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Unformatted text preview: CHAPTER 6 DEMAND [n the last chapter we presented the basic model of consumer choice: how
maximizing utility subject to a budget constraint yields optimal choices.
We saw that the optimal choices of the consumer depend on the consumer’s
income and the prices of the goods, and we worked a few examples to see
what the optimal choices are for some simple kinds of preferences. The consumer’s demand functions give the optimal amounts of each
of the goods as a function of the prices and income faced by the consumer.
We write the demand functions as 3:1 = w1(p1,p2,m) 532 = $2(P1,P2,m) The left—hand side of each equation stands for the quantity demanded. The
righthand side of each equation is the function that relates the prices and
income to that quantity. In this chapter we will examine how the demand for a good changes as
prices and income change. Studying how a choice responds to changes in the
economic environment is known as comparative statics, which we ﬁrst
described in Chapter 1. “Comparative” means that We want to compare 96 DEMAND (Ch. 6) two situations: before and after the change in the. economic eiivironment.
“Statics” means that we are not concerned with any adjustment process
that may be involved in moving from one choice to another; rather we will
only examine the equilibrium choice. In the case of the consumer, there are only two things in our model
that affect the optimal choice: prices and income. The comparative statics
questions in consumer theory therefore involve investigating how demand
changes when prices and income change. 6.1 Normal and Inferior Goods We start by considering how a consumer’s demand for a good changes
as his income changes. We want to know how the optimal choice at one
income compares to the optimal choice at another level of income. During
this exercise, we will hold the prices ﬁxed and examine only the change in
demand due to the income change. We know how an increase in money income affects the budget line when
prices are ﬁxedriit shifts it outward in a parallel fashion. So how does this
affect demand? We would normally think that the demand for each good would increase
when income increases, as shown in Figure 6.1. Economists, with a singular
lack of imagination, call such goods normal goods. If good 1 is a normal
good, then the demand for it increases when income increases, and de—
creases when income decreases. For a normal good the quantity demanded
always changes in the same way as income changes: If something is called normal, you can be sure that there must be a
possibility of being abnormal. And indeed there is. Figure 6.2 presents
an example of nice, well—behaved indifference curves where an increase of
income results in a reduction in the consumption of one of the goods. Such
a good is called an inferior good. This may be “abnormal,” but when
you think about it, inferior goods aren’t all that unusual. There are many
goods for which demand decreases as income increases; examples might
include gruel, bologna, shacks, or nearly any kind of lowquality good. Whether a good is inferior or not depends on the income level that we
are examining. It might very well be that very poor people consume more
bologna as their income increases. But after a point, the consumption of
bologna would probably decline as income continued to increase. Since in
real life the consumption of goods can increase or decrease when income
increases, it is comforting to know that economic theory allows for both
possibilities. INCOME OFFER CURVES AND ENGEL CURVES 97 Indifference
' curves X1" Normal goods. The demand: for beth'geods increasesWhen
income increases, so. both goods are normal goods. _ ' 62 Income Offer Curves and Engel Curves We have seen that an increase in income corresponds to shifting the budget
line outward in a parallel manner. We can connect together the demanded
bundles that we get as we shift the budget line outward to construct the
income offer curve. This curve illustrates the bundles of goods that are
demanded at the different levels of income, as depicted in Figure 6.3A.
