CHAPTER 5: Choice
TRUE/FALSE
1. At a boundary optimum, a consumer’s indifference curve must be tangent to her budget line.
2. Max Gross has the utility function U(x, y) = maxx, y. If the price of x is the same as the price of y,
Max will buy equal amounts of x and y.
3. If a consumer does not have convex preferences, then a point of tangency between her indifference
curve and her budget line must be an optimal consumption point.
4. Sharon spends all of her income on peaches and strawberries. Peaches are a normal good for her. Her
income increased by 20 percent and prices did not change. Her consumption of strawberries could not
have increased by more than 20 percent.
5. Clara’s utility function is U(x, y) = (x + 2)(y + 1). If her consumption of both x and y are doubled, then
her marginal rate of substitution between x and y remains constant.
6. Charlie’s utility function is U(x, y) = xy2. His marginal rate of substitution between x and y does not
change if the amount of both goods doubles.
7. Ambrose’s utility function is U(x, y) = x + 4y1/2. The price of x is $1 and the price of y is $2. If his
income rises from $100 to $150, his consumption of y increases by more than 10% but less than 50%.
8. Linus has utility function U(x, y) = x + 2y. If the price of x is $1 and the price of y is .50 cents then
Linus must consume equal amounts of both goods in order to maximize his utility.
9. Mary Granola’s utility function is U(x, y) = minx + 2y, y + 2x. Mary maximizes her utility subject to
a budget constraint. If she chooses the bundle (5, 6), then the price of x must be exactly twice the price
of y.
10. Millie’s utility function is U(x, y) = minx, y. She maximizes her utility subject to a budget constraint.
The price of x is the same as the price of y. If the price of x rises and the price of y and her income
remain constant, then her consumption of y will certainly decrease.
11. Other things being equal, a lump sum tax is at least as good for a consumer as a sales tax that collects
the same revenue from him.
12. If a consumer doesn’t consume any snails but does consume Big Macs, then his marginal rate of
substitution between snails and Big Macs when his snail consumption is zero must be equal to the ratio
of the price of snails to the price of Big Macs.
MULTIPLE CHOICE
1. Hans has $27 which he decides to spend on x and y. Commodity x costs $16 per unit and commodity y
costs $10 per unit. He has the utility function U(x, y) = 5x2 + 2y2 and he can purchase fractional units
of x and y. Hans will choose
a.
only x.
b.
only y.
c.
some of each commodity but more y than x.
d.
some of each commodity but more x than y.
e.
equal amounts of the two commodities.
2. George has $49 which he decides to spend on x and y. Commodity x costs $5 per unit and commodity
y costs $11 per unit. He has the utility function U(x, y) = 3x2 + 6y2 and he can purchase fractional units
of x and y. George will choose
a.
only y.
b.
some of each commodity but more x than y.
c.
only x.
d.
some of each commodity but more y than x.
e.
equal amounts of the two commodities.
3. Wanda Littlemore’s utility function is U(x, y) = x + 46y − 2y2. Her income is $135. If the price of x is
$1 and the price of y is $18, how many units of good x will Wanda demand?
a.
5
b.
12
c.
16
d.
0
e.
9
4. Wanda Littlemore’s utility function is U(x, y) = x + 47y − 3y2. Her income is $107. If the price of x is
$1 and the price of y is $23, how many units of good x will Wanda demand?
a.
11
b.
19
c.
0
d.
18
e.
15
5. Henri’s utility function is minx, 5y + 2z. The price of x is $1, the price of y is $15, and the price of z
is $7. Henri’s income is $44. How many units of x does Henri demand?
a.
9.78
b.
11
c.
5
d.
3
e.
None of the above.
6. Leo’s utility function is minx, 3y + 2z. The price of x is $1, the price of y is $9, and the price of z is
$8. Leo’s income is $8. How many units of x does Leo demand?
a.
2
b.
1.60
c.
5
d.
7
e.
None of the above.
7. Peter consumes no commodities other than Miller Lite and Bud Light. His annual budget for these two
commodities is described by the equation 5x + 30y = 300, where x is sixpacks of Miller Lite and y is
cases of Bud Light. Peter considers 2 cases of Bud Light to be perfect substitutes for 6 sixpacks of
Miller Lite.
a.
