# Microeconomics_4__Besanko

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are ex-
pressed by the following equations:
and MUy \ufffd 1x\ufffd(21y).
MUx \ufffd 1y\ufffd(21x)
U \ufffd 1xy,
Marginal Utility
E
S
D
L E A R N I N G - B Y- D O I N G E X E R C I S E 3 . 1
Problem
(a) Show that a consumer with this utility function
believes that more is better for each good.
(b) Show that the marginal utility of food is diminishing
and that the marginal utility of clothing is diminishing.
c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 82
3.2 UTILITY FUNCTIONS 83
Learning-By-Doing Exercise 3.2 shows the two ways to determine whether the
marginal utility of a good is positive. First, you can look at the total utility function.
If it increases when more of the good is consumed, marginal utility is positive. Second,
you can look at the marginal utility of the good to see if it is a positive number. When
the marginal utility is a positive number, the total utility will increase when more of
the good is consumed.
Some utility functions satisfy the assump-
tion that more is better, but with a marginal utility that
is not diminishing. Suppose a consumer\u2019s preferences for
hamburgers and root beer can be represented by the
utility function where H measures the
number of hamburgers consumed and R the number of
root beers. The marginal utilities are
Problem
(a) Does the consumer believe that more is better for
each good?
(b) Does the consumer have a diminishing marginal
utility of hamburgers? Is the marginal utility of root beer
diminishing?
MUR \ufffd 1
MUH \ufffd
1
22H
U \ufffd 2H \ufffd R,
Marginal Utility That Is Not Diminishing
E
S
D
L E A R N I N G - B Y- D O I N G E X E R C I S E 3 . 2
Solution
(a) U increases whenever H or R increases, so more
must be better for each good. Also, MUH and MUR are
both positive, again indicating that more is better.
( b) As H increases, MUH falls, so the consumer\u2019s mar-
ginal utility of hamburgers is diminishing. However,
MUR \ufffd 1 (no matter what the value of R), so the con-
sumer has a constant (rather than a diminishing) marginal
utility of root beer (i.e., the consumer\u2019s utility always in-
creases by the same amount when he purchases another
root beer).
Similar Problem: 3.5
Solution
(a) By examining the utility function, we can see that U
increases whenever x or y increases. This means that the
consumer likes more of each good. Note that we can
also see that more is better for each good by looking at
the marginal utilities MUx and MUy, which must always
be positive because the square roots of x and y must
always be positive (all square roots are positive numbers).
This means the consumer\u2019s utility always increases when
he purchases more food and/or clothing.
(b) In both marginal utility functions, as the value of the
denominator increases (holding the numerator con-
stant), the marginal utility diminishes. Thus, MUx and
MUy are both diminishing.
Similar Problem: 3.4
Indifference Curves
To illustrate the trade-offs involved in consumer choice, we can reduce the three-
dimensional graph of Brandon\u2019s utility function in Figure 3.4 to a two-dimensional
graph like the one in Figure 3.5. Both graphs illustrate the same utility function
In Figure 3.5 each curve represents baskets yielding the same level of utility
to Brandon. Each curve is called an indifference curve because Brandon would be
U \ufffd 1xy.
indifference curve A
curve connecting a set of
yield the same level of sat-
isfaction to the consumer.
c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 83
84 CHAPTER 3 CONSUMER PREFERENCES AND THE CONCEPT OF UTIL ITY
equally satisfied with (or indifferent in choosing among) all baskets on that curve. For
example, Brandon would be equally satisfied with baskets A, B, and C because they all
lie on the indifference curve with the value U \ufffd 4. (Compare Figures 3.4 and 3.5 to see
how the indifference curve U \ufffd 4 looks in a three-dimensional and a two-dimensional
graph of the same utility function.) A graph like Figure 3.5 is sometimes referred to
as an indifference map because it shows a set of indifference curves.
Indifference curves on an indifference map have the following four properties.
1. When the consumer likes both goods (i.e., when MUx and MUy are both posi-
tive), all the indifference curves have a negative slope.
2. Indifference curves cannot intersect.
3. Every consumption basket lies on one and only one indifference curve.
4. Indifference curves are not \u201cthick.\u2019\u2019
We will now explore these properties in further detail.
1. When the consumer likes both goods (i.e., when MUx and MUy are both positive), all the
indifference curves will have a negative slope. Consider the graph in Figure 3.6. Suppose
the consumer currently has basket A. Since the consumer has positive marginal utility
for both goods, she will prefer any baskets to the north, east, or northeast of A. We
indicate this in the graph by drawing arrows to indicate preference directions. The
arrow pointing to the east reflects the fact that The arrow pointing to the
north reflects the fact that
Points to the northeast or southwest of A cannot be on the same indifference
curve as A because they will be preferred to A or less preferred than A, respectively.
Thus, points on the same indifference curve as A must lie either to the northwest or
southeast of A. This shows that indifference curves will have a negative slope when
both goods have positive marginal utilities.
MUy 7 0.
MUx 7 0.
U = 2
C
B
A
U = 4
U = 6
U = 8
x, units of food
y,

u
n
its
o
f c
lo
th
in
g
2
4
6
8
10
12
0
2 4 6 8 10 12
FIGURE 3.5 Indifference Curves for
the Utility Function U \ufffd
The utility is the same for all baskets
on a given indifference curve. For
example, the consumer is indifferent
between baskets A, B, and C in the
graph because they all yield the same
level of utility (U \ufffd 4).
1xy
c03consumerpreferencesandtheconceptofutility.qxd 6/14/10 2:54 PM Page 84
3.2 UTILITY FUNCTIONS 85
2. Indifference curves cannot intersect. To understand why, consider Figure 3.7,
which shows two hypothetical indifference curves (with levels of utility U1 and U2)
that cross. The basket represented by point S on U1 is preferred to the basket rep-
resented by point T on U2, as shown by the fact that S lies to the northeast of T;
thus, U1 \ufffd U2. Similarly, the basket represented by point R on U2 is preferred
to the basket represented by point Q on U1 (R lies to the northeast of Q); thus,
U2 \ufffd U1. Obviously, it cannot be true that U1 \ufffd U2 and that U2 \ufffd U1. This
logical inconsistency arises because U1 and U2 cross; therefore, indifference curves
cannot intersect.
3. Every consumption basket lies on one and only one indifference curve. This follows
from the property that indifference curves cannot intersect. In Figure 3.7, the basket
represented by point A lies on the two intersecting indifference curves (U1 and U2);
a point can lie on two curves only at a place where the two curves intersect. Since
indifference curves cannot intersect, every consumption basket must lie on a single
indifference curve.
4. Indifference curves are not \u201cthick.\u201d To see why, consider Figure 3.8, which shows a
thick indifference curve passing through distinct baskets A and B. Since B lies to the
northeast of A, the utility at B must be higher than the utility at A. Therefore, A
and B cannot be on the same indifference curve.
FIGURE 3.6 Slope of Indifference Curves
Suppose that goods x and y are both liked by the consumer
(MUx \ufffd 0 and MUy \ufffd 0, indicating that the consumer
prefers more of y and more of x). Points in the shaded
region to the northeast of A cannot be on the same indif-
ference curve as A since they will be preferred to A. Points
in the shaded region to the southwest of A also cannot