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Unformatted text preview: CHAPTER 3 PREFERENCES We saw in Chapter 2 that the economic model of consumer behavior is very
simple: people choOse the best things they can afford. The last chapter was
devoted to clarifying the meaning of “can afford,” and this chapter will be
devoted to clarifying the economic concept of “best things.” We call the objects of consumer choice consumption bundles. This
is a complete list of the goods and services that are involved in the choice
problem that we are investigating. The word “complete” deserves empha
sis: when you analyze a consumer’s choice problem, make sure that you
include all of the appropriate goods in the deﬁnition of the consumption
bundle. If we are analyzing consumer choice at the broadest level, we would want
not only a complete list of the goods that a consumer might consume, but
also a description of when, where, and under what circumstances they
would become available. After all, people care about how much food they
will have tomorrow as well as how much food they have today. A raft in the
middle of the Atlantic Ocean is very different from a raft in the middle of
the Sahara Desert. And an umbrella when it is raining is quite a diﬁerent
good from an umbrella on a sunny day. It is often useful to think of the 34 PREFERENCES (Ch. 3) “same” good available in different locations or circumstances as a different
good, since the consumer may value the good differently in those situations. However, when we limit our attention to a simple choice problem, the
relevant goods are usually pretty obvious. We’ll often adopt the idea de—
scribed earlier of using just two goods and calling one of them “all other
goods” so that we can focus on the tradeoff between one good and ev
erything else. In this way we can consider consumption choices involving
many goods and still use two—dimensional diagrams. So let us take our consumption bundle to consist of two goods, and let
x1 denote the amount of one good and .122 the amount of the other. The
complete consumption bundle is therefore denoted by (331,172). As noted
before, we will occasionally abbreviate this consumption bundle by X. 3.1 Consumer Preferences We will suppose that given any two consumption bundles, (171,332) and
(y1,y2), the consumer can rank them as to their desirability. That is, the
consumer can determine that one of the consumption bundles is strictly
better than the other, or decide that she is indifferent between the two
bundles. We will use the symbol > to mean that one bundle is strictly preferred
to another, so that (331,332) >~ (y1,y2) should be interpreted as saying that
the consumer strictly prefers ($1,332) to (gr/1,3,2), in the sense that she
definitely wants the x—bundle rather than the y—bundle. This preference
relation is meant to be an operational notion. If the consumer prefers
one bundle to another, it means that he or she would choose one over the
other, given the opportunity. Thus the idea of preference is based on the
consumer’s behavior. In order to tell whether one bundle is preferred to
another, we see how the consumer behaves in choice situations involving
the two bundles. If she always chooses (£131,332) when (y1,y2) is available,
then it is natural to say that this consumer prefers ($1,172) to (yhyg). If the consumer is indifferent between two bundles of goods, we use
the symbol ~ and write (921,332) ~ (ghyg). Indifference means that the
consumer would be just as satisﬁed, according to her own preferences,
consuming the bundle (3:1, 2:2) as she would be consuming the other bundle,
(3/1, yzl If the consumer prefers or is indifferent between the two bundles we say
that she weakly prefers ($1, $2) to (y1,y2) and write (sham) t (91,312). These relations of strict preference, weak preference, and indifference
are not independent concepts; the relations are themselves related! For
example, if (931,;132) : (311,112) and (311,112) : (231,132) we can conclude that
(271,332) ~ (yhyg). That is, if the consumer thinks that ($1,232) is at least
as good as (311,312) and that (yl, 312) is at least as good as (331,132), then the
consumer must be indifferent between the two bundles of goods. ASSUMPTIONS ABOUT PREFERENCES 35 Similarly, if ($1,332) t ($11,312) but We know that it is not the case that
(£131,352) N (yhyg), we can conclude that we must have (331,302) > (yhyg).
