CHAPTER 4: Utility
TRUE/FALSE
1. With quasilinear preferences, the slope of indifference curves is constant along all rays through the
origin.
2. Wanda Lott has the utility function U(x, y) = maxx, y. Wanda’s preferences are convex.
3. If someone has a utility function U = 2minx, y, then x and y are perfect complements for that person.
4. Maximilian consumes two goods, x and y. His utility function is U(x, y) = maxx, y. Therefore x and
y are perfect substitutes for Max.
5. A person with the utility function U(x, y) = y + x2 has convex preferences.
6. Mr. Surly consumes only two goods and hates them both. His utility function is U(x, y) = 2maxx, y.
Mr. Surly has (weakly) convex preferences.
7. Angela’s utility function is U(x1, x2) = (x1 + x2)3. Her indifference curves are downward-sloping,
parallel straight lines.
8. Henrietta’s utility function is U(x1, x2) = x1x2. She has diminishing marginal rate of substitution
between goods 1 and 2.
9. Alice’s utility function is U(x, y) = x2y. Steve’s utility function is U(x, y) = x2y + 2x. Alice and Steve
have the same preferences since Steve’s utility function is a monotonic transformation of Alice’s.
10. Jeans utility function is U(x, y) = x + y 2 − y. If we draw her indifference curves with x on the
horizontal axis and y on the vertical axis, then these indifference curves are everywhere downward
sloping and get flatter as one moves from left to right.
11. The utility function U(x1, x2) = 2lnx1 + 3lnx2 represents Cobb-Douglas preferences.
12. Fiery Demon is a rotgut whisky made in Kentucky. Smoothy is an unblended malt whisky imported
from Scotland. Ed regards these brands as perfect substitutes. When he goes into a bar, he sometimes
buys only Fiery Demon. Other times he buys only Smoothy. This shows that Ed has unstable
preferences.
13. Mark strictly prefers consumption bundle A to consumption bundle B and weakly prefers bundle B to
bundle A. These preferences can be represented by a utility function.
14. A consumer has preferences represented by the utility function U(x1, x2) = 10(x21 + 2x1x2 + x22) − 50.
For this consumer, goods 1 and 2 are perfect substitutes.
15. A person with utility function U(x, y) = 5 + y2 + 2x has nonconvex preferences.
16. A person with the utility function U(x, y) = 10 + y2 + x has convex preferences.
17. A person with the utility function U(x1, x2) = minx1 + 2x2, 2x1 + x2 has convex but not strictly convex
preferences.
18. If one utility function is a monotonic transformation of another, then the former must assign a higher
utility number to every bundle than the latter.
19. Quasilinear preferences are homothetic when the optimal amount of good 1 is not affordable.
MULTIPLE CHOICE
1. Ike’s utility function is U(x, y) = 25xy. He has 12 units of good x and 8 units of good y. Ben’s utility
function for the same two goods is U(x, y) = 4x + 4y. Ben has 9 units of x and 13 units of y.
a.
Ike prefers Ben’s bundle to his own bundle, but Ben prefers his own bundle to Ike’s.
b.
Ben prefers Ike’s bundle to his own, but Ike prefers his own bundle to Ben’s.
c.
Each prefers the other’s bundle to his own.
d.
Neither prefers the other’s bundle to his own.
e.
Since they have different preferences, there is not enough information to determine who
envies whom.
2. Ernie’s utility function is U(x, y) = 32xy. He has 10 units of good x and 8 units of good y. Waldo’s
utility function for the same two goods is U(x, y) = 3x + 5y. Waldo has 9 units of x and 13 units of y.
a.
Waldo prefers Ernie’s bundle to his own, but Ernie prefers his own bundle to Waldo’s.
b.
Each prefers the other’s bundle to his own.
c.
Neither prefers the other’s bundle to his own.
d.
Ernie prefers Waldo’s bundle to his own bundle, but Waldo prefers his own bundle to
Ernie’s.
e.
