Mfe 510Advanced Microeconomics Weekend Question Paper
Mfe 510Advanced Microeconomics Weekend
Course:Masters Of Science In Finance
Institution: Kca University question papers
Exam Year:2014
UNIVERSITY EXAMINATIONS: 2013/2014
EXAMINATION FOR THE MASTERS OF SCIENCE (MSC) FINANCE AND
INVESTMENT/ACCOUNTING/ECONOMICS
MFE 510ADVANCED MICROECONOMICS WEEKEND
DATE: AUGUST, 2014
TIME: 3 HOURS
INSTRUCTIONS: Answer Question One and Any Other Three Questions
QUESTION ONE (31 MARKS)
1.
Answer whether each of the following statements is true or false. You do NOT need to explain
the reason.
a)
[5 Marks]
If a game has finite number of players and strategies, there always exists at least one
dominant strategy for some player
b)
If a strategy is played in some Nash equilibrium, this strategy will not be eliminated by
the iterated elimination of strictly dominated strategies
c) Every sub-game perfect Nash equilibrium is a Nash equilibrium
d) In the Stackelberg game the leader can ALWAYS earn a higher profit than the follower
e) If a static game is played repeatedly, then some outcome other than static Nash
equilibria can possibly be achieved as a sub-game perfect Nash equilibrium
2.
The following payoff matrix shows the game of "Rock-Paper-Scissors."
Rock Paper Scissors
Rock (0,0) (-1,1) (1,-1)
Paper (1,-1) (0,0) (-1,1)
Scissors (-1,1) (1,-1) (0,0)
1
This is a traditional zero-sum game whose rule can be explained as follows: Two players
simultaneously chose one action from {Rock, Paper and Scissors}. Rock wins Scissors,
Scissors wins Paper and Paper wins Rock. A winner gets a payoff of 1 while a loser receives -1.
If the players choose the same action, both receive 0.
Verify that the following mixed-strategy constitutes a Nash equilibrium: both players randomly
choose each strategy with equal probability (=1/3 each).
3.
[5 Marks]
See the following game tree.
1
B
A
2
1
D
F
C E
(0, 3)
(1, 0)
(5, 5)
(6, 2)
a) [3 Marks]
b) Find all pure-strategy Nash equilibria. How many are there? [2 Marks]
c)
4.
Translate the game into normal-form by drawing the payoff bi-matrix Solve the game by backward induction [2 Marks]
Consider the following static game:
P1/P2 D
C (4,3) (0,5)
D
a)
C (5,0) (1,2)
Find all the pure-strategy Nash Equilibrium. Are there any dominant strategies in this
game?
b)
[2 Marks]
Now consider a dynamic game in which the above static game is played twice. Then,
how many sub-games (including the entire game) does this game have?
c)
5.
[2 Marks]
Solve the sub-game perfect Nash equilibrium of the dynamic game in (b) [3 Marks]
Consider a duopoly game in which two firms simultaneously and independently select prices,
and
The firm''s products are differentiated. After the prices are set, consumers demand is
units of the good that firm produces. Assume that each firm''s marginal cost is 1,
and the payoff for each firm is equal to the firm''s profit.
a)
Write the payoff functions of the firms (as a function of their strategies
and
)
[3 Marks]
2
b) Is this a game of "strategic complements" or "strategic substitutes"? [2 Marks]
c) Solve the (pure-strategy) Nash equilibrium. [2 Marks]
QUESTION TWO (23 MARKS)
Suppose two firms produce an identical good. The inverse demand curve for the good is:
, where Q is the total quantity produced by the two firms. Each firm has a constant
marginal cost 1 of producing the good.
a)
Find the Cournot Nash equilibrium of this game. What quantity will each firm produce? what
will be the market price? What would be the profits of each firm?
b)
[5 Marks]
Suppose the firms formed a cartel: each firm produced the same output and maximized joint
profits. What quantity would each firm produce? What would be the market price? what would
each firm produce?
c)
[5 Marks]
Now, suppose firm decides how much to produce first; firm 2 chooses only after observing firm
1''s choice. Find the sub-game Nash equilibrium (Stackelberg equilibrium) of this game. What
quantities will they produce? What would be the Market Price? What would be the profit of
each firm?
d)
[5 Marks]
Now suppose that the firms interact indefinitely through time. They discount future profits at a
discount factor ?. For what value of ? is there an equilibrium where firms follow the "trigger
strategies". Thus they produce cartel output as long ad the other firm has always produced
cartel output and otherwise they produce Cournot Nash output?
[8 Marks]
QUESTION THREE (23 MARKS)
Consider the game three in the figure below:
3
X
(2,0,0)
A
M
U
(3, 3,1)
Y
2
(0, 0,0)
B
3
1
X
(0, 0,0)
Y
(1, 1,3)
D
X
2
(4, 2,2)
A
Y
(3, 0,3)
3
B
X
(3, 3,0)
Y
a) (1, 1,1)
How many information sets are there (including the one with the initial node)? [5 Marks]
b) How many sub-games are there (including the entire game)? [5 Marks]
c) Find all sub-game perfect Nash equilibria [13 Marks]
QUESTION FOUR (23 MARKS)
A monopolist faces two kinds of consumers: Students and non-students. The demand curve of each
student is
. The demand of each non-student is given as
There are 10
students and 10 non-students. There is a zero marginal cost of production.
a)
First, suppose that the monopolist must set a single price to sell to all consumers. What price
would the monopolist charge? How much would each student and each non-student consume?
[13 Marks]
b)
Now suppose that the monopolist can charge different prices to students and non-students.
What price will the monopolist charge in each market? How much would each student and each
non-student consume?
[10 Marks]
4
QUESTION FIVE (23 MARKS)
Suppose three cafe chain companies,
, are considering to open new shops near Ruaraka
tunnel (each company opens utmost one shop). They make decision independently and simultaneously.
A company receives 0 profit if it does not open a shop. If it opens, then each firm''s profits depend on
the number of shops which are given as follows:
Number of firms 1 2 3
Each firm''s profits 10 4 -2
a) Derive all pure-strategy Nash equilibria
b) Is there any mixed-strategy Nash equilibrium in which companies decides to open a shop with
the same probability
c)
If Yes, solve such .
[7 Marks]
[8 Marks]
Is there any equilibrium in which one company opens a shop for sure while other two firms
open with equal probability
If yes, solve such .
[8 Marks]
QUESTION SIX (23 MARKS)
Each of two individuals receives a ticket on which is an integer from 1 to 10 indicating the size of a
price ($) she may receive. Assume payoff of receiving the price $X is X. The individuals'' tickets are
assigned randomly and independently; the probability of an individual receiving each possible number
is 1/10. Each individual is given an option of exchanging her price for the other individual''s prize; the
individuals are given this option simultaneously. If both individuals wish to exchange , then the prizes
are exchanged; otherwise each individual receives her own prize. Each individual''s objective is to
maximize her expected monetary payoff.
a)
Consider the above situation as a Bayesian game. Then, what are the individuals'' strategies?
[6 Marks]
b)
If an individual has the ticket with $10, will she have an incentive to exchange or not? Explain
why?
c)
[6 Marks]
Solve for the Bayesian Nash equilibrium. Can exchange happen in equilibrium?
[11 Marks]
5
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