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Chapter 3
60
A Consumer’s Constrained Choice
If this is coffee, please bring me some tea; but if this is tea, please bring me some coffee. —Abraham Lincoln
icroeconomics provides powerful insights into the myriad questions and choices facing consumers. For example, does U.S. consumers’ purchases of relatively fewer SUVs and more small vehicles in the past few years reflect a change in tastes or a response to higher prices? How can we use information about con- sumers’ allocations of their budgets across various goods in the past to predict how a price change will affect their demands for goods today? Are consumers better off receiv- ing cash or a comparable amount in food stamps? Should you buy insurance or save your money? Work at home or in the marketplace? Have children? Invest in bonds or in stocks? To answer these and other questions about how consumers allocate their income over many goods, we use a model that lets us look at an individual’s decision making when faced with limited income and market-determined prices. This model allows us to derive the market demand curve that we used in our supply-and-demand model and to make a variety of predictions about consumers’ responses to changes in prices and income. Our model of consumer behavior is based on the following premises:
■ Individual tastes or preferences determine the amount of pleasure people derive from the goods and services they consume. ■ Consumers face constraints or limits on their choices. ■ Consumers maximize their well-being or pleasure from consumption, sub- ject to the constraints they face.
Consumers spend their money on the bundle of products that gives them the most pleasure. If you like music and don’t have much of a sweet tooth, you spend a lot of your money on concerts and CDs and relatively little on candy.^1 By contrast, your chocoholic friend with the tin ear may spend a great deal on Hershey’s Kisses and very little on music. All consumers must choose which goods to buy because limited incomes prevent them from buying everything that catches their fancy. In addition, government rules restrict what they may buy: Young consumers cannot buy alcohol or cigarettes legally, and laws prohibit people of all ages from buying crack cocaine and other recreational drugs (although, of course, enforcement is imperfect). Therefore, consumers buy the goods that give them the most pleasure, subject to the constraints that they cannot spend more money than they have and that they cannot spend it in ways that the gov- ernment prevents. In economic analyses that are designed to explain behavior (positive analysis—see Chapter 1) rather than to judge it (normative statements), economists assume that the
M
(^1) Microeconomics is the study of trade-offs: Should you save your money or buy that Superman Action Comic Number 1 you always wanted? Indeed, an anagram for microeconomics is income or comics.
consumer is the boss. If your brother gets pleasure from smoking, economists wouldn’t argue with him that it’s bad for him any more than they’d tell your sister, who likes read- ing Stephen King, that she should read Adam Smith’s Wealth of Nations instead.^2 Accepting each consumer’s tastes is not the same as condoning the resulting behaviors. Economists want to predict behavior. They want to know, for example, whether your brother will smoke more next year if the price of cigarettes decreases 10%. The follow- ing prediction is unlikely to be correct: “He shouldn’t smoke; therefore, we predict he’ll stop smoking next year.” A prediction based on your brother’s actual tastes is more likely to be correct: “Given that he likes cigarettes, he is likely to smoke more of them next year if the price falls.”
Preferences 61
- Preferences: We use five properties of preferences to predict which combinations, or bundle, of goods an individual prefers to other combinations.
- Utility: Economists summarize a consumer’s preferences using a utility function, which assigns a numerical value to each possible bundle of goods, reflecting the con- sumer’s relative ranking of these bundles.
- Budget Constraint: Prices, income, and government restrictions limit a consumer’s ability to make purchases by determining the rate at which a consumer can trade one good for another.
- Constrained Consumer Choice: Consumers maximize their pleasure from consuming various possible bundles of goods given their income, which limits the amount of goods they can purchase.
In this chapter, we examine four main topics:
3.1 Preferences
Do not do unto others as you would that they would do unto you. Their tastes may not be the same. —George Bernard Shaw
We start our analysis of consumer behavior by examining consumer preferences. Using four assumptions, we can make many predictions about preferences. Once we know about consumers’ preferences, we can add information about the con- straints that consumers face so that we can answer many questions, such as the ones posed at the beginning of the chapter, or derive demand curves, as we do in the next chapter. As a consumer, you choose among many goods. Should you have ice cream or cake for dessert? Should you spend most of your money on a large apartment or rent a single room and use the money you save to pay for trips and concerts? In short, you must allocate your money to buy a bundle ( market basket or combina- tion) of goods.