He will consume 60 sixpacks of Miller Lite per year.
b.
He will consume 10 cases of Bud Light per year.
c.
He will consume 14 cases of Bud Light per year.
d.
He will consume 12 sixpacks of Miller Lite per year.
e.
He is indifferent between any two bundles that use up his entire income.
8. Roger consumes no commodities other than Miller Lite and Bud Light. His annual budget for these
two commodities is described by the equation 5x + 25y = 300, where x is sixpacks of Miller Lite and y
is cases of Bud Light. Roger considers 2 cases of Bud Light to be perfect substitutes for 6 sixpacks of
Miller Lite.
a.
He will consume 12 cases of Bud Light per year.
b.
He will consume 12 sixpacks of Miller Lite per year.
c.
He will consume 16 cases of Bud Light per year.
d.
He will consume 60 sixpacks of Miller Lite per year.
e.
He is indifferent between any two bundles that use up his entire income.
9. Paul’s utility function is minx + 3y, 3x + y. Simon’s utility function is min3x + 9y, 9x + 3y. Paul
and Simon have the same income and face the same prices.
a.
Paul and Simon will demand the same amount of good x.
b.
Paul will demand more of good y than Simon.
c.
Simon will demand more of good y than Paul.
d.
Each will prefer the other’s consumption bundle to his own.
e.
None of the above.
10. Mary Granola consumes tomatoes and nectarines. Mary’s indifference curves are kinky. When she is
consuming more tomatoes than nectarines, she is just willing to trade 3 tomatoes for 1 nectarine. When
she is consuming more nectarines than tomatoes, she is just willing to trade 4 nectarines for 1 tomato.
Let P1 be the price of nectarines, and P2 the price of tomatoes. Mary maximizes her utility subject to
her budget constraint. (Hint: Sketch one of her indifference curves.)
a.
When P1 P2, she must consume only tomatoes.
b.
When P1 P2, she must consume 3 times as many tomatoes as nectarines.
c.
When P1 3P2, she must consume only tomatoes.
d.
When 4P1 P2, she must consume only nectarines.
e.
She must consume equal numbers of both.
11. Mary Granola consumes apples and uglifruits. Mary’s indifference curves are kinky. When she is
consuming more apples than uglifruits, she is just willing to trade 3 apples for 1 uglifruit. When she is
consuming more uglifruits than apples, she is just willing to trade 4 uglifruits for 1 apple. Let P1 be the
price of uglifruits and P2 the price of apples. Mary maximizes her utility subject to her budget
constraint. (Hint: Sketch one of her indifference curves.)
a.
When P1 3P2, she must consume only apples.
b.
When P1 P2, she must consume 3 times as many apples as uglifruits.
c.
When 4P1 P2, she must consume only uglifruits.
d.
When P1 P2, she must consume only apples.
e.
She must consume equal numbers of both.
12. Badger Madison consumes only beer and sausages. His income is $100. Beer costs him $.50 per can
and sausages cost $1 each. Where x is the number of cans of beer and y the number of sausages he
consumes per week, Badger’s utility function is U(x, y) = −[(x − 50)2 + (y − 40)2].
a.
Badger must always be unhappy since whatever he consumes, his utility is negative.
b.
He has monotonic preferences.
c.
If his income increases, he won’t change the commodity bundle that he buys.
d.
If the price of beer goes down, he will buy more beer.
e.
More than one of the above statements is true.
13. Janet consumes two commodities x and y. Her utility function is minx + 2y, y + 2x. She chooses to
buy 10 units of good x and 20 units of good y. The price of good x is $1. Janet’s income is
a.
$40.
b.
$50.
c.
$30.
d.
$20.
e.
There is not enough information in the problem to determine her income because we are
not told the price of good y.
14. Martha’s utility function is U(x, y) = minx + 2y, 2x + y. George’s utility function is
U(x, y) = min2x + 4y, 4x + 2y. If George and Martha have the same income and face the same prices
for the goods x and y,
a.
