Chapter 4
Utility
True-False
Topic: Utility
% Correct Responses: 85
Correct Answer: False
Diculty: 2
Discrimination Index: 10
4.1 With quasilinear preferences, the slope of indi erence curves is constant along all rays through
the origin.
Topic: Utility
% Correct Responses: 82
Correct Answer: False
Diculty: 2
Discrimination Index: 45
4.2 Wanda Lott has the utility function U(x; y) = maxfx; yg. Wanda's preferences are convex.
Topic: Utility
% Correct Responses: 96
Correct Answer: True
Diculty: 1
Discrimination Index: 20
4.3 If someone has a utility function U = 2minfx; yg; then x and y are perfect complements for
that person.
Topic: Utility
% Correct Responses: 66
Correct Answer: False
Diculty: 1
Discrimination Index: 20
4.4 Maximilian consumes two goods x and y. His utility function is U(x; y) = maxfx; yg. Therefore x and y are perfect substitutes for Max.
Topic: Utility
% Correct Responses: 36
Correct Answer: False
Diculty: 3
Discrimination Index: 11
4.5 A person with the utility function U(x; y) = y + x has convex preferences.
2
Topic: Utility
% Correct Responses: 25
Correct Answer: True
Diculty: 3
Discrimination Index: 47
4.6 Mr. Surly consumes only two goods and hates them both. His utility function is U(x; y) =
,maxfx; yg. Mr. Surly has (weakly) convex preferences.
CHAPTER 4 Utility
Topic: Utility
% Correct Responses: 71
Correct Answer: True
42
Diculty: 2
Discrimination Index: 43
4.7 Angela's utility function is U(x ; x ) = (x + x ) . Her indi erence curves are downward1
sloping, parallel straight lines.
2
1
Topic: Utility
% Correct Responses: 81
Correct Answer: True
2
3
Diculty: 2
Discrimination Index: 0
4.8 Henrietta's utility function is U(x ; x ) = x x . She has diminishing marginal rate of substi1
tution between goods 1 and 2.
2
1 2
Topic: Utility
% Correct Responses: 66
Correct Answer: False
Diculty: 2
Discrimination Index: 8
4.9 Alice's utility function is U(x; y) = x y. Steve's utility function is U(x; y) = x y + 2x. Alice
2
2
and Steve have the same preferences since Steve's utility function is a monotonic transformation of
Alice's.
Topic: Utility
% Correct Responses: 78
Correct Answer: False
Diculty: 3
Discrimination Index: 29
4.10 Jean's utility function is U(x; y) = x + y , y. If we draw her indi erence curves with x
2
on the horizontal axis and y on the vertical axis, then these indi erence curves are everywhere
downward-sloping and get atter as one moves from left to right.
Topic: Preferences
% Correct Responses: 99
Correct Answer: True
Diculty: 1
Discrimination Index: 4
4.11 The utility function U(x ; x ) = 2ln x + 3ln x represents Cobb-Douglas preferences.
1
2
1
2
TRUE-FALSE
Topic: Preferences
% Correct Responses: 92
Correct Answer: False
43
Diculty: 1
Discrimination Index: 29
4.12 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.
Topic: Utility
% Correct Responses: 93
Correct Answer: False
Diculty: 1
Discrimination Index: 16
4.13 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.
Topic: Utility
% Correct Responses: 63
Correct Answer: True
Diculty: 3
Discrimination Index: 80
4.14 A consumer has preferences represented by the utility function U(x ; x ) = 10(x + 2x x +
1
x ) , 50. For this consumer, goods 1 and 2 are perfect substitutes.
2
2
Topic: Utility
% Correct Responses: 52
Correct Answer: True
2
2
1
1 2
Diculty: 1
Discrimination Index: 38
4.15 A person with utility function U(x; y) = 5 + y + 2x has non-convex preferences.
2
Topic: Utility
% Correct Responses: 64
Correct Answer: False
Diculty: 2
Discrimination Index: 26
4.16 A person with the utility function U(x; y) = 10 + y + x has convex preferences.
2
Topic: Utility
% Correct Responses: 43
Correct Answer: True
Diculty: 2
Discrimination Index: 29
4.17 A person with the utility function U(x ; x ) = minfx + 2x ; 2x + x g has convex, but not
strictly convex preferences.
