ECON7002 – Problem Set 2

Rodrigo Guimar„es

October 26, 2015

Exercise 1 (Market completeness, Arrow-Debreu securities) Suppose there are 3 possible outcomes (states

of the world) for next period and that 3 di§erent assets in the economy are described by the following payo§ s

in each state

Asset Payo§ s (xt+1) Price

s1 s2 s3

1 1 1 1 0.968

2 1 1 0.8 0.953

3 1 1.5 0 0.993

_________

Pr (si) 0.7 0.25 0.05

1. Create as many Arrow-Debreu securities as you can using these assets by buying/selling any combination

of these assets.

2. Is the market complete?

3. What is the price of each Arrow-Debreu security you constructed?

4. Given the price of the Arrow-Debreu securities, can you price an asset paying (0.5,0.2,1)?

5. Would you be able to construct an SDF able to price any asset in this economy? Either argue why

you could or why not (or construct the SDF).

Exercise 2 (Arrow-Debreu equilibria and aggregate risk) An economy has two agents, two dates, and two

states in the second date. There is perfect symmetry in this economy: both agents have the same preferences

(and are risk averse), each have half of the endowments on each period and state, total endowment is the

same across states, they can trade Arrow-Debreu securities for each state and both states are equally likely.

1. What can you say about the Arrow-Debreu equilibrium in this economy?

2. What would happen if the only thing that changed was that each agent owned all of the endowment

in one of the states (so each hold nothing in one state and all in the other state)?

Exercise 3 (Arrow-Debreu and risk sharing – see also additional exercises) Suppose as above there are two

dates, two states in the second date, two agents with identical preferences: U (ci;t; ci;t+1 (1); ci;t+1 (2)) =

ci;t + 0:9

P

s=1;2

s ln (ci;t+1 (s)). Endowments for agent 1 and 2 are given by (6; 8; 0) and (6; 0; 8), repectively.

1

1. If markets are complete (i.e. they can trade an Arrow-Debreu security for each state), what will be

the consumption allocations in each state in period t + 1?

2. If 1 = 2 =

1

2

what will be the state prices and consumption allocations? Based on your answer

what would be the price of an asset that pays (2; 1) in date t + 1?

3. What if 1 =

3

4

; 2 =

1

4

(what are the contingent prices, consumption and the price of asset paying

(2; 1))?

4. If agents can only trade an Arrow-Debreu security for state 1, could you price an asset (that is not

traded) that pays (2; 1) in date t + 1?

5. What would happen if agents can only trade an Arrow-Debreu security for state 1 and an asset that

pays (2; 1)?

Exercise 4 (Law of one price and no-arbitrage pricing) Suppose there are two states in t + 1 and there

are two assets traded in the economy with payo§ s x1;t+1 = (2; 2); x2;t+1 = (1; 3) and their prices are

p1 = 2; p2 = 2:1. If a third asset with payo§ s x3;t+1 = (4; 2) were o§ered to you at the price p3 = 3, what

would you do if you could buy or sell any of these assets at these prices? Can this be an equilibrium for

the price of the 3rd asset (taking the Örst two as given)?

Exercise 5 (no-arbitrage and borrowing constraints) Suppose there are two states in period t+ 1, and that

there are two assets traded in the economy with payo§ s x1;t+1 = (1; 4); x2;t+1 = (2; 3) and their prices are

p1 = 2:4; p2 = 2:3. Suppose you can borrow or invest at the gross risk free rate between t and t + 1 of

R

f

t = 1:1 (so r

f

t = 10%).

1. is there an arbitrage opportunity in this economy?

2. what if you are not allowed to borrow at this risk free rate (but can still invest at this rate)?

2

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