Besanko & Braeutigam – Microeconomics, 4th edition
Solutions Manual
Chapter 3
Preferences and Utility
Solutions to Review Questions
1.
What is a basket (or a bundle) of goods?
A basket is a collection of goods and services that an individual might consume.
2.
What does the assumption that preferences are complete mean about the
consumer’s ability to rank any two baskets?
By requiring preferences to be complete, economists are ensuring that consumers will not
respond indecisively when asked to compare two baskets. A consumer will always be able to
state that either A is preferred B, B is preferred to A, or that she is indifferent between A and B.
3.
Consider Figure 3.1. If the more is better assumption is satisfied, is it possible to say
which of the seven baskets is least preferred by the consumer?
Unfortunately, it is impossible to say definitively whether D, H, or J is the least preferred basket.
Since more is better, baskets to the northeast are more preferred and baskets to the southwest are
less preferred. In this case, H has more clothing but less food than D, while J has more food but
less clothing than D. Without more information regarding how the consumer feels about
clothing relative to food, we cannot state which of these baskets is the least preferred.
4.
Give an example of preferences (i.e., a ranking of baskets) that do not satisfy the
assumption that preferences are transitive.
If a consumer states that A is preferred to B and that B is preferred to C, but then states that C is
preferred to A, she will be violating the assumption of transitivity. The third statement is
inconsistent with the first two.
5.
What does the assumption that more is better imply about the marginal utility of a
good?
If more is better, then the marginal utility of a good must be positive. That is, total utility must
increase if the consumer consumes more of the good.
6.
What is the difference between an ordinal ranking and a cardinal ranking?
An ordinal ranking simply orders the baskets, but does not give any indication as to how much
better one basket is when compared with another; only that one is better. A cardinal ranking not
only orders the baskets, but also provides information regarding the intensity of the preferences.
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For example, a cardinal ranking might indicate that one basket is twice as good as another
basket.
7.
Suppose Debbie purchases only hamburgers. Assume that her marginal utility is
always positive and diminishing. Draw a graph with total utility on the vertical axis and the
number of hamburgers on the horizontal axis. Explain how you would determine marginal
utility at any given point on your graph.
Utility
Slope of this line
measures marginal
utility at this level of
consumption, H’
Total Utility
H’
Hamburgers
Marginal utility would be measured as the slope of a line tangent to the total utility curve in the
graph above.
8.
Why can’t you plot the total utility and marginal utility curves on the same graph?
The two cannot be plotted on the same graph because utility and marginal utility are not
measured in the same dimensions. Total utility has the dimension U , while marginal utility has
the dimension of utility per unit, or U / y where y is the number of units purchased.
9.
Adam consumes two goods: housing and food.
a) Suppose we are given Adam’s marginal utility of housing and his marginal utility of food
at the basket he currently consumes. Can we determine his marginal rate of substitution of
housing for food at that basket?
b) Suppose we are given Adam’s marginal rate of substitution of housing for food at the
basket he currently consumes. Can we determine his marginal utility of housing and his
marginal utility of food at that basket?
a)
Yes, we can determine the MRS as
MU h
MU f
b)
No, when we know the MRS, all we know is the ratio of the marginal utilities. We cannot
“undo” that ratio to determine the individual marginal utilities. For example, if we know that
MRSh,f = 5, it could be the case that MUh = 5 and MUf = 1, but it could equivalently be the case
that MUh = 10 and MUf = 2. Clearly, there are countless combinations of MUh and MUf that
MRS h, f 
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could lead to some particular value of MRSh,f, and we have no way of inferring which is the right
one.
10.
