Besanko & Braeutigam – Microeconomics, 4th edition
Solutions Manual
Chapter 4
Consumer Choice
Solutions to Review Questions
1.
If the consumer has a positive marginal utility for each of two goods, why will the
consumer always choose a basket on the budget line?
Relative to any point on the budget line, when the consumer has a positive marginal utility for all
goods she could increase her utility by consuming some basket outside the budget line.
However, baskets outside the budget line are unaffordable to her, so she is constrained (as in
“constrained optimization”) to choosing the most preferred basket that lies along the budget line.
2.
How will a change in income affect the location of the budget line?
An increase in income will shift the budget line away from the origin in a parallel fashion
expanding the set of possible baskets from which a consumer may choose. A decrease in income
will shift the budget line in toward the origin in a parallel fashion, reducing the set of possible
baskets from which a consumer may choose.
3.
How will an increase in the price of one of the goods purchased by a consumer affect
the location of the budget line?
If the price of one of the goods increases, the budget line will rotate inward on the axis for the
good with the price increase. The budget line will continue to have the same intercept on the
other axis. For example, suppose someone buys two goods, cups of coffee and doughnuts, and
suppose the price of a cup of coffee increases. Then the budget line will rotate as in the
following diagram:
Doughnuts
BL2
BL1
Coffee
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4.
What is the difference between an interior optimum and a corner point optimum in
the theory of consumer choice?
With an interior optimum the consumer is choosing a basket that contains positive quantities of
all goods, while with a corner point optimum the consumer is choosing a basket with a zero
quantity for one of the goods. The tangency condition usually does not apply at corner optima.
5.
At an optimal interior basket, why must the slope of the budget line be equal to the
slope of the indifference curve?
If the optimum is an interior solution, the slope of the budget line must equal the slope of the
indifference curve. If these slopes are not equal at the chosen interior basket then the “bang for
the buck” condition will not hold. This condition states that at the optimum the extra utility
gained per dollar spent on good x must be equal to the extra utility gained per dollar spent on
good y . If this condition does not hold at the chosen basket, then the consumer could reallocate
his income to purchase more of the good with the higher “bang for the buck” and increase his
total utility while remaining within the given budget. Thus, if these slopes are not equal the
basket cannot be optimal assuming an interior solution.
6.
At an optimal interior basket, why must the marginal utility per dollar spent on all
goods be the same?
At an interior optimum, the slope of the budget line must equal the slope of the indifference
curve. This implies
MU x Px
MRS x, y 

MU y Py
This can be rewritten as
MU x MU y

Px
Py
which is known as the “bang for the buck” condition. If this condition does not hold at the
chosen interior basket, then the consumer can increase total utility by reallocating his spending to
purchase more of the good with the higher “bang for the buck” and less of the other good.
7.
Why will the marginal utility per dollar spent not necessarily be equal for all goods
at a corner point?
The “bang for the buck” condition will not necessarily hold at a corner solution optimum. The
consumer could theoretically increase total utility by reallocating his spending to purchase more
of the good with the higher “bang for the buck” and less of the other good. Since the basket is a
corner point, however, he is already purchasing zero of one of the goods. This implies that he
cannot purchase less of the good with a zero quantity (since negative quantities make no sense)
and therefore cannot reallocate spending.
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8.
Suppose that a consumer with an income of $1,000 finds that basket A maximizes
utility subject to his budget constraint and realizes a level of utility U1. Why will this
basket also minimize the consumer’s expenditures necessary to realize a level of utility U1?
In the utility maximization problem, the consumer maximizes utility subject to a fixed budget
constraint. At the optimum the slope of the budget line will equal the slope of the indifference
curve. If we now hold that indifference curve fixed, we can solve an expenditure minimization
problem in which we ask what is the minimum expenditure necessary to achieve that fixed level
of utility. Since the slope of the budget line and indifference curve have not changed, when the
expenditure is minimized the budget line and indifference curve will be tangent at the same point
as in the utility maximization problem. The same basket is optimal in both problems.
9.
What is a composite good?
First, consumers typically allocate income to more than two goods. Second, economists often
want to focus on the consumer‟s response to purchases of a single good or service. In this case it
is useful to present the consumer choice problem using a two-dimensional graph. Since there are
more than two goods the consumer is purchasing, however, an economist would need more than
two dimensions to show the problem graphically. To reduce the problem to two dimensions,
economists often group the expenditures on all other goods besides the one in question into a
single good termed a “composite good.” When the problem is shown graphically, one axis
represents the composite good while the other axis represents the single good in question. By
creating this composite good, the problem can be illustrated using a two-dimensional graph.
10.
How can revealed preference analysis help us learn about a consumer’s preferences
without knowing the consumer’s utility function?
By employing revealed preference analysis one can make inferences regarding a consumer‟s
preferences without knowing what the consumer‟s indifference map looks like. For example, if a
consumer chooses basket A over basket B when basket B costs at least as much as basket A, we
know that basket A is at least as preferred as basket B. If the consumer chooses basket C, which
is more expensive than basket D, then we know the consumer strictly prefers basket C to basket
D. By observing enough of these choices, one can determine how the consumer ranks baskets
even without knowing the exact shape of the consumer‟s indifference map.
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Solutions to Problems
4.1
Pedro is a college student who receives a monthly stipend from his parents of $1,000.
He uses this stipend to pay rent for housing and to go to the movies (you can assume that
all of Pedro’s other expenses, such as food and clothing have already been paid for). In the
town where Pedro goes to college, each square foot of rental housing costs $2 per month.
The price of a movie ticket is $10 per ticket. Let x denote the square feet of housing, and let
y denote the number of movie tickets he purchases per month.
a) What is the expression for Pedro’s budget constraint?
b) Draw a graph of Pedro’s budget line.
c) What is the maximum number of square feet of housing he can purchase given his
monthly stipend?
d) What is the maximum number of movie tickets he can purchase given his monthly
stipend?
e) Suppose Pedro’s parents increase his stipend by 10 percent. At the same time, suppose
that in the college town he lives in, all prices, including housing rental rates and movie
ticket prices, increase by 10 percent. What happens to the graph of Pedro’s budget line?
a) 2x + 10y ≤ 1000
b)
c) The maximum amount of housing Pedro can purchase is his budget divided by the price of
housing: $1,000/$2 per square feet = 500 square feet.
d) The maximum number of movie tickets Pedro can purchase is his budget divided by the price
of a movie ticket: $1,000/$10 per tickets = 100 tickets.
e) His budget line does not change at all.
