 billion. And the fact that the average cost per launch is only $5 billion simply does not tell us anything about the marginal cost of the fifth launch.
Suppose, for example, that the relationship between the number of rockets launched and the total cost of the program is as described in Table 1.1. The average cost per launch (third column) when there are four launches would then be $20 billion/4 = $5 billion per launch, just as the accountants reported. But note in the second column of the table that adding a fifth launch would raise costs from $20 billion to $32 billion, making the marginal cost of the fifth launch $12 billion. So if the benefit of an additional launch is $6 billion, increasing the number of launches from four to five would make absolutely no economic sense.
The following example illustrates how to apply the Cost-Benefit Principle correctly in this case.
EXAMPLE 1.6Focusing on Marginal Costs and Benefits
How many rockets should SpaceX launch?
SpaceX must decide how many rockets to launch. The benefit of each launch is estimated to be $6 billion, and the total cost of the program again depends on the number of launches as shown in Table 1.1. How many rockets should SpaceX launch?
SpaceX should continue to launch its jumbo rockets as long as the marginal benefit of the program exceeds its marginal cost. In this example, the marginal benefit is constant at $6 billion per launch, regardless of the number of shuttles launched. SpaceX should thus keep launching shuttles as long as the marginal cost per launch is less than or equal to $6 billion.
Applying the definition of marginal cost to the total cost entries in the second column of Table 1.1 yields the marginal cost values in the third column of Table 1.2. (Because marginal cost is the change in total cost that results when we change the number of launches by one, we place each marginal cost entry midway between the rows showing the corresponding total cost entries.) Thus, for example, the marginal cost of increasing the number of launches from one to two is $4 billion, the difference between the $7 billion total cost of two launches and the $3 billion total cost of one launch.
As we see from a comparison of the $6 billion marginal benefit per launch with the marginal cost entries in the third column of Table 1.2, the first three launches satisfy the cost-benefit test, but the fourth and fifth launches do not. SpaceX should thus launch three rockets.
CONCEPT CHECK 1.4
If the marginal benefit of each launch had been not $6 billion but $9 billion, how many rockets should SpaceX have launched?
The cost-benefit framework emphasizes that the only relevant costs and benefits in deciding whether to pursue an activity further are marginal costs and benefits—measures that correspond to the increment of activity under consideration. In many contexts, however, people seem more inclined to compare the average cost and benefit of the activity. As Example 1.5 made clear, increasing the level of an activity may not be justified, even though its average benefit at the current level is significantly greater than its average cost.
CONCEPT CHECK 1.5
Should a basketball team’s best player take all the team’s shots?
A professional basketball team has a new assistant coach. The assistant notices that one player scores on a higher percentage of his shots than other players. Based on this information, the assistant suggests to the head coach that the star player should take all the shots. That way, the assistant reasons, the team will score more points and win more games.
On hearing this suggestion, the head coach fires his assistant for incompetence. What was wrong with the assistant’s idea?
RECAP
THREE IMPORTANT DECISION PITFALLS
1. The pitfall of measuring costs or benefits proportionally. Many decision makers treat a change in cost or benefit as insignificant if it constitutes only a small proportion of the original amount. Absolute dollar amounts, not proportions, should be employed to measure costs and benefits.
2. The pitfall of ignoring implicit costs. When performing a cost-benefit analysis of an action, it is important to account for all relevant costs, including the implicit value of alternatives that must be forgone in order to carry out the action. A resource (such as a frequent-flyer coupon) may have a high implicit cost, even if you originally got it “for free,” if its best alternative use has high value. The identical resource may have a low implicit cost, however, if it has no good alternative uses.
3. The pitfall of failing to think at the margin. When deciding whether to perform an action, the only costs and benefits that are relevant are those that would result from taking the action. It is important to ignore sunk costs—those costs that cannot be avoided even if the action isn’t taken. Even though a ticket to a concert may have cost you $100, if you’ve already bought it and cannot sell it to anyone else, the $100 is a sunk cost and shouldn’t influence your decision about whether to go to the concert. It’s also important not to confuse average costs and benefits with marginal costs and benefits. Decision makers often have ready information about the total cost and benefit of an activity, and from these it’s simple to compute the activity’s average cost and benefit. A common mistake is to conclude that an activity should be increased if its average benefit exceeds its average cost. The Cost-Benefit Principle tells us that the level of an activity should be increased if, and only if, its marginal benefit exceeds its marginal cost.
Some costs and benefits, especially marginal costs and benefits and implicit costs, are important for decision making, while others, like sunk costs and average costs and benefits, are essentially irrelevant. This conclusion is implicit in our original statement of the Cost-Benefit Principle (an action should be taken if, and only if, the extra benefits of taking it exceed the extra costs). When we encounter additional examples of decision pitfalls, we will flag them by inserting the icon for the Cost-Benefit Principle as shown here.
Cost-Benefit
NORMATIVE ECONOMICS VERSUS POSITIVE ECONOMICS
The examples discussed in the preceding section make the point that people sometimes choose irrationally. We must stress that our purpose in discussing these examples was not to suggest that people generally make irrational choices. On the contrary, most people appear to choose sensibly most of the time, especially when their decisions are important or familiar ones. The economist’s focus on rational choice thus offers not only useful advice about making better decisions, but also a basis for predicting and explaining human behavior. We used the cost-benefit approach in this way when discussing how rising faculty salaries have led to larger class sizes. And as we will see, similar reasoning helps to explain human behavior in virtually every other domain.