George and Martha will both demand the same amount of y.
b.
Martha will always prefer George’s consumption bundle to her own.
c.
George will always prefer Martha’s consumption bundle to his own.
d.
George will demand more x than Martha demands.
e.
None of the above.
15. Ollie has a utility function U(x, y) = (x + 2)(y + 3). The price of x is $1 and the price of y is $1. When
he maximizes his utility subject to his budget constraint, he consumes positive amounts of both goods.
Ollie consumes
a.
exactly as many units of x as of y.
b.
1 more unit of x than he consumes of y.
c.
1 more unit of y than he consumes of x.
d.
2 more units of x than he consumes of y.
e.
None of the above.
16. Charlie has a utility function U(x, y) = (x + 3)(y + 4). The price of x is $1 and the price of y is $1.
When he maximizes his utility subject to his budget constraint, he consumes positive amounts of both
goods. Charlie consumes
a.
2 more units of x than he consumes of y.
b.
1 more unit of x than he consumes of y.
c.
exactly as many units of x as of y.
d.
1 more unit of y than he consumes of x.
e.
None of the above.
17. Danny Featherweight is taking a tough course in law school. His professor agreed to give him a course
grade of max2x, 3y where x and y are the number of answers he gets right on the first and second
midterms, respectively. Danny needs a course grade of 150 to pass. He finds that for the first midterm,
for every A minutes of study, he will get one more answer right. For the second midterm, for every B
minutes that he studies, he will get one more answer right. If he doesn’t study at all, Danny will get
nothing right on either exam. All Danny cares about is passing. He doesn’t want to waste any time
getting a higher grade than he needs.
a.
If A/B 2/3, then Danny will not study for the second exam.
b.
The ratio of the time Danny spends on exam 1 to the time he spends on exam 2 will be
2A/3B.
c.
The ratio of the time Danny spends on exam 2 to the time he spends on exam 1 will be
3A/2B.
d.
If A B, then Danny will not study for the first exam.
e.
Danny will spend 150/(2A + 3B) minutes studying for each exam.
18. Isobel consumes positive quantities of both jam and juice. The price of jam is 5 cents per unit and the
price of juice is 10 cents per unit. Her marginal utility of jam is 10 and her marginal utility of juice is
5.
a.
Without changing her total expenditures, she could increase her utility by consuming more
jam and less juice.
b.
Without changing her total expenditures, she could increase her utility by consuming more
juice and less jam.
c.
Without changing her total expenditures on jam and juice, she could not increase her
utility.
d.
She should spend more money on both jam and juice.
e.
We can’t tell whether any of the other statements are true or false without knowing the
quantities she consumes.
19. Harold’s utility function is U(x, y) = (x + 3)(y + 2). The price of x is $1. The price of y is $2. At all
incomes for which Harold consumes positive amounts of both goods, he will consume
a.
more than twice as many units of x as of y.
b.
more than twice as many units of y as of x.
c.
1 more unit of x than of y.
d.
1 more unit of y than of x.
e.
2/3 as many units of y as of x.
20. Jane’s utility function is U(x, y) = x + 2y, where x is her consumption of good X and y is her
consumption of good Y. Her income is $2. The price of Y is $2. The cost per unit of X depends on how
many units she buys. The total cost of x units of X is the square root of x.
a.
The bundle (1/4, 3/4) is Jane’s utility maximizing choice, given her budget.
b.
The bundle (1, 1/2) is Jane’s utility maximizing choice, given her budget.
c.
Given her budget, Jane would maximize her utility by spending all of her income on good
X.
d.
Given her budget, Jane would maximize her utility by spending all of her income on good
Y.
e.
None of the above.
21. As you may know, Mungoites each have two left feet and one right foot. Their preferences for left and
right shoes display perfect complementarity. Mungoites find shoes useful only in trios of two lefts and
a right. The price of each type of shoe is $10 a shoe, and Mungoites consume nothing other than shoes.
A Mungoite’s Engel curve for right shoes has the equation
a.
R = m/30.
b.
R = m − 10.
c.