Since they have different preferences, there is not enough information to determine who
envies whom.
3. Tim has preferences represented by the utility function U(x, y) = min6x + y, x + 2y. If x is on the
horizontal axis and y is on the vertical axis, what is the slope of his indifference curve at the point
(8,9)?
a.
−6
b.
−2/6
c.
−1/2
d.
−1/6
e.
−8/9
4. Jean-Pierre has preferences represented by the utility function U(x, y) = min2x + y, x + 6y. If x is on
the horizontal axis and y is on the vertical axis, what is the slope of his indifference curve at the point
(7,7)?
a.
−1/2
b.
−6/2
c.
−1/6
d.
−2
e.
−7/7
5. Doreen has preferences represented by the utility function U(x, y) = 10x + 5y. She consumes 10 units
of good x and 9 units of good y. If her consumption of good x is lowered to 1, how many units of y
must she have in order to be exactly as well off as before?
a.
20
b.
30
c.
27
d.
18
e.
None of the above.
6. Angela has preferences represented by the utility function U(x, y) = 2x + 2y. She consumes 10 units of
good x and 6 units of good y. If her consumption of good x is lowered to 4, how many units of y must
she have in order to be exactly as well off as before?
a.
14
b.
13
c.
12
d.
15
e.
None of the above.
7. Mac’s utility function is U(x, y) = max2x − y, 2y − x.
a.
Mac’s preferences are quasilinear.
b.
If Mac has more x than y, any increase in his consumption of y would lower his utility.
c.
If Mac has more x than y, a decrease in his consumption of y would raise his utility.
d.
Mac always prefers more of each good to less.
e.
Goods x and y are perfect substitutes.
8. Charles’s utility function is U(x, y) = xy. Anne’s utility function is U(x, y) = 1,000xy. Diana’s utility
function is −xy. Elizabeth’s utility function is U(x, y) = −1/(xy + 1). Fergie’s utility function is xy
−10,000. Margaret’s utility function is x/y. Philip’s utility function is x(y + 1). (The goods x and y are
two very expensive goods. We leave you to speculate about what they are.) Which of these persons
have the same preferences as Charles?
a.
Everybody except Diana
b.
Anne and Fergie
c.
Anne, Fergie, and Elizabeth
d.
None of them
e.
All of them
9. Raymond’s preferences are represented by the utility function U(x, y) = x/y if y 0 and U(x, y) = 0 if y
= 0.
a.
Raymond has indifference curves that are rectangular hyperbolas.
b.
Raymond prefers more of each good to less.
c.
Raymond has quasilinear preferences.
d.
Raymond has a bliss point.
e.
Raymond has indifference curves that are upward-sloping straight lines if y 0.
10. Molly’s utility function is U(x, y) = y + 4x.5. She has 25 units of x and 12 units of y. If her consumption
of x is reduced to 0, how many units of y would she need in order to be exactly as well off as before?
a.
48
b.
37
c.
32
d.
112
e.
None of the above.
11. Waldo’s utility function is U(x, y) = xy. Waldo consumes 5 units of x and 25 units of y.
a.
Waldo would be willing to make small exchanges of x for y in which he would give up 5
units of x for every unit of y he got.
b.
Waldo would be willing to trade away all of his x for y as long as he got more than 5 units
of y for every unit of x he gave up.
c.
Waldo likes x and y equally well so he is always willing to exchange 1 unit of either good
for more than 1 unit of the other.
d.
Waldo will always be willing to make trades at any price if he does not have equal
amounts of the two goods.
e.
None of the above.
12. Ike’s utility function is U(x, y) = xy. Ike consumes 2 units of x and 8 units of y.
a.
Ike would be willing to make small exchanges of x for y in which he would give up 4 units
of x for every unit of y he got.
b.
Ike would be willing to trade away all of his x for y as long as he got more than 4 units of y
for every unit of x he gave up.
c.
Ike will always be willing to make trades at any price if he does not have equal amounts of
the two goods.
d.
