Exam June Correction

8/2/2019 Exam June Correction

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MICROECONOMICS II 2005

Pierre Courtois, Lan Yao

Name: ___________________________________________________

NIA: _______________

Instructions: The duration of this exam is of 2 Hours. It is made of two parts. The

first is constituted by exercises while the second is more related to what we did in

the theoretic course. Always think to explain what you did even if it is in

castellano ! The Barem is over 120 points.

Part I. Exercises

1.(25 points) Consider a market with two firms, firm 1 and firm 2, both producing a

homogeneous good and facing inverse demand function

p(Y) = 30 2Y,

where Y = y1 + y2, and y1, y2 are the production levels of the two firms. Suppose the

firms have the following cost functions

C1 (y1) =2

1y1

2 and C2 (y2) = 4 y2

(a) (5Points) Suppose the two firms engage in Cournot competition. Find the

equilibrium price, the quantities produced by each firm as well as each firms

profits.

Profit is benefit minus cost. Since firms are in duopoly the quantity they produce affects

the price of the market.

Firm 1 profit is : =1=P(y1+y2)y1-c1(y1)=(30-2(y1+y2))y1-21 y1

2;

Firm 2 profit is : =2= P(y1+y2)y2-c2(y2)=(30-2(y1+y2))y2-4y2.

The two firms engage in Cournot competition. They therefore decide simultaneously the

quantity they produce. Both firm maximizes profit given what is doing its counterpart.

Solving first order conditions allow us to derive reaction functions of the firms:


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2 21: 1 6

5

13 12 : 2

2

yBR y

yBR y

=

=

, these two equations (so called best response function or reaction

functions) tell us how much both firms will produce given what is produced by the otherfirms.

We have a Counot Nash equilibrium if solving the system of those two equations admits

a solution. Substituting y2 in the first equation, we deduce that:

y1= 17/4 ; y2=35/8.

In order to find the equilibrium price, it suffices to replace y1 and y2 by their value in

P(y1+y2)=30-2(y1+y2), which gives pcn=51/4 where pcn denotes the price of the

cournot-nash equilibrium.

We deduce the profit of firm 1 and 2, replacing price and quantities by their value:

:=11445

32 :=2

1225

32

(b) (5 Points) Suppose the two firms engage in Stackelberg competition, where firm 1

acts as the (quantity) leader and firm 2 as the (quantity) follower. Find the

equilibrium price, the quantities produced and each firms profits.

If they engage in Stackelberg competition then one firm decides first the quantity she will

produce given what she expects about the other firm will do next.

To solve a Stackelberg competition case, we need first to figure out what will be doing

the follower, i.e. firm 2. In fact, the follower will decide the quantity produced y2

according to what will be produced by firm 1, y1. In other words, firm 2 will simply useher reaction function in order to estimate her best response to what is doing firm 1. As we

saw in question (a), firm 2 best response is:13 1

2 : 22

yBR y

= .

Knowing that firm 2 will follow her best response, firm 1 will maximize her profit given

BR2.

Mathematically, the maximization program of firm 1 will be :

Max=1=[ 30-2(y1+13 12

y)]y1-

2

1y1

2

First order condition allows us to deduce the quantity y1 which maximizes firm 1s

profit. Solving this first order derivative gives us : -3y1+17=0 and therefore y1=17/3.

Replacing y1 by its value in BR2, we deduce that firm 2 produces y2=11/3.

Replacing y1 and y2 in the inverse demand function allow us to deduce the price of the

market in a stackelberg competition:

P(y1+y2)=pst=34/3, where pst denotes the stackelberg price.

Replacing price and quantities by their value in the profit functions gives us:

:=1289

6:=2

242

9

(c) (5 Points) Explain the difference between Cournot competition and Stackelberg

competition. Represent graphically both solutions.


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The principal difference between cournot and stackelberg competition lies on the

order of actions. While in Cournot competition firms choose simultaneously the

quantity they produce, in Stackelberg competition, firms are deciding sequentially. As

you can notice in those examples, when deciding sequentially, the leader firm has a

strategic advantage since she knows how will react the follower. It follows that in

Stackelberg competition the leader gets higher profit than in cournot while it is thereverse for the follower.

(d) (5 Points) Suppose the two firms form a cartel and maximize joint profits. Find

the equilibrium price and the quantities produced by each firm. Represent thiscollusion agreement in the previous graph.

If the two firms form a cartel, then they will maximize the joint profit and decide jointly

the quantity they produce in order to maximize their global profit. Such agreement will

allow them to get higher profit than in the Cournot case.