1
2
1
2
1
2
CHAPTER 4 Utility
Topic: Utility
% Correct Responses: 0
Correct Answer: False
44
Diculty: 0
Discrimination Index: 0
4.18 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.
Topic: Preferences
% Correct Responses: 0
Correct Answer: False
Diculty: 0
Discrimination Index: 0
4.19 Quasilinear preferences are homothetic when the optimal amount of good 1 is not a ordable.
Utility
Multiple Choice
Topic: Utility
% Correct Responses: 28
Correct Answer: A
Diculty: 2
Discrimination Index: 17
4.1 Ike's utility function is U(x; y) = 25xy. He has 12 units of good x and 8 units of 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 guy's bundle to his own.
(d) Neither prefers the other guy's bundle to his own.
(e) Since they have di erent preferences, there is not enough information to determine who envies whom.
Topic: Utility
% Correct Responses: 28
Correct Answer: A
Diculty: 2
Discrimination Index: 17
4.2 Nick's utility function is U(x; y) = 33xy. He has 12 units of good x and 6 units of y. Boris's
utility function for the same two goods is U(x; y) = 2x+5y. Boris has 9 units of x and 13 units of y.
(a) Nick prefers Boris's bundle to his own bundle, but Boris prefers his own bundle to Nick's.
(b) Boris prefers Nick's bundle to his own, but Nick prefers his own bundle to Boris's.
(c) Each prefers the other guy's bundle to his own.
(d) Neither prefers the other guy's bundle to his own.
(e) Since they have di erent preferences, there is not enough information to determine who envies whom.
Topic: Utility
% Correct Responses: 58
Correct Answer: C
Diculty: 2
Discrimination Index: 27
4.3 Tim has preferences represented by the utility function: U(x; y) = minf6x + y; x + 2yg. If x is
on the horizontal axis and y is on the vertical axis, what is the slope of his indi erence curve at
the point (8; 9)?
(a) The slope is ,6.
(b) The slope is ,2=6.
(c) The slope is ,1=2.
(d) The slope is ,1=6.
(e) The slope is ,8=9.
CHAPTER 4 Utility
Topic: Utility
% Correct Responses: 58
Correct Answer: C
46
Diculty: 2
Discrimination Index: 27
4.4 Chen has preferences represented by the utility function: U(x; y) = minf4x + y; x + 6yg. If x
is on the horizontal axis and y is on the vertical axis, what is the slope of his indi erence curve at
the point (9; 4)?
(a) The slope is ,4.
(b) The slope is ,6=4.
(c) The slope is ,1=6.
(d) The slope is ,1=4.
(e) The slope is ,9=4.
Topic: Utility
% Correct Responses: 95
Correct Answer: C
Diculty: 1
Discrimination Index: 9
4.5 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 o as before?
(a) 30 units of good y
(b) 30 units of good y
(c) 27 units of good y
(d) 18 units of good y
(e) None of the above
Topic: Utility
% Correct Responses: 95
Correct Answer: C
Diculty: 1
Discrimination Index: 9
4.6 Sheila has preferences represented by the utility function U(x; y) = 8x + 4y. She consumes 12
units of good x and 3 units of good y. If her consumption of good x is lowered to 10, how many
units of y must she have in order to be exactly as well o as before?
(a) 12 units of good y
(b) 10 units of good y
(c) 7 units of good y
(d) 5 units of good y
(e) None of the above
MULTIPLE CHOICE
Topic: Utility
% Correct Responses: 0
Correct Answer: B
47
Diculty: 3
Discrimination Index: 0
4.7 Mac's utility function is U(x; y) = maxf2x , y; 2y , xg.
(a) Mac's preferences are quasi-linear.