Initially, the budget line (with x on the horizontal axis and y on the vertical axis) has a horizontal
intercept equal to 1000/2 = 500 and a vertical intercept equal to 1000/10 = 100. The slope of the
budget line is -2/10 = - 0.20 (the price of housing divided by the price of movie tickets).
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With the increase in Pedro‟s stipend and the increases in prices we have:
 Horizontal intercept of budget line: 1000(1.10)/(2(1.10)) = 500
 Vertical intercept of budget line: 1000(1.10)/(10(1.10)) = 100
 Slope of budget line: -2(1.10)/(10(1.10)) = - 0.20.
These are the same as before and thus the budget line does not change.
4.2
Sarah consumes apples and oranges (these are the only fruits she eats). She has
decides that her monthly budget for fruit will be $50. Suppose that one apple costs $0.25,
while one orange costs $0.50. Let x denote the quantity of apples and y denote the quantity
of oranges that Sarah purchases.
a. What is the expression for Sarah’s budget constraint?
b. Draw a graph of Sarah’s budget line.
c. Show graphically how Sarah’s budget line changes if the price of apples increases to
$0.50.
d. Show graphically how Sarah’s budget line changes if the price of oranges decreases to
$0.25.
e. Suppose Sarah decides to cut her monthly budget for fruit in half. Coincidentally, the
next time she goes to the grocery store, she learns that oranges and apples are on sale for
half price, will remain so for the next month, i.e., the price of apples falls from $0.25 per
apple to $0.125 per apple and the price of oranges falls from $0.50 per orange to $0.25
per orange. What happens to the graph of Sarah’s budget line?
a) 0.25x + 0.50y ≤ 50.
b)
c)
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d)
e) Sarah‟s budget line would not change.
 Horizontal intercept of the budget line: (0.5)$50/((0.5)(0.25) = 200
 Vertical intercept of the budget line: (0.5)$50/((0.5)(0.50) = 100
 Slope of the budget line = -(0.5)(0.25)/((0.5)(0.50)) = 0.50
These are the same as before, and thus the budget line does not change.
4.3
In Problem 3.7 of Chapter 3, we considered Julie’s preferences for food F and
clothing C. Her utility function was U(F, C) = FC. Her marginal utilities were MUF = C
and MUC = F. You were asked to draw the indifference curves U = 12, U = 18, and U = 24,
and to show that she had a diminishing marginal rate of substitution of food for clothing.
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Suppose that food costs $1 a unit and that clothing costs $2 a unit. Julie has $12 to spend on
food and clothing.
a) Using a graph (and no algebra), find the optimal (utility-maximizing) choice of food and
clothing. Let the amount of food be on the horizontal axis and the amount of clothing be on
the vertical axis.
b) Using algebra (the tangency condition and the budget line), find the optimal choice of
food and clothing.
c) What is the marginal rate of substitution of food for clothing at her optimal basket?
Show this graphically and algebraically.
d) Suppose Julie decides to buy 4 units of food and 4 units of clothing with her $12 budget
(instead of the optimal basket). Would her marginal utility per dollar spent on food be
greater than or less than her marginal utility per dollar spent on clothing? What does this
tell you about how she should substitute food for clothing if she wanted to increase her
utility without spending any more money?
a)
30
Clothing
25
20
15
Optimum at F=6, C=3.
10
5
0
0
5
10
15
20
25
30
35
Food
b)
The tangency condition implies that
MU F PF

MU C PC
Plugging in the known information results in
C 1

F 2
2C  F
Substituting this result into the budget line, F  2C  12 , yields
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2C  2C  12
4C  12
C 3
Finally, plugging this result back into the tangency condition implies F  6 . At the optimum the
consumer choose 6 units of food and 3 units of clothing.
c)
At the optimum, MRSF ,C  C / F  3 / 6  1/ 2 . Note that this is equal to the ratio of the
price of food to the price of clothing. The equality of the price ration and MRSF,C is seen in the
graph above as the tangency between the budget line and the indifference curve for U  18 .
d)
If the consumer purchases 4 units of food and 4 units of clothing, then
MU C 4
MU F 4
 4
  2.
1
2
PF
PC
This implies that the consumer could reallocate spending by purchasing more food and less
clothing to increase total utility. In fact, at the basket (4, 4) total utility is 16 and the consumer
spent $12. By giving up one unit of clothing the consumer saves $2 which can than be used to
purchase two units of food (they each cost $1). This will result in a new basket (6,3), total utility
of 18, and spending of $12. By reallocating spending toward the good with the higher “bang for
the buck” the consumer increased total utility while remaining within the budget constraint.
4.4
The utility that Ann receives by consuming food F and clothing C is given by U(F,
C) = FC + F. The marginal utilities of food and clothing are MUF = C + 1 and MUC = F.
Food costs $1 a unit, and clothing costs $2 a unit. Ann’s income is $22.
a) Ann is currently spending all of her income. She is buying 8 units of food. How many
units of clothing is she consuming?
b) Graph her budget line. Place the number of units of clothing on the vertical axis and the
number of units of food on the horizontal axis. Plot her current consumption basket.
c) Draw the indifference curve associated with a utility level of 36 and the indifference
curve associated with a utility level of 72. Are the indifference curves bowed in toward the
origin?
d) Using a graph (and no algebra), find the utility maximizing choice of food and clothing.
e) Using algebra, find the utility-maximizing choice of food and clothing.
f ) What is the marginal rate of substitution of food for clothing when utility is maximized?
Show this graphically and algebraically.
g) Does Ann have a diminishing marginal rate of substitution of food for clothing? Show
this graphically and algebraically.
a)
If Ann is spending all of her income then
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F  2C  22
8  2C  22
2C  14
C 7
b)
12
Clothing
10
8
6
4
2
0
0
5
10
15
20
25
30
35
Food
c)
Yes, the indifference curves are convex, i.e., bowed in toward the origin. Also, note that
they intersect the F-axis.