(^2) As the ancient Romans put it: “De gustibus non est disputandum”—there is no disputing about (accounting for) tastes. Or, as it was put in the movie Grand Hotel (1932), “Have caviar if you like, but it tastes like herring to me.”
Preferences 63
More Is Better. The more-is-better property (the economics jargon is nonsatiation ) holds that, all else the same, more of a commodity is better than less of it. 5 Indeed, economists define a good as a commodity for which more is preferred to less, at least at some levels of consumption. In contrast, a bad is something for which less is pre- ferred to more, such as pollution. We now concentrate on goods. Although the completeness and transitivity properties are crucial to the analysis that follows, the more-is-better property is included to simplify the analysis—our most important results would follow even without this property. So why do economists assume that the more-is-better property holds? The most compelling reason is that it appears to be true for most people.^6 A second reason is that if consumers can freely dispose of excess goods, consumers can be no worse off with extra goods. (We examine a third reason later in the chapter: We observe consumers buying goods only where this condition is met.)
Continuity. Loosely, the continuity property holds that if a consumer prefers Bundle a to Bundle b , then the consumer prefers Bundle c to b if c is very close to a. The pur- pose of this assumption is to rule out sudden preference reversals in response to small changes in the characteristics of a bundle. This assumption is technical and allows us later in this chapter to develop the mathematical theory concerning utility functions.
Strict Convexity. Strict convexity of preferences means that consumers prefer aver- ages to extremes. For example, if Bundle a and Bundle b are distinct bundles and the consumer prefers both of these bundles to Bundle c , then the consumer prefers a weighted average of a and b , b a (1 – b) b (where 0 < b < 1), to Bundle c. This con- dition is a technical one, which we will usually assume holds.^7
PREFERENCE MAPS
Surprisingly enough, with just the first three properties, we can tell a lot about a consumer’s preferences. One of the simplest ways to summarize information about a consumer’s preferences is to create a graphical interpretation—a map—of them. For simplicity, we concentrate on choices between only two goods, but the model can be generalized to handle any number of goods.
(^5) When teaching microeconomics to Wharton MBAs, I told them about a cousin of mine who had just joined a commune in Oregon. His worldly possessions consisted of a tent, a Franklin stove, enough food to live on, and a few clothes. He said that he didn’t need any other goods—that he was satiated. A few years later, one of these students bumped into me on the street and said, “Professor, I don’t remember your name or much of anything you taught me in your course, but I can’t stop thinking about your cousin. Is it really true that he doesn’t want anything else? His very existence is a repudiation of my whole way of life.” Actually, my cousin had given up his ascetic life and was engaged in telemarketing, but I, for noble pedagogical reasons, responded, “Of course he still lives that way—you can’t expect everyone to have the tastes of an MBA.” (^6) How much wealth do you need to live comfortably? In a survey of wealthy people ( Business Week , February 28, 2005, p. 13), those with a net worth of over $1 million said that they needed $2.4 million to live comfortably, those with at least $5 million in net worth said that they need $10.4 million, and those with at least $10 million wanted $18.1 million. Apparently, people never have enough. (^7) This condition helps ensure that the second-order condition holds in the consumer maximiza- tion problem.
64 CHAPTER 3 A Consumer’s Constrained Choice
B, Burritos per semester
(a)
15 25 30 Z, Pizzas per semester
25
20
15
10
5
d
a b
e
c
f
A
B
B, Burritos per semester
(b)
15 25 30 Z, Pizzas per semester
25
20
15
10 d
a b I^1
e
c
f
B, Burritos per semester
(c)
15 25 30 Z, Pizzas per semester
25
20
15
10 d
I^0
I^1
e I^2
c
f
Figure 3.1 Bundles of Pizzas and Burritos Lisa Might Consume. (a) Lisa prefers more to less, so she prefers Bundle e to any bundle in area B , includ- ing d. Similarly, she prefers any bundle in area A , such as f , to e. (b) The indifference curve, I^1 , shows a set of bundles (including c , e , and a ) among which she is indifferent. (c) The three indifference curves, I^1 , I^2 , and I^3 , are part of Lisa’s preference map, which summarizes her preferences.
