reallocating expenditures.5You may wish to try several other combinations ofX andYthat this person can afford to show that all of them provide a lower utility level than does combinationC. That is whyCis a point of tangency—it is the only affordable combination that allows this person to reachU2. For a point of non- tangency (sayB), a person can always get more utility because the budget constraint passes through the indifference curve (seeU1 in the figure). In Application 2.4:
Ticket Scalping, we examine a case in which people do not have such complete freedom in how they spend their incomes.
A P P L I C A T I O N 2 . 4 Ticket Scalping
Tickets to major concerts or sporting events are not usually auctioned off to the highest bidder. Instead, promoters tend to sell most tickets at ‘‘reasonable’’ prices and then ration the resulting excess demand either on a first-come-first-served basis or by limiting the number of tickets each buyer can purchase. Such rationing mechanisms create the possibility for further selling of tickets at much higher prices in the secondary market—that is, ticket ‘‘scalping.’’
A Graphical Interpretation
Figure 1 shows the motivation for ticket scalping for, say, Super Bowl tickets. With this consumer’s income and the quoted price of tickets, he or she would prefer to purchase four tickets (pointA). But the National Football League has decided to limit tickets to only one per customer. This limita- tion reduces the consumer’s utility fromU2(the utility he or she would enjoy with tickets freely available) toU1. Notice that this choice of one ticket (pointB) does not obey the
tangency rule for a utility maximum—given the actual price of tickets, this person would prefer to buy more than one. In fact, this frustrated consumer would be willing to pay more than the prevailing price for additional Super Bowl tickets.
He or she would not only be more than willing to buy a second ticket at the official price (since pointCis aboveU1) but also be willing to give up an additional amount of other goods (given by distanceCD) to get this ticket. It appears that this person would be more than willing to pay quite a bit to a ‘‘scalper’’ for the second ticket. For example, tickets for major events at the 1996 Atlanta Olympics often sold for five times their face prices, and resold tickets for the 2005 Super Bowl went for upwards of $2,000 to die-hard Patriots fans.
Antiscalping Laws
Most economists hold a relatively benign view of ticket scalping. They look at the activity as being a voluntary trans- action between a willing buyer and a willing seller. State and local governments often seem to see things differently, how- ever. Many have passed laws that seek either to regulate the prices of resold tickets or to outlaw ticket selling in locations near the events. The generally cited reason for such laws is that scalping is ‘‘unfair’’—perhaps because the ‘‘scalper’’
makes profits that are ‘‘not deserved.’’ This value judgment seems excessively harsh, however. Ticket scalpers provide a valuable service by enabling transactions between those who place a low value on their tickets and those who would value them more highly. The ability to make such transac- tions can itself be valuable to people whose situations change. Forbidding these transactions may result in wasted resources if some seats remain unfilled. The primary gainer from antiscalping laws may be ticket agencies who can gain a monopoly-like position as the sole source of sought-after tickets.
POLICYCHALLENGE
Antiscalping laws are just one example of a wide variety of laws that prevent individuals from undertaking voluntary transactions. Other examples include banning the sale of certain drugs, making it illegal to sell one’s vote in an elec- tion, or forbidding the selling of human organs. One reason often given for precluding certain voluntary transactions is that such transactions may harm third parties. Is that a good reason for banning such transactions? Does the possibility for harmful third-party effects seem to explain the various examples mentioned here? If not, why are such transactions banned?
FIGURE 1Rationing of Tickets Leads to Scalping
Other goods
Super Bowl tickets Income
1 2 3 4
A
U2 U1 D
C B
5
Given this consumer’s income and the price of tickets, he or she would prefer to buy four. With only one available, utility falls toU1. This person would pay up to distanceCDin other goods for the right to buy a second ticket at the original price.
Chevron is the more expensive of the two brands, so this person opts to buy only Exxon. Because the goods are identical, the utility-maximizing decision is to buy only the less expensive brand. People who buy only generic versions of prescription drugs or who buy all their brand-name household staples at a discount supermarket are exhibiting a similar type of behavior.