The income offer curve is also known as the income expansion path. If
both goods are normal goods, then the income expansion path will have a
positive slope, as depicted in Figure 6.3A. For each level of income, m, there will be some optimal choice for each
of the goods. Let us focus on good 1 and consider the optimal choice at
each set of prices and income, 1:1(p1,p2,m). This is simply the demand
function for good 1. If we hold the prices of goods 1 and 2 ﬁxed and look
at how demand changes as we change income, we generate a curve known
as the Engel curve. The Engel curve is a graph of the demand for one of
the goods as a function of income, with all prices being held constant. For
an example of an Engel curve, see Figure 6.38. 98 DEMAND (Ch‘ 6) lndifférence
”curves ' Optimal _
_ choices An inferior go'od. Good 1 is an inferior good, which means
thatthe demand for it decreases thn' incume increases. indifference
' curvcs  IA Income offercurve I I . B Engelcurve How demand changes as {incomé Changes. ”The income of
fer curve (or income expansion path) shbwn in panel A depicts ' the Optimal chcice at differéxit levels of" income and constant
prices. When we plot the optimal _ch0icé_0f'g00d 1 against in
come, m, we get the Engél curve"? "depicted in panel B. SOME EXAMPLES 99 6.3 Some Examples Let’s consider some of the preferences that we examined in Chapter 5 and
see what their income oﬁer curves and Engel curves look like. Perfect Substitutes The case of perfect substitutes is depicted in Figure 6.4. If 331 < p2, so
that the consumer is specializing in consuming good 1, then if his income
increases he will increase his consumption of good 1. Thus the income offer
curve is the horizontal axis, as shown in Figure 6.4A. ‘ indifference
curves ' Typical
budget
line x I x
1 1
A Income offer curve B Engel curve Perfect substitutes. The income offer curve (A) and an Engel
curve (B) in'the case of perfect substitutes. Since the demand for good 1 is 3:1 : m/p1 in this case, the Engel curve
will be a straight line with a slope of p1, as depicted in Figure 6.4B. (Since
m is on the vertical axis, and 331 on the horizontal axis, we can write
m = 301331, which makes it clear that the slope is p1.) Perfect Complements The demand behavior for perfect complements is shown in Figure 6.5. Since
the consumer will always consume the same amount of each good, no matter 100 DEMAND {(3116} what. the income offer curve is the diagonal line through the origin as
depicted in Figure 6.52%. We have seen that the demand for good '1 is
.1‘1 2 m/(m + 112), so the Engel culve is a straight line with a slope of
pl +11; as shown in Figuze 6 5B Figure
6.5 ' obb—Douglas Prefe'renees For the case of CobbDouglas preferences it is easier to look 11116 aigeba'alc
form of the demand functions to see what the graphs 111111001: like. .If
11(531 ,xg) — 3:13:2_“ the CobbDouglas demand for £11061 1 has the form
.11— “ (mi/pl. For a ﬁxed value of 1);, this” 13 a linear function of m. Thus
doubling m. will doub1e demand tripling 112 will triple demand, and so 011.
In fact multiplying m. by any positive number t will just multiply demand
bv the same amount. T he demand for good 2 13 1:2— .— (1 mahn /p2. and this IS 3.130 cleaxly 1inear.
The fact that the demand functioue for both goods are;l'1ne_ar functions
of income means that the income expansion paths will. be _straight lines
through the origin, as (lepicted in Figure 6.6151. The Engel'ejurve for good 1
will be a straight line with a slope of 3113/11.. as depicted inFigure 6.613. SOME EXAMPLES 101 Figure
6.6 is a necessary sued
_._the ease where the dem6nd for :8. good goes up by we emined above What aspect of the Consumer’ s preferences leads to
this behavior? "'  ' Suppose that the" consumer s preferences only depend on the ratio of
good 1 to good 2; ..This means that if the consumer prefers ($1 2’2} to
(33,332), their sheautomatically prefers (23:1, 2:62) to (2341, 292), (Sm1,332)
to (33,11,339), and soon, since the ratio of good 1 to good 2 is the same for
all 13f these bundles, In fact, the consumer prefers (ta:1 , tag) to (tyb tyg) for
any positive velueof t. Preferences that have this property are known as
homothetie preferences. It is not hard to show that the three examples
of preferencesgiven above—perfect substitutes, perfect complements, and
CobbLDouglasésle' 6H homothetie preferences. 102 DEMAND (Ch. 6) If the consumer has homothetic preferences, then the income offer curves
are all straight lines through the origin, as shown in Figure 6.7. More
speciﬁcally, if preferences are homothetic, it means that when income is
scaled up or down by any amount it > 0, the demanded bundle scales up