(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.
Topic: Utility
% Correct Responses: 30
Correct Answer: C
Diculty: 2
Discrimination Index: 29
4.8 Charles' 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
Topic: Utility
% Correct Responses: 55
Correct Answer: E
Diculty: 3
Discrimination Index: 67
4.9 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 indi erence curves that are rectangular hyperbolas.
(b) Raymond prefers more of each good to less.
(c) Raymond has quasi-linear preferences.
(d) Raymond has a bliss point.
(e) Raymond has indi erence curves that are upward sloping straight lines if y > 0.
CHAPTER 4 Utility
Topic: Utility
% Correct Responses: 92
Correct Answer: C
48
Diculty: 2
Discrimination Index: 17
4.10 Molly's utility function is U(x; y) = y + 4x: . She has 25 units of x and 12 units of y. If her
5
consumption of x is reduced to 0, how many units of y would she need in order to be exactly as
well o as before?
(a) 48 units.
(b) 37 units.
(c) 32 units.
(d) 112 units.
(e) None of the above.
Topic: Utility
% Correct Responses: 21
Correct Answer: E
Diculty: 2
Discrimination Index: 13
4.11 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 gives up 5 units of x for every
unit of y he gets.
(b) Waldo would be willing to trade away all of his x for y so long he gets more than 5 units of y for every
unit of x he gives up.
(c) Waldo likes x and y equally well so he is always willing to exchange 1 unit of either good for more than
one 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.
MULTIPLE CHOICE
Topic: Utility
% Correct Responses: 21
Correct Answer: E
49
Diculty: 2
Discrimination Index: 13
4.12 Ernie's utility function is U(x; y) = xy. Ernie consumes 4 units of x and 16 units of y.
(a) Ernie would be willing to make small exchanges of x for y in which he gives up 4 units of x for every
unit of y he gets.
(b) Ernie would be willing to trade away all of his x for y so long he gets more than 4 units of y for every
unit of x he gives up.
(c) Ernie likes x and y equally well so he is always willing to exchange 1 unit of either good for more than
one unit of the other.
(d) Ernie 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.
Topic: Utility
% Correct Responses: 16
Correct Answer: B
Diculty: 2
Discrimination Index: 0
4.13 Henry's utility function is x + 16xw + 64w where x is his consumption of x and w is his
consumption of w.
2
2
(a) Henry's preferences are nonconvex.
(b) Henry's indi erence curves are straight lines.
(c) Henry has a bliss point.
(d) Henry's indi erence curves are hyperbolas.
(e) None of the above.
Topic: Utility
% Correct Responses: 92
Correct Answer: D
Diculty: 1
Discrimination Index: 37
4.14 Josephine's utility function is U(x; y) = y + 5x: . She has one unit of x and two units of y. If
5
her consumption of x is reduced to zero, how much y must she have in order to be exactly as well
o as before?
(a) 14 units.
(b) 9 units.
(c) 11 units.
(d) 7 units.
(e) None of the above.
CHAPTER 4 Utility
Topic: Utility
% Correct Responses: 10
Correct Answer: C
50
Diculty: 2
Discrimination Index: 0
4.15 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 indi erence 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) There is no truth in any of the above statements.
Topic: Utility
% Correct Responses: 74
Correct Answer: A
Diculty: 1
Discrimination Index: 15
4.16 Harmon's utility function is U(x ; x ) = x x . His income is $100, the price of good 2 is
1
2
1
2
p2 = 4. Good 1 is priced as follows. The rst 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.
Topic: Utility
% Correct Responses: 61
Correct Answer: E
Diculty: 1
Discrimination Index: 37
4.17 Janet consumes x and x together in xed proportions. She always consumes 2 units of x
1
2
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) = minf2x1; x2g
(e) U(x1; x2) = minfx1 ; 2x2g
1
MULTIPLE CHOICE
Topic: Utility
% Correct Responses: 14
Correct Answer: C
51
Diculty: 3
Discrimination Index: 36
4.18 Oswald Odd consumes only goods 1 and 2. His utility function is U(x ; x ) = x + x +
minfx1 ; x2g. Each of Oswald's indi erence curves is:
1
2
1
2
(a) L-shaped.