The Cost-Benefit Principle is an example of a normative economic principle, one that provides guidance about how we should behave. For example, according to the Cost-Benefit Principle, we should ignore sunk costs when making decisions about the future. As our discussion of the various decision pitfalls makes clear, however, the Cost-Benefit Principle is not always a positive, or descriptive,economic principle, one that describes how we actually will behave. As we saw, the Cost-Benefit Principle can be tricky to implement, and people sometimes fail to heed its prescriptions.
That said, we stress that knowing the relevant costs and benefits surely does enable us to predict how people will behave much of the time. If the benefit of an action goes up, it is generally reasonable to predict that people will be more likely to take that action. And conversely, if the cost of an action goes up, the safest prediction will be that people will be less likely to take that action. This point is so important that we designate it as the Incentive Principle.
The Incentive Principle: A person (or a firm or a society) is more likely to take an action if its benefit rises, and less likely to take it if its cost rises. In short, incentives matter.
Incentive
The Incentive Principle is a positive economic principle. It stresses that the relevant costs and benefits usually help us predict behavior, but at the same time does not insist that people behave rationally in each instance. For example, if the price of heating oil were to rise sharply, we would invoke the Cost-Benefit Principle to say that people should turn their thermostats down, and invoke the Incentive Principle to predict that average thermostat settings will in fact go down.
ECONOMICS: MICRO AND MACRO
By convention, we use the term microeconomics to describe the study of individual choices and of group behavior in individual markets. Macroeconomics, by contrast, is the study of the performance of national economies and of the policies that governments use to try to improve that performance. Macroeconomics tries to understand the determinants of such things as the national unemployment rate, the overall price level, and the total value of national output.
Our focus in this chapter is on issues that confront the individual decision maker, whether that individual confronts a personal decision, a family decision, a business decision, a government policy decision, or indeed any other type of decision. Further on, we’ll consider economic models of groups of individuals such as all buyers or all sellers in a specific market. Later still we’ll turn to broader economic issues and measures.
No matter which of these levels is our focus, however, our thinking will be shaped by the fact that, although economic needs and wants are effectively unlimited, the material and human resources that can be used to satisfy them are finite. Clear thinking about economic problems must therefore always take into account the idea of trade-offs—the idea that having more of one good thing usually means having less of another. Our economy and our society are shaped to a substantial degree by the choices people have made when faced with trade-offs.
THE APPROACH OF THIS TEXT
Choosing the number of students to register in each class is just one of many important decisions in planning an introductory economics course. Another, to which the Scarcity Principle applies just as strongly, concerns which topics to include on the course syllabus. There’s a virtually inexhaustible set of issues that might be covered in an introductory course, but only limited time in which to cover them. There’s no free lunch. Covering some inevitably means omitting others.
Scarcity
All textbook authors are forced to pick and choose. A textbook that covered all the issues would take up more than a whole floor of your campus library. It is our firm view that most introductory textbooks try to cover far too much. One reason that each of us was drawn to the study of economics is that a relatively short list of the discipline’s core ideas can explain a great deal of the behavior and events we see in the world around us. So rather than cover a large number of ideas at a superficial level, our strategy is to focus on this short list of core ideas, returning to each entry again and again, in many different contexts. This strategy will enable you to internalize these ideas remarkably well in the brief span of a single course. And the benefit of learning a small number of important ideas well will far outweigh the cost of having to ignore a host of other, less important ones.
So far, we’ve already encountered three core ideas: the Scarcity Principle, the Cost-Benefit Principle, and the Incentive Principle. As these core ideas reemerge in the course of our discussions, we’ll call your attention to them. And shortly after a new core idea appears, we’ll highlight it by formally restating it.
A second important element in our philosophy is a belief in the importance of active learning. In the same way that you can learn Spanish only by speaking and writing it, or tennis only by playing the game, you can learn economics only by doing economics. And because we want you to learn how to do economics, rather than just to read or listen passively as the authors or your instructor does economics, we’ll make every effort to encourage you to stay actively involved.
For example, instead of just telling you about an idea, we’ll usually first motivate the idea by showing you how it works in the context of a specific example. Often, these examples will be followed by concept checks for you to try, as well as applications that show the relevance of the idea to real life. Try working the concept checks before looking up the answers (which are at the back of the corresponding chapter).
Think critically about the applications: Do you see how they illustrate the point being made? Do they give you new insight into the issue? Work the problems at the end of the chapters and take extra care with those relating to points that you don’t fully understand. Apply economic principles to the world around you. (We’ll say more about this when we discuss economic naturalism below.) Finally, when you come across an idea or example that you find interesting, tell a friend about it. You’ll be surprised to discover how much the mere act of explaining it helps you understand and remember the underlying principle. The more actively you can become engaged in the learning process, the more effective your learning will be.
ECONOMIC NATURALISM
With the rudiments of the cost-benefit framework under your belt, you are now in a position to become an “economic naturalist,” someone who uses insights from economics to help make sense of observations from everyday life. People who have studied biology are able to observe and marvel at many details of nature that would otherwise have escaped their notice. For example, on a walk in the woods in early April, the novice may see only trees. In contrast, the biology student notices many different species of trees and understands why some are already in leaf while others still lie dormant. Likewise, the novice may notice that in some animal species males are much larger than females, but the biology student knows that pattern occurs only in species in which males take several mates. Natural selection favors larger males in those species because their greater size helps them prevail in the often bloody contests among males for access to females. In contrast, males tend to be roughly the same size as females in monogamous species, in which there is much less fighting for mates.