This just says that if the consumer thinks that (9:1,:r2) is at least as good
as (y1,y2), and she is not indifferent between the two bundles, then it must
be that she thinks that (5131,13) is strictly better than (y1,y2). 3.2 Assumptions about Preferences Economists usually make some assumptions about the “consistency” of
consumers’ preferences. For example, it seems unreasonable——not to say
contradictory—to have a situation where (111,.CE2) > (311,312) and, at the
same time, (311,312) >— ($1,113). For this would mean that the consumer
strictly prefers the x—bundle to the y—bundle . . . and vice versa. So we usually make some assumptions about how the preference relations
work. Some of the assumptions about preferences are so fundamental that
we can refer to them as “axioms” of consumer theory. Here are three such
axioms about consumer preference. Complete. We assume that any two bundles can be compared. That is,
given any x—bundle and any y—bundle, we assume that (:51, 362) i (y1,y2),
or (y1,y2) : ($1,372), or both, in which case the consumer is indifferent
between the two bundles. Reﬂexive. We assume that any bundle is at least as good as itself:
($1,302) t ($1,562) Transitive. If (231,:c2) i (yhyg) and (y1,y2) t (21,22), then we assume
that (:51, 232) t (21, 22). In other words, if the consumer thinks that X is at
least as good as Y and that Y is at least as good as Z, then the consumer thinks that X is at least as good as Z. The ﬁrst axiom, completeness, is hardly objectionable, at least for the
kinds of choices economists generally examine. To say that any two bundles
can be compared is simply to say that the consumer is able to make a choice
between any two given bundles. One might imagine extreme situations
involving life or death choices where ranking the alternatives might be
difﬁcult, or even impossible, but these choices are, for the most part, outside
the domain of economic analysis. The second axiom, reﬂexivity, is trivial. Any‘ bundle is certainly at least
as good as an identical bundle. Parents of small children may occasionally
observe behavior that violates this assumption, but it seems plausible for
most adult behavior. The third axiom, transitivity, is more problematic. It isn’t clear that
transitivity of preferences is necessarily a property that preferences would
have to have. The assumption that preferences are transitive doesn’t seem 36 PREFERENCES (Ch. 3) compelling on grounds of pure logic alone. In fact it’s not. Transitivity is
a hypothesis about people’s choice behavior, not a statement of pure logic.
Whether it is a basic fact of logic or not isn’t the point: it is whether or not
it is a reasonably accurate description of how peOple behave that matters. What would you think about a person who said that he preferred a
bundle X to Y, and preferred Y to Z, but then also said that he preferred
Z to X 7 This would certainly be taken as evidence of peculiar behavior. More importantly, how would this consumer behave if faced with choices
among the three bundles X, Y, and Z? If we asked him to choose his most
preferred bundle, he would have quite a problem, for whatever bundle he
chose, there would always be one that was preferred to it. If we are to have
a theory where people are making “best” choices, preferences must satisfy
the transitivity axiom or something very much like it. If preferences were
not transitive there could well be a set of bundles for which there is no best
choice. 3.3 Indifference Curves It turns out that the whole theory of consumer Choice can be formulated
in terms of preferences that satisfy the three axioms described above, plus
a few more technical assumptions. However, we will ﬁnd it convenient to
describe preferences graphically by using a construction known as indif—
ference curves. Consider Figure 3.1 where we have illustrated two axes representing a
consumer’s consumption of goods 1 and 2. Let us pick a certain consump—
tion bundle (3:1,1'2) and shade in all of the consumption bundles that are
weakly preferred to ($1, 332). This is called the weakly preferred set. The
bundles on the boundary of this set—the bundles for which the consumer
is just indifferent to (x1,a:2)——form the indifference curve. We can draw an indifference curve through any consumption bundle we
want. The indifference curve through a consumption bundle consists of all
bundles of goods that leave the consumer indifferent to the given bundle. One problem with using indifference curves to describe preferences is
that they only show you the bundles that the consumer perceives as being
indifferent to each other—they don’t show you which bundles are better
and which bundles are worse. It is sometimes useful to draw small arrows
on the indifference curves to indicate the direction of the preferred bundles.
We won’t do this in every case, but we will do it in a few of the examples
where confusion might arise. If we make no further assumptions about preferences, indifference curves
can take very peculiar shapes indeed. But even at this level of generality,
we can state an important principle about indifference curves: indiﬂerence
curves representing distinct levels of preference cannot cross. That is, the
situation depicted in Figure 3.2 cannot occur. EXAMPLES OF PREFERENCES 37 Weakly preferred set:
bundles weakly
preferred to (x1, x2) Indifference
curve:
bundles
indifferent “3 (x1, x2) Weakly preferred set. The shaded area consists of all bun—
dles that are at least as good as the bundle ($1,332). In order to prove this, let us choose three bundles of goods, X, Y, and
Z, such that X lies only on one indifference curve, Y lies only on the other
indifference curve, and Z lies at the intersection of the indifference curves.
