Exercise 1 A monopolist has an inverse demand curve given ... - IDEA

Review exercises and questions for the exam (December 2004)
Exercise 1
A monopolist has an inverse demand curve given by p(y) = 12 − y and a
cost curve given by c(y) = y 2 .
a) What will be its profit-maximizing level of output
b) Suppose the government decides to put a tax on this monopolist so
that for each unit it sells it has to pay the government 2 euros. What will
be its output under this form of taxation
c) Suppose now that the government puts a lump sum tax of 10 euros on
the profits of the monopolist. What will be its output
d) What conclusion can you draw from these two types of taxation
Answers
a) Profit function: π(y) = p(y).y − c(y) = 12y − y 2 − y 2 = 12y − 2y 2
Max of profits: ∆π = 0⇒ 12 − 4y = 0, 12 = 4y[MR = MC] ⇒ y = 3
∆y
b) π(y) = p(y).y − c(y) − t(y) = 12y − y 2 − y 2 − 2y = 10y − 2y 2
Max of profits: ∆π = 0⇒ 10 − 4y = 0 ⇒ y = 2.5
∆y
c) π(y) = p(y).y − c(y) − t = 12y − y 2 − y 2 − 10 = 12y − 2y 2 − 10
Max of profits: ∆π = 0⇒ 12 − 4y = 0 ⇒ y = 3
∆y
d) Unlike the tax per unit of output, the lump sum tax is not a distortionary
tax, i.e. does not affect the marginal unit. Remember that prices
should reveal scarcity of the good, a distortionary tax affects this revelation.
Exercise 2
A monopoly faces a an inverse demand curve, p(y) = 100 − 2y, and has
constant marginal costs of 20.
a) What is its profit-maximizing level of output
b) What is its profit-maximizing price
c) What is the socially optimal price for this firm
d) What is the socially optimal level of output for this firm
e) What is the deadweight loss due to the monopolistic behavior of this
firm
f) Suppose this monopolist could operate as a perfectly discriminating
monopolist and sell each unit of output at the highest price it would fetch.
The deadweight loss in this case would be ... .
Answers
a) Profit function: π(y) = p(y).y − c(y) = 100y − 2y 2 − 20y = 80y − 2y 2
Profit-maximizing output: ∆π = 0⇒ 80 − 4y = 0 ⇒ 80 = 4y[MR = MC]
∆y
⇒ y = 20
b) Profit-maximizing price: p(y) = p(20) = 100 − 2 × 20 = 60
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c) Socially optimal price: p = MC ⇒ p = 20
d) Socially optimal output: 20 = 100 − 2y ⇒ y = 40
e) deadweight loss: 1(p 2 M −p C )×(y C −y M ) = 0.5(60−20)×(40−20) = 400.
The deadweight loss is 400 euros.
f) Perfect discrimination deadweight loss: 0. Remember that perfect
discrimination means that the monopolist charges a price for each consumer
and charges at the highest willingness to pay of the consumer. He produces
a level of output up to a point where the price is equal to the marginal cost
as in the competitive case, all the surplus going to the monopolist. At the
marginal point he sells to the consumer who has the lowest willingness to pay,
there is no surplus made on this consumer. In the competitive case, there is
a unique price for all the consumers. At this unique price, some consumers
are happy because they would be willing to pay more for the good. The
producer loses the surplus to the benefit of those consumers.
Exercise 3
Consider a market for a homogeneous good with two firms and demand
function:
Y = 30 − 3p (1)
where Y = y 1 + y 2 . Suppose the firms have the following cost functions:
C(y 1 ) = 3y 1
C(y 2 ) = 4y 2
a) Suppose the two firms engage in Cournot competition. Find the two
reaction functions and graph them.
b) Find the equilibrium price, the quantities produced by each firm as
well as each firm’s profits.
c) Suppose the two firms engage in Stackelberg competition, where firm 1
acts first and firm 2 acts when firm 1 quantity is known. Find the equilibrium
price, the quantities produced and each firm’s profits.
d) Suppose the two firms engage in Bertrand competition. Find the
equilibrium price of each firm, as well as quantities produced and profits.
e) Suppose the two firms form a cartel and maximize joint profits. Find
the equilibrium price and the quantities produced by each firm.