To solve such problem, the program to maximize is:

Max==P(y1+y2)(y1+y2)-c1(y1)-c2(y2)y1,y2

We need therefore to evaluate first order conditions. Partial derivatives are:30-5y1-4y2 and 26-4y1-4y2.

CN

Stackelberg

BR1

BR2

col


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For profit to be maximized, these two partial derivative need to be equal to zero. Solving

the system allows us to deduce that y1=4 and y2=5/2. Replacing those values into the

inverse demand gives us the collusion price pcol=17.

(e) (5 Points) What quantity will firm 1 produce if it believes that firm 2 will notdeviate from the equilibrium? (Hint: Assume y2 = y2

cartel)

We did not ask it in the previous question but if you want to check you can show that

both firm gets a profit higher than the one of the CN by signing this collusive agreement.

The problem is that both firm has also an incentive to deviate from this agreement. In

other words, firms have an incentive to sign the agreement to then not respect it hoping

the other firm will. This exactly what we study in this question.

If firm 1 deviates from the agreement thinking that firm 2 will respect it, she will simply

use her best response function in order to maximize profit. As we say, according to the

agreement, firm 2 should produce y2=5/2. We know from question (a) that firm 1 best

response function is :2 2

1: 1 65

yBR y = .

Replacing y2 by its value, we deduce that firm 1 will produce y1=5

2. (15 points) In the private center economicsworld.org, students use to love

management course believing thats the best to become wealthy. The demand for it is

D(Pd)=100-2Pd. The supply curve for that course is S(Ps)=3Ps.

(a) (5 Points) What are the equilibrium price and quantity?

We have an equilibrium when the market clear: D(P)=S(P).

Solving 100-2P=3P, we deduce that Pd=Ps=20. It follows that D(20)=S(20)=60.

(b) (5 Points) A tax of 10euros per courses is imposed on students. Write an equation

that relates the price paid by students to the price received by the suppliers. Write

an equation that states that supply equals demand.

If we impose a 10euros tax per course on students then Pd=Ps+10. At the equilibrium we

will have 100-2Pd=3Ps.

(c) (5 Points) Solve these two equations for the two unknowns Ps and Pd. With the10euros tax, what is the equilibrium price Pd paid by students and the total

number of lessons given ?

Solving these two equations we get 100-2(Ps+10)=3Ps. It follows Ps=16, Pd=26 and the

number of courses is 48.

3. (25 points) AirBaba is a new airline company enjoying a monopoly over the route

Barcelona-Valencia. AirBaba faces two different demand functions: PB = 600 YBfor business travellers and PT = 300 0.25 YT for tourists, where YB and YT are the


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demanded quantities of flight tickets by business travellers and tourists respectively,

and PB and PT are the prices of flight tickets for the corresponding type of

passengers. The monopolist AirBaba has a cost function given by: C(Y) = 60Y. The

objective of AirBaba is to maximize its profits.

(a) (2 Points) What is the aggregate demand function ? What is the inverse aggregate

demand function ?

The direct demand functions are Yb=600-Pb and Yt=1200-4Pt. The direct aggregate

demand function is therefore Y=1800-5P

The indirect aggregate demand function is P(Y)=1800

5

Y=360-

5

Y

(b) (5 Points) Suppose that AirBaba cannot discriminate between business travellers

and tourists. Find the equilibrium price of a flight ticket, the quantities demanded

for each type of consumers and the profits of the firm.

If AirBaba cannot discriminate, the profit of the firm is :

=(Y)=P(Y)Y-C(Y)=( 360-5

Y)Y-60Y

First order derivative is +2 Y

5300

Equalizing it to zero gives Y=750. It follows that P(750)=210.

Replacing P and Y in the profit function, we deduce that := 112500

Using the direct demand functions defined above and assuming Pb=Pt=P=210, we

deduce that Yb=390 while Yt=360.

(c) (4 Points) Draw this solution in a graph


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(d) (2 Points) What would be the socially optimal price of a flight ticket and the

socially optimal quantities demanded? (this socially optimal price and quantity

would be the one obtained in a perfectly competitive market)

At the social optimum P=MC=60. In such a case Yb=540 and Yt=960. AirBaba

would therefore sell 1500 tickets overall and her profit would be 86400. No surprise

the profit is lower than in the case of monopoly .

(e) (5 Points) What is the deadweight loss due to the monopolistic behaviour of

AirBaba? Locate it on the graph. Locate also the surplus of the consumer and ofthe producer.

The deadweight loss (DWL) is due to the monopolistic behavior. The firm wants to

maximize her profit and will produce a low quantity in order the price of tickets to be

high. There would be people willing to take the plane for a price higher than the marginal

cost of AirBaba. All those tickets which are not sold represent the inefficiency of

monopoly. Graphically, the DWL is the orange triangle underneath. Consumer surplus

and producer surplus are represented in green and blue.

