# Microeconomics_4__Besanko

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In reality this assumption is not always true. Let\u2019s return to the example of con-
suming hamburgers. Sarah may find that her total utility increases as she eats the first,
second, and third hamburgers each week. For these hamburgers, her marginal utility
B¿
¢U\ufffd¢y \ufffd 0.25
¢U\ufffd¢y
¢U\ufffd¢y
principle of diminish-
ing marginal utility
The principle that after
some point, as consump-
tion of a good increases,
the marginal utility of that
good will begin to fall.
c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 79
80 CHAPTER 3 CONSUMER PREFERENCES AND THE CONCEPT OF UTIL ITY
is positive, even though it may be diminishing with each additional hamburger she
eats. But presumably at some point she will find that an additional hamburger will
bring her no more satisfaction. For example, she might find that the marginal utility
of the seventh hamburger per week is zero, and the marginal utility of the eighth or
ninth hamburgers might even be negative.
Figure 3.3 depicts the total and marginal utility curves for this case. Initially (for
values of y \ufffd 7 hamburgers), total utility rises as consumption increases, and the slope
of the utility curve is positive (e.g., note that the segment RS, which is tangent to the
utility curve at point A when Sarah is purchasing her second hamburger, has a posi-
tive slope); thus, the marginal utility is positive (as depicted at point A\ufffd). However, the
marginal utility is diminishing as consumption increases, and at a consumption level
of seven hamburgers, Sarah has purchased so much of the good that the marginal util-
ity is zero (point B\ufffd). Since the marginal utility is zero, the slope of the total utility
curve is zero. (The segment MN, which is tangent to the utility curve at point B, has a
slope of zero.) If Sarah were to buy more than seven hamburgers, her total satisfaction
U
(y)
,

to
ta
l u
til
ity
o
f h
am
bu
rg
er
s
y, weekly consumption of hamburgers
0 2 7 9
S
U = ( y )
R A
B
M
C
N K
L
M
U
y,

m
a
rg
in
al
u
tili
ty
o
f h
am
bu
rg
er
s
y, weekly consumption of hamburgers
0 2 7 9
MUy
A'
B'
C'
(a)
(b)
FIGURE 3.3 Marginal Utility May Be Negative
The utility curve U(y) is shown in panel (a), and the
corresponding marginal utility curve is illustrated in
panel (b). The slope of the utility curve in the top panel
is positive at A; thus, the marginal utility is positive, as
indicated at point A\ufffd in panel (b). At point B the slope
of the utility curve is zero, meaning that the marginal
utility is zero, as shown at point B\ufffd. At point C the
slope of the utility function is negative; therefore, the
marginal utility is negative, as indicated at point C \ufffd.
c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 80
3.2 UTILITY FUNCTIONS 81
would decline (e.g., the slope of the total utility curve at point C is negative, and thus
the marginal utility is negative, as indicated at point C\ufffd).
Although more may not always be better, it is nevertheless reasonable to assume
that more is better for amounts of a good that a consumer might actually purchase.
For example, in Figure 3.3 we would normally only need to draw the utility function
for the first seven hamburgers. The consumer would never consider buying more than
seven hamburgers because it would make no sense for her to spend money on ham-
burgers that reduce her satisfaction.
PREFERENCES WITH MULTIPLE GOODS:
MARGINAL UTILITY, INDIFFERENCE CURVES,
AND THE MARGINAL RATE OF SUBSTITUTION
Let\u2019s look at how the concepts of total utility and marginal utility might apply to a
more realistic scenario. In real life, consumers can choose among myriad goods and
services. To study the trade-offs a consumer must make in choosing his optimal bas-
ket, we must examine the nature of consumer utility with multiple products.
We can illustrate many of the most important aspects of consumer choice
among multiple products with a relatively simple scenario in which a consumer,
Brandon, must decide how much food and how much clothing to purchase in a
given month. Let x measure the number of units of food and y measure the number
of units of clothing purchased each month. Further, suppose that Brandon\u2019s utility
for any basket (x, y) is measured by A graph of this consumer\u2019s utility
function is shown in Figure 3.4. Because we now have two goods, a graph of
Brandon\u2019s utility function must have three axes. In Figure 3.4 the number of units
of food consumed, x, is shown on the right axis, and the number of units of cloth-
ing consumed, y, is represented on the left axis. The vertical axis measures
Brandon\u2019s level of satisfaction from purchasing any basket of goods. For example,
U \ufffd 1xy.
FIGURE 3.4 Graph of the
Utility Function U \ufffd
The level of utility is shown on the
vertical axis, and the amounts of
food (x) and clothing (y) are shown,
respectively, on the right and left
axes. Contours representing lines of
constant utility are also shown. For
example, the consumer is indifferent
between baskets A, B, and C
because they all yield the same level
of utility (U \ufffd 4).
1xyA
B C
U = 2
2
2
2
4
4
4
6
6
6
8
8
8
10
10
10
12
12
12
0
U = 10
U = 8
U = 6
U = 4
x, units of fo
od
y
, units of clothing
U
,

le
ve
l o
f u
tili
ty
c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 81
82 CHAPTER 3 CONSUMER PREFERENCES AND THE CONCEPT OF UTIL ITY
basket A contains two units of food (x \ufffd 2) and eight units of clothing ( y \ufffd 8).
Thus, Brandon realizes a level of utility of with basket A. As the
graph indicates, Brandon can achieve the same level of utility by choosing other bas-
The concept of marginal utility is easily extended to the case of multiple goods.
The marginal utility of any one good is the rate at which total utility changes as the
level of consumption of that good rises, holding constant the levels of consumption of all
other goods. For example, in the case in which only two goods are consumed and the
utility function is U(x, y), the marginal utility of food (MUx) measures how the level
of satisfaction will change (\ufffdU ) in response to a change in the consumption of food
(\ufffdx), holding the level of y constant:
(3.2)
Similarly, the marginal utility of clothing (MUy) measures how the level of satis-
faction will change (\ufffdU ) in response to a small change in the consumption of clothing
(\ufffdy), holding constant the level of food (x):
(3.3)
One could use equations (3.2) and (3.3) to derive the algebraic expressions for MUx and
MUy from U(x, y).3 When the total utility from consuming a bundle (x, y) is
the marginal utilities are and So, at basket A (with
x \ufffd 2 and y \ufffd 8), and
Learning-By-Doing Exercise 3.1 shows that the utility function satisfies
the assumptions that more is better and that marginal utilities are diminishing. Because
these are widely regarded as reasonable characteristics of consumer preferences, we will
often use this utility function to illustrate concepts in the theory of consumer choice.
U \ufffd 1xy
MUy \ufffd 12\ufffd(218) \ufffd 1\ufffd4.MUx \ufffd 18\ufffd(212) \ufffd 1
MUy \ufffd 1x\ufffd(21y).MUx \ufffd 1y\ufffd(21x)
U \ufffd 1xy,
MUy \ufffd
¢U
¢y
2
x is held constant
MUx \ufffd
¢U
¢x
2
y is held constant
U \ufffd 1(2)(8) \ufffd 4
3Learning-By-Doing Exercise A.7 in the Mathematical Appendix shows how to derive the equations of
MUx and MUy in this case.
Let\u2019s look at a utility function that satisfies
the assumptions that more is better and that marginal
utilities are diminishing. Suppose a consumer\u2019s prefer-
ences between food and clothing can be represented by
the utility function where x measures the
number of units of food and y the number of units of
clothing, and the marginal utilities for x and y