or down by the same amount. This can be established rigorously, but it is
fairly clear from looking at the picture. If the indifference curve is tangent
to the budget line at (sf, 233), then the indifference curve through (tmi‘, 25.273)
is tangent to the budget line that has 15 times as much income and the same
prices. This implies that the Engel curves are straight lines as well. If you
double income, you just double the demand for each good. X2 . . ' m
,_ indifference
curves _
Engel
curve
Budget
lines "'
Income
offer curve
. . "1 X:
A Income offer curve . B Engel curve Homothetic preferences. An income offer curve (A) and an
Engel curve (B) inthe case .of homothetic preferences. Homothetic preferences are very convenient since the income effects are
so simple. Unfortunately, homothetic preferences aren’t very realistic for
the same reason! But they will often be of use in our examples. Quasilinear Preferences Another kind of preferences that generates a special form of income offer
curves and Engel curves is the case of quasilinear preferences. Recall the
deﬁnition of quasilinear preferences given in Chapter 4. This is the case
where all indifference curves are “shifted” versions of one indifference curve SOME EXAMPLES 103 as in Figure 6.8. Equivalently, the utility function for these preferences
takes the form u(;t1,:1:2) : 12(181) +Cv2. What happens if we shift the budget
line outward? In this case, if an indifference curve is tangent to the budget
line at a bundle (171%), then another indifference curve must also be
tangent at (.r’f , Lt§+k) for any constant k. Increasing income doesn’t change
the demand for good 1 at all, and all the extra income goes entirely to the
consumption of good 2. If preferences are quasilinear, we sometimes say
that there is a “zero income effect” for good 1. Thus the Engel curve for
good 1 is a vertical linefas you change income, the demand for good 1
remains constant. lndifférence  _
_ curves X
1
A Income offer curve ' _ . B' Engeicu'rve I. " Quasilinear preferences. An income oﬂ‘er curve (A) and an
Engel curve (B) with quasilinear preferences. _ What would be a reallife situation where this kind of thing might occur?
Suppose good 1 is pencils and good 2 is money to spend on other goods.
Initially I may spend my income only on pencils, but when my income
gets large enough7 I stop buying additional pencilseeall of my extra income
is spent on other goods. Other examples of this sort might be salt or
toothpaste. When we are examining a choice between all other goods and
some single good that isn’t a very large part of the consumer’s budget, the
quasilinear assumption may well be plausible, at least when the consumer’s
income is sufﬁciently large. 104 DEMAND (Ch. 6) 6.4 Ordinary Goods and Giffen Goods Let us now consider price changes. Suppose that we decrease the price of
good 1 and hold the price of good 2 and money income ﬁxed. Then what
can happen to the quantity demanded of good 1? Intuition tells us that
the quantity demanded of good 1 should increase when its price decreases.
Indeed this is the ordinary case, as depicted in Figure 6.9. I indifference .
II.CUI'V€5 '_ ' Optimal
choices _‘ Price.
"w decrease X1 An ordinary good. _' Ordinarily; thedemand for a good in
creases when itsprice'decreases, as is thecase here. When the price of good 1 decreases, the budget line becomes ﬂatter. Or
said another way, the vertical intercept is ﬁxed and the horizorltal intercept
moves to the right. In Figure 6.9, the optimal choice of good 1 moves to
the right as well: the quantity demanded of good 1 has increased. But we
might wonder whether this always happens this way. Is it always the case
that, no matter what kind of preferences the consumer has, the demand
for a good must increase when its price goes down? As it turns out, the answer is no. It is logically possible to ﬁnd well
behaved preferences for which a decrease in the price of good 1 leads to a
reduction in the demand for good 1. Such a good is called a Giﬁ'en good, ORDINARY GOODS AND GIFFEN GOODS 105 .‘ . Indifference
"2.. curves Optimal
choices decrease “Reduction X, in demand
for good 1 A Giﬁ‘en good. Good 1 is a Giﬁen good, since the demand
for it decreases when its price decreases. WM after the nineteenth—century economist who ﬁrst noted the possibility. An
example is illustrated in Figure 6.10. What is going on here in economic terms? What kind of preferences
might give rise to the peculiar behavior depicted in Figure 6.10? Suppose
that the two goods that you are consuming are gruel and milk and that
you are currently consuming 7 bowls of gruel and 7 cups of milk a week.