Ike likes x and y equally well so he is always willing to exchange 1 unit of either good for
more than 1 unit of the other.
e.
None of the above.
13. Henry’s utility function is x2 + 16xw + 64w2, where x is his consumption of x and w is his consumption
of w.
a.
Henry’s preferences are nonconvex.
b.
Henry’s indifference curves are straight lines.
c.
Henry has a bliss point.
d.
Henry’s indifference curves are hyperbolas.
e.
None of the above.
14. Josephine’s utility function is U(x, y) = y + 5x.5. She has 1 unit of x and 2 units of y. If her consumption
of x is reduced to zero, how much y must she have in order to be exactly as well off as before?
a.
14 units
b.
9 units
c.
11 units
d.
7 units
e.
None of the above.
15. Jim’s utility function is U(x, y) = xy. Jerry’s utility function is U(x, y) = 1,000xy + 2,000. Tammy’s
utility function is U(x, y) = xy(1 − xy). Oral’s utility function is −1/(10 + xy. Billy’s utility function is
U(x, y) = x/y. Pat’s utility function is U(x, y) = −xy.
a.
No two of these people have the same preferences.
b.
They all have the same preferences except for Billy.
c.
Jim, Jerry, and Pat all have the same indifference curves, but Jerry and Oral are the only
ones with the same preferences as Jim.
d.
Jim, Tammy, and Oral all have the same preferences.
e.
None of the above.
16. Harmon’s utility function is U(x1, x2) = x1x2. His income is $100. The price of good 2 is p2 = 4. Good 1
is priced as follows. The first 15 units cost $4 per unit and any additional units cost $2 per unit. What
consumption bundle does Harmon choose?
a.
(12.5, 12.5)
b.
(25, 12.5)
c.
(12.5, 25)
d.
(15, 10)
e.
None of the above.
17. Janet consumes x1 and x2 together in fixed proportions. She always consumes 2 units of x1 for every
unit x2. One utility function that describes her preferences is
a.
U(x1, x2) = 2x1x2.
b.
U(x1, x2) = 2x1 + x2.
c.
U(x1, x2) = x1 + 2x2.
d.
U(x1, x2) = min2x1, x2.
e.
U(x1, x2) = minx1, 2x2.
18. Oswald Odd consumes only goods 1 and 2. His utility function is U(x1, x2) = x1 + x2 + minx1, x2.
Each of Oswald’s indifference curves is
a.
L-shaped.
b.
made up of three line segments with slopes −2, −1, and −1/2.
c.
made up of two line segments with slopes −2 and −1/2.
d.
is smooth and has no kinks.
e.
is a diamond-shaped figure consisting of four line segments.
19. The absolute value of Mar’s MRS at his current consumption bundle is greater than 3. (That is,
MU1/MU2 3). Mars has convex preferences and is currently consuming positive amounts of both
goods.
a.
Taking away some of good 1 and giving Mars 3 units of good 2 for each unit of good 1
taken away will necessarily make him worse off.
b.
Taking away some of good 1 and giving Mars 3 units of good 2 for each unit of good 1
taken away will necessarily make him better off.
c.
Giving Mars some of good 1 and taking away 3 units of good 2 for each unit of good 1 he
is given will necessarily make him worse off.
d.
Giving Mars some of good 1 and taking away 3 units of good 2 for each unit of good 1 he
is given will necessarily make him better off.
e.
More than one of the above is true.
20. Isabella’s utility function is U(x, y) = 4minx, y + y. If we draw her indifference curves with x on the
horizontal axis and y on the vertical axis, these indifference curves are
a.
L-shaped with kinks where x = y.
b.
made up of two line segments that meet where x = y. One of these line segments is
horizontal and the other has slope −4.
c.
L-shaped with kinks where x = 5y.
d.
made up of two line segments that meet where x = 5y. One of these line segments is
vertical and the other has slope −1.
e.
V-shaped with kinks where x = 4y.