Each semester, Lisa, who lives for fast food, decides how many pizzas and burritos to eat. The various bundles of pizzas and burritos she might consume are shown in panel a of Figure 3.1, with (individual-size) pizzas per semester, q 1 , on the horizontal axis and burritos per semester, q 2 , on the vertical axis. At Bundle e , for example, Lisa consumes 25 pizzas and 15 burritos per semester. By the more-is-better property, all the bundles that lie above and to the right (area A ) are preferred to Bundle e because they contain at least as much of both pizzas and burritos as Bundle e. Thus Bundle f (30 pizzas and 20 burritos) in that region is preferred to e. By the same reasoning, Lisa prefers e to all the bundles that lie in area B , below and to the left of e , such as Bundle d (15 pizzas and 10 burritos). Bundles such as b (30 pizzas and 10 burritos), in the region below and to the right of e , or c (15 pizzas and 25 burritos), in the region above and to the left, may or may
66 CHAPTER 3 A Consumer’s Constrained Choice
B, Burritos per semester
(a) Crossing
Z, Pizzas per semester
I^1 I^0
a
b
e
B, Burritos per semester B, Burritos per semester
(b) Upward Sloping
Z, Pizzas per semester
I
a a
b
b
(c) Thick
Z, Pizzas per semester
I
Figure 3.2 Impossible Indifference Curves. (a) Suppose that the indifference curves cross at Bundle e. Lisa is indifferent between e and a on indifference curve I^0 and between e and b on I^1. If Lisa is indifferent between e and a and she is indifferent between e and b , she must be indifferent between a and b by transitivity. But b has more of both pizzas and burritos than a , so she must prefer a to b. Because of this contradiction, indifference curves cannot cross. (b) Suppose that indifference curve I slopes upward. The consumer is
indifferent between b and a because they lie on I but prefers b to a by the more-is-better assumption. Because of this contradiction, indifference curves can- not be upward sloping. (c) Suppose that indifference curve I is thick enough to contain both a and b. The consumer is indifferent between a and b because both are on I but prefers b to a by the more-is-better assumption because b lies above and to the right of a. Because of this contradiction, indifference curves can- not be thick.
both are on I^1. By transitivity, if Lisa is indifferent between e and a and she is indiffer- ent between e and b , she must be indifferent between a and b. But that’s impossible! Bundle b is above and to the right of Bundle a , so Lisa must prefer b to a by the more- is-better property. Thus because preferences are transitive and more is better than less, indifference curves cannot cross. Fourth, we show that indifference curves must be downward sloping. Suppose, to the contrary, that an indifference curve sloped upward, as in panel b of Figure 3.2. The consumer is indifferent between Bundles a and b because both lie on the same indif- ference curve, I. But the consumer prefers b to a by the more-is-better property: Bundle a lies strictly below and to the left of Bundle b. Because of this contradiction— the consumer cannot both be indifferent between a and b and strictly prefer b to a — indifference curves cannot be upward sloping.
Can indifference curves be thick? Answer Draw an indifference curve that is at least two bundles thick, and show that a pref- erence property is violated : Panel c of Figure 3.2 shows a thick indifference curve, I , with two bundles, a and b , identified. Bundle b lies above and to the right of a :
SOLVED PROBLEM 3.
Utility 67
3.2 Utility
Underlying our model of consumer behavior is the belief that consumers can compare various bundles of goods and decide which bundle gives them the greatest pleasure. We can summarize a consumer’s preferences by assigning a numerical value to each pos- sible bundle to reflect the consumer’s relative ranking of these bundles. Following the terminology of Jeremy Bentham, John Stuart Mill, and other nineteenth- century British utilitarianism economist-philosophers, economists apply the term utility to this set of numerical values that reflect the relative rankings of various bun- dles of goods. The statement that “Bonnie prefers Bundle x to Bundle y ” is equivalent to the statement that “Consuming Bundle x gives Bonnie more utility than consuming Bundle y .” Bonnie prefers x to y if Bundle x gives Bonnie 10 utils— units of utility— and Bundle y gives her 8 utils.
