ECONOMICS Assignment

1

Answer all of them

1. Suppose that a risk-averse individual has a wealth level Wo . This indi- vidual 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 with probability 1 and (1 + rl ) in a bad state with probability 1 . The expected

2

return from the risky asset is:

1

2

(1 + rl ) +

2

1

2 (1 + rh ) (2)

We assume that the expected return on the risky asset is higher:

1

2 (1 + rl ) +

1

2 (1 + rh ) > (1 + rf ) (3)

1 1

2 rl + 2 rh > rf (4)

(a) Show that a risk-averse person will always invest some amount in the risky asset.

(b) What will be the amount invested in the risky asset if 1 rl + 1 rh = rf ?

2 2

2. Consider the following Prisoner’s dilemma game played simultaneously:

•

NC (2)

C (2)

NC (1)

-2, -2

-10, -1

C (1)

-1, -10

-5, -5

The first pay off is that of player 1 and the second that of player 2. If player 1 moves first 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 ver- sion?

(e) Explain why it is the subgame perfect Nash Equilibrium in the sequen- tial version?

3. Consider the cost function of a firm for LR is:

C (q) = q3 − 20q2 + 110q (5)

and the demand curve of the market is:

QD = 3, 000 − 3P (6)

(a) What is the LR equilibrium P and Q? (b) How many firms will exist in the LR?

4. Consider two individuals a and b with the utility functions:

Ua = (xc ).5 (yc ).5 (7)

a a

Ub = (xc ).25 (yc ).75 (8)

b b

(xc , yc ) are the amount that consumers consume. In order to purchase these

i 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 : (xe , ye ) = (500, 500) (9)

a a

Eb : (xe , ye ) = (200, 800) (10)

b b

These endowments can be sold in the market. The total amount of endow- ments in the economy can be expressed as:

X = xe + xe = 700 (11)

a b

Y = ye + ye = 1300 (12)

a b

as:

(a) Find the equilibrium demand for good X. (b) What is the equilibrium Px ?

P

y

[Hint: If prices of goods are: Px , Py , the budget constraints can be written

Ia = Px xe + Py ye

(13)

a a

Ib = Px xe + Py ye

(14)

b b

]

5. Consider the market demand curve to be:

P = 1000 − Q (15) The cost function of a firm to be:

C (q) = 10q + q2 (16)

(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 firms?