80
70
U=72
Clothing
60
50
40
30
U=36
20
10
0
0
5
10
15
20
25
30
35
Food
d)
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80
70
U=72
Clothing
60
50
40
30
U=36
Optimum at F=12, C=5
20
10
0
0
5
10
15
20
25
30
35
Food
e)
The tangency condition requires that
MU F PF

MU C PC
Plugging in the known information yields
C 1 1

F
2
2C  2  F
Substituting this result into the budget line, F  2C  22 results in
(2C  2)  2C  22
4C  20
C 5
Finally, plugging this result back into the tangency condition implies that F  2(5)  2  12 . At
the optimum the consumer chooses 5 units of clothing and 12 units of food.
f)
MRS F ,C 
C 1 5 1 1


The marginal rate of substitution is equal to the price ratio.
12
2
F
g)
Yes, the indifference curves do exhibit diminishing MRSF ,C . We can see this in the
graph in part c) because the indifference curves are bowed in toward the origin. Algebraically,
MRSF ,C  C 1 F . As F increases and C decreases along an isoquant, MRSF ,C diminishes.
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4.5
Consider a consumer with the utility function U(x, y) = min(3x, 5y), that is, the two
goods are perfect complements in the ratio 3:5. The prices of the two goods are Px = $5 and
Py = $10, and the consumer’s income is $220. Determine the optimum consumption basket.
This question cannot be solved using the usual tangency condition. However, you can see from
the graph below that the optimum basket will necessarily lie on the “elbow” of some indifference
curve, such as (5, 3), (10, 6) etc. If the consumer were at some other point, he could always move
to such a point, keeping utility constant and decreasing his expenditure. The equation of all these
“elbow” points is 3x = 5y, or y = 0.6x. Therefore the optimum point must be such that 3x = 5y.
The usual budget constraint must hold of course. That is, 5x  10 y  220 . Combining these two
conditions, we get (x, y) = (20, 12).
y
(20,12)
(10,6)
(5,3)
x
4.6
Jane likes hamburgers (H) and milkshakes (M). Her indifference curves are bowed
in toward the origin and do not intersect the axes. The price of a milkshake is $1 and the
price of a hamburger is $3. She is spending all her income at the basket she is currently
consuming, and her marginal rate of substitution of hamburgers for milkshakes is 2. Is she
at an optimum? If so, show why. If not, should she buy fewer hamburgers and more
milkshakes, or the reverse?
From the given information we know that PH  3 , PM  1 , and MRSH , M  2. Comparing the
MRSH,M to the price ratio,
P
3
MRSH , M  2  H 
PM 1
Since these are not equal Jane is not currently at an optimum. In addition, we can say that
MU H
PH
 MRSH , M 
MU M
PM
which is equivalent to
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MU M MU H

PM
PH
That is, the “bang for the buck” from milkshakes is greater than the “bang for the buck” from
hamburgers. So Jane can increase her total utility by reallocating her spending to purchase fewer
hamburgers and more milkshakes.
4.7
Ray buys only hamburgers and bottles of root beer out of a weekly income of $100.
He currently consumes 20 bottles of root beer per week, and his marginal utility of root
beer is 6. The price of root beer is $2 per bottle. Currently, he also consumes 15
hamburgers per week, and his marginal utility of a hamburger is 8. Is Ray maximizing
utility at his current consumption basket? If not, should he buy more hamburgers each
week, or fewer?
Compare MUH/PH with MUR/PR, where the subscripts “H” and “R” refer respectively to
hamburgers and root beer. We have all the information to make this comparison except for the
price of a hamburger. But we can determine the price of a hamburger from Sam‟s budget
constraint:
PHH + PRR = Income, or PH(15) + 2(20) = 100.
So PH = $4 per hamburger.
Now we can see that MUH/PH = 8/4 = 2 and MUR/PR = 6/2 = 3.
Since the “bang for the buck” is higher for root beer than for hamburgers, he should buy fewer
hamburgers (and more root beer).
4.8
Dave currently consumes 10 hot dogs and 6 sodas each week. At his current
consumption basket, his marginal utility for hot dogs is 5 and his marginal utility for sodas
is 3. If the price of one hot dog is $1 and the price of one soda is $0.50, is Dave currently
maximizing his utility? If not, how should he reallocate his spending in order to increase
his utility?
To determine if this situation is optimal, determine if the tangency condition holds.
Is
MUH MUS

?
PH
PS
That is, is
MUH MUS
5
3


? No (5  6). So
.
PH
PS
1 0.50
Since the tangency condition does not hold, Dave is not currently maximizing his utility. To
increase his utility he should purchase more soda and fewer hot dogs (since the „bang for the
buck‟ for sodas is higher).
4.9
Helen’s preferences over CDs (C) and sandwiches (S) are given by U(S, C) = SC +
10(S + C), with MUC = S + 10 and MUS = C + 10. If the price of a CD is $9 and the price of
a sandwich is $3, and Helen can spend a combined total of $30 each day on these goods,
find Helen’s optimal consumption basket.
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See the graph below. The fact that Helen‟s indifference curves touch the axes should
immediately make you want to check for a corner point solution.
To see the corner point optimum algebraically, notice if there was an interior solution, the
tangency condition implies (S + 10)/(C +10) = 3, or S = 3C + 20. Combining this with the budget
constraint, 9C + 3S = 30, we find that the optimal number of CDs would be given by 18C  30
which implies a negative number of CDs. Since it‟s impossible to purchase a negative amount of
something, our assumption that there was an interior solution must be false. Instead, the
optimum will consist of C = 0 and Helen spending all her income on sandwiches: S = 10.
Graphically, the corner optimum is reflected in the fact that the slope of the budget line is steeper
than that of the indifference curve, even when C = 0. Specifically, note that at (C, S) = (0, 10)
we have P C / P S = 3 > MRSC,S = 2. Thus, even at the corner point, the marginal utility per dollar
spent on CDs is lower than on sandwiches. However, since she is already at a corner point with
C = 0, she cannot give up any more CDs. Therefore the best Helen can do is to spend all her
income on sandwiches: (C, S) = (0, 10). [Note: At the other corner with S = 0 and C = 3.3, P C /
P S = 3 > MRSC,S = 0.75. Thus, Helen would prefer to buy more sandwiches and less CDs, which
is of course entirely feasible at this corner point. Thus the S = 0 corner cannot be an optimum.]