Suppose Michael purchases only two goods, hamburgers (H) and Cokes (C).
a) What is the relationship between MRSH,C and the marginal utilities MUH and MUC ?
b) Draw a typical indifference curve for the case in which the marginal utilities of both
goods are positive and the marginal rate of substitution of hamburgers for Cokes is
diminishing. Using your graph, explain the relationship between the indifference curve and
the marginal rate of substitution of hamburgers for Cokes.
c) Suppose the marginal rate of substitution of hamburgers for Cokes is constant. In this
case, are hamburgers and Cokes perfect substitutes or perfect complements?
d) Suppose that Michael always wants two hamburgers along with every Coke. Draw a
typical indifference curve. In this case, are hamburgers and Cokes perfect substitutes or
perfect complements?
a)
MRS H ,C 
MU H
MU C
b)
C
H
The indifference curve in this case will be convex toward the origin. The marginal rate of
substitution is measured as the absolute value of the slope of a line tangent to the indifference
curve. As can be seen in the graph above, this slope becomes less negative as we move down the
indifference curve, implying a diminishing MRS.
c)
If the MRS was constant, this would imply that at any consumption level the consumer
would be willing to trade a fixed amount of one good for a fixed amount of the other. This
occurs with perfect substitutes.
d)
If the consumer wishes to always consume goods in a fixed ratio, then the goods are
perfect complements. In this case, the indifference curves will be L-shaped.
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C
1
2
H
11.
Suppose a consumer is currently purchasing 47 different goods, one of which is
housing. The quantity of housing is measured by H. Explain why, if you wanted to measure
the consumer’s marginal utility of housing (MUH) at the current basket, the levels of the
other 46 goods consumed would be held fixed.
Marginal utility is defined as the change in total utility relative to a change in consumption for a
particular good. In order to accurately measure the change in total utility, the levels of the other
goods would need to be held constant. If they were not, the change in total utility would occur as
a result of multiple goods changing and it would be impossible to determine what portion of the
change in total utility should be assigned to each good.
Solutions to Problems
3.1
Bill has a utility function over food and gasoline with the equation U = x2y, where x
measures the quantity of food consumed and y measures the quantity of gasoline. Show
that a consumer with this utility function believes that more is better for each good.
By plugging in ever higher numerical values of x and ever higher numerical values of y, it can be
verified that U increases whenever x or y increases.
3.2
Consider the single-good utility function U(x) = 3x2, with a marginal utility given by
MUx = 6x. Plot the utility and marginal utility functions on two separate graphs. Does this
utility function satisfy the principle of diminishing marginal utility? Explain.
The two graphs are shown below. It can be seen from both graphs that this function does not
satisfy the law of diminishing marginal utility. The first figure shows that utility increases with x,
and moreover, that it increases at an increasing rate. For example, an increase in x from 2 to 3,
increases utility from 12 to 27 (an increase of 15), while an increase in x from 3 to 4 induces an
increase in utility from 27 to 48 (an increase of 21).
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This fact is easier to see in the second figure. The marginal utility is an increasing function of x.
Higher values of x imply a greater marginal utility. Therefore this function exhibits increasing
marginal utility.
U(x) = 3x2
MUx = 6x
3.3
Jimmy has the following utility function for hot dogs: U(H) = 10H − H2, with MUH =
10 − 2H
a) Plot the utility and marginal utility functions on two separate graphs.
b) Suppose that Jimmy is allowed to consume as many hot dogs as he likes and that hot
dogs cost him nothing. Show, both algebraically and graphically, the value of H at which he
would stop consuming hot dogs.
The first figure below shows Jimmy’s utility function for hotdogs. You can see that the point at
which H = 5 corresponds to the flat portion of the utility function, i.e. the point at which the
marginal utility of hotdogs is zero, and beyond which the marginal utility is negative.
Alternatively using the second graph it is clear that the point H = 5 is when the marginal utility
intersects the x-axis, and beyond which it is negative. Both graphs tell you that to maximize his
utility Jimmy should only consume 5 hotdogs and not more.
To answer this question algebraically, you should first recognize from the marginal utility
function that Jimmy has a diminishing marginal utility of hotdogs. Therefore the point at which
he should stop consuming hotdogs is the point at which MU H  0, or 10  2H  0. This gives H
= 5.