Now the price of gruel declines. If you consume the same 7 bowls of gruel
a week, you will have money left over with which you can purchase more
milk. In fact, with the extra money you have saved because of the lower
price of gruel, you may decide to consume even more milk and reduce your
consumption of gruel. The reduction in the price of gruel has freed up some
extra money to be spent on other thingSabut one thing you might want to
do with it is reduce your consumption of gruel! Thus the price change is to
some extent like an income change. Even though money income remains
constant, a change in the price of a good will change purchasing power,
and thereby change demand. So the Giffen good is not implausible purely on logical grounds, although
Giﬁen goods are unlikely to be encountered in realworld behavior. Most
g00ds are ordinary goods—when their price increases, the demand for them
declines. We’ll see why this is the ordinary situation a little later. 106 DEMAND (Ch. 6) Incidentally, it is no accident that we used gruel as an example of both
an inferior good and a Giffen good. It turns out that there is an intimate
relationship between the two which we will explore in a later chapter. But for now our exploration of consumer theory may leave you with
the impression that nearly anything can happen: if income increases the
demand for a good can go up or down, and if price increases the demand can
go up or down. Is consumer theory compatible with any kind of behavior?
Or are there some kinds of behavior that the economic model of consumer
behavior rules out? It turns out that there are restrictions on behavior
imposed by the maximizing model. But we’ll have to wait until the next
chapter to see what they are. 6.5 The Price Offer Curve and the Demand Curve Suppose that we let the price of good 1 change while we hold p2 and income
ﬁxed. Geometrically this involves pivoting the budget line. We can think of
connecting together the optimal points to construct the price oﬂ'er curve
as illustrated in Figure 6.11A. This curve represents the bundles that would
be demanded at different prices for good 1. ' . Indifference
curves .
 . 'Pnce
offer A Price offer curve I . . ' '_ B Demand curve The price oﬂ‘er curve "and. demand curve. Panel A contains
a prise offer curVe which depicts the optimal choices as the price
' of good 1 changes. Panel B centains the. associated demand
I curve, which depicts a plot of the optimal chaise of good 1 as a
' function of its price. _ SOME EXAMPLES 107 We can depict this same information in a different way. Again, hold
the price of good 2 and money income ﬁxed, and for each different value
of p1 plot the optimal level of consumption of good 1. The result is the
demand curve depicted in Figure 6.11B. The demand curve is a plot
of the demand function, x1(p1,p2,m), holding p2 and m ﬁxed at some
predetermined values. Ordinarily, when the price of a good increases, the demand for that
good will decrease. Thus the price and quantity of a good will move in
opposite directions, which means that the demand curve will typically have
a negative slope. In terms of rates of change, we would normally have A
£1. < 0,
13191 which simply says that demand curves usually have a negative slope. However, we have also seen that in the case of Giffen goods, the demand
for a good may decrease when its price decreases. Thus it is possible, but
not likely, to have a demand curve with a positive slope. 6.6 Some Examples Let’s look at a few examples of demand curves, using the preferences that
we discussed in Chapter 3. Perfect Substitutes The offer curve and demand curve for perfect substituteSrthe red and blue
pencils exampleﬂare illustrated in Figure 6.12. As we saw in Chapter 5,
the demand for good 1 is zero when p1 > 192, any amount on the budget
line when p1 2 p2, and m/p1 when p1 < 192. The offer curve traces out
these possibilities. In order to find the demand curve, we fix the price of good 2 at some
