Intermediate Microeconomics
ECNS 301
Fall 2015
Exam #: 1
Version B
Tuesday October 6, 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 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 a 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 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 #: 1, Version B
Due: 10/6/2015
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.
(a) The demand curve is inelastic for inferior goods and elastic for normal goods.
Solution: False, demand for both inferior and normal goods is downwards sloping so there is an elastic and inelastic portion on each.
(b) If the demand curve for a good is given by the equation q = 2/p, where q is the
quantity and p is the price, then at any positive price, the elasticity of demand will
be -1.
Solution: True, the price elasticity of demand is
elasticity of demand is − p22 p2 = −1.
∂q p ∂q
.
∂p q ∂p
= − p22 and the price
p
(c) There are two goods. You know how much of good 1 a consumer can afford if
she spends all of her income on good 1. If you knew the ratio of the prices of
the two goods, then you could draw the consumer’s budget line without any more
information.
Solution: True, you have the intercept on the good 1 axis and the slope of the
line is the ratio of the prices.
(d) If preferences are transitive, more is always preferred to less.
Solution: False, transitivity means that you make choices that are logically
consistent. Less could still be more with transitive preferences. What makes
more more is the assumption of more is better.
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ECNS 301
Exam #: 1, Version B
Due: 10/6/2015
Short Answer/Numerical
2. The market supply and demand functions for a particular market are as follows.
Q = 50 − p
Q = 2p − 4
The government is considering a per-unit tax of τ to be levied on sellers.
(a) What are the equilibrium prices and quantity with the tax expressed as a function
of τ ?
Solution: With a per-unit tax of τ levied on sellers, the price sellers get is p−τ
and the price buyers pay is p. To find the equilibrium, set D(p) = S(p − τ ) and
solve for p.
D(p) = S(p − τ )
50 − p = 2(p − τ ) − 4
3p = 54 + 2τ
2
p = 18 + τ
3
The price buyers pay is p = 18 + 32 τ . The price sellers get is p = 18 − 13 τ . The
market quantity is:
2
Q = 50 − (18 + τ ) = 32 −
3
1
Q = 2(18 − τ ) − 4 = 32 −
3
2
τ
3
2
τ.
3
(b) Show that as the tax rate increases the price buyers pay increases, the price sellers
get decreases and the equilibrium quantity decreases.
Solution: Let pB be the price buyers pay and pS be the price sellers get.
dpB
2
= >0
dτ
3
dpS
1
=− <0
dτ
3
dQ
2
=− <0
dτ
3
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ECNS 301
Exam #: 1, Version B
Due: 10/6/2015
(c) What value of τ maximizes tax revenue and how much tax revenue is generated?
Solution: For tax revenue, the government gets τ for each good sold, so tax
2
revenue is τ Q. This reduces to 32τ − 2τ3 and the optimization problem is as
follows.
2τ 2
max 32τ −
τ
3
The first order condition is
4
32 − τ = 0
3
so the optimal tax rate is τ = 24. When τ = 24, Q = 16 and tax revenue is 384.
(d) Based on your answers above, what do we know about the relative price elasticity
of supply and demand?
Solution: By examining the prices, we know that buyers pay 2/3 of the tax
and sellers pay 1/3 of the tax. Since buyers have a larger tax incidence, we
know that demand is more inelastic than supply or that supply is more elastic
than demand.
3. You decide to quit school, travel back in time, and work at a mine. Your mining job
pays $50 per month. The company store only sells cans of beans for $1 and bottles of
whiskey for $5. Every month you buy 15 cans of beans and 7 bottles of whiskey.
(a) What’s an equation describing your budget line and what’s the slope of your budget
line?
Solution: Let Qb be the number of cans of beans consumed and let Qw be the
number of the bottles of whiskey consumed. An equation for the budget line is
as follows.
50 = Qb + 5Qw
(b) Assuming prices don’t change, the government imposes a 10% tax on whiskey, but
decides to give poor miners (that’s you) a subsidy of $3.50 per month. What’s an
equation describing your new budget line?
Solution:
50 + 3.5 = Qb + 5Qw (1 + 0.1)
or
53.5 = Qb + 5.5Qw
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ECNS 301
Exam #: 1, Version B
Due: 10/6/2015
(c) Do you prefer the status quo or the new policy with the whiskey tax and subsidy,
and why?
Solution: With the status quo, you bought 15 cans of beans and 7 bottles of
whiskey. With the new prices,
15 + 5.5(7) = 53.5
so you can still afford your old bundle. Since you can still afford the same bundle
as before, the new policy doesn’t make you worse off. If with the new policy,
you can purchase a more preferred bundle, then you like the new policy better.
(d) The next election year comes around and the government now decides to give everybody a subsidy worth 10% of their income. An unintended consequence of this
policy is that the inflation rate increases and prices are 8% higher. Are consumers
better off or worse off and why?
Solution: Consumers are better off because their income has grown by more
than prices so their budget constraint has shifted out.
Consider any budget constraint for income m and goods x and y
m = px x + py y.
Now if income increases by 10% and prices increase by 8%, we have
1.1m = 1.08px x + 1.08py y
1.1m = 1.08(px x + py y)
1.1
m = px x + py y
1.08
Since
1.1
1.08
> 1 income has increased by more than the price increase.
4. There are two goods: x and y. A consumer’s utility function is u(x, y) = 5x2 y + 10x3
and that consumer has income of m = 189.
(a) What is the marginal rate of substitution for this consumer?
Solution: The MRS is the ratio of the marginal utilities. The marginal utilities
are
∂u(x, y)
= 10xy + 30x2
∂x
∂u(x, y)
M Uy =
= 5x2
∂y
M Ux =
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ECNS 301
Exam #: 1, Version B
and the MRS is
M RS =
Due: 10/6/2015
2y
+ 6.
x
(b) Write out the Lagrangian associated with the consumer’s problem.
Solution:
max L(x, y, λ) = 5x2 y + 10x3 − λ (px x + py y − m)
x,y,λ
(c) If the price of good y is py = 3 and the price of good x is px = 20, find the quantities
demanded by the consumer.
Solution: Look at the next part. It will take less time if you leave px as a
variable and use the values for m and py . In this case the Lagrangian is
max L(x, y, λ) = 5x2 y + 10x3 − λpx x − λ3y + λ189
x,y,λ
and the first order conditions are as follows.
∂L
= 10xy + 30x2 − λpx = 0
∂x
∂L
= 5x2 − 3λ = 0
∂y
∂L
= −px x − 3y + 189 = 0
∂λ
Now solve the first two equations for λ, then set them equal to one another.
10xy + 30x2
px
5x2
3
10xy + 30x2
px
30xy + 90x2
30y
=λ
=λ
5x2
3
= 5px x2
= x(5px − 90)
x(5px − 90)
y=
p 30 x
y=x
−3
6
=
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ECNS 301
Exam #: 1, Version B
Due: 10/6/2015
Now substitute the value of y into the last first order condition (the budget
constraint).
px x + 3y = 189
p
x
px x + 3x
− 3 = 189
6
1
px x + px x − 9x = 189
2
2px x + px x − 18x = 378
x(3px − 18) = 378
378
(3px − 18)
126
x=
(px − 6)
Now when px = 20, x = 126/14 = 9 and y = 9 20
−
3
= 3.
6
x=
(d) How does you answer to the part above change in the price of good x is px = 24?
Solution: Now when px = 24, x = 126/18 = 7 and y = 7
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24
6
− 3 = 7.