Learning a few simple economic principles broadens our vision in a similar way. It enables us to see the mundane details of ordinary human existence in a new light. Whereas the uninitiated often fail even to notice these details, the economic naturalist not only sees them, but becomes actively engaged in the attempt to understand them. Let’s consider a few examples of questions economic naturalists might pose for themselves.
The Economic Naturalist 1.1
Why do many hardware manufacturers include more than $1,000 worth of “free” software with a computer selling for only slightly more than that?
The software industry is different from many others in the sense that its customers care a great deal about product compatibility. When you and your classmates are working on a project together, for example, your task will be much simpler if you all use the same word-processing program. Likewise, an executive’s life will be easier at tax time if her financial software is the same as her accountant’s.
The implication is that the benefit of owning and using any given software program increases with the number of other people who use that same product. This unusual relationship gives the producers of the most popular programs an enormous advantage and often makes it hard for new programs to break into the market.
Recognizing this pattern, Intuit Corp. offered computer makers free copies of Quicken, its personal financial-management software. Computer makers, for their part, were only too happy to include the program because it made their new computers more attractive to buyers. Quicken soon became the standard for personal financial-management programs. By giving away free copies of the program, Intuit “primed the pump,” creating an enormous demand for upgrades of Quicken and for more advanced versions of related software. Thus, TurboTax, Intuit’s personal income-tax software, has become the standard for tax-preparation programs.
Inspired by this success story, other software developers have jumped onto the bandwagon. Most hardware now comes bundled with a host of free software programs. Some software developers are even rumored to pay computer makers to include their programs!
The Economic Naturalist 1.1 illustrates a case in which the benefit of a product depends on the number of other people who own that product. As the next Economic Naturalist demonstrates, the cost of a product may also depend on the number of others who own it.
The Economic Naturalist 1.2
Why don’t auto manufacturers make cars without heaters?
Virtually every new car sold in the United States today has a heater. But not every car has a satellite navigation system. Why this difference?
One might be tempted to answer that, although everyone needs a heater, people can get along without navigation systems. Yet heaters are of little use in places like Hawaii and southern California. What is more, cars produced as recently as the 1950s did not all have heaters. (The classified ad that led one young economic naturalist to his first car, a 1955 Pontiac, boasted that the vehicle had a radio, heater, and whitewall tires.)
Although heaters cost extra money to manufacture and are not useful in all parts of the country, they do not cost much money and are useful on at least a few days each year in most parts of the country. As time passed and people’s incomes grew, manufacturers found that people were ordering fewer and fewer cars without heaters. At some point it actually became cheaper to put heaters in all cars, rather than bear the administrative expense of making some cars with heaters and others without. No doubt a few buyers would still order a car without a heater if they could save some money in the process, but catering to these customers is just no longer worth it.
Similar reasoning explains why certain cars today cannot be purchased without a satellite navigation system. Buyers of the 2018 BMW 750i, for example, got one whether they wanted it or not. Most buyers of this car, which sells for more than $75,000, have high incomes, so the overwhelming majority of them would have chosen to order a navigation system had it been sold as an option. Because of the savings made possible when all cars are produced with the same equipment, it would have actually cost BMW more to supply cars for the few who would want them without navigation systems.
Buyers of the least-expensive makes of car have much lower incomes on average than BMW 750i buyers. Accordingly, most of them have more pressing alternative uses for their money than to buy navigation systems for their cars, and this explains why some inexpensive makes continue to offer navigation systems only as options. But as incomes continue to grow, new cars without navigation systems will eventually disappear.
The insights afforded by The Economic Naturalist 1.2 suggest an answer to the following strange question:
The Economic Naturalist 1.3
Why do the keypad buttons on drive-up automated teller machines have Braille dots?
Braille dots on elevator buttons and on the keypads of walk-up automated teller machines enable blind people to participate more fully in the normal flow of daily activity. But even though blind people can do many remarkable things, they cannot drive automobiles on public roads. Why, then, do the manufacturers of automated teller machines install Braille dots on the machines at drive-up locations?
Why do the keypad buttons on drive-up automated teller machines have Braille dots?
The answer to this riddle is that once the keypad molds have been manufactured, the cost of producing buttons with Braille dots is no higher than the cost of producing smooth buttons. Making both would require separate sets of molds and two different types of inventory. If the patrons of drive-up machines found buttons with Braille dots harder to use, there might be a reason to incur these extra costs. But since the dots pose no difficulty for sighted users, the best and cheapest solution is to produce only keypads with dots.
The preceding example was suggested by Cornell student Bill Tjoa, in response to the following assignment:
There is probably no more useful step you can take in your study of economics than to perform several versions of the assignment in Concept Check 1.6. Students who do so almost invariably become lifelong economic naturalists. Their mastery of economic concepts not only does not decay with the passage of time, but it actually grows stronger. We urge you, in the strongest possible terms, to make this investment!
CONCEPT CHECK 1.6
In 500 words or less, use cost-benefit analysis to explain some pattern of events or behavior you have observed in your own environment.