By assumption the indifference curves represent distinct levels of prefer
ence, so one of the bundles, say X, is strictly preferred to the other bundle,
Y. We know that X N Z and Z N Y, and the axiom of transitivity there—
fore implies that X N Y. But this contradicts the assumption that X > Y.
This contradiction establishes the result~—indifference curves representing
distinct levels of preference cannot cross. What other properties do indifference curves have? In the abstract, the
answer is: not many. Indifference curves are a way to describe preferences.
Nearly any “reasonable” preferences that you can think of can be depicted
by indifference curves. The trick is to learn what kinds of preferences give
rise to what shapes of indifference curves. 3.4 Examples of Preferences Let us try to relate preferences to indifference curves through some exam—
ples. We’ll describe some preferences and then see what the indifference
curves that represent them look like. 38 PREFERENCES (Ch. 3} . Alle. ed
indi erence
curves X: Indifference curves cannot cross. If they did, X, Y, and
Z would all have to be indifferent to each other and thus could
_ not lie on distinct .indiﬁerence curves. There is a general procedure for constructing indifference curves given
a “verbal” description of the preferences. First plop your pencil down on
the graph at some consumption bundle (1:1, 132). Now think about giving a
little more of good 1, A331, to the consumer, moving him to (:31 + A331,:L'2).
Now ask yourself how would you have to change the consumption of 2:2
to make the consumer indifferent to the original consumption point? Call
this change A502. Ask yourself the question “For a given change in good
1, how does good 2 have to change to make the consumer just indifferent
between (331 + A$1,$2 + A332) and ($1,552)?” Once you have determined
this movement at one consumption bundle you have drawn a piece of the
indifference curve. Now try it at another bundle, and so on, until you
develop a clear picture of the overall shape of the indifference curves. Perfect Substitutes Two goods are perfect substitutes if the consumer is Willing to substitute
one good for the other at a constant rate. The simplest case of perfect
substitutes occurs when the consumer is willing to substitute the goods on
a onetoone basis. Suppose, for example, that we are considering a choice between red pen
cils and blue pencils, and the consumer involved likes pencils, but doesn’t
care about color at all. Pick a consumption bundle, say (10,10). Then for
this consumer, any other consumption bundle that has 20 pencils in it is EXAMPLES OF PREFERENCES 39 just as good as (10,10). Mathematically speaking, any consumption bun
dle (331,132) such that $1 + x2 _—_ 20 will be on this consumer’s indifference
curve through (10, 10). Thus the indifference curves for this consumer are
all parallel straight lines with a slope of —1, as depicted in Figure 3.3.
Bundles with more total pencils are preferred to bundles with fewer total
pencils, so the direction of increasing preference is up and to the right, as
illustrated in Figure 3.3. How does this work in terms of general procedure for drawing indifference
curves? If we are at (10, 10), and we increase the amount of the first good
by one unit to 11, how much do we have to change the second good to get
back to the original indifference curve? The answer is clearly that we have
to decrease the second good by 1 unit. Thus the indifference curve through
(10,10) has a slope of —1. The same procedure can be carried out at any
bundle of goods with the same resultskin this case all the indifference
curves have a constant slope of —1. ;._ . 3 j _ " indifferEnce curves . . . .. . . ' 9r,” '
Perfect substitutes The consumer only cares about the total number of pencﬁs not about their colors. This the indifference
curves are straight lines with a siope (cf —1    The important fact about perfect substitutes is that the indifference
curves have a constant slope. Suppose, for example, that we graphed blue
pencils on the vertical axis and pairs of red pencils on the horizontal axis.