UTILITY FUNCTION
The utility function is the relationship between utility measures and every possible bundle of goods. If we know the utility function, we can summarize the information in indifference maps succinctly. Suppose that the utility, U , that Lisa gets from pizzas and burritos is
From this function, we know that the more Lisa consumes of either good, the greater her utility. Using this function, we can determine whether she would be happier if she had Bundle x with 16 pizzas and 9 burritos or Bundle y with 13 of each. The utility she gets from x is utils. The utility she gets from y is utils. Therefore, she prefers y to x. The utility function is a concept that economists use to help them think about con- sumer behavior; utility functions do not exist in any fundamental sense. If you asked your mother what her utility function is, she would be puzzled—unless, of course, she is an economist. But if you asked her enough questions about her choices of bundles of goods, you could construct a function that accurately summarizes her preferences. For example, by questioning people, Rousseas and Hart (1951) constructed indiffer- ence curves between eggs and bacon, and MacCrimmon and Toda (1969) constructed indifference curves between French pastries and money (which can be used to buy all other goods). Typically, consumers can easily answer questions about whether they prefer one bundle to another, such as “Do you prefer a bundle with one scoop of ice cream and two pieces of cake to another bundle with two scoops of ice cream and one piece of cake?” But they have difficulty answering questions about how much more they prefer
U = (^2) q 1 q 2.
Bundle b has more of both burritos and pizzas. Thus by the more-is-better prop- erty, Bundle b must be strictly preferred to Bundle a. But the consumer must be indifferent between a and b because both bundles are on the same indifference curve. Because both relationships between a and b cannot be true, there is a con- tradiction. Consequently, indifference curves cannot be thick. (We illustrate this point by drawing indifference curves with very thin lines in our figures.)
Utility 69
corner (origin) along either wall or into the room, where Lisa has more q 1 or q 2 or both. Lisa’s utility or hill of happiness rises as you move away from the corner. What is the relationship between Lisa’s utility and one of her indifference curves, those combinations of q 1 and q 2 that give Lisa a particular level of utility? Imagine that the hill of happiness is made of clay. If you were to cut the hill parallel to the floor at a particular height on the wall—a given level of utility—you’d get a smaller hill above the cut. Now suppose that you place that smaller hill directly on the floor and trace the outside edge of the hill. Looking down at the floor, the traced outer edge of the hill rep- resents an indifference curve on the two-dimensional floor. Making other parallel cuts in the hill of happiness, placing the smaller hills on the floor, and tracing their outside edges, you could obtain a map of indifference curves on which each indifference curve reflects a different level of utility. In short, an indifference curve consists of all those bundles that correspond to a par- ticular utility measure. If Lisa’s utility function is U ( q 1 , q 2 ), then the expression for one of her indifference curves is
(3.1)
This expression determines all those bundles of q 1 and q 2 that give her utils of
pleasure. For example, if the utility function is , then the indifference
curve includes any ( q 1 , q 2 ) bundles such that q 1 q 2 16, including the bundles (4, 4), (2, 8), (8, 2), (1, 16), and (16, 1).
WILLINGNESS TO SUBSTITUTE BETWEEN GOODS
To analyze how consumers make choices when faced with limited resources, it is useful to know the slope of an indifference curve at a particular bundle of goods. Economists call the slope at a point of an indifference curve the marginal rate of substitution ( MRS ), because it is the maximum amount of one good that a consumer will sacrifice (trade) to obtain one more unit of another good. Lisa is willing to trade one good for more of another good. The downward slope of her indifference curve in Figure 3.3 shows that Lisa is willing to give up some burritos for more pizzas and vice versa. Because the indifference curve is downward sloping, the MRS is a negative number. We can use calculus to determine the MRS at a point on Lisa’s indifference curve in Equation (3.1). We will show that the MRS depends on how much extra utility Lisa gets from a little more of each good. We call the extra utility that a consumer gets from con- suming the last unit of a good the marginal utility. Given that Lisa’s utility function is U ( q 1 , q 2 ), the extra or marginal utility that she gets from a little more pizza, holding the quantity of burritos fixed, is
Similarly, the marginal utility from more burritos is U 2 0 U / 0 q 2 , where we hold the amount of pizza constant. We determine the slope of Lisa’s indifference curve, the MRS , by ascertaining the changes in q 1 and q 2 that leave her utility unchanged, keeping her on her original indif- ference curve: U ( q 1 , q 2 ). Let q 2 ( q 1 ) be the implicit function that shows how much q 2 it takes to keep Lisa’s utility at U given that she consumes q 1. We want to know how
U
marginal utility of pizza =
0 U
0 q 1
= U 1.
4 = U = (^2) q 1 q 2
U = (^2) q 1 q 2
U
U = U ( q 1 , q 2 ).