price pg and graph the demand for good 1 versus the price of good 1 to get
the shape depicted in Figure 6.12B. Perfect Complements The case of perfect complements—the right and left shoes exampleris
depicted in Figure 6.13. We know that whatever the prices are, a consumer
will demand the same amount of goods 1 and 2. Thus his offer curve will
be a diagonal line as depicted in Figure 6.13A. We saw in Chapter 5 that the demand for good 1 is given by
m 101 + P2 ‘
If we ﬁx m and p2 and plot the relationship between $1 and p1, we get the
curve depicted in Figure 6.13B. 331: 108 DEMAND (Ch. 6) Indifference
curves X1 "VPI = m/P'z' X1 A Price offer curve 8 Demand curve Perfect substitutes. Price offer curve (A) and demand curve
(B) in the case of perfect substitutes. x . p
2 indifference Pr“ e 1
curves ofigr
Demand
curve
X1
A Price offer curve B Demand curve Perfect complements. Price offer curve (A) and demand
curve (B) in the case of perfect compiements. A Discrete Good Suppose that good 1 is a discrete good. If p1 is very high then the consumer
will strictly prefer to consume zero units; if p1 is low enough the consumer
will strictly prefer to consume one unit. At some price 7'1, the consumer will
be indifferent between consuming good 1 or not consuming it. The price SOME EXAMPLES 109 at which the consumer is just indifferent to consuming or not consuming
1 the good is called the reservation price. The indifference curves and demand curve are depicted in Figure 6.14. Optimal
_ bundles
 at 1'2 Slope = —r2 I Optimal
'bundies
arr1
31:00:: _ '__1' 2"oo‘oo
A Optimal bundles at differentprices B Demand curve A discrete good. As the price. of. good 1 decreases there will_
be some price, the reservation price, at which the consumer is
just indifferent between consuming good 1 or not consuming it.
As the price decreases further, more units of the discrete good
will be demanded. M It is clear from the diagram that the demand behavior can be described
by a sequence of reservation prices at which the consumer is just willing
to purchase another unit of the good. At a price of T1 the consumer is
willing to buy 1 unit of the good; if the price falls to T2, he is willing to
buy another unit, and so on. These prices can be described in terms of the original utility function.
For example, T1 is the price where the consumer is just indifferent between
consuming 0 or 1 unit of good 1, so it must satisfy the equation u(0, m) : u(1, m — T1). (6.1)
Similarly r2 satisﬁes the equation u(1,m — 7'2) 2 u(2, m — 27'2). (6.2) 1 The term reservation price comes from auction markets When someone wanted to
sell something in an auction he would typically state a minimum price at which he
was willing to sell the good. If the best price offered was below this stated price, the
seller reserved the right to purchase the item himself. This price became known as
the seller’s reservation price and eventually came to be used to describe the price at
which someone was just willing to buy or sell some item. 110 DEMAND (Ch. 6) The lefthand side of this equation is the utility from consuming one unit of
the good at a price of r;;. The right—hand side is the utility from consuming
two units of the good, each of which sells for m. If the utility function is quasilinear, then the formulas describing the
reservation prices become somewhat simpler. If u(5c1,a:2) = v(a:1) + 3:2,
and 11(0) = 0, then we can write equation (6.1) as v(0)+m=m=v(1)+m—r1.
Since 11(0) 2 0, we can solve for 7‘1 to ﬁnd
r1 2 11(1). (6.3)
Similarly, we can write equation (6.2) as
1)(1) + m _.. 1‘2 = 11(2) + m  2T2.
Canceling terms and rearranging, this expression becomes
T2 2 11(2) — 12(1). Proceeding in this manner, the reservation price for the third unit of con
sumption is given by
1‘3 = 12(3) — 11(2) and so on. In each case, the reservation price measures the increment in utility nec
essary to induce the consumer to choose an additional unit of the good.