4.10 The utility that Corey obtains by consuming hamburgers (H) and hot dogs (S) is
given by
. The marginal utility of hamburgers is
and the
marginal utility of steaks is equal to
a) Sketch the indifference curve corresponding to the utility level U = 12.
b) Suppose that the price of hamburgers is $1 per hamburger, and the price of steak is $8
per steak. Moreover, suppose that Corey can spend $100 per month on these two foods.
Sketch Corey’s budget line for hamburgers and steak given this budget.
c) Based on your answer to parts (a) and (b), what is Corey’s optimal consumption basket
given his budget?
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a) Some points on the U = 12 indifference curve include
S
0
5
12
21
32
45
60
H
100
81`
64
49
36
25
16
U
12
12
12
12
12
12
12
Connecting these points gives us the U = 12 indifference curve:
b) The equation of the budget line is H + 8S = 100
Graphing this on the same axes as the U = 12 indifference curve gives us:
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c) The optimal consumption basket is S = 0, H = 100, i.e., point R in the figure below. There are
several ways to see this. One way is to sketch a few more indifference curves (each
corresponding to a different level of utility). This picture strongly suggests that the point of
maximum utility occurs at point R.
Another way is to compare the marginal utility per dollar of spent on hamburger and the
marginal utility per dollar spent on steak at point R. From the information given in the statement
and
and so at point R
of the problem,
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Thus, at point R, the marginal utility per dollar spent on hamburger is greater than the marginal
utility per dollar spent on steak, and so the consumer would like to purchase more hamburger
and less steak. However, at point R, no further reduction in the quantity of steak is possible, and
thus R is the optimal consumption basket.
4.11 This problem will help you understand what happens if the marginal rate of
substitution is not diminishing. Dr. Strangetaste buys only french fries (F) and hot dogs (H)
out of his income. He has positive marginal utilities for both goods, and his MRSH,F is
increasing. The price of hot dogs is PH, and the price of french fries is PF .
a) Draw several of Dr. Strangetaste’s indifference curves, including one that is tangent to
his budget line.
b) Show that the point of tangency does not represent a basket at which utility is
maximized, given the budget constraint. Using the indifference curves you have drawn,
indicate on your graph where the optimal basket is located.
a)
F
Preference
Directions
A
B
C
H
b)
At point A, Dr. Strangetaste‟s indifference curve, which is bowed out from the origin, is
tangent to his budget line. This point is not an optimum because, for example, Dr. Strangetaste
could move to point B on his budget line and achieve a higher level of total utility. Point B,
though, is not an optimum either because Dr. Strangetaste could move to point C, a corner point,
to achieve an even higher level of total utility. When the MRS is increasing, a corner point
optimum will occur (with F = 0 in this picture, though it could equivalently be with H = 0 for
another set of indifference curves).
4.12 Julie consumes two goods, food and clothing, and always has a positive marginal
utility for each good. Her income is 24. Initially, the price of food is 2 and the price of
clothing is 2. After new government policies are implemented, the price of food falls to 1
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and the price of clothing rises to 4. Suppose, under the initial budget constraint, her
optimal choice is 10 units of food and 2 units of clothing.
a) After the prices change, can you predict whether her utility will be higher, lower, or the
same as under the initial prices?
b) Does your answer require that there be a diminishing marginal rate of substitution of
food for clothing? Explain.
As given, Julie consumes F = 10 and C = 2 with an income of 24.
Initially (with PF = PC = 2) she spends all her income: PFF + PCC = 2(10) + 2(2) = 24.
To buy her initial basket at the new prices, she would only need to spend
PFF + PCC = 1(10) + 4(2) = 18.
Thus, her initial basket lies inside her new budget constraint (assuming her income stays at 24).
With her new budget line she would be able to choose a new basket to the “northeast” of (i.e., a
basket involving more food and clothing than) her initial basket, making her better off.
4.13 Toni likes to purchase round trips between the cities of Pulmonia and Castoria and
other goods out of her income of $10,000. Fortunately, Pulmonian Airways provides air
service and has a frequent-flyer program. Around trip between the two cities normally
costs $500, but any customer who makes more than 10 trips a year gets to make additional
trips during the year for only $200 per round trip.
a) On a graph with round trips on the horizontal axis and “other goods” on the vertical
axis, draw Toni’s budget line. (Hint: This problem demonstrates that a budget line need
not always be a straight line.)
b) On the graph you drew in part (a), draw a set of indifference curves that illustrates why
Toni may be better off with the frequent-flyer program.
c) On a new graph draw the same budget line you found in part (a). Now draw a set of
indifference curves that illustrates why Toni might not be better off with the frequent-flyer
program.
a)
The budget line will have a kink where round trips = 10 and other goods = 5,000.
Northwest of the kink, the budget line‟s slope will be –500 . Southeast of the kink, the slope will
be –200.
Other
10,000
5,000
C
B
Round Trips
10 20
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b)
With the indifference curves drawn on the above graph, Toni is better off with the
frequent flyer program (at point B) than she would be without it (at point C). Without the
frequent flyer program the best she could achieve is point C, which lies on the hypothetical
budget line where the price of round trips is always $500.
c)
With the indifference curves drawn on graph below, Toni is no better off with the
frequent flyer program than she would be without it (at point A). At this point, her indifference
curve is tangent to a portion of the budget line where the frequent flyer program does not apply
(less than 10 round trips).
Other
10,000
A
5,000
Round Trips
10
20
35
4.14 A consumer has preferences between two goods, hamburgers (measured by H) and
milkshakes (measured by M). His preferences over the two goods are represented by the
utility function U = √H + √M. For this utility function MUH = 1/(2√H) and MUM =
1/(2√M).
a) Determine if there is a diminishing MRSH,M for this utility function.
b) Draw a graph to illustrate the shape of a typical indifference curve. Label the curve U1.
Does the indifference curve intersect either axis? On the same graph, draw a second
indifference curve U2, with U2 > U1.
c) The consumer has an income of $24 per week. The price of a hamburger is $2 and the
price of a milkshake is $1. How many milkshakes and hamburgers will he buy each week if
he maximizes utility? Illustrate your answer on a graph.
