Intermediate Microeconomics
ECNS 301
Spring 2015
Exam #: 2
Version A
Wednesday April 1, 2015
Name:
Instructions:
You must answer all of the following questions. Each question is worth the same amount.
You have the class period to complete the exam.
Answer each question clearly and concisely. You must show your work to receive credit.
This exam is given under the rules of the Montana State University. By printing your
name above you acknowledge the University’s Honor Code and agree to comply with the
provisions of the Honor Code. You may not use notes or receive any assistance. There is to
be no talking during the exam. You may use a calculator, but are never allowed to use device
allowing you to take photographs or transmit over a network. No notes, no assistance,
no talking, no cell phones, but you can use a calculator.
Clearly print your name above, in the space provided on the next page and in your blue
book(s). You must turn in your blue book(s). There are two versions of the exam. Indicate
your exam version on your blue book. It is your responsibility to make sure your
version of the exam is different from the students next to you. If you have the same version
as any of the students next to you, you will be asked to move.
ECNS 301
Exam #: 2, Version A
Due: 4/1/2015
True/False/Uncertain Plus Explanation
1. For each of the following, state whether it is true, false or uncertain and explain your
answer. No points are given without explanation.
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(a) If inputs into production cannot be substituted for each other but have to be employed in fixed-proportions isoquants are straight, downward-sloping lines.
Solution: False. Isoquants are straight lines when inputs are perfect substitutes. When inputs are used in fixed-proportions, isoquants are right angles.
(b) A firm may express increasing, constant and decreasing returns to scale for various
levels of output.
Solution: True. As output increases a firm can experience all types of returns
to scale.
(c) If the government wishes to increase the utility of consumers by a specific amount,
it is less expensive to do that through a cash gift than through a price subsidy on
a commonly purchased good (such as food).
Solution: True, with a cash gift there is no substitution effect, but with a price
subsidy there is a substitution effect.
(d) Unlike indifference curves, isoquants can intersect.
Solution: False, if isoquants intersected then one isoquant wouldn’t describe
efficient production.
Short Answer/Numerical
2. Consumer’s consume food and other goods. The amount of food consumed is denoted
f with price pf and the amount of other goods is denoted y with price py . In order
to support farmers (and low income consumers), the state of Montana is considering
subsidizing the price of food so that the quantity of food consumed by every consumer is
30. With the price subsidy the price of food becomes p0f = pf − τ where τ is the amount
of the per unit subsidy. There are 1 million people in Montana and each person has the
following preferences.
U (f, y) = min{f, 4y}
py is normalized to 1, pf = 6, income is m = 100, and the price subsidy considered is
τ = 3.75.
(a) How does the price subsidy change the optimal consumption bundle of each consumer? What was it before the subsidy and after?
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ECNS 301
Exam #: 2, Version A
Due: 4/1/2015
Solution: The preferences described by the utility function exhibit perfect complements. In equilibrium we know that f = 4y, and the budget constraint is
m = pf f + py y. Combining the two equations, we get m = pf 4y + py y and
solving for y yields the demand function y ∗ = 4pfm+py . The demand function for
good f is f ∗ = 4p4m
.
f +py
Before the price subsidy, py = 1, pf = 6 and m = 100. The consumption
bundles of each consumer before the price subsidy are as follows.
100
m
=
=4
4pf + py
4(6) + 1
4m
4(100)
f∗ =
=
= 16
4pf + py
4(6) + 1
y∗ =
After the price subsidy, py = 1, pf = 2.25 and m = 100. The consumption
bundles of each consumer after the price subsidy are as follows.
m
100
= 10
=
4pf + py
4(2.25) + 1
4m
4(100)
= 40
f∗ =
=
4pf + py
4(2.25) + 1
y∗ =
(b) Will the food subsidy achieve it’s objective?
Solution: The objective was to have f ∗ = 30 and with the policy f ∗ = 40 so
the policy leads to the consumption of too much food.
(c) If instead of a subsidy on the price food, consider an income subsidy which costs
the government just as much as the price subsidy did. How much does the income
subsidy cost the government and what are the optimal consumption bundles of each
consumer with the income subsidy?
Solution: First we figure out how much the price subsidy costs. With the price
subsidy f ∗ = 40, so for every person 40 units of food are subsidized at a rate of
τ = 3.75. The price subsidy costs τ f ∗ = 3.75(40) = 150 per person.