Loosely speaking, the reservation prices measure the marginal utilities as
sociated with different levels of consumption of good 1. Our assumption
of convex preferences implies that the sequence of reservation prices must
decrease: M > T2 > 13   . Because of the special structure of the quasilinear utility function, the
reservation prices do not depend on the amount of good 2 that the consumer
has. This is certainly a special case, but it makes it very easy to describe
demand behavior. Given any price p, we just ﬁnd where it falls in the list
of reservation prices. Suppose that 1) falls between 1‘6 and T7, for example.
The fact that 1’6 > p means that the consumer is willing to give up p dollars
per unit bought to get 6 units of good 1, and the fact that p > 117 means
that the consumer is not willing to give up p dollars per unit to get the
seventh unit of good 1. This argument is quite intuitive, but let’s look at the math just to make
sure that it is clear. Suppose that the consumer demands 6 units of good 1.
We want to show that we must have 7‘5 2 p 2 1‘7. If the consumer is maximizing utility, then we must have v(6) +m—6p 2 11(1L‘1)+m—pa:1 SUBSTITUTES AND COMPLEMENTS 111 for all possible choices of :51. In particular, we must have that
v(6)+m—6p2v(5)+m—5p.
Rearranging this equation we have
7‘6 = 71(6) ~ 11(5) 2 p, which is half of what we wanted to show.
By the same logic, v(6)+m—6p2v(7)+m—7p.
Rearranging this gives us
17 2 71(7)  0(6) = m, which is the other half of the inequality we wanted to establish. 6.7 Substitutes and Complements We have already used the terms substitutes and complements, but it is now
appropriate to give a formal deﬁnition. Since we have seen perfect substi
tutes and perfect complements several times already, it seems reasonable
to look at the imperfect case. Let’s think about substitutes ﬁrst. We said that red pencils and blue
pencils might be thought of as perfect substitutes, at least for someone who
didn’t care about color. But what about pencils and pens? This is a case
of “imperfect” substitutes. That is, pens and pencils are, to some degree,
a substitute for each other, although they aren’t as perfect a substitute for
each other as red pencils and blue pencils. Similarly, we said that right shoes and left shoes were perfect comple~
ments. But what about a pair of shoes and a pair of socks? Right shoes
and left shoes are nearly always consumed together, and shoes and socks
are usually consumed together. Complementary goods are those like shoes
and socks that tend to be consumed together, albeit not always. Now that we’ve discussed the basic idea of complements and substitutes,
we can give a precise economic deﬁnition. Recall that the demand function
for good 1, say, will typically be a function of the price of both good 1 and
good 2, so we write 271(p1, 192, m). We can ask how the demand for good 1
changes as the price of good 2 changes: does it go up or down? If the demand for good 1 goes up when the price of good 2 goes up, then
we say that good 1 is a substitute for good 2. In terms of rates of change,
good 1 is a substitute for good 2 if Al'l —— > 0.
A192 112 DEMAND (Ch. 6) The idea is that when good 2 gets more expensive the consumer switches to
consuming good 1: the consumer substitutes away from the more expensive
good to the less expensive good. On the other hand, if the demand for good 1 goes down when the price
of good 2 goes up, we say that good 1 is a complement to good 2. This
means that A561 A172 < 0.
Complements are goods that are consumed together, like coffee and sugar,
so when the price of one good rises, the consumption of both goods will
tend to decrease. The cases of perfect substitutes and perfect complements illustrate these
points nicely. Note that A5131 /Apg is positive (or zero) in the case of perfect
substitutes, and that A$1/Ap2 is negative in the case of perfect comple
ments. A couple of warnings are in order about these concepts. First, the two—
good case is rather special when it comes to complements and substitutes.