Answers
a)Y = 30 − 3p ⇒ y 1 + y 2 = 30 − 3p ⇒ p = 10 − 1(y 3 1 + y 2 )
profit function π 1 (y 1 , y2) e = p(y 1 +y2)y e 1 −C(y 1 ) = [10− 1(y 3 1 +y2)]y e 1 −3y 1
⇒ π 1 (y 1 , y2) e = 7y 1 − 1 3 y2 1 − 1y 3 1y2
e
Profit-maximizing output: ∆π 1(y 1 ,y2 e)
∆y 1
= 0 ⇒ 7 − 2y 3 1 − 1 3 ye 2 = 0
Reaction curve: y 1 = 10.5 − 0.5y2
e
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profit function π 2 (y1, e y 2 ) = p(y1 e +y 2 )y 2 −C(y 2 ) = [10− 1 3 (ye 1 +y 2 )]y 2 −4y 2
⇒ π 2 (y1, e y 2 ) = 6y 2 − 1 3 y2 2 − 1 3 ye 1y 2
Profit-maximizing output: ∆π 2(y 1 ,y2 e)
∆y 2
= 0 ⇒ 6 − 2y 3 2 − 1 3 ye 1 = 0
Reaction curve: y 2 = 9 − 0.5y1
e
b) y 2 = 9 − 0.5y 1 y 1 = 10.5 − 0.5y 2
y 2 = 5 y 1 = 8
y 1 + y 2 = 30 − 3p ⇒ 5 + 8 = 30 − 3p ⇒ p = 5.66
π 1 = 21, 3
π 2 = 8.3
c) Firm 1 is the leader. Its profit function:
π 1 (y 1 , y 2 ) = p(y 1 + y 2 )y 1 − C(y 1 ) = [10 − 1 3 (y 1 + y 2 )]y 1 − 3y 1 = 7y 1 − 1 3 y2 1 −
1
y 3 1y 2 ,
and the reaction curve of firm 2 is: y 2 = 9 − 0.5y 1 .
Substituting it into the profit function of firm 1:
π 1 (y 1 , y 2 ) = 7y 1 − 1 3 y2 1 − 1y 3 1(9 − 0.5y 1 ) = 4y 1 − 1 6 y2 1
Max of profit: ∆π 1
∆y 1
= 0 ⇒ 4 − 1 3 y 1 = 0
y 1 = 12
Plugging it into the reaction curve of firm 2:
y 2 = 9 − 0.5 ∗ 12 = 3
Equilibrium price: p = 10 − 1 3 (y 1 + y 2 ) = 10 − 4 − 1 = 5
π 1 = 24
π 2 = 3
d) Bertrand competition:
price=marginal cost. For firm 1, p = 3 and for firm 2, p = 4. The
equilibrium price will be p = 3.99 and the firm 1 will get all the demand,
since at this price firm 2 is not profitable:
y 2 = 0
y 1 = Y = 30 − 3p = 30 − 3 × 3.99 = 18.03
π 1 = 3.99 × 18.03 − 3 × 18.03 = 17.85
e) The profit maximization of a cartel is to maximize joint profits:
The profit maximization of a cartel is to maximize joint profits:
profit function:
π(y 1 , y 2 ) = p(y 1 + y 2 )[y 1 + y 2 ] − c(y 1 ) − c(y 2 )
= [10 − 1 3 (y 1 + y 2 )][y 1 + y 2 ] − 3y 1 − 4y 2
= 10(y 1 + y 2 ) − 1 3 (y 1 + y 2 ) 2 − 3y 1 − 4y 2
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∆π(y 1 ,y 2 )
∆(y 1 +y 2 ) = 0 ⇒ 10 − 2 3 y 1 − 2 3 y 2 − 3 = 0 ⇒ y 1 + y 2 = 21 2
∆π(y 1 ,y 2 )
= 0 ⇒ 10 − 2y ∆(y 1 +y 2 ) 3 2 − 2y 3 1 − 4 = 0 ⇒ y 1 + y 2 = 9
Therefore, only firm 1 will produce a quantity of 10.5. Firm 2 will close
down.
p = 10 − 1(y 3 1 + y 2 ) = 10 − 1 (10.5) = 6.5.
