## (solution) Prof. M. Magill Econ 659 - Problem 3 given: Thursday October 2:

how do you know if a financial market is complete? From the attached document

Prof. M. Magill Econ 659 - Problem 3 given: Thursday October 2: Fall 2014 due: Thursday October 9. Subject: Computing Sequential Market Equilibria In this problem set you will use Matlab to compute sequential market equilibria for the economy

studied in Problem set # 1. We want to compare sequential market equilibria with the contingent

market equilibria of the previous problem set. Read pages 37-40 and Sections 8 and 9 in MQ. I

have also posted MMLecture-6 on Blackboard: the last 8 pages contain an introduction to sequential

markets which you will find useful. As I point out in the notes you will need to modify the budget

equations of the agents to allow for their ownership shares of equity.

Consider the economy of Problem set #1 with 3 agents (i = 1, 2, 3), one good and two dates

(t = 0, 1) with 4 possible states at date 1. When applicable in each of the problems that follow,

use the following notation: for agent i let z0i = holding of bond , ?1i = share of equity of firm 1 ,

?2i = share of equity of firm 2; let ??3 = (0, 0) (initial share holdings of agent 3), ??1 = (1, 0) (initial

share holdings of agent 1), ??2 = (0, 1) (initial share holdings of agent 2). In a sequential market

equilibrium we must have

3

X z3i =0 i=1 3

X ?ki = 1, k = 2, 3 i=1 C. Sequential Market Equilibrium

(1) Let pr(k, i) = 1/2 for all k and i, ai = 0 for all i and ? = 2, b1 = b2 = 500, b3 = 1000 :

suppose also that the only traded security is the riskless bond with payoff (1, 1, 1, 1) and that

agents have no initial holdings of the bond (agents do not inherit debts from the past and

the bond is in zero net supply).

(a) Assume, as in question 1 of B, that there is no risk: ?1 = ?2 = 0. Calculate the

SM equilibrium. Compare the allocation and the interest rate r with that of the AD

equilibrium obtained in question 1 of B (of course in the AD equilibrium we mean the

interest rate rAD implied by the AD price of the income stream (1, 1, 1, 1)). Explain the

intuition of the result. 1 (b) Suppose now that we add risk to the economy, as in question 2 of B, by setting e1 = 100,

e2 = 200. Calculate the SM equilibrium. Compare the interest rate with the interest rate

in (a) (here you are comparing the effect of adding risk with the same market structure).

Also compare the interest rate with that obtained in question 2 of B (here you are

comparing how agents cope with the same risk with two different market structures).

What do agents try to do when they can only use the bond to cope with date 1 risk?

Does this help you in explaining why you find that rSM &lt; rAD ? You may find it easier

to understand what?s going on by examining what happens to the ?bond price? rather

than the ?interest rate?.

(2) We continue to assume that pr(k, i) = 1/2 for all k and i, ai = 0 for all i and ? = 2, e1 = 100,

e2 = 200 (as in question 2 of B). But we assume that two new securities are added by letting

agents 1 and 2 issue ownership shares to their (firms) income streams: thus there are three

securities: the riskless bond in zero net supply, the equity of firm 1 whose payoff is the date

1 endowment of agent 1, and the equity of firm 2 whose payoff is the date 1 endowment of

agent 2: the latter two securities are both in positive net supply. Agent 1 (2) is full initial

owner of his equity and can sell part of it to the other agents on the equity market.

(a) Are the financial markets complete?

(b) Find the SM equilibrium and compare it to the AD equilibrium found in question 2 of

B. Is there a surprise here?

(c) Use question 2(b) of B to explain the result found in (b).

(d) Note that in the SM equilibrium that you have just found in (b) none of the agents use

the bond. It can be shown that this comes from the fact that the three agents have the

same constant-relative-risk-aversion. If there were trade on the bond, one agent would

have to leverage his portfolio and would be in a relatively riskier position than the others.

Suppose that we keep all the parameters the same except for the coefficient ai of agent

3. Compute the SM equilibrium for a3 = 500, a3 = 10, 000: note that in the latter case

the consumption streams of agents 1 and 2 and the bond price are very close to those in

the AD equilibrium of the riskless economy in 1(a) in B. Explain what?s happening in

these equilibria by examining the agents? portfolios, consumption streams and the bond

price, and explain what you find.

(3) Let?s return to the case where ai = 0 for all i, and lets continue to assume that agents 1 and

2 face idiosyncratic shocks, e1 = 100, e2 = 200. Suppose now that the agents have different

assessments of the probabilities of success of the firms given by

2 pr(1, 1) = 2/3 pr(2, 1) = 2/3

pr(1, 2) = 1/3 pr(2, 2) = 1/2

pr(1, 3) = 3/4 pr(2, 3) = 1/4

Suppose that the securities are the riskless bond and the equities of the two firms. Find the

SM equilibrium and compare it to the AD equilibrium of the same economy.

(4) Suppose you keep the economy the same as in the previous question, except that we add a

fourth security with payoff stream (0, 0, 0, 100) across the states at date 1. Find the SM equilibrium and compare it to the AD equilibrium. Explain the outcome. Suppose that instead

of adding the stream (0, 0, 0, 100) we add a call option on firm 2?s equity with strike price

500: explain why the SM equilibrium allocation does not coincide with the AD equilibrium.

Try to show that if instead we add an option on an appropriate weighted sum of the two

equity contracts (i.e. an option on an index) then we could get back to the AD equilibrium

allocation. 3

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