MU H 1/(2 H )
M


MU M 1/(2 M )
H
This utility function has a diminishing marginal rate of substitution since MRSH , M declines as H
increases and M decreases.
a)
MRSH , M 
b)
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40.00
35.00
30.00
U2
M
25.00
20.00
15.00
U1
10.00
5.00
0.00
0
5
10
15
20
25
30
35
H
Since it is possible to have U > 0 if either H = 0 (and M > 0) or M = 0 (and H > 0), the
indifference curves will intersect both axes.
c)
We know from the tangency condition that
M 2

H 1
M  4H
Substituting this into the budget line, 2H  M  24 , yields
2 H  4 H  24
H 4
Finally, plugging this back into the tangency condition implies M  4(4)  16 . At the optimum
the consumer will choose 4 hamburgers and 16 milkshakes. This can be seen in the graph above.
4.15 Justin has the utility function U = xy, with the marginal utilities MUx = y and MUy =
x. The price of x is 2, the price of y is py, and his income is 40. When he maximizes utility
subject to his budget constraint, he purchases 5 units of y. What must be the price of y and
the amount of x consumed?
When Justin maximizes utility, his optimal consumption basket will be on the budget constraint
and satisfy the tangency condition.
Any basket on the budget line will satisfy pxx + pyy = I, or 2x + 5py = 40.
The tangency condition requires that MUx / px = MUy / py, or that 5 / 2 = x / py. This implies that
5py = 2x.
Putting these two equations together reveals that 5py + 5py = 40; thus py = 4.
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4.16 A student consumes root beer and a composite good whose price is $1. Currently,
the government imposes an excise tax of $0.50 per six-pack of root beer. The student now
purchases 20 six-packs of root beer per month. (Think of the excise tax as increasing the
price of root beer by $0.50 per six-pack over what the price would be without the tax.) The
government is considering eliminating the excise tax on root beer and, instead, requiring
consumers to pay $10.00 per month as a lump sum tax (i.e., the student pays a tax of $10.00
per month, regardless of how much root beer is consumed). If the new proposal is adopted,
how will the student’s consumption pattern (in particular, the amount of root beer
consumed) and welfare be affected? (Assume that the student’s marginal rate of
substitution of root beer for other goods is diminishing.)
Composite Good
BL1
B
BL2
A
Root Beer
20
Assume the student is initially at an interior optimum, point A. Denote the initial price of root
beer by P and the student‟s income as M. Point A then consists of RA = 20 units of root beer and
YA = M – 20P units of the composite good. The effect of the proposal is to rotate the budget line
outward (the price change) and then shift it inward (the lump sum tax), for a total movement
from BL1 to BL2. Notice that BL2 intersects BL1 exactly at point A: under the proposal, (RA, YA)
costs the student 20(P – 0.5) + M – 20P = M – 10, which is equal to her income under the
proposal.
Because A was initially optimal, MRSR,Y = P at point A. Yet the price ratio along BL2 is (P –
0.5). Hence MRSR,Y > P R / P Y, so the student can increase her utility by purchasing more root
beer and less of the composite good, at a point such as B depicted in the graph above. Thus, the
proposal will make the student better off.
4.17 When the price of gasoline is $2.00 per gallon, Joe consumes 1,000 gallons per year.
The price increases to $2.50, and to offset the harm to Joe, the government gives him a cash
transfer of $500 per year. Will Joe be better off or worse off after the price increase and
cash transfer than he was before? What will happen to his gasoline consumption? (Assume
that Joe’s marginal rate of substitution of gasoline for other goods is diminishing.)
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Other Goods
B
A
BL2
BL1
1000
Gas
Assume Joe is initially at an interior optimum, point A, and that the price of other goods is $1.
Let Joe‟s income be M. Point A then consists of GA = 1000 units of root beer and YA = M – 2000
units of other goods. The effect of the proposal is to rotate the budget line inward (the price
change) and then shift it outward (the cash transfer), for a total movement from BL1 to BL2.
Notice that BL2 intersects BL1 exactly at point A: after the price increase, (GA, YA) costs Joe
1000*2.50 + M – 2000 = M + 500, which is equal to his income after the cash transfer.
Because A was initially optimal, MRSG,Y = 2 at point A. Yet the price ratio along BL2 is 2.5.
Hence MRSG,Y < P G / P Y, so Joe can increase his utility by purchasing less gas and more of the
composite good, at a point such as B depicted in the graph above. Thus, the proposal will make
Joe better off.
4.18 Paul consumes only two goods, pizza (P) and hamburgers (H) and considers them to
be perfect substitutes, as shown by his utility function: U(P, H) = P + 4H. The price of pizza
is $3 and the price of hamburgers is $6, and Paul’s monthly income is $300. Knowing that
he likes pizza, Paul’s grandmother gives him a birthday gift certificate of $60 redeemable
only at Pizza Hut. Though Paul is happy to get this gift, his grandmother did not realize
that she could have made him exactly as happy by spending far less than she did. How
much would she have needed to give him in cash to make him just as well off as with the
gift certificate?
Paul‟s initial budget constraint is the line AC, allowing him to purchase at most 50 hamburgers
or at most 100 pizzas. The $60 cash certificate shifts out his budget constraint without changing
the maximum number of hamburgers that he can buy. The new budget constraint is ABD and he
can now buy a maximum of 120 pizzas.
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Hamburgers
60
55
50
B
A
20
C
100
D
120
Pizza
Initially, Paul‟s optimal basket contains all hamburgers and no pizza, at point A where (P, H) =
(0, 50), because MUH /P H = 4/6 > MUP / P P = 1/3. His utility level at point A is U(0, 50) = 200.
When he gets the gift certificate, Paul‟s optimal basket is at point B, spending all of his regular
income on hamburgers and the $60 gift certificate on pizza. So point B is where (P, H) = (20,
50) with a utility of U(20, 50) = 220.
However, Paul could also achieve a utility of 220 by consuming 220/4 = 55 hamburgers. To buy
the extra 5 hamburgers he would require 5*6 = $30. So, if he had received a cash gift of $30 it
would have made Paul exactly as well off as the $60 gift certificate for pizzas.
4.19 Jack makes his consumption and saving decisions two months at a time. His income
this month is $1,000, and he knows that he will get a raise next month making his income
$1,050. The current interest rate (at which he is free to borrow or lend) is 5 percent.