70 CHAPTER 3 A Consumer’s Constrained Choice
q^2
, Burritos per semester
q 1 , Pizzas per semester
I
e
Figure 3.3 Marginal Rate of Substitution. Lisa’s marginal rate of substitution, MRS d q 2 /d q 1 , at Bundle e is the slope of indifference curve I at that point. The marginal rate of substitution, at e is the same as the slope of the line that is tangent to I at e.
much q 2 must change if we increase q 1 , d q 2 /d q 1 , given that we require her utility to remain constant. To answer this question, we differentiate U ( q 1 , q 2 ( q 1 )) with respect to q 1 :
(3.2)
Because is a constant, d /d q 1 0. Since Lisa derives pleasure from both goods, if we increase one of the goods, we must decrease the other to hold her utility constant and keep her on her indifference curve. Rearranging the terms in Equation (3.2), we find that her marginal rate of sub- stitution is
(3.3)
Thus the slope of her indifference curve is the negative of the ratio of her marginal util- ities. We can give a graphical interpretation of the slope of the indifference curve. The slope of her indifference curve I at Bundle e is the same as the slope of the line that is tangent to the indifference curve at that point.
MRS =
d q 2 d q 1
0 U > 0 q 1 0 U > 0 q 2
U 1
U 2
U U
d U d q 1
0 U ( q 1 , q 2 ( q 1 )) 0 q 1
0 U ( q 1 , q 2 ( q 1 )) 0 q 2
d q 2 d q 1
= U 1 + U 2
d q 2 d q 1
U
Suppose that Jackie has what is known as a Cobb-Douglas utility function:^8 (3.4) where a is a positive constant, q 1 is the number of music CDs she buys a year, and q 2 is the number of movie DVDs she buys. What is her marginal rate of substitution?
U = q 1^ aq 21 -^ a ,
SOLVED PROBLEM 3.
(^8) The Cobb-Douglas utility function may be written more generally as U Aq 1 c (^) q 2 d. However, we can always transform that utility function into this simpler one through a monotonic transforma- tion: q 1^ aq 21 a^ F ( A q 1^ c^ q 2^ d ), where F ( x ) x 1/( c^ ^ d )/ A , so that a c /( c d ).
Coke, Cans per week
(a) Perfect Substitutes
1 2 3 4 Pepsi, Cans per week
1
0
2
3
4
I^1 I^2 I^3 I^4
Ice cream, Scoops per week
(b) Perfect Complements
1 2 3 Pie, Slices per week
1
2
3
0
I^1
I^2
I^3
a
d
e c
b B, Burritos per semester
(c) Imperfect Substitutes
Z, Pizzas per semester
I
Figure 3.4 Perfect Substitutes, Perfect Complements, Imperfect Substitutes. (a) Ben views Coke and Pepsi as perfect substitutes. His indifference curves are straight, parallel lines with a marginal rate of substitu- tion (slope) of −1. Ben is willing to exchange one can of Coke for one can of Pepsi. (b) Maureen likes pie à la mode but does not like pie or ice cream by itself: She
views ice cream and pie as perfect complements. She will not substitute between the two; she consumes them only in equal quantities. (c) Lisa views burritos and pizza as imperfect substitutes. Her indifference curve lies between the extreme cases of perfect substi- tutes and perfect complements.