Finally, the utility-maximizing situation illu- strated in Figure 2.9(d) shows that this person will buy shoes only in pairs. Any departure from this pattern would result in buying extra left or right shoes, which alone provide no utility. In similar circumstances involving complementary goods, peo- ple also tend to purchase those goods together.
Other items of apparel (gloves, earrings, socks, and so forth) are also bought mainly in pairs. Most peo- ple have preferred ways of concocting the beverages they drink (coffee and cream, gin and vermouth) or of making sandwiches (peanut butter and jelly, ham and cheese); and people seldom buy automobiles, stereos, or washing machines by the part. Rather, they consume these complex goods as fixed packages made up of their various components.
Overall then, the utility-maximizing model of choice provides a very flexible way of explaining why people make the choices that they do. Because
F I G U R E 2 . 8
D i f f e r en ce s i n Pr efe r en ce s Re s ul t in Di f f e r i ng C ho i ce s
(a) Hungry Joe Hamburgers
per week
8
U0 U1 U2
Soft drinks per week 0 4
(b) Thirsty Teresa Hamburgers
per week
Income Income Income
2
U0U1U2
Soft drinks per week
0 16
(c) Extra-Thirsty Ed Hamburgers
per week
U0 U1U2
Soft drinks per week
0 20
The three individuals illustrated here all have the same budget constraint. They have $30 to spend, hamburgers cost $3, and soft drinks cost $1.50. These people choose very different consumption bundles because they have differing preferences for the two goods.
M i c r o Q u i z 2 . 5
Figure 2.8 and Figure 2.9 show that the condi- tion for utility maximization should be amended sometimes to deal with special situations.
1. Explain how the condition should be changed for ‘‘boundary’’ issues such as those shown in Figure 2.8(c) and 2.9(c), where people buy zero amounts of some goods. Use this to explain why your authors never buy any lima beans.
2. How do you interpret the condition in which goods are perfect complements, such as those shown in Figure 2.9(d)? If left and right shoes were sold separately, could any price ratio make you depart from buying pairs?
people are faced with budget constraints, they must be careful to allocate their incomes so that they provide as much satisfaction as possible. Of course, they will not explicitly engage in the kinds of graphic analyses shown in the figures for this chapter. But this model seems to be a good way of making precise the notion that people ‘‘do the best with what they’ve got.’’ We look at how this model can be used to illustrate a famous court case in Application 2.5: What’s a Rich Uncle’s Promise Worth?
A Few Numerical Examples
Graphs can be helpful in conceptualizing the utility maximization process, but to solve problems, you will probably need to use algebra. This section provides a few ideas on how to solve such problems.
F I G U R E 2 . 9
U t i l i t y - M a x i mi z i n g C h o i c e s f o r S pe c i al Ty p e s o f G o o d s
(a) A useless good Smoke
grinders per week
U1
E
E
E E U2 U3
Food per week Income
Income
Income
Income
0 10
(b) An economic bad Houseflies
per week
U1 U2 U3
Food per week
0 10
(c) Perfect substitutes Gallons
of Exxon per week
U1 U2 U3
Gallons of Chevron per week
0
(d) Perfect complements Right shoes
per week
U3
U1 U2
Left shoes per week 0
2
2
The four panels in this figure repeat the special indifference curve maps from Figure 2.5. The resulting utility-maximizing positions (denoted byEin each panel) reflect the specific relationships among the goods pictured.
A P P L I C A T I O N 2 . 5 What’s a Rich Uncle’s Promise Worth?
One of the strangest legal cases of the nineteenth century was the New York case ofHamer v. Sidway, in which nephew Willie sued his uncle for failing to carry through on the promise to pay him $5,000 if he did not smoke, drink, or gamble until he reached the age of 21. No one in the case disagreed that the uncle had made this deal with Willie when he was about 15 years old. The legal issue was whether the uncle’s promise was a clear ‘‘contract,’’ enforceable in court.
An examination of this peculiar case provides an instructive illustration of how economic principles can help to clarify legal issues.