Denoting this month’s consumption by x and next month’s by y, for each of the following
utility functions state whether Jack would choose to borrow, lend, or do neither in the first
month. (Hint: In each case, start by assuming that Jack would simply spend his income in
each month without borrowing or lending money. Would doing so be optimal?)
a) U(x, y) = xy2, MUx = y2, MUy = 2xy
b) U(x, y) = x2 y, MUx = 2xy, MUy = x2
c) U(x, y) = xy, MUx = y, MUy = x
a)
In this case, MRS x, y 
MU x
y
. If Jack neither borrows nor lends, then MRSx,y =

MU y 2 x
1050/(2*1000) = 0.525. Recall that if the interest rate is r, the slope of the budget line is –(1+r)
= –1.05. Thus, if he neither borrows nor lends it will be the case that MRSx,y < 1 + r. That is, the
“bang for the buck” for spending this month (good x) is less than that for spending next month
(good y). Thus, Jack should lend some of his income this month (so x < 1000) in order to earn
interest and have higher spending next month (y > 1050).
b)
Now MRSx,y = 2y/x. If Jack neither borrows nor lends, MRSx,y = 2.1 > (1 + r). Thus, Jack
could increase his utility by borrowing in the first month (so that x > 1000 and y < 1050).
c)
Now MRSx,y = y/x. If Jack neither borrows nor lends, MRSx,y = 1.05 = (1 + r). Thus,
Jack‟s utility is maximized when he neither borrows nor lends, simply spending all of his income
in each month: (x, y) = (1000, 1050).
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4.20 The figure below shows a budget set for a consumer over two time periods, with a
borrowing rate rB and a lending rate rL, with rL < rB. The consumer purchases C1 units of
a composite good in period 1 and C2 units in period 2. The following is a general fact about
consumers making consumption decisions over two time periods: Let A denote the basket
at which a consumer spends exactly his income each period (the point at the kink of the
budget line). Then a consumer with a diminishing MRSC1,C2 will choose to borrow in the
first period if at basket A MRSC1,C2 > 1 + rB and will choose to lend if at basket A
MRSC1,C2 < 1 + rL. If the MRS lies between these two values, then he will neither borrow
nor lend. (You can try to prove this if you like. Keep in mind that diminishing MRS plays
an important role in the proof.) Using this rule, consider the decision of Meg, who earns
$2,000 this month and $2,200 the next with a utility function given by U(C1, C2) = C1C2,
where the C’s denote the value of consumption in each month. Suppose rL=0.05 (the
lending rate is 5 percent) and rB = 0.12 (the borrowing rate is 12 percent). Would Meg
borrow, lend, or do neither this month? What if the borrowing rate fell to 8 percent?
The utility function implies that MRSC1,C2 = C2 / C1. At point A, MRSC1,C2 = 1.10, which lies
between (1 + r L) = 1.05 and (1 + r B) = 1.12. Therefore Meg will neither borrow nor lend and will
simply spend her entire income each month.
If the borrowing rate falls to 8%, then the lower part of the budget line pivots outward, as
depicted in the graph below. Then at point A, MRSC1,C2 > (1+ r B) > (1 + r L) since 1.10 > 1.08 >
1.05. So Meg can increase her utility by moving away from point A to a point like B, where she
borrows money, spending more than her income this month (C1 > 2000) and less than her income
next month (C2 < 2200).
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C2
2200+2000(1.05)
Slope= –1.05
2200
A
B
Slope = –1.08
2000
2000+2200/1.08
C1
4.21 Sally consumes housing (denote the number of units of housing by h) and other
goods (a composite good whose units are measured by y), both of which she likes. Initially
she has an income of $100, and the price of a unit of housing (Ph) is $10. At her initial
basket she consumes 2 units of housing. A few months later her income rises to $120;
unfortunately, the price of housing in her city also rises, to $15. The price of the composite
good does not change. At her later basket she consumes 1 unit of housing. Using revealed
preference analysis (without drawing indifference curves), what can you say about how she
ranks her initial and later baskets?
Other
B
C
A
BL2
BL1
H
With the initial budget line, BL1, Sally chooses point A, where (xA, yA) = (2, 80). When her
income and the price of housing increase, the budget line becomes BL2 and she chooses point B,
where (xB, yB) = (1, 105). Importantly, because the equation for BL1 is 10x+ y = 100 and that for
BL2 is 15x + y = 120, we can solve to see that these lines intersect at x = 4, to the right of point
A on BL1.
Consider a hypothetical basket C on BL2 but northeast of A. We can then deduce that B  A ,
because (i) B is at least as preferred as C (since B was chosen when C was affordable), and (ii) C
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is strictly preferred to A (since C lies to the northeast of A). By transitivity, B must be strictly
preferred to A.
4.22 Samantha purchases food (F) and other goods (Y ) with the utility function U = FY.
Her income is 12. The price of a food is 2 and the price of other goods 1.
a) How many units of food does she consume when she maximizes utility?
b) The government has recently completed a study suggesting that, for a healthy diet, every
consumer should consume at least F = 8 units of food. The government is considering
giving a consumer like Samantha a cash subsidy that would induce her to buy F = 8. How
large would the cash subsidy need to be? Show her optimal basket with the cash subsidy on
an optimal choice diagram with F on the horizontal axis and Y on the vertical axis. c) As an
alternative to the cash subsidy in part (b), the government is also considering giving
consumers like Samantha food stamps, that is, vouchers with a cash value that can only be
redeemed to purchase food. Verify that if the government gives her vouchers worth $16,
she will choose F = 8. Illustrate her optimal choice on an optimal choice diagram. (You may
use the same graph you drew in part (b).)