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U(H) = 10H – H2
MUH = 10 – 2H
3.4
Consider the utility function U(x, y) = y√x with the marginal utilities MUx = y/(2√x)
and MUy = √x.
a) Does the consumer believe that more is better for each good?
b) Do the consumer’s preferences exhibit a diminishing marginal utility of x? Is the
marginal utility of y diminishing?
3.4
a)
Since U increases whenever x or y increases, more of each good is better. This is
also confirmed by noting that MUx and MUy are both positive for any positive values of x and
y.
b)
Since MU x  y 2 x , as x increases (holding y constant), MU x falls. Therefore the
marginal utility of x is diminishing. However, MU y  x . As y increases, MUy does not
change. Therefore the preferences exhibit a constant, not diminishing, marginal utility of y.
3.5
Carlos has a utility function that depends on the number of musicals and the
number of operas seen each month. His utility function is given by U = xy2, where x is the
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number of movies seen per month and y is the number of operas seen per month. The
corresponding marginal utilities are given by: MUx = y2 and MUy = 2xy.
a) Does Carlos believe that more is better for each good?
b) Does Carlos have a diminishing marginal utility for each good?
a)
By plugging in ever higher numerical values of x and ever higher numerical values of y, it
can be verified that Carlos’ utility goes up whenever x or y increases.
b)
First consider the marginal utility of x, MUx. Since x does not appear anywhere in the
formula for MUx, MUx is independent of x. Hence, the marginal utility of movies is independent
of the number of movies seen, and so the marginal utility of movies does not decrease as the
number of movies increases. Next consider the marginal utility of y, MUy. Notice that MUy is an
increasing function of y. Hence, the marginal utility of operas does not decrease in the number of
operas seen. In this case, neither good, movies or operas, exhibits diminishing marginal utility.
3.6
For the following sets of goods draw two indifference curves, U1 and U2, with U2 >
U1. Draw each graph placing the amount of the first good on the horizontal axis.
a) Hot dogs and chili (the consumer likes both and has a diminishing marginal rate of
substitution of hot dogs for chili)
b) Sugar and Sweet’N Low (the consumer likes both and will accept an ounce of Sweet’N
Low or an ounce of sugar with equal satisfaction)
c) Peanut butter and jelly (the consumer likes exactly 2 ounces of peanut butter for every
ounce of jelly)
d) Nuts (which the consumer neither likes nor dislikes) and ice cream (which the consumer
likes)
e) Apples (which the consumer likes) and liver (which the consumer dislikes)
a)
Chili
U2
U1
Hot Dogs
b)
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Sweet’N Low
Slopes = –1
U1
U2
Sugar
c)
Jelly
U2
2
U1
1
2
Peanut
Butter
4
d)
Ice Cream
U2
U1
Nuts
e)
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Liver
U1
U2
Apples
3.7
Alexa likes ice cream, but dislikes yogurt. If you make her eat another gram of
yogurt, she always requires two extra grams of ice cream to maintain a constant level of
satisfaction. On a graph with grams of yogurt on the vertical axis and grams of ice cream
on the horizontal axis, graph some typical indifference curves and show the directions of
increasing utility.
Grams
of
Yogurt
3
U3>U2>U1
U2
U1
U3
Directions of
Increased
Satisfaction
2
1
1
2
3
4
5
6
Grams of
Ice Cream
3.8
Joe has a utility function over hamburgers and hot dogs given by U = x + √y , where
x is the quantity of hamburgers and y is the quantity of hot dogs. The marginal utilities for
this utility function are MUx = 1 and MUy = 1/(2√y ). Does this utility function have the
property that MRSx,y is diminishing?
This utility function does have the property of diminishing MRSx,y. One way to verify this is to
graph several indifference curves. Another way to tell is to use algebra. Recall that
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MRS x , y 
Solutions Manual
MU x
. Applying that general formula to this case gives us MRS x , y  2 y . As we
MU y
move “down” the indifference curve, x increases and y decreases. As y decreases, 2 y
decreases. Thus, MRSx,y decreases.