(b) made up of 3 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 gure consisting of 4 line segments.
Topic: Utility
% Correct Responses: 4
Correct Answer: A
Diculty: 3
Discrimination Index: 0
4.19 The absolute value of Mars' 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 o .
(b) Taking away some Good 1 and giving Mars 3 units of Good 2 for each unit of Good 1 taken away will
necessarily make him better o .
(c) Giving Mars some Good 1 and taking away 3 units of Good 2 for each unit of Good 1 he is given will
necessarily make him worse o .
(d) Giving Mars some Good 1 and taking away 3 units of Good 2 for each unit of Good 1 he is given will
necessarily make him better o .
(e) More than one of the above is true.
CHAPTER 4 Utility
Topic: Utility
% Correct Responses: 48
Correct Answer: B
52
Diculty: 3
Discrimination Index: 84
4.20 Isabella's utility function is U(x; y) = 4minfx; yg + y. If we draw her indi erence curves with
x on the horizontal axis and y on the vertical axis, these indi erence 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.
Topic: Utility
% Correct Responses: 48
Correct Answer: B
Diculty: 3
Discrimination Index: 84
4.21 Penelope's utility function is U(x; y) = 2minfx; yg + y. If we draw her indi erence curves
with x on the horizontal axis and y on the vertical axis, these indi erence 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 ,2.
(c) L-shaped with kinks where x = 3y.
(d) made up of two line segments that meet where x = 3y. One of these line segments is vertical and the
other has slope ,1.
(e) V-shaped with kinks where x = 2y.
Topic: Utility
% Correct Responses: 94
Correct Answer: E
Diculty: 1
Discrimination Index: 20
4.22 Charlie has the utility function U(xA; xB ) = xAxB . His indi erence 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.
MULTIPLE CHOICE
Topic: Utility
% Correct Responses: 94
Correct Answer: E
53
Diculty: 1
Discrimination Index: 20
4.23 Charlie has the utility function U(xA; xB ) = xAxB . His indi erence curve passing through
10 apples and 35 bananas will also pass through the point where he consumes 2 apples and:
(a) 35 bananas.
(b) 70 bananas.
(c) 177 bananas.
(d) 182 bananas.
(e) 175 bananas.
Topic: Utility
% Correct Responses: 95
Correct Answer: B
Diculty: 0
Discrimination Index: 47
4.24 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 indi erence
curve at his current consumption is:
(a) ,15.
(b) ,6.
(c) ,12.
(d) ,1=6.
(e) ,1=12.
Topic: Utility
% Correct Responses: 95
Correct Answer: B
Diculty: 0
Discrimination Index: 47
4.25 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 25 apples and 100 bananas, if
we put apples on the horizontal axis and bananas on the vertical axis, the slope of his indi erence
curve at his current consumption is:
(a) ,25.
(b) ,4.
(c) ,8.
(d) ,1=4.
(e) ,1=8.
CHAPTER 4 Utility
Topic: Utility
% Correct Responses: 53
Correct Answer: C
54
Diculty: 2
Discrimination Index: 49
4.26 Ambrose has the utility function U(x ; x ) = 4x = + x . If Ambrose is initially consuming
1
2
1 2
1
2
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
Topic: Utility
% Correct Responses: 53
Correct Answer: C
Diculty: 2
Discrimination Index: 49
4.27 Ambrose has the utility function U(x ; x ) = 4x = + x . If Ambrose is initially consuming 4
1
2
1 2
1
2
units of nuts and 18 units of berries, then what is the largest number of berries that he would be
willing to give up in return for an additional 32 units of nuts.