R = 2m.
d.
R = 10m.
e.
R = m/10.
22. Howard has the utility function U(x, y) = x − (1/y). His income is $30.
a.
Howard does not like good y.
b.
Howard has a bliss point.
c.
If the price of good x is $4 and the price of good y is $1, Howard will buy 2 units of y.
d.
Howard will buy good y only if it is cheaper than good x.
e.
None of the above.
23. Minnie Applesauce is shopping for a summer lake cottage. Minnie hates mosquito bites, but the
cheapest lake cottages have the most mosquitos. The price of a lake cabin is related to b, the number of
mosquito bites you can expect per hour, according to the formula p = $20,000 − 100b. Minnie’s utility
function is u = x − 5b2, where x is her expenditure on all goods other than her lake cabin. If Minnie
makes her best choice of lake cabin, how many mosquito bites per hour will she get?
a.
10
b.
5
c.
20
d.
25
e.
None of the above.
24. The prices of goods x and y are each $1. Jane has $20 to spend and is considering choosing 10 units of
x and 10 units of y. Jane has nice convex preferences and more of each good is better for her. Where x
is drawn on the horizontal axis and y is drawn on the vertical axis, the slope of her indifference curve
at the bundle (10, 10) is −2.
a.
The bundle (10, 10) is the best she can afford.
b.
She would be better off consuming more of good x and less of good y.
c.
She would be better off consuming more of good y and less of good x.
d.
She must dislike one of the goods.
e.
More than one of the above is true.
25. Which of the following could possibly change a rational consumer’s demand?
a.
Changing his utility function by cubing it
b.
Changing his utility function by multiplying it by 3 and subtracting 100 from it
c.
Increasing all prices and his income by $3
d.
Multiplying all prices and income by 2.2
e.
More than one of the above
26. Coke and Pepsi are perfect substitutes for Mr. Drinker and the slope of his indifference curves is
− 1. One day he bought 2 cans of Coke and 20 cans of Pepsi. (The cans of both drinks are the same
size.)
a.
Coke is less expensive than Pepsi.
b.
Coke is more expensive than Pepsi.
c.
Coke and Pepsi cost the same.
d.
Mr. Drinker prefers Pepsi to Coke.
e.
None of the above.
27. Ed and Al both consume only bread and cheese. Both of them always choose to have some bread and
some cheese, and both have strictly convex preferences. However, Ed likes to have a great deal of
bread with a little cheese, and Al likes lots of cheese with a little bread. Both face the same prices for
both goods and have chosen bundles to maximize their utilities subject to their budgets.
a.
Al’s marginal rate of substitution is larger in absolute value than Ed’s.
b.
Ed’s marginal rate of substitution is larger in absolute value than Al’s.
c.
Their marginal rates of substitution are the same.
d.
Who has the larger marginal rate of substitution depends on income levels.
e.
There is not enough information for us to be able to tell who has the larger marginal rate
of substitution.
28. If his wage rate increases, then a utility maximizing consumer will necessarily
a.
increase (or leave constant) his labor supply.
b.
increase (or leave constant) his labor supply if leisure is a normal good but otherwise
might reduce his labor supply.
c.
increase (or leave constant) his labor supply if leisure is an inferior good but otherwise
might reduce his labor supply.
d.
decrease (or leave constant) his labor supply.
e.
none of the above.
29. Angela consumes only two goods, x and y. Her income doubles and the prices of the two goods remain
unchanged. Assuming that she is a utility maximizer and likes both goods,
a.
she will consume more of both goods.
b.
the ratio of her consumption of x to y remains constant.
c.
her utility doubles.
d.
if her preferences are convex, she must consume more x.
e.
None of the above.
30. Arthur’s preferences are defined over two basic food groups, beer, x1, and ice cream, x2. His utility
function is u(x1, x2) = x21 + x2. He has $100 to spend, and each of these goods costs $10 per quart.
a.
Arthur will consume 5 quarts of ice cream and 5 quarts of beer.
b.
Arthur will find that 10 quarts of beer and no ice cream is the best bundle.
c.