750

210


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I did not ask to calculate the DWL. You could however have done so since you know the

quantity sold by the monopolist Ym, the optimal social quantity Y*, the price of the

monopolist Pm and the social optimal price P*. DWL=(pm-p*)(Y*-Ym)/2

(f) (7 Points) Suppose now that AirBaba discriminates between both types of

consumers (third degree price discrimination). Find the equilibrium price of each

type of flight tickets, the quantities demanded and the profits of the firm. Is

AirBaba better off ? Are the consumers better off ?

If she discriminates, AirBaba will sell tickets at different prices.

We should write again AirBabas profit function:

=(Yb+Yt)=P(Yb)Yb+P(Yt)Yt-C(Yb+Yt)=( 300-4

Yt)Yt+(600-Yb)Yb-60(Yt+Yb)

In order to find quantities Yb and Yt which maximizes AirBabas profit we need to

evaluate the first order derivatives.

First order conditions are : -1/2*Yt+240=0 and -2*Yb+540=0

Solving these two equations give the quantity sold to both types of consumers Yt=480

and Yb=270

We deduce the price Pt and Pb using inverse demand functions :Pb=330 and Pt=180.

It follows that profit of AirBaba is then := 130500 , the firm is better off than any of

the cases seen above.

The consumers are worse off than in the social optimal case. They are however better off

than in the single pricing monopoly case.

Note that to check that out result are right, it suffices to evaluate that with these values

MRb=MRt.

P

CS

PS

MC

Q

DWL

Ym Y*

P*

Pm


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Part II Theory

1. (10 points) In a monopolistic situation, we have a power on the market. What is this

power ? To answer this question you will explain what is the profit maximizationprogram of a monopolist and of a firm choosing the quantity she produces in a

competitive market. You will deduce what is the main difference between the two

situations and will explain what does it mean in terms of market power.

A monopoly means that a single firm if producing a product. Quantity produced by the

monopolist therefore affects the price of that product. For instance if you are the only one

producing cellular phones, the price wont be the same if you produce 100 phones or

10000 phones. Indeed, the more you produce the lower will be the price.

This relation between price and quantity does not exist if many firms are competing.

Taking the same example, if you reduce your phone production from 10000 to 100 while

many other firms are producing phones, your choice wont affect the market price of

telephone. We say in that case that the firm is price taker.

When the monopolist firm maximizes her profit, she takes that power into account and

therefore decides the quantity she produces, knowing the price will be affected. The

maximization program of the firm is then:

Max Profit =P(Y).Y-C(Y).

Alternatively the competitive firm knows she is price taker. She knows her decision

wont affect price and will therefore perform the following maximization:

Max Profit=P.Y-C(Y).

The power of the monopolist is therefore a power on price. We could say that the

monopolist is price maker. This allows her to get higher profit than competitive marketfirms.


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2. (25 points) Imagine your boat sinks beside a deserted inhabited island. You are with

another person which survived from the boat accident and the plan is to survive the

best you can. There is nothing on the island besides you two, some oranges and

coconuts. Since you want to survive, both of you individually collect all the oranges

and the coconuts available on the island.

24

(a) (3 Points) Given that W is the initial endowment (i.e. the amount of coconuts

and oranges both of you managed to collect), how many coconuts and how many

oranges were on that island ? What are the endowments of both of you ?

I have 15 oranges and 24 Coconuts while the other person has 45 oranges and 8

coconuts. I deduce that overall it was initially 60 oranges and 34 coconuts on that

island. I denote my endowment Wa: (15,24) and the other person endowment Wb :

(45,8).

(b) (3 Points) Put on the Edgeworth Box an arrow indicating the direction of your

preferences as well as an arrow indicating the other person preferences. Do you

think those preferences are well behaved ? Explain why.

Note that the arrow indicating the direction of my preferences is in blue while of the

arrow indicating the other preferences is in black. Both individual preferences are

well behaved, they are monotonic and convex. Indeed both individual prefer more toless and they also prefer mixture of the two goods rather that extrems.

You

The other

person

Oranges

Coconuts

15

45

8

W

X

YV

Contract

curve

This is the GE


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(c) (3 Points) By trading, can both of you pareto improve his well being ? Explain

what is a Pareto improving situation and locate the set of pareto improving

allocations on the box.

Reaching a Pareto improving situation means that both players are getting better off.We can see in the box that there are situations which can make me and the otherhappier than in the situation W. For instance, the point V, allows both of us to

reach an indifference curve which is better than the one we were on at first. The set

of Pareto improving situation is the lens shape in red. Any of these situations allow

to Pareto improve our welfare.