SUMMARY
· Economics is the study of how people make choices under conditions of scarcity and of the results of those choices for society. Economic analysis of human behavior begins with the assumption that people are rational—that they have well-defined goals and try to achieve them as best they can. In trying to achieve their goals, people normally face trade-offs: Because material and human resources are limited, having more of one good thing means making do with less of some other good thing. (LO1)
· Our focus in this chapter has been on how rational people make choices among alternative courses of action. Our basic tool for analyzing these decisions is cost-benefit analysis. The Cost-Benefit Principle says that a person should take an action if, and only if, the benefit of that action is at least as great as its cost. The benefit of an action is defined as the largest dollar amount the person would be willing to pay in order to take the action. The cost of an action is defined as the dollar value of everything the person must give up in order to take the action. (LO2)
· In using the cost-benefit framework, we need not presume that people choose rationally all the time. Indeed, we identified three common pitfalls that plague decision makers in all walks of life: a tendency to treat small proportional changes as insignificant, a tendency to ignore implicit costs, and a tendency to fail to think at the margin—for example, by failing to ignore sunk costs or by failing to compare marginal costs and benefits. (LO3)
· Often the question is not whether to pursue an activity but rather how many units of it to pursue. In these cases, the rational person pursues additional units as long as the marginal benefit of the activity (the benefit from pursuing an additional unit of it) exceeds its marginal cost (the cost of pursuing an additional unit of it). (LO4)
· Microeconomics is the study of individual choices and of group behavior in individual markets, while macroeconomics is the study of the performance of national economics and of the policies that governments use to try to improve economic performance.
CORE PRINCIPLES
Scarcity
The Scarcity Principle (also called the No-Free-Lunch Principle)
Although we have boundless needs and wants, the resources available to us are limited. So having more of one good thing usually means having less of another.
Cost-Benefit
The Cost-Benefit Principle
An individual (or a firm or a society) should take an action if, and only if, the extra benefits from taking the action are at least as great as the extra costs.
Incentive
The Incentive Principle
A person (or a firm or a society) is more likely to take an action if its benefit rises, and less likely to take it if its cost rises. In short, incentives matter.
KEY TERMS
average benefit
average cost
economic surplus
economics
macroeconomics
marginal benefit
marginal cost
microeconomics
normative economic principle
opportunity cost
positive economic principle
rational person
sunk cost
REVIEW QUESTIONS
1. 1.A friend of yours on the tennis team says, “Private tennis lessons are definitely better than group lessons.” Explain what you think he means by this statement. Then use the Cost-Benefit Principle to explain why private lessons are not necessarily the best choice for everyone. (LO2)
2. 2.True or false: Your willingness to drive downtown to save $30 on a new appliance should depend on what fraction of the total selling price $30 is. Explain. (LO3)
3. 3.Why might someone who is trying to decide whether to see a movie be more likely to focus on the $10 ticket price than on the $20 she would fail to earn by not babysitting? (LO3)
4. 4.Many people think of their air travel as being free when they use frequent-flyer coupons. Explain why these people are likely to make wasteful travel decisions. (LO3)
5. 5.Is the nonrefundable tuition payment you made to your university this semester a sunk cost? How would your answer differ if your university were to offer a full tuition refund to any student who dropped out of school during the first two months of the semester? (LO3)
PROBLEMS
1. 1.Suppose the most you would be willing to pay to have a freshly washed car before going out on a date is $6. The smallest amount for which you would be willing to wash someone else’s car is $3.50. You are going out this evening and your car is dirty. How much economic surplus would you receive from washing it? (LO2)
2. 2.To earn extra money in the summer, you grow tomatoes and sell them at a local farmers’ market for 30 cents per pound. By adding compost to your garden, you can increase your yield as shown in the accompanying table. If compost costs 50 cents per pound and your goal is to make as much profit as possible, how many pounds of compost should you add? (LO2)
3. 3.*You and your friend Joe have identical tastes. At 2 p.m., you go to the local Ticketmaster outlet and buy a nonrefundable $30 ticket to a basketball game to be played that night in Syracuse, 50 miles north of your home in Ithaca. Joe plans to attend the same game, but because he cannot get to the Ticketmaster outlet, he plans to buy his ticket at the game. Tickets sold at the game cost only $25 because they carry no Ticketmaster surcharge. (Many people nonetheless pay the higher price at Ticketmaster, to be sure of getting good seats.) At 4 p.m., an unexpected snowstorm begins, making the prospect of the drive to Syracuse much less attractive than before (but ensuring the availability of good seats). If both you and Joe are rational, is one of you more likely to attend the game than the other? (LO2)
4. 4.Tom is a mushroom farmer. He invests all his spare cash in additional mushrooms, which grow on otherwise useless land behind his barn. The mushrooms double in weight during their first year, after which time they are harvested and sold at a constant price per pound. Tom’s friend Dick asks Tom for a loan of $200, which he promises to repay after one year. How much interest will Dick have to pay Tom in order for Tom to recover his opportunity cost of making the loan? Explain briefly. (LO3)
5. 5.Suppose that in the last few seconds you devoted to question 1 on your physics exam you earned 4 extra points, while in the last few seconds you devoted to question 2 you earned 10 extra points. You earned a total of 48 and 12 points, respectively, on the two questions, and the total time you spent on each was the same. If you could take the exam again, how—if at all—should you reallocate your time between these questions? (LO3)
6. 6.Martha and Sarah have the same preferences and incomes. Just as Martha arrived at the theater to see a play, she discovered that she had lost the $10 ticket she had purchased earlier. Sarah also just arrived at the theater planning to buy a ticket to see the same play when she discovered that she had lost a $10 bill from her wallet. If both Martha and Sarah are rational and both still have enough money to pay for a ticket, is one of them more likely than the other to go ahead and see the play anyway? (LO3)