Since income is being held ﬁxed, if you spend more money on good 1, you’ll
have to spend less on good 2. This puts some restrictions on the kinds of
behavior that are possible. When there are more than two goods, these
restrictions are not so much of a problem. Second, although the deﬁnition of substitutes and complements in terms
of consumer demand behavior seems sensible, there are some difﬁculties
with the deﬁnitions in more general environments. For example, if we use
the above deﬁnitions in a situation involving more than two goods, it is
perfectly possible that good 1 may be a substitute for good 3, but good 3
may be a complement for good 1. Because of this peculiar feature, more
advanced treatments typically use a somewhat different deﬁnition of sub
stitutes and complements. The deﬁnitions given above describe concepts
known as gross substitutes and gross complements; they will be suf—
ﬁcient for our needs. 6.8 The Inverse Demand Function If we hold p2 and m ﬁxed and plot p1 against 3:1 we get the demand
curve. As suggested above, we typically think that the demand curve
slopes downwards, so that higher prices lead to less demand, although the
Giffen example shows that it could be otherwise. As long as we do have a downward—sloping demand curve, as is usual,
it is meaningful to speak of the inverse demand function. The inverse
demand function is the demand function viewing price as a function of
quantity. That is, for each level of demand for good 1, the inverse demand
function measures what the price of good 1 would have to be in order for
the consumer to choose that level of consumption. So the inverse demand THE INVERSE DEMAND FUNCTiON 113 function measures the same relationship as the direct demand function, but
just from another point of view. Figure 6.15 depicts the inverse demand
functi0nor the direct demand function} depending on your point of view. Inverse demand
curve [310(1) Xi
Inverse demand curve. If you view the demand curve as Figure
measuring price as a function of quantity, you have an inverse 6.15 demand function.
W Recall, for example, the CobbDouglas demand for good 1, :51 : am / p1.
We could just as well write the relationship between price and quantity as
p1 : cam/£231. The ﬁrst representation is the direct demand function; the second is the inverse demand function.
The inverse demand function has a useful economic interpretation. Recall that as long as both goods are being consumed in positive amounts, the
optimal choice must satisfy the condition that the absolute value of the MRS equals the price ratio: MRS : 31—.
P2
This says that at the optimal level of demand for good 1, for example, we
must have
P1 :P2lMRSi (64)
Thus, at the optimal level of demand for good 1, the price of good 1
is proportional to the absolute value of the MRS between good 1 and good 2. 114 DEMAND (Ch. 6) Suppose for simplicity that the price of good 2 is one. Then equation
(6.4) tells us that at the optimal level of demand, the price of good 1
measures how much the consumer is willing to give up of good 2 in order
to get a little more of good 1. In this case the inverse demand func—
tion is simply measuring the absolute value of the MRS. For any opti
mal level of 331 the inverse demand function tells how much of good 2
the consumer would want to have to compensate him for a small reduc—
tion in the amount of good 1. Or, turning this around, the inverse de
mand function measures how much the consumer would be willing to sac
rifice of good 2 to make him just indifferent to having a little more of
good 1. If we think of good 2 as being money to spend on other goods, then we
can think of the MRS as being how many dollars the individual would be
willing to give up to have a little more of good 1. We suggested earlier that
in this case, we can think of the MRS as measuring the marginal willingness
to pay. Since the price of good 1 is just the MRS in this case, this means
that the price of good 1 itself is measuring the marginal willingness to
pay. At each quantity 391, the inverse demand function measures how many
dollars the consumer is willing to give up for a little more of good 1; or,
said another way, how many dollars the consumer was willing to give up for
the last unit purchased of good 1. For a small enough amount of good 1,
they come down to the same thing. Looked at in this way, the downward—sloping demand curve has a new
meaning. When 1131 is very small, the consumer is willing to give up a lot of
money—that is, a lot of other goods, to acquire a little bit more of good 1.