(d) (3 Points) Explain what is a pareto efficient situation and locate all the pareto

efficient allocations on the box. How do we call this set of allocations?

A Pareto efficient situation is a situation which cannot be improved upon. For

instance the situation V described in the previous question is pareto improvingcompared to W but is not Pareto efficient since another allocation can still make both

players better off. We say that an allocation is Pareto efficient when we cannot

improve the well being of a player without harming another player. The set of pareto

efficient outcomes is the contract curve, the point on this curve are the tangency

points between both players indifference curves (in green on the box).

(e) (3 Points) Given W, locate all the pareto efficient allocations which are pareto

improving. How do we call this set?

The set of all Pareto efficient allocations which are also Pareto improving compared

to W are called the Core of the game. This is the part of the contract curve which

pertain to the set of pareto improving allocations.

(f) (3 Points) Given W, can the allocation X be a general equilibrium ? Explain

why. What about the allocation Y?

If the allocation X is a general equilibrium, then the market must clear. In other

words both consumers should then spend their all budget when choosing allocation

X. This means that if X is a general equilibrium, it must pertain to both individualbudget constraints. Those budget constraints have the same slope (i.e. the relative

prices (values) of coconuts and oranges), both passes by the initial endowment W

which defines the budget of both players. In fact these budget constraint are a single

line we draw in yellow in box. One ca notice that if that budget constraint passes by

W and X then indifference curve of both players are not tangent to the budget

constraint in X. If relative prices are such, then there is an excess demand of

coconuts and an excess supply of oranges. This tell us that X is not a general

equilibrium.

(g) (3 Points) Given W and assuming that your preferences are well behaved, do

you think that many general equilibrium can be attained ? Is the general


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equilibrium always a pareto efficient outcome ? After having explained where

should be the general equilibrium of this game, locate it in the box.

By definition a General Equilibrium (GE) is Pareto efficient. The result of trade

pertains to the core of the game. There is only one point where both indifferencecurves are tangent to the budget constraint. It means that there is only one GE in thebox. That GE is the tangency point between the two indifference curve and the

budget constraint.

(h) (3 Points) Given an endowment and assuming preferences are well behaved, can

we say that a general equilibrium always exist ? If so is this allocation always

pareto efficient ?

If preferences are well behaved, we know thanks to the first theorem of welfare that

a GE always exists and that this GE is a pareto efficient allocation.

(i) (3 Points) Imagine that I am the owner of the island, I could then decide to give

you and the other shipwrecked man (hombre naufragado) an amount of coconuts

and oranges different than W. Can I choose to give both of you an initial

endowment which will lead allocation X to be a general equilibrium ? How do

we call this result in economics ?

This result is known in economics as the second theorem of welfare. As soon as

preferences are well behaved, any pareto efficient allocations of a game can be a

general equilibrium given the right endowment is chosen. As a consequence, the

answer to the question is yes, if we choose another initial endowement, theallocation X, which pertains to the contract curve can eventually be a GE of the

game.

3. (5 points) Explain what is the own price elasticity of demand. How do we calculate it

?

The own price elasticity of demand is the sensitivity of the variation of the demand to a

variation of price:

%

%

Qd

P

=

To calculate it, we evaluate :

p dp

q dq

4. (15 points) Using the underneath graph, explain second degree price discrimination.

5.

equal


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We have here two demand curves representing two kinds of individuals : kind 1 and kind

2. The firm wants to maximize profit. Knowing that there are two kinds of individuals,

the objective of the firm is to sell her product at a different price to the two kinds. Using

the left graph, one could imagine that the firm is selling quantity0

1X for price A to the

first kind of individuals and the quantity0

2

X for price A+B+C to the second kind. In such

case the firm would be able to capture the whole surplus of the consumers and could not

be better off. The problem is that such solution is not possible since kind 2 individuals

would then prefer to buy quantity0

1X for price A and therefore get a surplus of B. A

choice for the firm could then to propose to sell0

1X for price A and0

2X for price A+C.

The kind 2 of consumer would then get a surplus of B and it would allow the firm to

capture the surplus C. The middle graph shows us that following this reasoning the firm

to maximize profit should reduce the quantity of good offered to firm 1. A would then be

reduced but C would be increased more that proportionally. This leads us the graph

upright which is the optimal second part tariff of the firm : sell package 1m

X for price A

and package 02X for price A+C. In such situation, The firm gets the part A+C+D of the

surplus and the kind 2 consumer the part B of it.

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Exam June Correction