Now instead of the price subsidy, the government gives an income subsidy of
$150. With the income subsidy, income changes from $100 to $250. The optimal
consumption bundle of each consumer with the income subsidy is as follows.
m
250
y∗ =
=
= 10
4pf + py
4(6) + 1
4m
4(250)
f∗ =
=
= 40
4pf + py
4(6) + 1
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ECNS 301
Exam #: 2, Version A
Due: 4/1/2015
(d) What policy should the government implement and what policy is favored by consumers? Why?
Solution: Because the two goods are perfect complements and will always be
consumed in equal proportions, the two policies have the same effect.
3. A firm’s production function is as follows.
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q = 2KL2 + 13L + 4
(a) What is the marginal product of labor and the average product of labor?
Solution:
∂q
= 4KL + 13
∂L
q
4
APL = = 2KL + 13 +
L
L
M PL =
(b) Find the marginal rate of technical substitution as a function of just K and L.
Solution: The MRTS is the ratio of the marginal products. You found M PL
above. Now find M PK = 2L2 . Taking the ratio of the marginal products, the
MRTS is
M RT S =
If you found MRTS as M RT S =
4KL + 13
M PL
=
.
M PK
2L2
M PK
,
M PL
that works too.
(c) Find the value of L that minimizes the average product of labor.
Solution: The APL = 2KL + 13 + L4 (from above) and if you want to find the
value of L that minimizes this, then take the derivative and set it equal to zero.
min 2KL + 13 +
L
0 = 2K −
4
L2
2
K
12
2
L=
K
L2 =
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4
L
ECNS 301
Exam #: 2, Version A
Due: 4/1/2015
(d) At what value of K does the average product of labor equal the marginal product
of labor?
Solution: At K = L22 as the average product of labor equals the marginal
product of labor where the average product of labor is minimized. If you want
to set APL = M PL and solve for K, we have:
APL = M PL
4
2KL + 13 + = 4KL + 13
L
4
= 2KL
L
2
K= 2
L
2
1
4. Consider the following utility function: U = x13 x23 . The consumer has income of M , the
price of x1 is P1 , and the price of x2 is P2 .
(a) What is the marginal rate of substitution?
Solution: The marginal rate of substitution is the slope of the indifference
curve which is also the ratio of the marginal utilities. The utility function is
2
1
U = x13 x23
and the marginal utilities/MRS are
∂U
2 −1 1
= x1 3 x23
∂x1
3
∂U
1 2 −2
= x13 x2 3
∂x2
3
−1
M RS =
1
2x1 3 x23
2
−2
x13 x2 3
2x2
M RS =
.
x1
(b) What is the equation for the budget constraint?
Solution: The budget constraint is
M = P1 x 1 + P2 x 2 .
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ECNS 301
Exam #: 2, Version A
Solving for x2 we get
x2 =
Due: 4/1/2015
P1
M
− x1 .
P2 P 2
(c) What is the optimal consumption bundle if M = 54, P1 = 2 and P2 = 3? What
level of utility is achieved?
Solution: This is the solution to the consumer’s problem. The optimization
problem is
2
1
max U = x13 x23
x1 ,x2
subject to M = P1 x1 + P2 x2 .
Write the Lagrangian as
2
1
max L(x1 , x2 , λ) = x13 x23 + λ (M − P1 x1 − P2 x2 )
x1 ,x2 ,λ
and the first order conditions are as follows.
2 −1 1
∂L
= x1 3 x23 − P1 λ = 0
∂x1
3
1 2 −2
∂L
= x13 x2 3 − P2 λ = 0
∂x2
3
∂L
= 100 − P1 x1 − P2 x2 = 0
∂λ
Combine the first two first order conditions to get
2x1 x2
x2
= 1
P1
P2
2P2 x2 = P1 x1 .
Now plug this into the last first order condition (the budget constraint) to get
M = P1 x 1 + P2 x 2
M = (2P2 x2 ) + P2 x2
M = 3P2 x2
M
x∗2 =
3P2
2P2 x2
2M
x∗1 =
=
P1
3P1
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ECNS 301
Exam #: 2, Version A
Due: 4/1/2015
If M = 54, P1 = 2 and P2 = 3, then
M
54
=
=6
3P2
3(3)
2M
2(54)
x∗1 =
=
= 18
3P1
3(2)
x∗2 =
2
1
2
1
U = x13 x23 = (18) 3 (6) 3
(d) Derive the demand function for good x1 with income of M , the price of x1 is P1 ,
and the price of x2 is P2 (general, not specific).
Solution: We did this in the last part.
x∗1 =
2M
3P1
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