3.9
Julie and Toni consume two goods with the following utility functions:
UJulie = (x + y)2, MUJuliex = 2(x + y), MUJuliey = 2(x + y)
UToni = x + y, MUTonix = 1, MUToniy = 1
a) Graph an indifference curve for each of these utility functions.
b) Julie and Toni will have the same ordinal ranking of different baskets if, when basket A
is preferred to basket B by one of the functions, it is also preferred by the other. Do Julie
and Toni have the same ordinal ranking of different baskets of x and y? Explain.
Indifference curves corresponding to U = 2 are shown for both Julie and Toni in the graph
below. Notice that the indifference curves are parallel everywhere – indeed, MRSx,y = 1 for both
Julie and Toni, for all values of x and y. Toni’s indifference curve for the utility level UToni = 2 is
the same as Julie’s indifference curve for the utility level UJulie = 4. So whenever Julie ranks
bundle A higher than bundle B, Toni would have the same ranking, and vice-versa. So Julie and
Toni will have the same ordinal ranking of bundles of x and y. (Julie will associate each bundle
with a higher utility level than Toni will, but that is a cardinal ranking.)
UToni = 2
UJulie = 2
3.10 The utility that Julie receives by consuming food F and clothing C is given by U(F,
C) = FC. For this utility function, the marginal utilities are MUF = C and MUC = F.
a) On a graph with F on the horizontal axis and C on the vertical axis, draw indifference
curves for U = 12, U = 18, and U = 24.
b) Do the shapes of these indifference curves suggest that Julie has a diminishing marginal
rate of substitution of food for clothing? Explain.
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c) Using the marginal utilities, show that the MRSF,C = C/F. What is the slope of the
indifference curve U = 12 at the basket with 2 units of food and 6 units of clothing? What is
the slope at the basket with 4 units of food and 3 units of clothing? Do the slopes of the
indifference curves indicate that Julie has a diminishing marginal rate of substitution of
food for clothing? (Make sure your answers to parts (b) and (c) are consistent!)
a)
14
12
Clothing
10
8
6
4
2
0
0
5
10
15
Food
b)
Yes, since the indifference curves are bowed in toward the origin we know that MRSF,C
declines as F increases and C decreases along an indifference curve.
MU F C
MRS F ,C 

c)
MU C F
When F = 2 and C = 6, MRSF,C = 3. The slope of the indifference curve is –3. When F = 4 and
C = 3, MRSF,C = 0.75, so the slope of the indifference curve is –0.75. Since the marginal rate of
substitution falls as F increases and C decreases, she has a diminishing marginal rate of
substitution.
3.11 Sandy consumes only hamburgers (H) and milkshakes (M). At basket A, containing
2 hamburgers and 10 milkshakes, his MRSH,M is 8. At basket B, containing 6 hamburgers
and 4 milkshakes, his MRSH,M is 1/2. Both baskets A and B are on the same indifference
curve. Draw the indifference curve, using information about the MRSH,M to make sure that
the curvature of the indifference curve is accurately depicted.
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Milkshakes
Slope = –8
A
10
Slope = –½
B
4
Hamburgers
2
6
3.12 Adam likes his café latte prepared to contain exactly 1/3 espresso and 2/3 steamed
milk by volume. On a graph with the volume of steamed milk on the horizontal axis and
the volume of espresso on the vertical axis, draw two of his indifference curves, U1 and U2,
with U1 > U2.
Volume
of
Espresso
3
U2
2
U1
1
1
2
3
4
5
6
Volume of
Steamed Milk
3.13 Draw indifference curves to represent the following types of consumer preferences.
a) I like both peanut butter and jelly, and always get the same additional satisfaction from
an ounce of peanut butter as I do from 2 ounces of jelly.
b) I like peanut butter, but neither like nor dislike jelly.
c) I like peanut butter, but dislike jelly.
d) I like peanut butter and jelly, but I only want 2 ounces of peanut butter for every ounce
of jelly.