7. 7.Residents of your city are charged a fixed weekly fee of $6 for garbage collection. They are allowed to put out as many cans as they wish. The average household disposes of three cans of garbage per week under this plan. Now suppose that your city changes to a “tag” system. Each can of garbage to be collected must have a tag affixed to it. The tags cost $2 each and are not reusable. What effect do you think the introduction of the tag system will have on the total quantity of garbage collected in your city? Explain briefly. (LO4)
8. 8.Once a week, Smith purchases a six-pack of cola and puts it in his refrigerator for his two children. He invariably discovers that all six cans are gone on the first day. Jones also purchases a six-pack of cola once a week for his two children, but unlike Smith, he tells them that each may drink no more than three cans per week. If the children use cost-benefit analysis each time they decide whether to drink a can of cola, explain why the cola lasts much longer at Jones’s house than at Smith’s. (LO4)
9. 9.*For each long-distance call anywhere in the continental United States, a new phone service will charge users 30 cents per minute for the first 2 minutes and 2 cents per minute for additional minutes in each call. Tom’s current phone service charges 10 cents per minute for all calls, and his calls are never shorter than 7 minutes. If Tom’s dorm switches to the new phone service, what will happen to the average length of his calls? (LO4)
10. 10.*The meal plan at University A lets students eat as much as they like for a fixed fee of $500 per semester. The average student there eats 250 pounds of food per semester. University B charges $500 for a book of meal tickets that entitles the student to eat 250 pounds of food per semester. If the student eats more than 250 pounds, he or she pays $2 for each additional pound; if the student eats less, he or she gets a $2 per pound refund. If students are rational, at which university will average food consumption be higher? Explain briefly. (LO4)
ANSWERS TO CONCEPT CHECKS
1. 1.1The benefit of buying the game downtown is again $10 but the cost is now $12, so your economic surplus would be $2 smaller than if you’d bought it at the campus store. (LO2)
2. 1.2Saving $100 is $10 more valuable than saving $90, even though the percentage saved is much greater in the case of the Chicago ticket. (LO3)
3. 1.3Since you now have no alternative use for your coupon, the opportunity cost of using it to pay for the Cancun trip is zero. That means your economic surplus from the trip will be $1,350 − $1,000 = $350 > 0, so you should use your coupon and go to Cancun. (LO3)
4. 1.4The marginal benefit of the fourth launch is $9 billion, which exceeds its marginal cost of $8 billion, so the fourth launch should be added. But the fifth launch should not, because its marginal cost ($12 billion) exceeds its marginal benefit ($9 billion). (LO3)
5. 1.5If the star player takes one more shot, some other player must take one less. The fact that the star player’s average success rate is higher than the other players’ does not mean that the probability of making his next shot (the marginal benefit of having him shoot once more) is higher than the probability of another player making his next shot. Indeed, if the best player took all his team’s shots, the other team would focus its defensive effort entirely on him, in which case letting others shoot would definitely pay. (LO3)
APPENDIX
Working with Equations, Graphs, and Tables
Although many of the examples and most of the end-of-chapter problems in this book are quantitative, none requires mathematical skills beyond rudimentary high school algebra and geometry. In this brief appendix, we review some of the skills you’ll need for dealing with these examples and problems.
One important skill is to be able to read simple verbal descriptions and translate the information they provide into the relevant equations or graphs. You’ll also need to be able to translate information given in tabular form into an equation or graph, and sometimes you’ll need to translate graphical information into a table or equation. Finally, you’ll need to be able to solve simple systems with two equations and two unknowns. The following examples illustrate all the tools you’ll need.
USING A VERBAL DESCRIPTION TO CONSTRUCT AN EQUATION
We begin with an example that shows how to construct a long-distance telephone billing equation from a verbal description of the billing plan.
EXAMPLE 1A.1A Verbal Description
Your long-distance telephone plan charges you $5 per month plus 10 cents per minute for long-distance calls. Write an equation that describes your monthly telephone bill.
An equation is a simple mathematical expression that describes the relationship between two or more variables, or quantities that are free to assume different values in some range. The most common type of equation we’ll work with contains two types of variables: dependent variables and independent variables. In this example, the dependent variable is the dollar amount of your monthly telephone bill and the independent variable is the variable on which your bill depends, namely, the volume of long-distance calls you make during the month. Your bill also depends on the $5 monthly fee and the 10 cents per minute charge. But, in this example, those amounts are constants, not variables. A constant, also called a parameter, is a quantity in an equation that is fixed in value, not free to vary. As the terms suggest, the dependent variable describes an outcome that depends on the value taken by the independent variable.
Once you’ve identified the dependent variable and the independent variable, choose simple symbols to represent them. In algebra courses, X is typically used to represent the independent variable and Y the dependent variable. Many people find it easier to remember what the variables stand for, however, if they choose symbols that are linked in some straightforward way to the quantities that the variables represent. Thus, in this example, we might use B to represent your monthly bill in dollars and T to represent the total time in minutes you spent during the month on long-distance calls.
Having identified the relevant variables and chosen symbols to represent them, you are now in a position to write the equation that links them:
B = 5 + 0.10T,(1A.1)
where B is your monthly long-distance bill in dollars and T is your monthly total long-distance calling time in minutes. The fixed monthly fee (5) and the charge per minute (0.10) are parameters in this equation. Note the importance of being clear about the units of measure. Because B represents the monthly bill in dollars, we must also express the fixed monthly fee and the per-minute charge in dollars, which is why the latter number appears in Equation 1A.1 as 0.10 rather than 10. Equation 1A.1 follows the normal convention in which the dependent variable appears by itself on the left-hand side while the independent variable or variables and constants appear on the right-hand side.