Arthur will find that 10 quarts of ice cream and no beer is the best bundle.
d.
Arthur is indifferent between any two points on the line that connects (5, 5) and (10, 10).
e.
Arthur will spend 2/3 of his income on beer and 1/3 of his income on ice cream.
31. Andrew’s utility function is U(x1, x2) = 4x21 + x2. Andrew’s income is $32, the price of good 1 is $16
per unit, and the price of good 2 is $1 per unit. What happens if Andrew’s income increases to $80 and
prices do not change? (Hint: Does he have convex preferences?)
a.
He will consume 48 more units of good 2 and the same number of units of good 1 as
before.
b.
He will increase his consumption of both goods.
c.
He will reduce his consumption of good 2.
d.
He will consume the same number of units of good 2 and 3 more units of good 1 as before.
e.
None of the above.
32. Lorenzo lives on x and y alone. His utility function is U(x, y) = min3x + 4y, 7y. The prices of both
goods are positive. He will
a.
never buy more x than y.
b.
buy equal amounts of the two goods.
c.
always buy more y than x.
d.
always buy more x than y.
e.
More than one of the above is true.
33. Mort’s utility function is U(x1, x2) = x1x2. His income is $100. The price of good 2 is $10. Good
1 is priced as follows. The first 6 units cost $10 per unit and any additional units cost $5 per unit. What
consumption bundle does Mort choose?
a.
(5, 5)
b.
(7, 3.5)
c.
(9, 3)
d.
(6, 4)
e.
None of the above.
34. Clara’s utility function is U(x, y) = (x + 2)(y + 1). If her marginal rate of substitution is −4 and she is
consuming 14 units of good x, how many units of good y must she be consuming?
a.
30
b.
68
c.
18
d.
63
e.
9
35. Clara’s utility function is U(x, Y) = (x + 2)(Y + 1). If her marginal rate of substitution is −3 and she is
consuming 12 units of good x, how many units of good Y must she be consuming?
a.
15
b.
46
c.
24
d.
41
e.
7
36. Elmer’s utility function is U(x, y) = minx, y2. If the price of x is $25 and the price of y is $15 and if
Elmer chooses to consume 7 units of y, what must his income be?
a.
$2,660
b.
$280
c.
$1,430
d.
$1,330
e.
There is not enough information to determine his income.
37. Elmer’s utility function is U(x, y) = minx, y2. If the price of x is $10 and the price of y is $15 and if
Elmer chooses to consume 4 units of y, what must his income be?
a.
$220
b.
$100
c.
$320
d.
$440
e.
There is not enough information to determine his income.
38. Justin consumes goods x and y and has a utility function U(x, y) = x2 + y. The price per unit of x is px
and the price per unit of y is py. He has enough money so that he can afford at least 1 unit of either
good. When he chooses his best affordable bundle,
a.
his budget line must be tangent to the indifference curve passing through this bundle.
b.
he must consume only x.
c.
he must consume only y if p2x / py exceeds his income.
d.
he must consume some of each good if px = py.
e.
he must consume some of each good if py = px /2.
39. Charlie has a utility function U(xA, xB) = xAxB, the price of apples is $1, and the price of bananas is $2.
If Charlie’s income were $120, how many units of bananas would he consume if he chose the bundle
that maximized his utility subject to his budget constraint?
a.
30
b.
15
c.
60
d.
6
e.
90
40. Charlie has a utility function U(xA, xB) = xAxB, the price of apples is $1, and the price of bananas is $2.
If Charlie’s income were $200, how many units of bananas would he consume if he chose the bundle
that maximized his utility subject to his budget constraint?
a.
25
b.
50
c.
10
d.
100
e.
150
41. Charlie’s utility function is U(xA, xB) = xAxB. If Charlie’s income is $40, the price of apples is $4, and
the price of bananas is $2, how many apples are there in the best bundle that Charlie can afford?
a.
10
b.
12
c.
8
d.
9
e.
5
42. Ambrose’s utility function is U(x1, x2) = 4x1/21 + x2. If the price of nuts (good 1) is $1, the price of
berries (good 2) is $6, and his income is $264, how many units of nuts will Ambrose choose?
a.