indifference curves over commodities are convex. When people have a lot of one good, they are willing to give up a relatively large amount of it to get a good of which they have relatively little. However, after that first trade, they are willing to give up less of the first good to get the same amount more of the second good. It is hard to imagine that Lisa’s indifference curves are concave to the origin. If her indifference curve were strictly concave, Lisa would be willing to give up more burri- tos to get one more pizza, the fewer the burritos she has. Two extreme versions of downward-sloping, convex indifference curves are plausible: straight-line or right- angle indifference curves. One extreme case is perfect substitutes, goods that a consumer is completely indif- ferent as to which to consume. Because Ben cannot taste any difference between Coca- Cola and Pepsi-Cola, he views them as perfect substitutes: He is indifferent between one additional can of Coke and one additional can of Pepsi. His indifference curves for these two goods are straight, parallel lines with a slope of –1 everywhere along the curve, as in panel a of Figure 3.4. Thus Ben’s MRS is 1 at every point along these indifference curves. (His marginal utility from each good is identical, so the MRS U 1 / U 2 1.) The slope of indifference curves of perfect substitutes need not always be –1; it can be any constant rate. For example, Amos knows from reading the labels that Clorox bleach is twice as strong as a generic brand. As a result, Amos is indifferent between one cup of Clorox and two cups of the generic bleach. Amos’s utility function over Clorox, C , and the generic bleach, G , is U ( C , G ) = iC + jG , (3.7)
72 CHAPTER 3 A Consumer’s Constrained Choice
Utility 73
where both goods are measured in cups, i 2, and j 1. His indifference curves are straight lines with a slope or MRS of i / j 2, where the generic bleach is on the vertical axis. 9 The other extreme case is perfect complements : goods that a consumer is interested in consuming only in fixed proportions. Maureen doesn’t like apple pie, A , by itself or vanilla ice cream, V , by itself but loves apple pie à la mode: a slice of pie with a scoop of vanilla ice cream on top. Her utility function is
(3.8)
where i j 1 and the min function says that the utility equals the smaller of the two arguments, iA or jV. Her indifference curves have right angles in panel b of Figure 3.4. If she has only one piece of pie, she gets as much pleasure from it and one scoop of ice cream, Bundle a , as from one piece and two scoops, Bundle d , or as from one piece and three scoops, Bundle e. For example, if she were at b , she would be unwilling to give up an extra slice of pie to get, say, two extra scoops of ice cream, as at point e. That is, she won’t eat the extra scoops because she does not have pieces of pie to go with the ice cream. The only condition where she doesn’t have an excess of either good is when iA = jV , or V/A i/j 1. Therefore, she consumes only bundles like a , b , and c in which pie and ice cream are in fixed (here, equal) proportions, because she is unwilling to sub- stitute more of one good for less of another. (The marginal utility is zero for each good, because increasing that good while holding the other one constant does not increase Maureen’s utility.) The standard-shaped, convex indifference curve in panel c of Figure 3.4 lies between these two extreme examples. Convex indifference curves show that a consumer views two goods as imperfect substitutes. A consumer with a Cobb-Douglas utility function 3.4 has convex indifference curves.
U ( A , V )^ =^ min( iA ,^ jV^ ),
Indifference Curves Between Food and Clothing Using the estimates of Eastwood and Craven (1981), the figure shows the indif- ference curves of the average U.S. consumer between food consumed at home and clothing. The food and clothing measures are weighted averages of various goods. At relatively low quantities of food and clothing, the indifference curves, such as I^1 , are nearly right angles: perfect complements. As we move away from the ori- gin, the indifference curves become flatter: closer to perfect substitutes. One interpretation of these indifference curves is that there are minimum levels of food and clothing necessary to support life. The consumer cannot trade one good for the other if it means having less than those critical levels. As the consumer obtains more of both goods, however, the consumer is increasingly willing to trade between
APPLICATION
(^9) Sometimes it is difficult to guess which goods are close substitutes. According to Harper’s Index 1994, flowers, perfume, and fire extinguishers rank 1, 2, and 3 among Mother’s Day gifts that Americans consider “very appropriate.”
Budget Constraint 75
B, Burritos per semester Opportunity set
50 = Y/p (^) Z
L^1 (p (^) Z= $1, Y= $50)
25 = Y/p (^) B
0 Z, Pizzas per semester
Figure 3.5 Budget Constraint. If Y $50, p 1 $1, and p 2 $2, Lisa can buy any bundle in the opportunity set, the shaded area, including points on the budget line , L , which has a slope of – 1 / 2.