Graphing the Uncle’s Offer
Figure 1 shows Willie’s choice between ‘‘sin’’ (that is, smok- ing, drinking, and gambling) on theX-axis and his spending on everything else on theY-axis. Left to his own devices, Willie would prefer to consume pointA—which involves some sin along with other things. This would provide him with utility ofU2. Willie’s uncle is offering him pointB—an extra $5,000 worth of other things on the condition that
sin¼0. In this graph, it is clear that the offer provides more utility (U3) than pointA, so Willie should take the offer and spend his teenage years sin-free.
When the Uncle Reneges
When Willie came to collect the $5,000 for his abstinence, his uncle assured him that he would place the funds in a bank account that Willie would get once he was ‘‘capable of using it wisely.’’ But the uncle died and left no provision for pay- ment in his will. So Willie ended up with no money. The consequences of being stiffed for the $5,000 can be shown in Figure 1 by pointC—this is the utility Willie would get by spending all his income on non-sin items.
Willie Goes to Court
Not willing to take his misfortune lying down, Willie took his uncle’s estate to court, claiming, in effect, that he had made a contract with his uncle and deserved to be paid. The primary legal question in the case concerned the issue of
‘‘consideration’’ in the purported contract between Willie and his uncle. In contract law the promise of party A to do something for party B is enforceable only if there is evidence that an actual bargain was reached. One sign that such an agreement has been reached is the payment of some form of consideration from B to A that seals the deal. Although there was no explicit payment from Willie to his uncle in this case, the court ultimately ruled that Willie’s 6 years of abstinence itself played that role here. Apparently the uncle derived pleasure from seeing a ‘‘sin-free’’ Willie so this was regarded as sufficient consideration in this case. After much wrangling, Willie finally got paid.
TOTHINKABOUT
1. Suppose that the uncle’s heirs had offered to settle by making Willie as well-off as he would have been by acting sinfully in his teenage years. In Figure 1, how could you show the amount they would have to pay?
2. Would the requirement that the uncle make Willie
‘‘whole’’ by paying the amount suggested in question 1 provide the right incentives for him to stick to the original deal?
FIGURE 1Willie’s Utility and His Uncle’s Promises Other
goods
C B
A
U3
Budget constraint
U2 U1
Sin Left to his own devices, Willie consumes pointAand gets utility U2. His uncle’s offer would increase utility toU3. But, when his uncle reneges, Willie getsU1(pointC).
Perfect Substitutes Problems involving perfect substitutes are the easiest to solve—all you have to do is figure out which good is least expensive given the utility provided. When the goods are identical (Exxon and Chevron), this is easy—
the consumer will choose to spend all of his or her budget on the good with the lowest price.6If Exxon costs $3 per gallon, and Chevron is $3.25, he or she will buy only Exxon. If the gasoline budget is $30, 10 gallons will be bought.
When goods are perfect substitutes, but not identical, the story is a bit more complicated. Suppose a person regards apple juice (A) and grape juice (G) as perfect substitutes for his or her thirst, but each ounce of apple juice provides 4 units of utility, whereas each ounce of grape juice provides 3 units of utility. In this case, the person’s utility function would be:
UðA,GÞ ¼4Aþ3G (2.9)
The fact that this utility function is linear means that its indifference curves will be straight lines as in Figure 2.9c. If the price of apple juice is 6 cents per ounce, and the price of grape juice is 5 cents per ounce, it might at first seem that this person will buy only grape juice. But that conclusion disregards the difference in utility provided by the drinks. To decide which drink is really least expensive, suppose this person has 30 cents to spend. If he or she spends it all on apple juice, 5 ounces can be bought, and Equation 2.9 shows that these will yield a utility of 20. If the person spends the 30 cents all on grape juice, 6 ounces can be bought, and utility will be 18. So, apple juice is actually the better buy after utility differences are taken into account.7If this person has $1.20 to spend on fruit juice, he or she will spend it all on apple juice, purchasing 20 ounces and receiving utility of 80.
Perfect Complements Problems involving perfect complements are also easy to solve so long as you keep in mind that the good must be purchased in a fixed ratio to one another. If left shoes and right shoes cost $10 each, a pair will cost $20, and a person will spend all of his or her shoe budget on pairs. With $60 to spend, three pairs will be bought.