20
b.
144
c.
288
d.
147
e.
72
43. Ambrose’s utility function is U(x1, x2) = 4x1/21 + x2. If the price of nuts (good 1) is $1, the price of
berries (good 2) is $5, and his income is $145, how many units of nuts will Ambrose choose?
a.
100
b.
200
c.
9
d.
103
e.
50
44. Our old friend, Edmund Stench, of Chapter 2 loves punk rock video tapes. He has no income and
therefore has to accept garbage in his backyard in return for money. Each video tape cost $2 and each
sack of garbage that he accepts brings him $1. His utility function is given by U(c, g) = min2c, 20 −
g, where c is the number of video tapes and g is the number of sacks of garbage that he gets per
month. Each month he will choose to accept
a.
20 sacks of garbage.
b.
no garbage.
c.
5 sacks of garbage.
d.
10 sacks of garbage.
e.
15 sacks of garbage.
45. Joseph’s utility function is given by UJ = xA + 2xB, where xA denotes his consumption of apples and xB
his consumption of bananas. Clara’s utility function is given by UC = 3xA + 2xB. Joseph and Clara shop
at the same grocery store.
a.
When we observe that Joseph leaves the store with some bananas, then we can deduce that
Clara also buys some bananas.
b.
There exist prices of apples and bananas such that both consumers buy strictly positive
amounts of both goods.
c.
When we observe that Joseph leaves the store with some apples and some bananas, then
we can deduce that Clara also buys some apples and some bananas.
d.
When we observe that Joseph leaves the store with some apples, then we can deduce that
Clara also buys some apples.
e.
Apples and bananas are perfect complements for Joseph.
46. If a consumer maximizes her preferences subject to her budget by choosing a consumption bundle
where the ratio of her marginal utilities of shelter and food, MUS/MUF, is greater than the ratio of the
prices of shelter and food, pS/pF, then she must
a.
be consuming food but no shelter.
b.
be consuming shelter but no food.
c.
believe that shelter is a “bad.”
d.
be consuming both food and shelter.
e.
not be spending all of her income.
PROBLEM
1. Max has the utility function U(x, y) = x(y + 1). The price of x is $2 and the price of y is $1. Income is
$10. How much x does Max demand? How much y? If his income doubles and prices stay unchanged,
will Max’s demand for both goods double?
2. Casper consumes cocoa and cheese. Cocoa is sold in an unusual way. There is only one supplier, and
the more cocoa you buy from him, the higher the price you have to pay per unit. In fact y units of
cocoa will cost Casper y2 dollars. Cheese is sold in the usual way at a price of 2 dollars per unit.
Casper’s income is 20 dollars and his utility function is U(x, y) = x + 2y, where x is his consumption of
cheese and y is his consumption of cocoa.
a. Sketch Casper’s budget set and shade it in.
b. Sketch some of his indifference curves and label the point that he chooses.
c. Calculate the amount of cheese and the amount of cocoa that Casper demands at these prices and
this income.
3. Is the following statement true or false? Briefly explain your answer. “A utility maximizer will always
choose a bundle at which his indifference curve is tangent to his budget line.”
4. Max has a utility function U(x, y) = 2xy + 1. The prices of x and y are both $1 and Max has an income
of $20.
a. How much of each good will he demand?
b. A tax is placed on x so that x now costs Max $2 while his income and the price of y stay the same.
How much of good x does he now demand?
c. Would Max be as well off as he was before the tax if when the tax was imposed, his income rose by
an amount equal to $1 times the answer to part (b)?
5. Harold consumes chardonnay and quiche. His utility function is U(c, q) = min{c, q2}. Draw a diagram
showing three or four of Harold’s indifference curves. If the price of chardonnay is $10 and the price
of quiche is $3 and if Harold is consuming 4 units of quiche, how many units of chardonnay is he
consuming?
6. Les has the utility function U(x, y) = (x + 1)(y + 4). The price of y is $1. Les spends all of his income to
buy 6 units of y and no x. What must the price of x be? Explain your answer and draw a diagram to
illustrate it.