21. Emily’s utility function is U(x, y) = 3minx, y + y. If we draw her indifference curves with x on the
horizontal axis and y on the vertical axis, these indifference curves are
a.
made up of two line segments that meet where x = 4y. One of these line segments is
vertical and the other has slope −1.
b.
L-shaped with kinks where x = 4y.
c.
L-shaped with kinks where x = y.
d.
made up of two line segments that meet where x = y. One of these line segments is
horizontal and the other has slope −3.
e.
V-shaped with kinks where x = 3y.
22. Charlie has the utility function U(xA, xB) = xAxB. His indifference curve passing through 32 apples and 8
bananas will also pass through the point where he consumes 4 apples and
a.
16 bananas.
b.
32 bananas.
c.
68 bananas.
d.
72 bananas.
e.
64 bananas.
23. Charlie has the utility function U(xA, xB) = xAxB. His indifference curve passing through 35 apples and
18 bananas will also pass through the point where he consumes 5 apples and
a.
131 bananas.
b.
137 bananas.
c.
21 bananas.
d.
42 bananas.
e.
126 bananas.
24. Charlie’s utility function is U(A, B) = AB, where A and B are the numbers of apples and bananas,
respectively, that he consumes. When Charlie is consuming 15 apples and 90 bananas, if we put apples
on the horizontal axis and bananas on the vertical axis, the slope of his indifference curve at his current
consumption is
a.
215.
b.
−6.
c.
−12.
d.
−1/6.
e.
−1/12.
25. Charlie’s utility function is U(A, B) = AB, where A and B are the numbers of apples and bananas,
respectively, that he consumes. When Charlie is consuming 20 apples and 80 bananas, if we put apples
on the horizontal axis and bananas on the vertical axis, the slope of his indifference curve at his current
consumption is
a.
−8.
b.
−1/4.
c.
−20.
d.
−4.
e.
−1/8.
26. Ambrose has the utility function U(x1, x2) = 4x1/21 + x2. If Ambrose is initially consuming 64 units of
nuts and 10 units of berries, then what is the largest number of berries that he would be willing to give
up in return for an additional 17 units of nuts?
a.
9
b.
19
c.
4
d.
2
e.
1
27. Ambrose has the utility function U(x1, x2) = 4x1/21 + x2. If Ambrose is initially consuming 25 units of
nuts and 17 units of berries, then what is the largest number of berries that he would be willing to give
up in return for an additional 39 units of nuts?
a.
8
b.
12
c.
25
d.
6
e.
3
28. Joe Bob’s cousin Leonard consumes goods 1 and 2. Leonard thinks that 2 units of good 1 is always a
perfect substitute for 3 units of good 2. Which of the following utility functions is the only one that
would not represent Leonard’s preferences?
a.
U(x1, x2) = 3x1 + 2x2 + 1,000.
b.
U(x1, x2) = 9x21 + 12x1x2 + 4x22.
c.
U(x1, x2) = min3x1, 2x2.
d.
U(x1, x2) = 30x1 + 20x2 − 10,000.
e.
More than one of the above does not represent Leonard’s preferences.
29. Joe Bob’s cousin Peter consumes goods 1 and 2. Peter thinks that 4 units of good 1 is always a perfect
substitute for 2 units of good 2. Which of the following utility functions is the only one that would not
represent Peter’s preferences?
a.
U(x1, x2) = min2x1, 4x2.
b.
U(x1, x2) = 20x1 + 40x2 − 10,000.
c.
U(x1, x2) = 2x1 + 4x2 + 1,000.
d.
U(x1, x2) = 4x21 + 16x1x2 + 16x22.
e.
More than one of the above does not represent Peter’s preferences.
30. Harry Mazzola has the utility function U(x1, x2) = minx1 + 2x2, 2x1 + x2. He has $40 to spend on corn
chips and french fries. If the price of corn chips is $3 per unit and the price of french fries is $4, then
Harry will
a.
definitely spend all of his income on corn chips.
b.
definitely spend all of his income on french fries.
c.
consume at least as many corn chips as french fries but might consume both.
d.
consume at least as many french fries as corn chips but might consume both.
e.
consume equal amounts of french fries and corn chips.