When the complementary relationship is not one-to-one, the calculations are slightly more complicated. Suppose, for example, a person always buys two bags of popcorn at $2.50 each at the movie theater. If the theater ticket itself costs $10, the combination ‘‘movieþpopcorn’’ costs $15. With a monthly movie budget of $30, this person will attend two movies each month.
Let’s look at the algebra of the movie situation. First, we need a way to phrase the utility function for movies (M) and popcorn (C). The way to do this is with the function:
UðM,CÞ ¼Minð2M,CÞ (2:10)
Where ‘‘Min’’ means that utility is given by the smaller of the two terms in parentheses. If, for example, this person attends a movie but buys no popcorn, utility is zero. If he or she attends a movie and buys three bags of popcorn, utility is 2—the extra bag of popcorn does not raise utility. To avoid such useless spending,
6If the goods cost the same, the consumer is indifferent as to which is bought. He or she might as well flip a coin.
7Another way to see this uses footnote 6. Here,MUA¼4,MUG¼3,PA¼6,PG¼5. Hence,MUA=PA¼4=6¼2=3, MUG=PG¼3=5. Since 2/3 > 3/5, apple juice provides more utility per dollar spent than does grape juice.
this person should only consume bundles for whichC¼2M—that is, two bags of popcorn for each movie. To find out how much will actually be bought, you can now substitute this into this person’s budget constraint:
30¼10Mþ2:5C or 10Mþ5M¼15M¼30 soM¼2,C¼4 (2:11) Notice that this solution assures utility maximization. Our graphical treatment (Figure 2.9d) showed that this person will only consume these two perfect comple- ments in a fixed ratio of two bags of popcorn to each movie. That fact allows us to treat movies and popcorn as a single item in the budget contraint, so finding the solution is easy.
A Middle-Ground Case Most pairs of goods are neither perfect substitutes nor perfect complements. Rather, the relationship between them allows some substitut- ability but not the sort of all-or-nothing behavior shown in the Exxon-Chevron example. One of the challenges for economists is to figure out ways of writing utility functions to cover these situations. Although this can become a very mathematical topic, here we can describe one simple middle-ground case. Suppose that a person consumes onlyXand Yand utility is given by the function we examined in the Appendix to Chapter 1:
U Xð ,YÞ ¼ ffiffiffiffiffiffiffiffiffiffiffi X·Y
p (2:12)
We know from our previous discussion that this function has reasonably shaped contour lines, so it may be a good example to study. To show utility maximization with this function, we need first to figure out how the MRS exhibited by an indifference curve depends on the quantities of each good consumed. Unfortu- nately, for most functions figuring out the slope of an indifference curve requires calculus. So, often you will given the MRS. In this case, the MRS is given by:8
MRSðX,YÞ ¼Y=X (2:13)
Utility maximization requires that Equation 2.8 hold. Let’s again assume thatY (hamburgers) costs $3, andX(soft drinks) costs $1.50. The utility maximization requires that:
MRSðX;YÞ ¼Y=X ¼PX=PY ¼$3=$1:50¼2 soY¼2X (2:14) To get the final quantities bought, we need to introduce the budget constraint, so let’s again assume that this person has $30 to spend on fast food. Substituting the utility-maximizing condition in Equation 2.13 into the budget constraint (Equation 2.5) yields:
30¼3Xþ1:5Y ¼3Xþ1:5ð2XÞ ¼6X soX ¼5;Y¼10 (2:15) One feature of this solution is that this person spends precisely half his or her budget ($15) onXand half onY. This will be true no matter what income is and no matter what the prices of the two goods are. Consequently, this utility function is a very special case and may not explain consumption patterns in
8This can be derived by noting that marginal utilities are just the (partial) derivatives of this function. Hence, MUX¼@U=@X¼0:5 ffiffiffiffiffiffiffiffiffiffi
pY=X
andMUY¼@U=@Y¼0:5 ffiffiffiffiffiffiffiffiffiffi pX=Y
. So,MRS X,Yð Þ ¼MUX=MUY¼Y=X.
the real world. The function (which, as we pointed out before, is called a ‘‘Cobb- Douglas’’ function) can be generalized a bit, as we show in Problem 2.10, but for most actual studies of consumer behavior, much more complicated functions are used.