As 51:1 is larger, the consumer is willing to give up less money, on the margin,
to acquire a little more of good 1. Thus the marginal willingness to pay,
in the sense of the marginal willingness to sacriﬁce good 2 for good 1, is
decreasing as we increase the consumption of good 1. Summary 1. The consumer’s demand function for a good will in general depend on
the prices of all goods and income. 2. A normal good is one for which the demand increases when income
increases. An inferior good is one for which the demand decreases when
income increases. 3. An ordinary good is one for which the demand decreases when its price
increases. A Giffen good is one for which the demand increases when its
price increases. APPENDIX 115 4. If the demand for good 1 increases when the price of good 2 increases,
then good 1 is a substitute for good 2. If the demand for good 1 decreases
in this situation, then it is a complement for good 2. 5. The inverse demand function measures the price at which a given quan
tity will be demanded. The height of the demand curve at a given level
of consumption measures the marginal willingness to pay for an additional
unit of the good at that consumption level. REVIEW QUESTIONS 1. If the consumer is consuming exactly two goods, and she is always spend—
ing all of her money, can both of them be inferior goods? 2. Show that perfect substitutes are an example of homothetic preferences.
3. Show that Cobb—Douglas preferences are homothetic preferences. 4. The income offer curve is to the Engel curve as the price offer curve is
to . . .7 5. If the preferences are concave will the consumer ever consume both of
the goods together? 6. Are hamburgers and buns complements or substitutes? 7. What is the form of the inverse demand function for good 1 in the case
of perfect complements? 8. True or false? If the demand function is 3:1 2 “p1, then the inverse
demand function is ac = ~1/p1.
APPENDIX If preferences take a special form, this will mean that the demand functions that
come from those preferences will take a special form. In Chapter 4 we described
quasilinear preferences. These preferences involve indifference curves that are all
parallel to one another and can be represented by a utility function of the form u(m1,m2) 2 11(21) + $2.
The maximization problem for a utility function like this is max 11(931) + :32
271,132 116 DEMAND (Ch. 6) st. plscl +p2m2 = m. Solving the budget constraint for :32 as a function of $1 and substituting into the
objective function, we have max 12(951) + m/p2  pin/P2 I:1
Differentiating gives us the ﬁrst—order condition I :0: P1
’U I — —.
( 1) [)2 This demand function has the interesting feature that the demand for good 1
must be independent of income—just as we saw by using indifference curves.
The inverse demand curve is given by 131(501) = v'(m1)p2. That is, the inverse demand function for good 1 is the derivative of the utility
function times p2. Once we have the demand function for good 1, the demand
function for good 2 comes from the budget constraint. For example, let us calculate the demand functions for the utility function u(:r1,m2) =lnx1 + :32.
Applying the ﬁrstorder condition gives 1:131 m1 102’
so the direct demand function for good 1 is x; = ﬂ,
P1 and the inverse demand function is 33 2 ——*.
:01( 1) $1 The direct demand function for good 2 comes from substituting $1 : p2/p1 into the budget constraint: m
IEQ=——1. P2
A warning is in order concerning these demand functions. Note that the de
mand for good 1 is independent of income in this example. This is a general
feature of a quasilinear utility function—the demand for good 1 remains con—
stant as income changes. However, this can only be true for some values of
income. A demand function can’t literally be independent of income for all val—
ues of income; after all, when income is zero, all demands are zero. It turns APPENDIX 117 out that the quasilinear demand function derived above is only relevant when a
positive amount of each good is being consumed. In this example, when m < p2, the optimal consumption of good 2 will be zero.
As income increases the marginal utility of consumption of good 1 decreases.
When m = p2, the marginal utility from spending additional income on good
1 just equals the marginal utility from spending additional income on good 2.
After that point, the consumer spends all additional income on good 2. So a better way to write the demand for good 2 is: _{0 whenmSpQ
m2_ m/p2—1 whenm>p2' For more on the properties of quasilinear demand functions see Hal R. Varian,
Microeconomic Analysis, 3rd ed. (New York: Norton, 1992). ...
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 Fall '09
 Franchsica

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