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In the following pictures, U2 > U1.
a)
Jelly
4
2
U1
U2
1
2
Peanut Butter
b)
Jelly
U1
U2
Peanut Butter
c)
U1
Jelly
U2
Peanut Butter
d)
Jelly
U1
U2
2
1
2
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4
Peanut Butter
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3.14 Dr. Strangetaste buys only food (F) and clothing (C) out of his income. He has
positive marginal utilities for both goods, and his MRSF,C is increasing. Draw two of Dr.
Strangetaste’s indifference curves, U1 and U2, with U2 > U1.
Clothing
U1
U2
Food
The following exercises will give you practice in working with a variety of utility functions and
marginal utilities and will help you understand how to graph indifference curves.
3.15 Consider the utility function U(x, y) = 3x + y, with MUx = 3 and MUy = 1.
a) Is the assumption that more is better satisfied for both goods?
b) Does the marginal utility of x diminish, remain constant, or increase as the consumer
buys more x? Explain.
c) What is MRSx, y?
d) Is MRSx, y diminishing, constant, or increasing as the consumer substitutes x for y along
an indifference curve?
e) On a graph with x on the horizontal axis and y on the vertical axis, draw a typical
indifference curve (it need not be exactly to scale, but it needs to reflect accurately whether
there is a diminishing MRSx, y). Also indicate on your graph whether the indifference curve
will intersect either or both axes. Label the curve U1.
f ) On the same graph draw a second indifference curve U2, with U2 > U1.
a)
Yes, the “more is better” assumption is satisfied for both goods since both marginal
utilities are always positive.
b)
The marginal utility of x remains constant at 3 for all values of x.
MRS x , y  3
c)
d)
The MRS x , y remains constant moving along the indifference curve.
e & f) See figure below
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Y
U1 U2
X
3.16 Answer all parts of Problem 3.15 for the utility function U(x, y) = √xy. The marginal
utilities are MUx = √y/(2√x) and MUy = √x/(2√y).
a)
Yes, the “more is better” assumption is satisfied for both goods since both marginal
utilities are always positive.
b)
The marginal utility of x diminishes as the consumer buys more x .
 y  2 y  y


c)
MRS x , y  
 2 x  x  x



d)
As the consumer substitutes x for y , the MRS x , y will diminish.
e & f) See figure below. The indifference curves will not intersect either axis.
450
400
U2
350
Y
300
250
200
150
100
U1
50
0
0
5
10
15
20
25
30
35
X
3.17 Answer all parts of Problem 3.15 for the utility function U(x, y) = xy + x. The
marginal utilities are MUx = y + 1 and MUy = x.
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Y
a)
Yes, the “more is better” assumption is satisfied for both goods since both marginal
utilities are always positive.
b)
The marginal utility of x remains constant as the consumer buys more x.
y 1
MRS x , y 
c)
x
d)
As the consumer substitutes x for y , the MRS x , y will diminish.
e & f) See figure below. The indifference curves intersect the x-axis, since it is possible that U
> 0 even if y = 0.
20
18
16
U2
14
12
10 U1
8
6
4
2
0
0
5
10
15
20
25
30
35
X
3.18 Answer all parts of Problem 3.15 for the utility function U(x, y) = x0.4y0.6. The
marginal utilities are MUx = 0.4 (y0.6/x0.6) and MUy = 0.6 (x0.4/y0.4).
a)
Yes, the “more is better” assumption is satisfied for both goods since both marginal
utilities are always positive.
b)
The marginal utility of x diminishes as the consumer buys more x .
.4( y 0.6 / x 0.6 ) 0.4 y
MRS x , y 

c)
.6( x 0.4 / y 0.4 ) 0.6 x
d)
As the consumer substitutes x for y , the MRS x , y will diminish.
e & f) See figure below. The indifference curves do not intersect either axis.