(a) 6
(b) 24
(c) 16
(d) 8
(e) 4
Topic: Utility
% Correct Responses: 60
Correct Answer: C
Diculty: 2
Discrimination Index: 52
4.28 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 + 1000.
(b) U(x1 ; x2) = 9x21 + 12x1x2 + 4x22.
(c) U(x1; x2) = Minf3x1; 2x2g.
(d) U(x1 ; x2) = 30x1 + 20x2 , 10; 000.
(e) More than one of the above does NOT represent Leonard's preferences.
MULTIPLE CHOICE
Topic: Utility
% Correct Responses: 60
Correct Answer: C
55
Diculty: 2
Discrimination Index: 52
4.29 Joe Bob's cousin Tim consumes goods 1 and 2. Tim 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 Tim's preferences?
(a) U(x1; x2) = 2x1 + 4x2 + 1000.
(b) U(x1 ; x2) = 4x21 + 16x1x2 + 16x22.
(c) U(x1; x2) = Minf2x1; 4x2g.
(d) U(x1 ; x2) = 20x1 + 40x2 , 10; 000.
(e) More than one of the above does NOT represent Tim's preferences.
Topic: Utility
% Correct Responses: 0
Correct Answer: E
Diculty: 0
Discrimination Index: 0
4.30 Harry Mazzola has the utility function U(x ; x ) = minfx + 2x ; 2x + x g. He has $40 to
1
2
1
2
1
2
spend on corn chips and french fries, if the price of corn chips is 3 dollar(s) per unit and the price
of french fries is 4 dollars per unit, then Harry will:
(a) de nitely spend all of his income on corn chips.
(b) de nitely spend all of his income on french fries.
(c) consume at least as much corn chips as french fries, but might consume both.
(d) consume at least as much french fries as corn chips, but might consume both.
(e) consume equal amounts of french fries and corn chips.
Topic: Utility
% Correct Responses: 0
Correct Answer: E
Diculty: 0
Discrimination Index: 0
4.31 Harry Mazzola has the utility function U(x ; x ) = minfx + 2x ; 2x + x g. He has $40 to
1
2
1
2
1
2
spend on corn chips and french fries, if the price of corn chips is 3 dollar(s) per unit and the price
of french fries is 5 dollars per unit, then Harry will:
(a) de nitely spend all of his income on corn chips.
(b) de nitely spend all of his income on french fries.
(c) consume at least as much corn chips as french fries, but might consume both.
(d) consume at least as much french fries as corn chips, but might consume both.
(e) consume equal amounts of french fries and corn chips.
CHAPTER 4 Utility
Topic: Utility
% Correct Responses: 0
Correct Answer: C
56
Diculty: 0
Discrimination Index: 0
4.32 Phil Rupp's sister Ethel has the utility function U(x; y) = minf4x + y; 5yg. Where x is measured on the horizontal axis and y on the vertical axis, her indi erence curves:
(a) consist of a vertical line segment and a horizontal line segment which meet in a kink along the line y = 4x.
(b) consist of a vertical line segment and a horizontal line segment which meet in a kink along the line
x = 4y.
(c) consist of a horizontal line segment and a negatively sloped line segment which meet in a kink along the
line x = y.
(d) consist of a positively sloped line segment and a negatively sloped line segment which meet along the
line x = y.
(e) consist of a horizontal line segment and a positively sloped line segment which meet in a kink along the
line x = 4y.
Topic: Utility
% Correct Responses: 0
Correct Answer: C
Diculty: 0
Discrimination Index: 0
4.33 Phil Rupp's sister Ethel has the utility function U(x; y) = minf5x + y; 6yg. Where x is measured on the horizontal axis and y on the vertical axis, her indi erence curves:
(a) consist of a vertical line segment and a horizontal line segment which meet in a kink along the line y = 5x.
(b) consist of a vertical line segment and a horizontal line segment which meet in a kink along the line
x = 5y.
(c) consist of a horizontal line segment and a negatively sloped line segment which meet in a kink along the
line x = y.
(d) consist of a positively sloped line segment and a negatively sloped line segment which meet along the
line x = y.
