Chapter 5 practice questions
3. Suppose that 75 units of X and 30 units of Y give a consumer the same utility as 60 units of X
and 60 units of Y. Over this range:
a. What is the marginal rate of substitution over this range of consumption?
b. If the consumer obtains one more unit of X, how many units of Y must be given up in
order to keep utility constant?
c. If the consumer obtains one more unit of Y, how many units of X must be given up in
order to keep utility constant?
5. A consumer buys only two goods, X and Y.
a. If the MRS between X and Y is 5 and the marginal utility of X is 10, what is the marginal
utility of Y?
b. If the MRS between X and Y is 16 and the marginal utility of Y is 40, what is the marginal
utility of X?
c. If a consumer chooses to consume more Y and less X while moving
___________(downward, upward) along an indifference curve, the marginal utility of X
will ___________ (increase, decrease, remain constant) and the marginal utility of Y will
___________ (increase, decrease, remain constant)? The MRS will ___________
(increase, decrease, remain constant).
6. Use the figure below to answer the following questions:
a. The equation of budget line sw is ______________________.
b. The equation of budget line ut is ______________________.
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c. The equation of budget line rv is ______________________.
d. The equation of budget line uw is ______________________.
e. If the relevant budget line is rv and the consumer’s income is $6,000, what are the prices
of X and Y? At the same income, if the budget line is sw, what are the prices of X and Y?
f. If the budget line is uw, P = $12.50 and P = $1,000, what is income?
x
y
7.
Suppose a consumer has the indifference map shown below. The relevant budget line is LZ. The
price of good Y is $200.
a. What is the consumer’s income?
b. What is the price of X?
c. Write the equation for the budget line LZ.
d. What combination of X and Y will the consumer choose? Why?
e. What is the marginal rate of substitution at this combination?
f. Explain in terms of the MRS why the consumer would not choose combinations
designated by A or B.
g. Suppose the budget line pivots to LM, income remaining constant. What is the new price
of X? What combination of X and Y is now chosen?
h. What is the new MRS?
11. Assume that an individual consumes three goods, X, Y, and Z. The marginal utility (assumed
measurable) of each good is independent of the rate of consumption of other goods. The
prices of X, Y, and Z are, respectively, $8, $9, and $10. The total income of the consumer is
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$144, and the marginal utility schedule is as follows:
Units of
good
Marginal
utility of X
Marginal
utility of Y
Marginal
utility of Z
1
50
36
120
2
45
33
100
3
40
30
80
4
35
27
60
5
30
24
40
6
24
21
30
7
20
18
25
8
8
15
20
9
5
12
15
10
2
9
10
a. Given a $144 income, how much of each good should the consumer purchase to
maximize utility?
b. Suppose income rises to $254 with the same set of prices; what combination will the
consumer choose?
c. Let income fall to $30 and prices remain the same. How does the consumer allocate
income now?
d. Given the solution in part c, what would you say if the consumer maintained that X and Y
are not purchased because she could no longer afford to buy X or Y?
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Answers
3. a.
MRS = 2 = -
æ +30 ö
DY
= -ç
DX
è -15 ÷ø
b. Utility remains constant if the consumer gives up 2 units of Y after obtaining 1 more unit
of X éë -(DY / DX ) = -(-2 / +1) = 2ùû .
c. Utility remains constant if the consumer gives up 1/2 unit of X after obtaining 1 more unit
of Y éë -(DY / DX ) = -(+1/ -1 2) = 2ùû .
5. a. MRS = MUx/MUy , thus 10/MUy = 5 and MUy must be equal to 2.
b. MRS = MUx/MUy , thus MUx/40 = 16 and MUx must be equal to 640.
c. upward; increase; decrease; increase
6. a. sw: Y = 20 – 0.005X
b. ut: Y = 50 – 0.0250X
c. rv: Y = 25 – 0.005X
d. uw: Y = 50 – 0.0125X
e. Use the Y- and X-intercepts of budget line rv, 25 and 5,000, respectively, to find the prices
of X and Y.
Find the price of X: $6,000/ Px = 5,000  Px = $1.20.
Find the price of Y: $6,000/ Py = 25  Py = $240.
Similarly, for budget line sw, when income is $6,000:
Find the price of X: $6,000/ Px = 4,000  Px = $1.50.
Find the price of Y: $6,000/ Py = 20  Py = $300.
f. Use the X- and Y-intercepts of budget line uw, 4,000 and 50, respectively, along with the
given prices of X and Y ($12.50 and $1,000, respectively) to find the income associated
with budget line uw (use either one of the following two equivalent solutions for income):
M/$12.50 = 4,000  M = $50,000
M/$1,000 = 50  M = $50,000
7. a. $100,000 (= 500  $200)
b. 400  Px = $100,000  Px = $250
c. For LZ: Y = 500 – 1.25X or 250X + 200Y = 100,000
d. The consumer will choose 200 units of X and 250 units of Y, where indifference curve II
is tangent to budget line LZ. No other combination costing $100,000 provides more
utility than X = 200 , Y = 250; i.e., no other bundle on LZ lies on a higher indifference
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curve than indifference curve II.
e. At the optimal choice, MRS = Px /Py = $250/$200 = 1.25.
f. At combination A: The consumer could trade (give up) MRS units of Y to get 1 more unit
of X and the consumer’s utility would be unchanged. The consumer must give up Px /Py
units of Y to get 1 more unit of X and remain on the budget line. By visual inspection of
slopes, MRS > Px /Py at combination A. Thus the consumer can buy 1 more X and give up
only Px /Py units of Y, which is less than the loss of Y that would leave utility unchanged
(i.e., MRS units of Y). Since the consumer gives up less Y than the amount that would
leave the consumer indifferent, trading Px /Py units of Y for 1 more X must increase
utility, and combination A would not be chosen by the consumer.
At combination B: By the definition of MRS, the consumer can give up 1 unit of X in
return for MRS more units of Y and the consumer’s utility will not change. With market
prices Px and Py, the consumer can buy Px /Py (= 1.25) more units of Y if 1 less unit of X
is purchased and remain on the budget line. Visual inspection of the slopes of the
indifference curve and budget line at point B shows that Px /Py. MRS at combination B.
The consumer can buy Px /Py more Y if 1 fewer unit of X is purchased, which is more Y
than would be needed to remain indifferent. Therefore, giving up 1 unit of X to get MRS
more units of Y must increase utility, and combination B would not be chosen by the
consumer.
g. 250  Px = $100,000; thus Px = $400. The consumer will now choose 300 units of Y and
100 units of X, where indifference curve I is tangent to budget line LM.
h. MRS = Px /Py = $400/$200 = 2
11. a. 6 X, 4 Y, 6 Z
b. 8 X, 10 Y, 10 Z
c. 0 X, 0 Y, 3 Z.
d. Since the price of X and the price of Y are both less than $30, the consumer can indeed
afford to buy some of both good X and good Y. The consumer chooses not to purchase X
or Y because MUx /Px and MUy /Py are both smaller than MUz /Pz when the consumer
optimally allocates her $30 of spending.
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