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Exercise 4
Answer T (True) or F (False):
1/ The point where (x 2 1, x 2 2) where MRS=2 and MRT=1 is not efficient because
both consumers could be made better off by increasing the production
of good 1. [True]
2/ Two firms form a cartel. Then, the firm producing more than the
other one does not have incentives to deviate from the equilibrium. [False]
3/ When there is a monopolist in a market, the production level is lower
than in the case of perfect competition. [True]
4/ When a monopolist is charged a tax, she transfers half of it to consumers.
[False]
5/ A monopolist that can discriminate perfectly will extract all of consumer’s
surplus. [True]
6/ Equal division is always fair. [False] [Remember: A fair allocation is
equitable and Pareto efficient].
7/ You like vanilla ice cream. I move to your town and demand so much
vanilla ice cream that its price rises. This is an example of a negative externality
from consumption. [False]
8/ In an economy with production, where the technology exhibits constant
returns to scale, the Second Welfare Theorem of Welfare Economics
holds. [True]
9/ A monopolist charging where the elasticity of demand |ɛ| equals one
is necessarily charging a perfectly competitive price. [False]
10/ A firm creating negative production externalities typically produces
more than the efficient quantity. [True]
Exercise 5: some topics to work...
1/ Compare the shape of the demand curve for a competitive firm and
for a monopolist. Graph them. Explain the difference.
A competitive firm faces a flat demand curve because its production,
which is negligible in the aggregate supply, does not influence the price.
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Since it cannot influence the price, it does not affect the demand it faces.
That’s why we say that the competitive firm faces a flat demand curve. For
a monopolistic firm, the demand curve is decreasing, because its production,
i.e., its supply affects the price and the demand for each corresponding
price. If its supply decreases, the price increases. When the price increases,
the demand decreases. That’s why we say that a monopolistic firm faces a
decreasing demand curve.
2/ What is the markup
The markup is the difference between the price of one unit of the good
and its marginal cost. Under perfect competition, the markup is 0.
3/ Explain the situation of a natural monopoly.
A natural monopoly occurs when a firm cannot operate at an efficient
level of output without losing money.
4/ What is a monopsonist Why will hire an inefficiently small amount
of the factor of production
5/ Why a cartel is unstable
6/ What does general equilibrium mean
GE refers to the study of how the economy can adjust to have demand
equal supply in all markets at the same time.
7/ Explain what Pareto efficiency means
A Pareto efficient allocation is one in which there is no feasible reallocation
of the goods that would make all consumers at least as well-off and at least
one consumer strictly better off.
7bis/ Walras’law states that the value of aggregate excess demand is zero
for all prices.
8/ The First Welfare Theorem states that a competitive equilibrium is
Pareto efficient.
9/ The Second Welfare Theorem states that as long as preferences are
convex, then every Pareto efficient allocation can be supported as a competitive
equilibrium.
10/ In an competitive economy with production, Pareto efficiency requires
that each individual’s marginal rate of substitution be equal to the marginal
rate of transformation.
11/ Arrow’s impossibility theorem shows that there is no ideal way to
aggregate individual preferences into social preferences.
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12/ As long as the welfare function is increasing in each individual’s utility,
a welfare maximum will be Pareto efficient. Furthermore, every Pareto
efficient allocation can be thought of as maximizing some welfare function.
13/ If externalities are present, the outcome of a competitive equilibrium
market is unlikely to be Pareto efficient.