Once we have the equation for the monthly bill, we can use it to calculate how much you’ll owe as a function of your monthly volume of long-distance calls. For example, if you make 32 minutes of calls, you can calculate your monthly bill by simply substituting 32 minutes for T in Equation 1A.1:
B = 5 + 0.10(32) = 8.20.(1A.2)
Your monthly bill when you make 32 minutes of calls is thus equal to $8.20.
CONCEPT CHECK 1A.1
Under the monthly billing plan described in Example 1A.1, how much would you owe for a month during which you made 45 minutes of long-distance calls?
GRAPHING THE EQUATION OF A STRAIGHT LINE
The next example shows how to portray the billing plan described in Example 1A.1 as a graph.
EXAMPLE 1A.2Graphing an Equation
Construct a graph that portrays the monthly long-distance telephone billing plan described in Example 1A.1, putting your telephone charges, in dollars per month, on the vertical axis and your total volume of calls, in minutes per month, on the horizontal axis.
The first step in responding to this instruction is the one we just took, namely, to translate the verbal description of the billing plan into an equation. When graphing an equation, the normal convention is to use the vertical axis to represent the dependent variable and the horizontal axis to represent the independent variable. In Figure 1A.1, we therefore put B on the vertical axis and T on the horizontal axis. One way to construct the graph shown in the figure is to begin by plotting the monthly bill values that correspond to several different total amounts of long-distance calls. For example, someone who makes 10 minutes of calls during the month would have a bill of B = 5 + 0.10(10) = $6. Thus, in Figure 1A.1 the value of 10 minutes per month on the horizontal axis corresponds to a bill of $6 per month on the vertical axis (point A). Someone who makes 30 minutes of long-distance calls during the month will have a monthly bill of B = 5 + 0.10(30) = $8, so the value of 30 minutes per month on the horizontal axis corresponds to $8 per month on the vertical axis (point C). Similarly, someone who makes 70 minutes of long-distance calls during the month will have a monthly bill of B = 5 + 0.10(70) = $12, so the value of 70 minutes on the horizontal axis corresponds to $12 on the vertical axis (point D). The line joining these points is the graph of the monthly billing Equation 1A.1.
As shown in Figure 1A.1, the graph of the equation B = 5 + 0.10T is a straight line. The parameter 5 is the vertical intercept of the line—the value of B when T = 0, or the point at which the line intersects the vertical axis. The parameter 0.10 is the slope of the line, which is the ratio of the rise of the line to the corresponding run. The ratio rise/run is simply the vertical distance between any two points on the line divided by the horizontal distance between those points. For example, if we choose points A and C in Figure 1A.1, the rise is 8 − 6 = 2 and the corresponding run is 30 − 10 = 20, so rise/run = 2/20 = 0.10. More generally, for the graph of any equation Y = a + bX, the parameter a is the vertical intercept and the parameter b is the slope.
FIGURE 1A.1 The Monthly Telephone Bill in Example 1A.1.The graph of the equation B = 5 + 0.10T is the straight line shown. Its vertical intercept is 5 and its slope is 0.10.
DERIVING THE EQUATION OF A STRAIGHT LINE FROM ITS GRAPH
The next example shows how to derive the equation for a straight line from a graph of the line.
EXAMPLE 1A.3Deriving an Equation from a Graph
Figure 1A.2 shows the graph of the monthly billing plan for a new long-distance plan. What is the equation for this graph? How much is the fixed monthly fee under this plan? How much is the charge per minute?
FIGURE 1A.2 Another Monthly Long-Distance Plan.The vertical distance between points A and C is 12 − 8 = 4 units, and the horizontal distance between points A and C is 40 − 20 = 20, so the slope of the line is 4/20 = 1/5 = 0.20. The vertical intercept (the value of B when T = 0) is 4. So the equation for the billing plan shown is B = 4 + 0.20T.
The slope of the line shown is the rise between any two points divided by the corresponding run. For points A and C, rise = 12 − 8 = 4 and run = 40 − 20 = 20, so the slope equals rise/run = 4/20 = 1/5 = 0.20. And since the horizontal intercept of the line is 4, its equation must be given by
B = 4 + 0.20T.(1A.3)
Under this plan, the fixed monthly fee is the value of the bill when T = 0, which is $4. The charge per minute is the slope of the billing line, 0.20, or 20 cents per minute.
CONCEPT CHECK 1A.2
Write the equation for the billing plan shown in the accompanying graph. How much is its fixed monthly fee? Its charge per minute?
CHANGES IN THE VERTICAL INTERCEPT AND SLOPE
The next two examples and concept checks provide practice in seeing how a line shifts with a change in its vertical intercept or slope.
EXAMPLE 1A.4Change in Vertical Intercept
Show how the billing plan whose graph is in Figure 1A.2 would change if the monthly fixed fee were increased from $4 to $8.
An increase in the monthly fixed fee from $4 to $8 would increase the vertical intercept of the billing plan by $4 but would leave its slope unchanged. An increase in the fixed fee thus leads to a parallel upward shift in the billing plan by $4, as shown in Figure 1A.3. For any given number of minutes of long-distance calls, the monthly charge on the new bill will be $4 higher than on the old bill. Thus 20 minutes of calls per month cost $8 under the original plan (point A) but $12 under the new plan (point A′). And 40 minutes cost $12 under the original plan (point C), $16 under the new plan (point C′); and 60 minutes cost $16 under the original plan (point D), $20 under the new plan (point D′).