The indifference curves for these two goods would have a slope of —2, since
the consumer would be willing to give up two blue pencils to get one more
pair of red pencils. 40 PREFERENCES (Ch. 3) In the textbook we’ll primarily consider the case where goods are perfect
substitutes on a onefor—one basis, and leave the treatment of the general
case for the workbook. Perfect Complements Perfect complements are goods that are always consumed together in
ﬁxed proportions. In some sense the goods “complement” each other. A
nice example is that of right shoes and left shoes. The consumer likes shoes,
but always wears right and left shoes together. Having only one out of a
pair of shoes doesn’t do the consumer a bit of good. Let us draw the indifference curves for perfect complements. Suppose
we pick the consumption bundle (10,10). Now add 1 more right shoe, so
we have (11,10). By assumption this leaves the consumer indifferent to
the original position: the extra shoe doesn’t do him any good. The same
thing happens if we add one more left Shoe: the consumer is also indifferent
between (10,11) and (10,10). Thus the indifference curves are Lshaped, with the vertex of the L oc
curring where the number of left shoes equals the number of right shoes as
in Figure 3.4. LEFT SHOES . . indifference
 curves RIGHT SHOES . Perfect complements. . The consumer always wants to con—
same the goods. in ﬁxed proportibnsi to each other. Thus the
. indifference curves are LrShaPGd. . . .  EXAMPLES OF PREFERENCES 41 Increasing both the number of left shoes and the number of right shoes
at the same time will move the consumer to a more preferred position,
so the direction of increasing preference is again up and to the right, as
illustrated in the diagram. The important thing about perfect complements is that the consumer
prefers to consume the goods in ﬁxed proportions, not necessarily that
the proportion is onetoone. If a consumer always uses two teaspoons of
sugar in her cup of tea, and doesn’t use sugar for anything else, then the
indifference curves will still be L—shaped. In this case the corners of the
L will occur at (2 teaspoons sugar, 1 cup tea), (4 teaspoons sugar, 2 cups
tea) and so on, rather than at (1 right shoe, 1 left shoe), (2 right shoes, 2
left shoes), and so on. In the textbook we’ll primarily consider the case where the goods are
conSumed in proportions of one—for—one and leave the treatment of the
general case for the workbook. Bads A bad is a commodity that the consumer doesn’t like. For example, sup—
pose that the commodities in question are now pepperoni and anchovies—
and the consumer loves pepperoni but dislikes anchovies. But let us suppose
there is some possible tradeoff between pepperoni and anchovies. That is,
there would be some amount of pepperoni on a pizza that would compen
sate the consumer for having to consume a given amount of anchovies. How
could we represent these preferences using indifference curves? Pick a bundle ($1,332) consisting of some pepperoni and some anchovies.
If we give the consumer more anchovies, what do we have to do with the
pepperoni to keep him on the same indifference curve? Clearly, we have
to give him some extra pepperoni to compensate him for having to put up
with the anchovies. Thus this consumer must have indifference curves that
slope up and to the right as depicted in Figure 3.5. The direction of increasing preference is down and to the rightﬁthat
is, toward the direction of decreased anchovy consumption and increased
pepperoni consumption, just as the arrows in the diagram illustrate. Neutrals A good is a neutral good if the consumer doesn’t care about it one way
or the other. What if a consumer is just neutral about anchovies?1 In this
case his indifference curves will be vertical lines as depicted in Figure 3.6. 1 Is anybody neutral about anchovies? 42 PREFERENCES (Ch. 3) mm AN‘CHOVIES ' indifference .
curves ' PEPPERONI Beds. Here anchovies are a “bad,” and pepperoni is a “good”
for this consumer. Thus the indifference curves have a positive
slope. ANCHOVIES Indifference
curves ' PEPPERONI A neutral good. The consumerlikespepperoni but 'isneutral
about anchovies, so the indifference Curves are vertical lines. He only cares about the amount of pepperoni he has and doesn’t care at
all about how many anchovies he has. The more pepperoni the better, but
adding more anchovies doesn’t affect him one way or the other. EXAMPLES OF PREFERENCES 43 Satiation We sometimes want to consider a situation involving satiation, where
there is some overall best bundle for the consumer, and the “closer” he is
to that best bundle, the better off he is in terms of his own preferences.
For example, suppose that the consumer has some most preferred bundle
of goods (Eljg), and the farther away he is from that bundle, the worse
off he is. In this case we say that ($1,252) is a satiation point, or a bliss
point. The indifference curves for the c0nsumer look like those depicted in
Figure 3.7. The best point is (5132) and points farther away from this
bliss point lie on “lower” indifference curves. Indifference
curves Satiation
point 21 I xi Satiated preferences. The bundle (351,352) is the satiation
point or bliss. point, and the indiﬁerence curves surround this
point. ' ' ' ' W.“ In this case the indifference curves have a negative slope when the con
sumer has “too little” or “too much” of both goods, and a positive slope
when he has “too much” of one of the goods. When he has too much of one
of the goods, it becomes a badﬁreducing the consumption of the bad good
moves him closer to his “bliss point.” If he has too much of both goods,
they both are bads, so reducing the consumption of each moves him closer
to the bliss point. ...
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 Fall '09
 Franchsica
 indifference curves, Discrete Goods

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