Exercise 6
Consider a pure exchange economy with two goods (1 and 2) and two
consumers (A and B) whose utility functions and initial endowments are
given by:
U A (x 1 , x 2 ) = √ x 1 + x 2 , w A = (3, 7)
U B (x 1 , x 2 ) = x 1 + 2x 2 , w B = (2, 4)
a) Find all the Pareto-efficient allocations (Contract curve).
b) Consider good 1 as the numéraire (p 1 = 1), what is the equilibrium
price of good 2 Find the equilibrium allocation.
c) Is the equilibrium a fair allocation
d) Find the fair allocations.
Answers
a) Any Pareto efficient allocation must satisfy: MRS A = MRS B and the
following feasibility conditions:
x A 1 + x B 1 = w1 A + w1 B = 3 + 2 = 5
x A 2 + x B 2 = w2 A + w2 B = 7 + 4 = 11
MRS A =
√
1/(2
x A 1 )
[
∂U A /∂x A 1
∂U A /∂x A 2
= 1/(2 √
]
x A 1 )
1
=
[
]
1
= ∂U B /∂x B 1
= MRS
2 ∂U B /∂x B B
2
= 1 ⇒ 1 2 xA 1 = 1
The contract curve (the set of Pareto efficient allocations) is given by
x 1 = 1 (This is a straight vertical line in the Edgeworth box). The Pareto
set is P = { (x A 1 , x A 2 ); x A 1 = 1 }
We know that MRS A = MRS B = p 1
p 2
Since p 1
p 2
= 1 and p 2 1 = 1, then p 2 must be equal to 2. There is only one
couple of prices (p 1 , p 2 ) that leads to Pareto efficiency.
b) Consumer A’s problem √
max U A (x A 1 , x A 2 ) = x A 1 + x A 2
subject to p 1 x A 1 + p 2 x A 2 = m A (with p 1 = 1 and m A = 3 + 7p 2 )
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B:
√
L = x A 1 + x A 2 − λ(x A 1 + p 2 x A 2 − m A )
∂L
∂x A 1
∂L
∂x A 2
1
2
√
√
= 1
2 x A 1
x A 1
− λ = 0
= 1 − λp 2 = 0
= 1 2 ⇒ xA 1 = 1 4 p2 2
Substituting it into the budget constraint x A 1 + p 2 x A 2 = 3 + 7p 2 :
1
4 p2 2 + p 2 x A 2 = 3 + 7p 2 ⇒ x A 2 = 7 + 3 p 2
− 1 4 p 2
Since p 2 = 2, x A 2 = 6.5
Using the feasibility conditions, we can find the allocation for consumer
x A 1 + x B 1 = 5 ⇒ x B 1 = 5 − 1 = 4
x A 2 + x B 2 = 11 ⇒ x B 2 = 11 − 6.5 = 4.5
The equilibrium allocation is A(1; 6.5) and B(4; 4.5).
c) Is this allocation fair
A fair allocation is equitable and Pareto efficient. We already know that
the allocation is Pareto efficient. Now, let’s check if it is equitable. To do so
we see if A envies B and vice versa.
Consumer A : with his own allocation : U A (1, 6.5) = √ 1 + 6.5 = 7.5
with B’s allocation: U A (4, 4.5) = √ 4 + 4.5 = 6.5
Consumer A does not envy B.
Consumer B : with his own allocation : U B (4, 4.5) = 4 + 9 = 13
with A’s allocation: U B (1, 6.5) = 1 + 13 = 14
Consumer B envies A. Therefore, the allocation is not equitable. So, this
is not a fair allocation.
d) There is no fair allocation. Since there is only one couple of prices
leading to Pareto efficiency and this allocation is not fair, there is no fair
allocation with these types of preferences and resource constraints.