BL
Cash
32
Y
UC
B
16
12
UB
R
6
BL
Food
Stamps
C
A
BL
No
subsidy
UA
S
3
8
F
a)
MUY = F and MUF = Y, so MRSF,Y = Y/F, which diminishes as F increases along an
indifference curve. Since the indifference curves do not intersect either axis, an optimal basket
will be interior. At such an optimum two conditions must be satisfied: (1) tangency: MRSF,Y = PF
/ PY, or Y = 2F, and (2) budget line (“BL No subsidy” in the graph): 2F + Y = 12. This F = 3 and
Y = 6. This optimum is depicted as point A in the graph.
b)
We need to find an interior optimum with F = 8. As income increases, the consumer
chooses a basket along the Income Consumption Curve, which consists of the tangency points Y
= 2F. So Y = 2(8) = 16. Total expenditure will then be 2F + Y = 2(8) + 16 = 32. So the consumer
needs an income of 32 (“BL Cash” in the graph). Since the consumer has an income of 12, she
needs an additional income of 20 (=32 – 12). So the subsidy needed is 20. This optimum is
shown as point B in the graph.
c)
From part (b) we see that, with no restrictions on how the government subsidy can be
spent, the consumer would like to buy 16 units of Y, more than her own budget (without
subsidy) would permit. So we expect that with food stamps, she will use the voucher to purchase
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the required 8 units of food and spend all of her own unrestricted income (12) on Y. In other
words, this consumer will be at point C on the graph, at the kink on the budget constraint RCS
(labeled “BL Food Stamps”).
We can verify that (F = 8, Y = 12) is her optimal choice by looking at the “bang for the
buck” condition at C. MUF/price of food = Y/2 = 12/2 = 6. MUY/price of Y = F/1 = 8. So the
consumer would like to substitute more Y for F, but cannot do so because at basket C she has
purchased all the other goods she can given her budget constraint.
4.23 As shown in the following figure, a consumer buys two goods, food and housing, and
likes both goods. When she has budget line BL1, her optimal choice is basket A. Given
budget line BL2, she chooses basket B, and with BL3, she chooses basket C.
a) What can you infer about how the consumer ranks baskets A, B, and C? If you can infer
a ranking, explain how. If you cannot infer a ranking, explain why not.
b) On the graph, shade in (and clearly label) the areas that are revealed to be less preferred
to basket B, and explain why you indicated these areas.
c) On the graph, shade in (and clearly label) the areas that are revealed to be (more)
preferred to basket B, and explain why you indicated these areas.
a)
From figure 4.21 we can infer that A B , that B  C , and therefore by transitivity that
A C .
Housing
A
D
B
E
C
BL1
BL2
BL3
Food
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First, A must be strictly preferred to B since A is at least as preferred as D and D must be strictly
preferred to B. Second, B must be strictly preferred to C since B is at least as preferred as E and
E is strictly preferred to C. So by transitivity A must be strictly preferred to C.
b)
A
Baskets strictly
preferred to B
B
C
Baskets less preferred
to B
B will be strictly preferred to everything inside BL2. In addition, since B is strictly preferred to
C, B will be strictly preferred to everything below BL3, including all the points along BL3 itself
(since C is weakly preferred to everything on BL3).
c)
See the graph in part (b). Everything to the northeast of B is strictly preferred to B. In
addition, since A is strictly preferred to B [see part (a)], everything to the northeast of A must
also be strictly preferred to B. Notice, however, that there are points on BL1 (both northwest and
southeast of A) about which we cannot infer anything.
4.24 The following graph shows the consumption decisions of a consumer over bundles of
x and y, both of which he likes. When faced with budget line BL1, he chose basket A, and
when faced with budget line BL2, he chose basket B. If he were to face budget line BL3,
what possible set of baskets could he choose in order for his behavior to be consistent with
utility maximization?
Let point E denote the intersection of BL1 and BL3, and point F denote the intersection of BL2
and BL3 (see the figure below).
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E
G
A
BL3
B
F
BL1
BL2
First, any point to the northwest of E, including E, is in the consumer‟s budget set when he faces
budget constraint BL1. The fact that he chose A over these points implies that A is at least as
preferred to E and strictly preferred to the points northwest of E. However, point G lies on BL3
and is northeast of A, so G  A. Therefore, by transitivity G is strictly preferred to E and all the
points on BL3 northwest of E.
Similarly, A is northeast of B so A B . Since B is at least as preferred as any point on BL2,
including F, by transitivity we know that A is strictly preferred to F and all points on BL3 to the
southeast of F. And since G  A, we know that G is strictly preferred to these points as well.
Therefore the consumer could choose any point between E and F on BL3, but neither E nor F
themselves.
4.25 Darrell has a monthly income of $60. He spends this money making telephone calls
home (measured in minutes of calls) and on other goods. His mobile phone company offers
him two plans:
• Plan A: pay no monthly fee and make calls for $0.50 per minute.
• Plan B: Pay a $20 monthly fee and make calls for $0.20 per minute.
Graph Darrell’s budget constraint under each of the two plans. If Plan A is better for him,
what is the set of baskets he may purchase if his behavior is consistent with utility
maximization? What baskets might he purchase if Plan B is better for him?
Let x denote the number of phone calls, and y denote spending on other goods. Under Plan A,
Darrell‟s budget line is JLM. Under Plan B, it is JKLN. These budget lines intersect at point L,
or about x = 67.
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y
60
40
J
L
K
M
67
120
N
200
x
If we know that Darrell chooses Plan A, his optimal bundle must lie on the line segment JL. No
point between L and M would be optimal under this plan because then Darrell could have chosen
a point under Plan B, between L and N, that would have given him more minutes, and left him
with more money to buy other goods. However, we cannot exclude point L itself (Darrell could,
for instance, have perfect complements preferences with an “elbow” at point L). Thus, if Darrell
chooses Plan A his optimal basket could be anywhere between J and L, including either of these
points.
Similarly, if he chose Plan B then his optimal basket must lie between L and N. Any point
between L and K (but not including point L) would be dominated by a point under Plan A
between L and J. Thus, if Darrell chooses Plan B his optimal basket could be anywhere between
L and N, including either of these points.
4.26 Figure 4.17 illustrates the case in which a consumer is better off with a quantity
discount. Can you draw an indifference map for a consumer who would not be better off
with the quantity discount?
Other Goods
A
Electricity
With this set of indifference curves, the tangency with the budget line occurs on the portion of
the budget line where the quantity discount has not taken effect. Therefore, the consumer does
not receive any benefit from the quantity discount.
