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Microeconomics, IB and IBP
RE-TAKE EXAM, January 2010
Open book, 4 hours
Question 1
Suppose that the market demand function for corn is Qd = 15−2P while
the market supply function for corn is QS = 5P − 2.5, both measured in
billions of bushels per year. Suppose that the government imposes a $1.40
tax on each bushel of corn.
1.1 What will be the effects on consumer, producer, and aggregate surplus,
and what will be the deadweight loss caused by the tax?
1.2 What is the economic incidence of taxes. Provide an intuition for the
resulting tax incidence.
Answer:
1.1 We start by inverting the functions given, so that it is easier to draw
them.
Qd = 15 − 2P =⇒ P = 7.5 − 0.5Qd
QS = 5P − 2.5 =⇒ P = 0.5 + 0.2QS
We calculate the equilibrium before taxes. Since demand equals supply, and
since at equilibrium Qd = QS = Q, we write:
7.5 − 0.5Q = 0.5 + 0.2Q =⇒
Q = 10
and thus P
= 2.5
At this price and quantity, the consumer surplus is calculated as the area
under the demand function and above the price, while the producer surplus
is the area above the supply functions and below the price:
CS =
PS =
(7.5 − 2.5)10
= 25
2
(2.5 − 0.5)10
= 10
2
1
We now introduce the tax t = 1.4. Since the market is competitive, it
does not matter where it applies to. Below, we assume that it applies to
consumers, and thus it shifts demand function inwards (although we could
easily have assumed that it applied to producers, in which case it would
have shifted the supply function to the left).
P + 1.4 = 7.5 − 0.5Qd =⇒ P = 6.1 − 05Qd
Solving for the equilibrium quantity, we get
6.1 − 05Qd = 0.5 + 0.2QS =⇒
Q = 8
Going back to the demand function, we can derive that P = 2.1 (the producer price) and P + 1.4 = 3.5 (the consumer price). Calculating the new
CS and PS, we have
CS =
PS =
(7.5 − 3.5)8
= 16
2
(2.1 − 0.5)18
= 6.4
2
At this tax equilibrium there are also tax revenues collected by the government which are equal to T R = tQ = 1.4 · 8 = 11.2.
Comparing the pre- and post-tax equilibria, we see that total welfare
has fallen from 25 + 10 = 35 to 16 + 6.4 + 11.2 = 33.6. The resulting loss of
1.4(= 35 − 33.6) is the so-called deadweight loss (DWL) of taxes.
1.2: Tax incidence: that is, who pays the tax: the consumer or the
producer?
From the above calculations we saw that the consumer paid 3.5 after
taxes, while she was paying 2.5 before taxes were applied. The producer
instead, got 2.1 for its product, while before taxes she was getting 2.5.
Clearly the biggest part of the 1.4 tax is paid by the consumer. More
precicely, the consumer pays
3.5 − 2.5
= 0.71, i.e. 71%
1.4
while the producer pays
2.5 − 2.1
= 0.29, i.e. 29%
1.4
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of the tax. The intuition for such a result builds on the fact that the agent
with the most inelastic function bears the tax heaviest. In the above example, it is the consumer that has the most inelastic function.
Question 2.
A local video rental monopoly faces the weekly demand function Q =
1000 − 50P. The marginal cost of a rental is $1. Suppose that the town
government places a $1 tax on a video rental.
2.1 What effects will the tax have on the price the monopolist charges?
2.2 What subsidy would persuade the monopolist to sell the same quantity
of rentals that would be sold in a competitive video rental industry?
Answer:
Q
.Given this
2.1: We first invert the demand function to P = 20 − 50
(inverted) demand function, the marginal revenue has the same slope and
Q
Q
= 20 − 25
.
double the slope, i.e. M R = 20 − 2 50
Since the monopolist always set M R = M C, the pre-tax equilibrium is
characterised by the following quantity and price:
Q
25
and thus P
20 −
= 1 =⇒ Q = 475
= 10.5
When taxes apply, the marginal cost of the monopolist will increase
by the amount of tax, i.e. M C = 2. (one can also assume that the tax
falls on the consumers, in which case the demand shrinks inward - such an
assumption is also fine - below, I continue with assuming that the tax falls
on the producer). The new equilibrium is now
Q
25
and thus P
20 −
= 2 =⇒ Q = 450
= 11
Thus, the price the monopolist charges increases.
2.2: First of all, we need to find how much it would be produced if the
market was perfectly competitive (and without taxes). In that case firms
will charge a price equal to their marginal cost, i.e. P = 1(= M C). It is
then easy to see that
Q = 1000 − 50 · 1 = 950.
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To induce such an output, a subsidy is considered. To find this subsidy,
we write
M R = M C − s =⇒
950
= 1 − s =⇒ s = 19.
20 −
25
Thus the subsidy will have to be $19 per rental before a monopoly is induced
to produce the competitive output.
Question 3
"Competitive firms can easily have profits – the real question is for how
long they can keep these profits". Discuss this statement: is it true or false
and why? Make sure to develop your arguments both with diagrams and
with economic intuition.
Answer:
The statement is true, if one assumes short run profits not disappearing
instantaneously. One should refer to short run (i.e. before entry) and long
run (after entry) equilibria that a competitive firm faces. Please see pp.350354 at the textbook.
It is important to discuss the fact that the theory of perfect competition
assumes that entry and exit occurs instantaneously, while in reality such
entry and exit takes time, and thus short run profits are not competed away
instantaneously.
Question 4
Noah and Naomi have decided to start a firm to produce garden tables.
If Noah and Noami have √
at least 500 square meters of garage space, their
weekly production is Q = L, where L is the amount of labour they hire, in
hours. The wage rate is $12 an hour, and a 500 square-meter garage rents
for $250 per week.
4.1 What is the weekly cost function for producing garden tables? Graph
it.
After some time Noah and Noami want to expand. They now want to
produce 100 garden benches per week in two production plants A and B.
Assume that the cost functions at the two plants are CA = 6000QA −3(QA )2
and CB = 650Q2 + 2(QB )2 .
4.2 What is the best assignment of output between the two plants? (that
is, how many to produce at A and how many at B).
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Answer:
4.1: The weekly cost function has two elements; the variable cost and
the fixed cost. The variable refers to the workers hired, while the fixed to
the rent paid for the garage space. Thus
TC = w · L + F =
= 12 · L + 250
To find the demand for labour (L), i.e. how many workers Noah and Naomi
√
1
will want to hire, we solve Q = L = L 2 as L = Q2 . Thus, the cost function
(i.e. total costs as a function of output) is
T C = 12 · Q + 250
We can easily graph such a function in the figure below.
TC
250
Q
4.2: There is typo in the exam question. The cost function of plan B
should be CB = 650Q2B +2(QB )2 and not CB = 650Q2 +2(QB )2 . While some
assumed that, some assumed that it should have been CB = 650QB +2(QB )2
and did their calculations based on that. No matter what, I graded the
calculations and reasoning given the assumption that the student has chosen,
and not the actual result. Below, I present the answer key for the CB =
650Q2B + 2(QB )2 .
In order to solve for the best assignment of outputs, we know that the
equilibrium should be characterised by M CA = M CB , i.e. the marginal
costs should be the same in both places. Thus we have
M CA = 6000 − 6QA = 1304QB = M CB
5
Given that QA + QB = 100, we write
6000 − 6(100 − QB ) = 1304QB
which implies that QB = 7.6 and QA = 92.4.
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