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160
140
U2
120
Y
100
80
U1
60
40
20
0
0
5
10
15
20
25
30
35
X
3.19 Answer all parts of Problem 3.15 for the utility function U = √x + 2√y. The marginal
utilities for x and y are, respectively, MUx = 1/(2√x ) and MUy = 1/√y .
a)
Yes, the “more is better” assumption is satisfied for both goods since both marginal
utilities are always positive.
b)
The marginal utility of x diminishes as the consumer buys more x .
y
1 /( 2 x )
MRS x , y 

c)
1/ y
2 x
d)
As the consumer substitutes x for y , the MRS x , y will diminish.
e & f) See figure below. Since it is possible to have U > 0 if either x = 0 (and y > 0) or y = 0
(and x > 0), the indifference curves intersect both axes.
120
100
Y
80
60
U2
40
U1
20
0
0
20
40
60
80
X
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3.20 Answer all parts of Problem 3.15 for the utility function U(x, y) = x2 + y2. The
marginal utilities are MUx = 2x and MUy = 2y.
a)
Yes, the “more is better” assumption is satisfied for both goods since both marginal
utilities are always positive.
b)
The marginal utility of x increases as the consumer buys more x .
2x x
c)
MRS x , y 

2y y
d)
As the consumer substitutes x for y , the MRS x , y will increase.
e & f) See figure below. Since it is possible to have U > 0 if either x = 0 (and y > 0) or y = 0
(and x > 0), the indifference curves intersect both axes.
25
U2
20
Y
15
U1
10
5
0
0
5
10
15
20
25
30
35
X
3.21 Suppose a consumer’s preferences for two goods can be represented by the Cobb–
Douglas utility function U = Axαyβ , where A, α, and β are positive constants. The marginal
utilities are MUx = αAxα−1yβ and MUy = βAxαyβ−1. Answer all parts of Problem 3.15 for this
utility function.
a)
Yes, the “more is better” assumption is satisfied for both goods since both marginal
utilities are always positive.
b)
Since we do not know the value of  , only that it is positive, we need to specify three
possible cases:
When   1 , the marginal utility of x diminishes as x increases.
When   1, the marginal utility of x remains constant as x increases.
When   1, the marginal utility of x increases as x increases.
Ax 1 y  y


Ax y  1 x
c)
MRS x , y
d)
As the consumer substitutes x for y , the MRS x , y will diminish.
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e & f) The graph below depicts indifference curves for the case where A  1 and     0.5.
Thus U ( x, y)  x 0.5 y 0.5 . Regardless, the indifference curves will never intersect either axis.
450
U2
400
350
Y
300
250
200
150
U1
100
50
0
0
5
10
15
20
25
30
35
X
3.22 Suppose a consumer has preferences over two goods that can be represented by the
quasi-linear utility function U(x, y) = 2√x + y. The marginal utilities are MUx = 1/√x and
MUy = 1.
a) Is the assumption that more is better satisfied for both goods?
b) Does the marginal utility of x diminish, remain constant, or increase as the consumer
buys more x? Explain.
c) What is the expression for MRSx,y?
d) Is the MRSx,y diminishing, constant, or increasing as the consumer substitutes more x for
y along an indifference curve?
e) On a graph with x on the horizontal axis and y on the vertical axis, draw a typical
indifference curve (it need not be exactly to scale, but it should accurately reflect whether
there is a diminishing MRSx,y). Indicate on your graph whether the indifference curve will
intersect either or both axes.
f) Show that the slope of every indifference curve will be the same when x = 4. What is the
value of that slope?
a)
Yes, the “more is better” assumption is satisfied for both goods since both marginal
utilities are always positive.
b)
The marginal utility of x increases as the consumer buys more x .
1
x 1
c)
MRS x , y 
x
1
d)
As the consumer substitutes x for y , the MRS x , y will diminish.
e)
Since it is possible to have U > 0 if either x = 0 (and y > 0) or y = 0 (and x > 0), the
indifference curves intersect both axes.