rewrite her budget constraint, Equation 3.9, as
According to Equation 3.10, she can buy more burritos with a higher income (d q 2 /d Y 1/ p 2 > 0), the purchase of fewer pizzas (d q 2 /d q 1 p 1 / p 2 < 0), or a lower price of burritos or pizzas [d q 2 /d p 2 –( Y – p 1 q 1 )/ p 22 – q 2 / p 2 < 0, d q 2 /d p 1 q 1 / p 2 < 0]. For example, if she has one more dollar of income ( Y ), she can buy 1/ p 2 more burritos. If p 1 $1, p 2 $2, and Y $50, Equation 3.10 is
This equation is plotted in Figure 3.5. This budget line shows the combinations of bur- ritos and pizzas that Lisa can buy if she spends all of her $50 on these two goods. As this equation shows, every two pizzas cost Lisa one burrito. How many burritos can she buy if she spends all her money on burritos? By setting q 1 0 in Equation 3.10, we find that q 2 Y / p 2 $50/$2 25. Similarly, if she spends all her money on pizzas, q 2 0 and q 1 Y / p 1 $50/$1 50. The budget constraint in Figure 3.5 is a smooth, continuous line. Implicitly, Lisa can buy fractional numbers of burritos and pizzas. Is that true? Do you know of a restau- rant that will sell you a quarter of a burrito? Probably not. Why, then, don’t we draw the opportunity set and the budget constraint as points (bundles) of whole numbers of burritos and pizzas? The reason is that Lisa can buy a burrito at a rate of one-half per time period. If Lisa buys one burrito every other week, she buys an average of one-half burrito every week. Thus it is plausible that she could purchase fractional amounts over time, and this diagram concerns her behavior over a semester. Lisa could, of course, buy any bundle that costs less than $50. The opportunity set consists of all the bundles a consumer can buy, including all the bundles inside the
q 2 =
$50 - ($1 * q 1 ) $
q 1.
q 2 =
Y - p 1 q 1 p 2
76 CHAPTER 3 A Consumer’s Constrained Choice
budget constraint and on the budget constraint (all those bundles of positive q 1 and q 2 such that p 1 q 1 p 2 q 2 ≤ Y ). Lisa’s opportunity set is the shaded area in the figure. For example, she could buy 10 burritos and 15 pizzas for $35, which falls inside her bud- get constraint. However, she can obtain more of the foods she loves by spending all of her budget and picking a bundle on the budget constraint rather than a bundle below the constraint. We call the slope of the budget line the marginal rate of transformation ( MRT ): the trade-off the market imposes on the consumer in terms of the amount of one good the consumer must give up to obtain more of the other good. It is the rate at which Lisa can trade burritos for pizzas in the marketplace, where the prices she pays and her income are fixed. Holding prices and income constant and differentiating Equation 3.10 with respect to q 1 , we find that the slope of the budget constraint, or the marginal rate of transfor- mation, is
(3.11)
Because the price of a pizza is half that of a burrito ( p 1 $1 and p 2 $2), the marginal rate of transformation that Lisa faces is
An extra pizza costs her half an extra burrito—or, equivalently, an extra burrito costs her two pizzas.
3.4 Constrained Consumer Choice
My problem lies in reconciling my gross habits with my net income. —Errol Flynn
Were it not for the budget constraint, consumers who prefer more to less would con- sume unlimited amounts of at least some goods. Well, they can’t have it all! Instead, consumers maximize their well-being subject to their budget constraints. To complete our analysis of consumer behavior, we have to determine the bundle of goods that maximizes well-being subject to the budget constraint. We first take a graphical approach and then use calculus.
THE CONSUMER’S OPTIMAL BUNDLE
Veni, vidi, Visa. (We came, we saw, we went shopping.) —Jan Barrett
We want to determine which bundle within the opportunity set gives the consumer the highest level of utility. That is, we are trying to solve a constrained maximization prob- lem, where a consumer maximizes utility subject to a budget constraint. To determine which bundle in the opportunity set gives Lisa the highest level of pleasure, we use her indifference curves in panel a of Figure 3.6. We will show that her optimal bundle lies on an indifference curve that touches the budget constraint, L , at only one point ( e on I^2 )—hence that indifference curve does not cross the constraint.
MRT = -
p 1 p 2
MRT =
d q 2 d q 1
p 1 p 2
78 CHAPTER 3 A Consumer’s Constrained Choice
the constraint. Only some of the bundles on indifference curve I^1 lie within the opportunity set: Bundles a and c and all the points on I^1 between them, such as d , can be purchased. Because I^1 crosses the budget constraint, the bundles between a and c on I^1 lie strictly inside the constraint, so there are bundles in the opportunity set (area A B ) that are preferable to these bundles on I^1 and are affordable. By the more-is-better prop- erty, Lisa prefers e to d because e has more of both pizzas and burritos than d. By tran- sitivity, Lisa prefers e to a , c , and all the other points on I^1 —even those, like g , that Lisa can’t afford. Because indifference curve I^1 crosses the budget constraint, area B con- tains at least one bundle that is preferred to—lies above and to the right of—at least one bundle on the indifference curve. Thus the optimal bundle must lie on the budget constraint and be on an indiffer- ence curve that does not cross it. Such a bundle is the consumer’s optimum. If Lisa is consuming this bundle, she has no incentive to change her behavior by substituting one good for another. There are two ways for an optimal bundle to lie on an indifference curve that touches the budget constraint but does not cross it. The first is an interior solution , in which the optimal bundle has positive quantities of both goods: The optimal bundle is on the budget line rather than at one end or the other. The other possibility is called a corner solution , where the optimal bundle is at one end or the other of the budget line: It is at a corner with one of the axes.
