Answer all of them1. Suppose that a risk-averse individual has a wealth level Wo . This individual has the option of inv...
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Answer all of them
1. Suppose that a risk-averse individual has a wealth level Wo . This individual has the option of investing in the risk-free asset with a return of
(1 + rf )
(1)
There is another asset that is risky with a return of (1 + rh ) in a good state
1
with probability 2 and (1 + rl ) in a bad state with probability 1 . The expected
2
return from the risky asset is:
1
1
(1 + rl ) + (1 + rh )
2
2
(2)
We assume that the expected return on the risky asset is higher:
1
1
(1 + rl ) + (1 + rh ) > (1 + rf )
2
2
1
1
rl + rh > rf
2
2
(3)
(4)
(a) Show that a risk-averse person will always invest some amount in the
risky asset.
1
(b) What will be the amount invested in the risky asset if 2 rl + 1 rh = rf ?
2
2. Consider the following Prisonerâs dilemma game played simultaneously:
â¢
NC (1)
C (1)
NC (2)
-2, -2
-1, -10
C (2)
-10, -1
-5, -5
The ?rst pay o? is that of player 1 and the second that of player 2.
If player 1 moves ?rst or we are in a sequential game now.
(a) What is normal form of the game in the sequential version?
(b) What are the Nash Equilibria in the sequential version?
(c) Draw the extensive form in the sequential version.
(d) What is the subgame perfect Nash Equilibrium in the sequential version?
(e) Explain why it is the subgame perfect Nash Equilibrium in the sequential version?
3. Consider the cost function of a ?rm for LR is:
C(q) = q 3 ? 20q 2 + 110q
(5)
and the demand curve of the market is:
QD = 3, 000 ? 3P
1
(6)
(a) What is the LR equilibrium P and Q?
(b) How many ?rms will exist in the LR?
4. Consider two individuals a and b with the utility functions:
Ua
Ub
.5
c
(xc ) (ya ).5
a
c .25 c .75
(xb ) (yb )
=
=
(7)
(8)
c
(xc , yi ) are the amount that consumers consume. In order to purchase these
i
bundles, they must also have something to sell. In this economy, the production
of goods has already taken place. Each consumer has the following endowments
of goods:
Ea
Eb
:
:
e
(xe , ya ) = (500, 500)
a
e
(xe , yb ) = (200, 800)
b
(9)
(10)
These endowments can be sold in the market. The total amount of endowments in the economy can be expressed as:
X
Y
= xe + xe = 700
a
b
e
e
= ya + yb = 1300
(11)
(12)
(a) Find the equilibrium demand for good X.
(b) What is the equilibrium Px ?
Py
[Hint: If prices of goods are: Px , Py , the budget constraints can be written
as:
Ia
Ib
e
= Px xe + Py ya
a
e
= Px xe + Py yb
b
(13)
(14)
5. Consider the market demand curve to be:
P = 1000 ? Q
(15)
The cost function of a ?rm to be:
C(q) = 10q + q 2
(a) What is the monopoly output?
(b) What is the output for perfectly discriminating monopolist?
(b) What is the Cournot output if there are 2 ?rms?
1. Suppose that a risk-averse individual has a wealth level Wo . This individual has the option of investing in the risk-free asset with a return of
(1 + rf )
(1)
There is another asset that is risky with a return of (1 + rh ) in a good state
1
with probability 2 and (1 + rl ) in a bad state with probability 1 . The expected
2
return from the risky asset is:
1
1
(1 + rl ) + (1 + rh )
2
2
(2)
We assume that the expected return on the risky asset is higher:
1
1
(1 + rl ) + (1 + rh ) > (1 + rf )
2
2
1
1
rl + rh > rf
2
2
(3)
(4)
(a) Show that a risk-averse person will always invest some amount in the
risky asset.
1
(b) What will be the amount invested in the risky asset if 2 rl + 1 rh = rf ?
2
2. Consider the following Prisonerâs dilemma game played simultaneously:
â¢
NC (1)
C (1)
NC (2)
-2, -2
-1, -10
C (2)
-10, -1
-5, -5
The ?rst pay o? is that of player 1 and the second that of player 2.
If player 1 moves ?rst or we are in a sequential game now.
(a) What is normal form of the game in the sequential version?
(b) What are the Nash Equilibria in the sequential version?
(c) Draw the extensive form in the sequential version.
(d) What is the subgame perfect Nash Equilibrium in the sequential version?
(e) Explain why it is the subgame perfect Nash Equilibrium in the sequential version?
3. Consider the cost function of a ?rm for LR is:
C(q) = q 3 ? 20q 2 + 110q
(5)
and the demand curve of the market is:
QD = 3, 000 ? 3P
1
(6)
(a) What is the LR equilibrium P and Q?
(b) How many ?rms will exist in the LR?
4. Consider two individuals a and b with the utility functions:
Ua
Ub
.5
c
(xc ) (ya ).5
a
c .25 c .75
(xb ) (yb )
=
=
(7)
(8)
c
(xc , yi ) are the amount that consumers consume. In order to purchase these
i
bundles, they must also have something to sell. In this economy, the production
of goods has already taken place. Each consumer has the following endowments
of goods:
Ea
Eb
:
:
e
(xe , ya ) = (500, 500)
a
e
(xe , yb ) = (200, 800)
b
(9)
(10)
These endowments can be sold in the market. The total amount of endowments in the economy can be expressed as:
X
Y
= xe + xe = 700
a
b
e
e
= ya + yb = 1300
(11)
(12)
(a) Find the equilibrium demand for good X.
(b) What is the equilibrium Px ?
Py
[Hint: If prices of goods are: Px , Py , the budget constraints can be written
as:
Ia
Ib
e
= Px xe + Py ya
a
e
= Px xe + Py yb
b
(13)
(14)
5. Consider the market demand curve to be:
P = 1000 ? Q
(15)
The cost function of a ?rm to be:
C(q) = 10q + q 2
(a) What is the monopoly output?
(b) What is the output for perfectly discriminating monopolist?
(b) What is the Cournot output if there are 2 ?rms?
2
(16)
(16)