FIGURE 1A.3 The Effect of an Increase in the Vertical Intercept.An increase in the vertical intercept of a straight line produces an upward parallel shift in the line.
CONCEPT CHECK 1A.3
Show how the billing plan whose graph is in Figure 1A.2 would change if the monthly fixed fee were reduced from $4 to $2.
EXAMPLE 1A.5Change in Slope
Show how the billing plan whose graph is in Figure 1A.2 would change if the charge per minute were increased from 20 cents to 40 cents.
Because the monthly fixed fee is unchanged, the vertical intercept of the new billing plan continues to be 4. But the slope of the new plan, shown in Figure 1A.4, is 0.40, or twice the slope of the original plan. More generally, in the equation Y = a + bX, an increase in b makes the slope of the graph of the equation steeper.
FIGURE 1A.4 The Effect of an Increase in the Charge per Minute.Because the fixed monthly fee continues to be $4, the vertical intercept of the new plan is the same as that of the original plan. With the new charge per minute of 40 cents, the slope of the billing plan rises from 0.20 to 0.40.
CONCEPT CHECK 1A.4
Show how the billing plan whose graph is in Figure 1A.2 would change if the charge per minute were reduced from 20 cents to 10 cents.
Concept Check 1A.4 illustrates the general rule that in an equation Y = a + bX, a reduction in b makes the slope of the graph of the equation less steep.
CONSTRUCTING EQUATIONS AND GRAPHS FROM TABLES
The next example and concept check show how to transform tabular information into an equation or graph.
EXAMPLE 1A.6Transforming a Table to a Graph
Table 1A.1 shows four points from a monthly long-distance telephone billing equation. If all points on this billing equation lie on a straight line, find the vertical intercept of the equation and graph it. What is the monthly fixed fee? What is the charge per minute? Calculate the total bill for a month with 1 hour of long-distance calls.
One approach to this problem is simply to plot any two points from the table on a graph. Because we are told that the billing equation is a straight line, that line must be the one that passes through any two of its points. Thus, in Figure 1A.5 we use A to denote the point from Table 1A.1 for which a monthly bill of $11 corresponds to 20 minutes per month of calls (second row) and C to denote the point for which a monthly bill of $12 corresponds to 40 minutes per month of calls (fourth row). The straight line passing through these points is the graph of the billing equation.
FIGURE 1A.5 Plotting the Monthly Billing Equation from a Sample of Points.Point A is taken from row 2, Table 1A.1, and point C from row 4. The monthly billing plan is the straight line that passes through these points.
Unless you have a steady hand, however, or use extremely large graph paper, the method of extending a line between two points on the billing plan is unlikely to be very accurate. An alternative approach is to calculate the equation for the billing plan directly. Because the equation is a straight line, we know that it takes the general form B = f + sT, where f is the fixed monthly fee and s is the slope. Our goal is to calculate the vertical intercept f and the slope s. From the same two points we plotted earlier, A and C, we can calculate the slope of the billing plan as s = rise/run = 1/20 = 0.05.
So all that remains is to calculate f, the fixed monthly fee. At point C on the billing plan, the total monthly bill is $12 for 40 minutes, so we can substitute B = 12, s = 0.05, and T = 40 into the general equation B = f + sT to obtain
12 = f + 0.05(40),(1A.4)
or
12 = f + 2,(1A.5)
which solves for f = 10. So the monthly billing equation must be
B = 10 + 0.05T.(1A.6)
For this billing equation, the fixed fee is $10 per month, the calling charge is 5 cents per minute ($0.05/minute), and the total bill for a month with 1 hour of long-distance calls is B = 10 + 0.05(60) = $13, just as shown in Figure 1A.5.
CONCEPT CHECK 1A.5
The following table shows four points from a monthly long-distance telephone billing plan.
If all points on this billing plan lie on a straight line, find the vertical intercept of the corresponding equation without graphing it. What is the monthly fixed fee? What is the charge per minute? How much would the charges be for 1 hour of long-distance calls per month?
SOLVING SIMULTANEOUS EQUATIONS
The next example and concept check demonstrate how to proceed when you need to solve two equations with two unknowns.
EXAMPLE 1A.7Solving Simultaneous Equations
Suppose you are trying to choose between two rate plans for your long-distance telephone service. If you choose Plan 1, your charges will be computed according to the equation
B = 10 + 0.04T,(1A.7)
where B is again your monthly bill in dollars and T is your monthly volume of long-distance calls in minutes. If you choose Plan 2, your monthly bill will be computed according to the equation
B = 20 + 0.02T.(1A.8)
How many minutes of long-distance calls would you have to make each month, on average, to make Plan 2 cheaper?
Plan 1 has the attractive feature of a relatively low monthly fixed fee, but also the unattractive feature of a relatively high rate per minute. In contrast, Plan 2 has a relatively high fixed fee but a relatively low rate per minute. Someone who made an extremely low volume of calls (for example, 10 minutes per month) would do better under Plan 1 (monthly bill = $10.40) than under Plan 2 (monthly bill = $20.20) because the low fixed fee of Plan 1 would more than compensate for its higher rate per minute. Conversely, someone who made an extremely high volume of calls (say, 10,000 minutes per month) would do better under Plan 2 (monthly bill = $220) than under Plan 1 (monthly bill = $410) because Plan 2’s lower rate per minute would more than compensate for its higher fixed fee.