Interior Solution. In panel a of Figure 3.6, Bundle e on indifference curve I^2 is the optimum bundle. It lies in the interior of the budget line away from the corners. Lisa prefers consuming a balanced diet, e , of 10 burritos and 30 pizzas, to eating only one type of food. For the indifference curve I^2 to touch the budget constraint but not cross it, it must be tangent to the budget constraint: The budget constraint and the indifference curve have the same slope at the point e where they touch. The slope of the indifference curve, the marginal rate of substitution, measures the rate at which Lisa is willing to trade burritos for pizzas: MRS U 1 / U 2 , Equation 3.3. The slope of the budget line, the marginal rate of transformation, measures the rate at which Lisa can trade her money for burritos or pizza in the market: MRT – p 1 / p 2 , Equation 3.11. Thus Lisa’s utility is maximized at the bundle where the rate at which she is willing to trade bur- ritos for pizzas equals the rate at which she can trade in the market:
(3.12)
Rearranging terms, this condition is equivalent to
(3.13)
Equation 3.13 says that U 1 / p 1 , the marginal utility of pizzas divided by the price of a pizza—the amount of extra utility from pizza per dollar spent on pizza—equals U 2 / p 2 , the marginal utility of burritos divided by the price of a burrito. Thus Lisa’s utility is maximized if the last dollar she spends on pizzas gets her as much extra utility as the last dollar she spends on burritos. If the last dollar spent on pizzas gave Lisa more extra utility than the last dollar spent on burritos, Lisa could increase her happiness by spending more on pizzas and less on burritos. Her cousin Spenser is a different story.
U 1
p 1
U 2
p 2
MRS = -
U 1
U 2
p 1 p 2
= MRT.
Constrained Consumer Choice 79
Corner Solution. Spenser’s indifference curves in panel b of Figure 3.6 are flatter than Lisa’s. His optimal bundle lies on an indifference curve that touches the budget line, L , only once, at the upper-left corner of the opportunity set, e , where he buys only burritos (25 burritos and 0 pizzas). Bundle e is the optimal bundle because the indifference curve does not cross the constraint into the opportunity set. If it did, another bundle would give Spenser more pleasure. Spenser’s indifference curve is not tangent to his budget line. It would cross the budget line if both the indifference curve and the budget line were continued into the “negative pizza” region of the diagram, on the other side of the burrito axis.
Nigel, a Brit, and Bob, a Yank, have the same tastes, and both are indifferent between a sport-utility vehicle (SUV) and a luxury sedan. Each has a budget that will allow him to buy and operate one vehicle for a decade. For Nigel, the price of owning and operating an SUV is greater than that for the car. For Bob, an SUV is a relative bargain because he benefits from an SUV tax break. Use an indifference curve–budget line analysis to explain why Nigel buys and operates a car while Bob chooses an SUV. Answer
- Describe their indifference curves : Because Nigel and Bob view the SUV and the car as perfect substitutes, each has an indifference curve for buying one vehicle that is a straight line with a slope of –1 and that hits each axis at 1 in the figure.
- Describe the slopes of their budget lines : Nigel faces a budget line, LN , that is flat- ter than the indifference curve, and Bob faces one, LB , that is steeper_._
- Use an indifference curve and a budget line to show why Nigel and Bob make dif- ferent choices : As the figure shows, LN^ hits the indifference curve, I, at 1 on the car axis, eN, and LB^ hits I at 1 on the SUV axis, eB. Thus Nigel buys the relatively inex- pensive car and Bob scoops up a relatively cheap SUV. Comment : If Nigel and Bob were buying a bundle of cars and SUVs for their large families or firms, the analysis would be similar—Bob would buy relatively more SUVs than would Nigel.
Cars per decade
SUVs per decade
1
LB
l
L N
eN
eB 1
SOLVED PROBLEM 3.
