Deck 3: Utility

Question 1

If someone has a utility function U = 2 min{x, y}, then x and y are perfect complements for that person.

Answer 1

True

Question 2

A person with the utility function U(x₁, x₂) = min{x₁ + 2x₂, 2x₁ + x₂} has convex but not strictly convex preferences.

Answer 2

False

Question 3

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.

Answer 3

False

Question 4

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.

Question 5

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.

Question 6

Mr. Surly consumes only two goods and hates them both. His utility function is U(x, y) = 2max{x, y}. Mr. Surly has (weakly) convex preferences.

Question 7

A person with the utility function U(x, y) = 10 + y² + x has convex preferences.

Question 8

A person with the utility function U(x, y) = y + x² has convex preferences.

Question 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.

Question 10

Angela's utility function is U(x₁, x₂) = (x₁ + x₂)³. Her indifference curves are downward-sloping, parallel straight lines.

Question 11

Quasilinear preferences are homothetic when the optimal amount of good 1 is not affordable.

Question 12

A person with utility function U(x, y) = 5 + y² + 2x has nonconvex preferences.

Question 13

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.

Question 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.

Question 15

The utility function U(x₁, x₂) = 2lnx₁ + 3lnx₂ represents Cobb-Douglas preferences.

Question 16

With quasilinear preferences, the slope of indifference curves is constant along all rays through the origin.

Question 17

Wanda Lott has the utility function U(x, y) = max{x, y}. Wanda's preferences are convex.

Question 18

Henrietta's utility function is U(x₁, x₂) = x₁x₂. She has diminishing marginal rate of substitution between goods 1 and 2.

Question 19

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.

Question 20

Maximilian consumes two goods, x and y. His utility function is U(x, y) = max{x, y}. Therefore x and y are perfect substitutes for Max.

Question 21

Jean-Pierre has preferences represented by the utility function U(x, y) = min{ 2x + 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

Question 22

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.

Question 23

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.

Question 24

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) 30
B) 30
C) 27
D) 18
E) None of the above.

Question 25

Emily's utility function is U(x, y) = 3min{x, 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.

Question 26

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 . Fergie's utility function is xy - 10,000. Margaret's utility function is . 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

Question 27

The absolute value of Mar's MRS at his current consumption bundle is greater than 3. (That is, ). 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.

Question 28

Mac's utility function is U(x, y) = max{ 2x - 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.

Question 29

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.

Question 30

Raymond's preferences are represented by the utility function U(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.

Question 31

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 - . Billy's utility function is . 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.

Question 32

Isabella's utility function is U(x, y) = 4min{x, 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.

Question 33

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.

Question 34

Tim has preferences represented by the utility function U(x, y) = min{ 6x + 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

Question 35

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.

Question 36

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.

Question 37

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.

Question 38

Oswald Odd consumes only goods 1 and 2. His utility function is U(x₁, x₂) = x₁ + x₂ + min{x₁, x₂}. Each of Oswald's indifference curves is

A) L-shaped.
B) made up of three line segments with slopes -2, -1, and -.
C) made up of two line segments with slopes -2 and -.
D) is smooth and has no kinks.
E) is a diamond-shaped figure consisting of four line segments.

Question 39

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) = min{ 2x₁, x₂}.
E) U(x₁, x₂) = min{x₁, 2x₂}.

Question 40

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.

Question 41

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.

Question 42

Phil Rupp's sister Ethel has the utility function U(x, y) = min{ 4x + 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.

Question 43

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 . 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) = . 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.

Question 44

Ambrose has the utility function . 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

Question 45

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) = min{ 3x₁, 2x₂}.
D) U(x₁, x₂) = 30x₁ + 20x₂ - 10,000.
E) More than one of the above does not represent Leonard's preferences.

Question 46

Use separate graphs to draw indifference curves for each of the following utility functions:
a. U(x, y) = min{2x + y,2y + x}.
b. U(x, y) = max{2x + y,2y + x}.
c. U(x, y) = x + min{x, y}.
In which of these cases are preferences convex?

Question 47

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.

Question 48

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) = min{ 2x₁, 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.

Question 49

Ambrose has the utility function . 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

Question 50

Charlie has the utility function U(x_A, x_B) = x_Ax_B. 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.

Question 51

Victor Finick likes to have the same amount of x as he has of y. His utility function is U(x, y) = min{2x- 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?

Question 52

Phil Rupp's sister Ethel has the utility function U(x, y) = min{ 5x + 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.

Question 53

Charlie has the utility function U(x_A, x_B) = x_Ax_B. 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.

Question 54

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) = min{x,2y}.
c. U(x, y) = max{x,2y}.

Question 55

Harry Mazzola has the utility function U(x₁, x₂) = min{x₁ + 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.

Question 56

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.

Question 57

Harry Mazzola has the utility function U(x₁, x₂) = min{x₁ + 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.