(e) consist of a horizontal line segment and a positively sloped line segment which meet in a kink along the
line x = 5y.
Utility
Essay
Topic: Utility
% Correct Responses: 0
Diculty: 1
Discrimination Index: 0
4.1 Jim's utility function is U(x; y) = xy. Jerry's utility function is U(x; y) = 1000xy + 2000.
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 + 1000). Pat's utility function is U(x; y) = :5xy , 10000.
Billy's utility function is U(x; y) = x=y. F Francis's utility function is U(x; y) = ,xy. (a) Who has
the same preferences as Jim? (b) Who had the same indi erence curves as Jim? (c) Explain why
the answers to (a) and (b) di er.
Answer: Jerry, Pat and Oral have the same preferences as Jim since their utility functions are
monotonic transformations of Jim's. Jerry, Pat, Oral, Tammy, and Francis have the same indifference curves as Jim, but Tammy and Francis have di erent preferences. Francis' utility function
is a decreasing transformation of Jim's, so he orders his indi erence curves in the opposite way.
Tammy's utility function is a transformation of Jim's but is sometimes increasing sometimes decreasing.
Topic: Utility
% Correct Responses: 0
Diculty: 3
Discrimination Index: 0
4.2 A consumer has a utility function of the form U(x; y) = xa + yb 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? (i) Preferences are quasi-linear, convex, and x is a normal good. (ii)
Preferences are homothetic. (iii) Preferences are homothetic and convex. (iv) Goods x and y are
perfect substitutes.
Answer: (i) a = 1 and b is between 0 and 1. (ii) a = b. (iii) a = b and a is between 0 and 1.(iv)
a = b = 1.
Topic: Utility
% Correct Responses: 0
Diculty: 3
Discrimination Index: 0
4.3 Victor Finick likes to have the same amount of x as he has of y. His utility function is
U(x; y) = minf2x , y; 2y , xg. a) Draw the indi erence curve for Victor that passes through the
bundle (0; 0) and the indi erence curve that passes through (4; 4). (Hint: Each indi erence 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 o ? c) Does Victor always prefer more
of either good to less?
Answer: Victor's indi erence curves are V-shaped. The one through the origin consists of the two
rays y = 2x and x = 2y. The one through (2; 2) has two rays going out from (2; 2) ,, one with slope
1=2; and the other with slope 2. b) Yes c) No. If x > y; then an increase in x by itself makes him
worse o and if y > x; an increase in y by itself makes him worse o .
CHAPTER 4 Utility
Topic: Utility
% Correct Responses: 0
58
Diculty: 1
Discrimination Index: 0
4.4 Use separate graphs to sketch two indi erence curves for people with each of the following
utility functions: a) U(x; y) = x + 2y b) U(x; y) = Minfx; 2yg c) U(x; y) = maxfx; 2yg
Answer: a) These are straight lines with slope ,1=2. b) These are L-shaped. The corners lie along
the locus x = 2y. c) A typical indi erence curve consists of a horizontal line from the y axis to the
locus x = 2y and then a vertical line to the y axis from the point where the horiontal line met the
line x = 2y.
Topic: Utility
% Correct Responses: 0
Diculty: 2
Discrimination Index: 0
4.5 Use separate graphs to draw indi erence curves for each of the following utility functions: a)
U(x; y) = minf2x + y; 2y + xg
b) U(x; y) = maxf2x + y; 2y + xg. c) U(x; y) = x + minfx; yg. In which of these cases are preferences
convex?
Answer: If you take a point on the line x = y and draw two lines through it, one with a slope
of ,1=2 and the other with a slope of ,2; the outer envelope of these lines will be an indi erence
curve for a) and the inner envelope will be an indi erence curve for b). The indi erence curves for
c) passing through a point on the line x = y consist of a line segment going down and to the right
with slope ,1 and a line segment going up and to the left with slope ,2. Cases a) and c) display
convex preferences and case b) does not.