Exercise 7
BCN Sports is a new sports club in Barcelona. A recent study shows that
the demand (in hours) is p M = 16 − q M for men and p W = 10 − q W /2 for
women. BCN Sports has o cost function given by:
C(q) = 2q (2)
a) Suppose that Joan, who owns BCN Sports, wants to charge only price
per hour both for women and men. What would that price be How many
hours would be demanded What would BCN Sports’ profits be
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) Joan’s nephew courses ADE/ECO at UPF and argues that given their
different demands, the two groups should be charged different prices. In this
case, what prices would he charge for men and women How many hours
would be demanded by each group What would BCN Sports’profits be
c) Joan’s niece also courses ADE/ECO at UP and she suggests him to
charge an entry fee in addition to the charge per hour. What prices would
he charge as entry and per hour for each group How many hours would be
demanded in each group What would BCN Sports’ profit would be
d) What option (among a), b) and c)) would Joan prefer What about
the men What about the women Explain.
Answers
a) q M = 16 − p M and q W = 20 − 2p W
q M + q W = q = 36 − 3p ⇒ p = 12 − q 3
π = 10q − 1 3 q2
∆π
∆q = 10 − 2 3 q = 0 ⇒ q = 15
p = 12 − 1 3 q ⇒ p = 7
π = pq − cq = 75
b) π M = p M q M − 2q M = (16 − q M )q M − 2q M = 14q M − q 2 M
∆π M
∆q M
= 14 − 2q M = 0 ⇒ q M = 7
p M = 16 − 7 = 9
π W = p W q W − 2q W = 8q W − 1 2 q2 W
∆π W
∆q W
= 8 − q W = 0 ⇒ q W = 8
p W = 10 − 4 = 6
π = π M + π W = 9 ∗ 7 − 2 ∗ 7 + 6 ∗ 8 − 2 ∗ 8 = 81
c) p∗ = MC = 2
q M = 16 − 2 = 14 ⇒ F M = 1 (16 − 2) ∗ 14 = 98 (surface of a triangle
2
= 1 × height × base)
2
q W = 10 − 2 = 8 ⇒ F W = 1 (10 − 2) ∗ 16 = 64
2
Profits=consumers’surplus + profits per hour. But profits per hour=0
(since p = MC). Therefore:
π M = 98
π W = 64
π = π M + π W = 98 + 64 = 162
d) which is the best option
From the table, it is straightforward to see that the option c) is the best
for the firm. For the men, it is a) and for the women it is b).
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Pricing strategy a) Pricing strategy b) Pricing strategy c)
Men p M = 7 p M = 9 F M = 98 and p M = 2
Women p W = 7 p W = 6 F W = 64 and p W = 2
Profits 75 81 162
of the firm
Exercise 8
The fishermen of a village on the Valencia’s coast are fishing sardines.
The production of sardines depends on the number of fishing boats and is
given by the function:
S = 1500b − 5b 2 (3)
where b is the number of fishing boats and S is the production (in kilos)
of sardines. Each boat has a cost of 1000 euros and the price of 1 kilo of
sardines is 2 euros.
a) If the fishermen decide to divide equally among themselves the profits
from their production of fish, how many boats will they use for fishing
b) If the council of the village wants to maximize the profits from fishing
of the village and can decide the number of fishing boats, how many boats
will be allowed
c) In which situation will the quantity be higher Why is so Which
situation leads to the efficient production of fish Explain.
Answers
a) Value of the production = 2S b
Production takes place up to the point: 2S b
= 1000
2(1500b−5b 2 )
b
= 1000 ⇒ b = 200
S = 1500 × 200 − 5 × (200 2 ) = 300000 − 200000 = 100000
b) π = p × S − c × b = 2(1500b − 5b 2 ) − 1000b
∆π
= 2000 − 20b = 0 ⇒ b = 100
∆b
S = 1500 × 100 − 5(100 2 ) = 150000 − 50000 = 100000
c) The quantity produced is the same. In the first situation, the addition
of a boat is still profitable but does not take into account the reduction of
the output per boat it will cause due to the limitation of resources. The
social cost is ignored in the first situation. In the second situation, the true
cost is taken into account and leads to a smaller number of boat. This is the
efficient production of fish.
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Buena suerte...
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