Our task here is to find the break-even calling volume, which is the monthly calling volume for which the monthly bill is the same under the two plans. One way to answer this question is to graph the two billing plans and see where they cross. At that crossing point, the two equations are satisfied simultaneously, which means that the monthly call volumes will be the same under both plans, as will the monthly bills.
In Figure 1A.6, we see that the graphs of the two plans cross at A, where both yield a monthly bill of $30 for 500 minutes of calls per month. The break-even calling volume for these plans is thus 500 minutes per month. If your calling volume is higher than that, on average, you will save money by choosing Plan 2. For example, if you average 700 minutes, your monthly bill under Plan 2 ($34) will be $4 cheaper than under Plan 1 ($38). Conversely, if you average fewer than 500 minutes each month, you will do better under Plan 1. For example, if you average only 200 minutes, your monthly bill under Plan 1 ($18) will be $6 cheaper than under Plan 2 ($24). At 500 minutes per month, the two plans cost exactly the same ($30).
FIGURE 1A.6 The Break-Even Volume of Long-Distance Calls.When your volume of long-distance calls is 500 minutes per month, your monthly bill will be the same under both plans. For higher calling volumes, Plan 2 is cheaper; Plan 1 is cheaper for lower volumes.
The question posed here also may be answered algebraically. As in the graphical approach just discussed, our goal is to find the point (T, B) that satisfies both billing equations simultaneously. As a first step, we rewrite the two billing equations, one on top of the other, as follows:
B = 10 + 0.04T (Plan 1).
B = 20 + 0.02T (Plan 2).
As you’ll recall from high school algebra, if we subtract the terms from each side of one equation from the corresponding terms of the other equation, the resulting differences must be equal. So if we subtract the terms on each side of the Plan 2 equation from the corresponding terms in the Plan 1 equation, we get
Finally, we solve the last equation (Plan 1 − Plan 2) to get T = 500.
Plugging T = 500 into either plan’s equation, we then find B = 30. For example, Plan 1’s equation yields 10 + 0.04(500) = 30, as does Plan 2’s: 20 + 0.2(500) = 30.
Because the point (T, B) = (500, 30) lies on the equations for both plans simultaneously, the algebraic approach just described is often called the method of simultaneous equations.
CONCEPT CHECK 1A.6
Suppose you are trying to choose between two rate plans for your long-distance telephone service. If you choose Plan 1, your monthly bill will be computed according to the equation
B = 10 + 0.10T (Plan 1),
where B is again your monthly bill in dollars and T is your monthly volume of long-distance calls in minutes. If you choose Plan 2, your monthly bill will be computed according to the equation
B = 100 + 0.01T (Plan 2).
Use the algebraic approach described in the preceding example to find the break-even level of monthly call volume for these plans.
KEY TERMS
constant
dependent variable
equation
independent variable
parameter
rise
run
slope
variable
vertical intercept
ANSWERS TO APPENDIX CONCEPT CHECKS
1. 1A.1To calculate your monthly bill for 45 minutes of calls, substitute 45 minutes for T in Equation 1A.1 to get B = 5 + 0.10(45) = $9.50.
2. 1A.2Calculating the slope using points A and C, we have rise = 30 − 24 = 6 and run = 30 − 15 = 15, so rise/run = 6/15 = 2/5 = 0.40. And since the horizontal intercept of the line is 18, its equation is B = 18 + 0.40T. Under this plan, the fixed monthly fee is $18 and the charge per minute is the slope of the billing line, 0.40, or 40 cents per minute.
3. 1A.3A $2 reduction in the monthly fixed fee would produce a downward parallel shift in the billing plan by $2.
4. 1A.4With an unchanged monthly fixed fee, the vertical intercept of the new billing plan continues to be 4. The slope of the new plan is 0.10, half the slope of the original plan.
5. 1A.5Let the billing equation be B = f + sT, where f is the fixed monthly fee and s is the slope. From the first two points in the table, calculate the slope s = rise/run = 10/10 = 1.0. To calculate f, we can use the information in row 1 of the table to write the billing equation as 20 = f + 1.0(10) and solve for f = 10. So the monthly billing equation must be B = 10 + 1.0T. For this billing equation, the fixed fee is $10 per month, the calling charge is $1 per minute, and the total bill for a month with 1 hour of long-distance calls is B = 10 + 1.0(60) = $70.
6. 1A.6Subtracting the Plan 2 equation from the Plan 1 equation yields the equation
0 = −90 + 0.09T (Plan 1 − Plan 2),
which solves for T = 1,000. So if you average more than 1,000 minutes of long-distance calls each month, you’ll do better on Plan 2.
1The examples in this section are inspired by the pioneering research of Daniel Kahneman and the late Amos Tversky. Kahneman was awarded the 2002 Nobel Prize in economics for his efforts to integrate insights from psychology into economics. You can read more about this work in Kahneman’s brilliant 2011 book, Thinking Fast and Slow (New York: Macmillan).
2Source: Doyle, Arthur Conan, “The Adventure of Silver Blaze,” The Memoirs of Sherlock Holmes. London, U.K.: George Newnes Ltd., 1893.
3 See, for example, Richard Thaler, “Toward a Positive Theory of Consumer Choice,” Journal of Economic Behavior and Organization 1, no. 1 (1980).
*Denotes more difficult problem.