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4.27 Angela has a monthly income of $120, which she spends on MP3s and a composite
good (whose price you may assume is $1 throughout this problem). Currently, she does not
belong to an MP3 club, so she pays the retail price of an MP3 of $2; her optimal basket
includes 20 MP3s monthly. For the past several months Asteroid, a media company, has
offered her the chance to join their “Premium Club”; to join the club she would need to
pay a membership fee of $60 per month, but then she could buy all the MP3s she wants at a
price of $0.50. She has decided not to join the club. Asteroid has now introduced an
“Economy Club”; to join, Angela would need to pay a membership fee of $30 per month,
but then she could buy all the MP3s she wants at a price of $1. Draw a graph illustrating
(1) Angela’s budget line and optimal basket when she joins no club, (2) the budget line she
would have faced had she joined the Premium Club, and (3) her budget line if she joins the
Economy Club. Will Angela surely want to join the Economy Club? If she were to join the
club, how many MP3s per month might she buy? Show how you arrive at your answers
using a revealed preference argument.
The budget lines for the “no club,” “Economy Club,” and “Premium Club” opportunities
available to Angela are drawn below.
Y
BL – no club
120
90
BL –
Economy Club
A(20,80)
R
B(30,60)
60
W
C(60,30)
60
BL1 –
Premium Club
S
Z
90
120
MP3s
She chooses A with “no club,” and chooses not to join the “Premium Club” when given the
chance. Thus, A is weakly preferred to any basket on WZ, and strongly preferred basket on the
segment CS (except for C). So, if she does join the Economy Club, she would not choose a
basket on CS (except possibly for C).
With “no club” she chooses A when she could have afforded B; thus A > B. Since the rest of the
segment RB lies inside the “no club” BL, these baskets are strongly inferior to A. So, if she does
join the Economy Club, she would not choose a basket on RB (except possibly for B).
To summarize, if she joins the Economy Club, she might consume any basket on the segment
BC, including B and C. So her consumption in that case would be 30 < MP3s < 60.
But she might not join the Economy Club at all! Although we have established that A > B and A
> C, we cannot say how she ranks A with the baskets between B and C.
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4.28 Alex buys two goods, food (F) and clothing (C). He likes both goods. His preferences
for the goods do not change from month to month. The following table shows his income,
the baskets he selected, and the prices of the goods over a two-month period.
Month PF
PC
Income Basket
Chosen
1
3
2
48
F=16, C=0
2
2
4
48
F=14, C=5
a) On the graph with F on the horizontal axis and C on the vertical axis, plot and clearly
label the budget lines and consumption baskets during these two weeks. Label the
consumption bundle in week 1 by point A on the graph and the consumption basket in
week 2 by point B. Using revealed preference analysis, what can you say about Alex’s
preferences for baskets A and B (i.e., how does he rank them)?
b) In month 3 Alex’s income rises to 57. The prices of food and clothing are both 3.
Assuming his preferences do not change, describe the set of baskets he might consume in
month 3 if he continues to maximize utility. Show this set of baskets in the graph.
a)
C
24
22
20
BL1
18
16
14
BL3
12
10
BL2
E =(10,9)
8
6
B=(14,5)
C
4
2
A
0
2
4
6
8
10
12
14
16
18
20
22
24
F
B is weakly preferred to C, which is strongly preferred to A. By transitivity, B is strongly
preferred to A.
b)
We know that he will choose a basket on BL3.
He will not choose any basket on or inside BL1. Why? He could choose B, and B>A (by part A).
Given BL1, he chose A instead of other baskets on BL1, all of which were affordable.
He will not choose a basket on BL3 to the southeast of B. Why? Given BL2 he chose B, and B is
strictly preferred to any basket inside BL3, including those on BL2 to the southeast of B.
Therefore, he might choose any basket on BL3 between B and E, not including E. This set of
baskets is illustrated with the heavy line in the graph.
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Besanko & Braeutigam – Microeconomics, 4th edition
Solutions Manual
4.29 Brian consumes units of electricity (E) and a composite good (Y), whose price is
always 1. He likes both goods. In period 1 the power company sets the price of electricity at
$7 per unit, for all units of electricity consumed. Brian consumes his optimal basket, 20
units of electricity and 70 units of the composite good. In period 2 the power company then
revises its pricing plan, charging $10 per unit for the first 5 units and $4 per unit for each
additional unit. Brian’s income is unchanged. Brian’s optimal basket with this plan
includes 30 units of electricity and 60 units of the composite good. In period 3 the power
company allows the consumer to choose either the pricing plan in period 1 or the plan in
period 2. Brian’s income is unchanged. Which pricing plan will he choose? Illustrate your
answer with a clearly labeled graph.
Y
210 R
160
Original BL
S
New BL
C
70
60
A
B
20
J
30
T
5
E
Brian‟s income is I = (7)(20) + 1(70) = $210.
His initial budget line is the solid curve RJ.
His initial optimal basket is A.
His new budget line is the dotted, piecewise linear curve RST. He chooses basket B, which costs
10(5) + 4(30 – 5) + 1(60) = 210.
In period 2 Basket A costs (10)(5) + (4)(20 – 5) + 1(70) = $180, so Basket A lies inside the new
budget line.
In period 3 Brian will choose the plan from the second period. B is weakly preferred to a basket
like C, which, in turn is strictly preferred to A (because C lies to the northeast of A). Thus he
strictly prefers B to A, and he can only reach B by choosing the plan from period 2.
4.30 Carina consumes two goods, X and Y, both of which she likes. In month 1 she
chooses basket A given budget line BL1. In month 2 she chooses B given budget line BL2,
and in month 3 she chooses C given budget lineBL3. Assume her indifference map is
unchanged over the three months. Use the theory of revealed preference to show whether
her choices are consistent with utility maximizing behavior. If so, show how she ranks the
three baskets. If it is not possible to infer how she ranks the baskets, explain why not.
Copyright © 2011 John Wiley & Sons, Inc.
Chapter 4 - 32
Besanko & Braeutigam – Microeconomics, 4th edition
Solutions Manual
From BL1 we would infer that A is weakly preferred to S, and that S is strongly preferred to C;
by transitivity we conclude that A is strongly preferred to C.
Y
BL2
B
BL3
R
A
BL1
S
C
X
From BL3 we would infer that C is weakly preferred to R, which is strongly preferred to A; by
transitivity we conclude that C is strongly preferred to A.
It cannot be simultaneously true that A is strongly preferred to C and C is strongly preferred to
A, for this would imply that A is strongly preferred to C, which is strongly preferred to A, or that
A is strongly preferred to itself. The preferences are either intransitive, or else the consumer is
failing to maximize utility in each of the three time periods.
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Chapter 4 - 33