Copyright © 2011 John Wiley & Sons, Inc.
Chapter 3 - 19
Besanko & Braeutigam – Microeconomics, 4th edition
Solutions Manual
25
U2
20
Y
15
U1
10
5
0
0
5
10
15
20
25
30
35
X
f)
The slope of a typical indifference curve at some basket ( x, y ) is the MRS x , y  1
x
. At
x  4 , MRS x, y  1
 0.5 . Note that this holds regardless of the value of y . Therefore, the
4
slope of any indifference curve at x  4 will be 0.5 .
3.23 Daniel and Will each consume two goods. When they consume the same basket,
Daniel’s marginal utility of each good is higher than Will’s. But at any basket they both
have the same marginal rate of substitution of one good for the other. Do they have the
same ordinal ranking of different baskets?
Since the two consumers have the same marginal rate of substitution at any basket, the slopes of
their indifference curves through that basket will always be the same. In other words, their
willingness to trade off one good for the other will be identical at any basket. So their
indifference curves will have the same shape. Their ordinal ranking for all baskets will be the
same.
3.24 Claire consumes three goods out of her income, food (F) shelter (S), and clothing
(C). At her current levels of consumption, her marginal utility of food is 3 and her marginal
utility of shelter is 6. Her marginal rate of substitution of shelter for clothing is 4. Do you
have enough information to determine her marginal rate of substitution of food for
clothing? If so, what is it? If not, why not?
We need to find MRSF,C = MUF/MUC
We are given: MUF = 3, MUS = 6.
We are also given MRSS,C = 4, so we know that MUS/MUC = 6/MUC = 4.
Thus MUC = 6/4 = 1.5. Now we can determine MRSF,C = MUF/MUC = 3/1.5 = 2.
Copyright © 2011 John Wiley & Sons, Inc.
Chapter 3 - 20
Besanko & Braeutigam – Microeconomics, 4th edition
Solutions Manual
3.25 Suppose a person has a utility function given by U = [xρ + yρ]1/ρ where ρ is a
number between -∞ and 1. This is called a constant elasticity of substitution (CES) utility
function. You will encounter CES functions in Chapter 6, where the concept of elasticity of
substitution will be explained. The marginal utilities for this utility function are given by
MUx = [xρ + yρ]1/ρ−1xρ−1
MUy = [xρ + yρ]1/ρ−1 yρ−1
Does this utility function exhibit the property of diminishing MRSx,y?
Recall that MRS x , y 
MU x
. Substituting in the marginal utilities given above yields
MU y
x  1
. Now, because  < 1, x - 1 decreases as x increases. By the same logic, y - 1
 1
y
increases as y decreases. As we “slide down” an indifference curve, x increases and y decreases,
so it follows that MRSx,y decreases. Thus, this utility function exhibits diminishing marginal rate
of substitution of x for y.
MRS x , y 
3.26 Annie consumes three goods out of her income, food (F) shelter (S), and clothing
(C). At her current levels of consumption, her marginal rate of substitution of food for
clothing is 2 and her marginal rate of substitution of clothing for shelter is 3.
a) Do you have enough information to determine her marginal rate of substitution of food
for shelter? If so, what is it? If not, why not?
b) Do you have enough information to determine her marginal utility of shelter? If so, what
is it? If not, why not?
a)We do have enough information to determine MRSF,S = MUF/MUS.
We are given: MRSF,C = MUF/MUC = 2 and MRSC,S = MUC/MUS =3.
From MUF/MUC = 2, we know that MUC = MUF / 2.
From MUC/MUS = 3, we know that MUC = 3MUS.
Thus MUF /2 = 3MUS, so MUF/MUS = 6. Thus MRSF,S = 6.
b) While the marginal rates of substitution provide quantitative information about the ratios of
the marginal utilities, alone they do not allow us to determine the individual marginal utilities of
any of the three goods.
Copyright © 2011 John Wiley & Sons, Inc.
Chapter 3 - 21