31. Harry Mazzola has the utility function U(x1, x2) = minx1 + 2x2, 2x1 + x2. He has $40 to spend on corn
chips and french fries. If the price of corn chips is $1 per unit and the price of french fries is $4, then
Harry will
a.
consume at least as many french fries as corn chips but might consume both.
b.
consume at least as many corn chips as french fries but might consume both.
c.
definitely spend all of his income on corn chips.
d.
definitely spend all of his income on french fries.
e.
consume equal amounts of french fries and corn chips.
32. Phil Rupp’s sister Ethel has the utility function U(x, y) = min4x + y, 5y. Where x is measured on the
horizontal axis and y on the vertical axis, her indifference curves consist of a
a.
vertical line segment and a horizontal line segment which meet in a kink along the line
y = 4x.
b.
vertical line segment and a horizontal line segment which meet in a kink along the line
x = 4y.
c.
horizontal line segment and a negatively sloped line segment which meet in a kink along
the line x = y.
d.
positively sloped line segment and a negatively sloped line segment which meet along the
line x = y.
e.
horizontal line segment and a positively sloped line segment which meet in a kink along
the line x = 4y.
33. Phil Rupp’s sister Ethel has the utility function U(x, y) = min5x + y, 6y. Where x is measured on the
horizontal axis and y on the vertical axis, her indifference curves consist of a
a.
horizontal line segment and a negatively sloped line segment which meet in a kink along
the line x = y.
b.
vertical line segment and a horizontal line segment which meet in a kink along the line
y = 5x.
c.
positively sloped line segment and a negatively sloped line segment which meet along the
line x = y.
d.
vertical line segment and a horizontal line segment which meet in a kink along the line
x = 5y.
e.
horizontal line segment and a positively sloped line segment which meet in a kink along
the line x = 5y.
PROBLEM
1. Jim’s utility function is U(x, y) = xy. Jerry’s utility function is U(x, y) = 1,000xy + 2,000. Tammy’s
utility function is U(x, y) = xy(1 − xy). Oral’s utility function is U(x, y) = −1/(10 + 2xy). Marjoe’s
utility function is U(x, y) = x(y + 1,000). Pat’s utility function is U(x, y) = 0.5xy − 10,000. Billy’s
utility function is U(x, y) = x/y. Francis’s utility function is U(x, y) = −xy.
a. Who has the same preferences as Jim?
b. Who has the same indifference curves as Jim?
c. Explain why the answers to (a) and (b) differ.
2. A consumer has a utility function of the form U(x, y) = x a + y b, where both a and b are nonnegative.
What additional restrictions on the values of the parameters a and b are imposed by each of the
following assumptions?
a. Preferences are quasilinear and convex, and x is a normal good.
b. Preferences are homothetic.
c. Preferences are homothetic and convex.
d. Goods x and y are perfect substitutes.
3. Victor Finick likes to have the same amount of x as he has of y. His utility function is
U(x, y) = min2x− y, 2y− x.
a. Draw the indifference curve for Victor that passes through the bundle (0, 0) and the indifference
curve that passes through (4, 4). (Hint: Each indifference curve is the intersection of two line
segments.)
b. If Victor has a bundle that he likes better than (0, 0) and his consumption of both goods is doubled,
is Victor better off?
c. Does Victor always prefer more of either good to less?
4. Use separate graphs to sketch two indifference curves for people with each of the following utility
functions:
a. U(x, y) = x + 2y.
b. U(x, y) = minx,2y.
c. U(x, y) = maxx,2y.
5. Use separate graphs to draw indifference curves for each of the following utility functions:
a. U(x, y) = min2x + y,2y + x.
b. U(x, y) = max2x + y,2y + x.
c. U(x, y) = x + minx, y.
In which of these cases are preferences convex?