CHAPTER 9.
9.1. a) Given preferences and endowments, it is clear that the allocation {(4, 2, 2) ,(4, 2, 2)} is
PO and feasible. In general, there is an infinity of PO allocations.
b) Yes, but only if one of the following securities is traded
1
1
sor
1
1
s21
c) Agents will be happy to store the commodity for two reasons : consumption smoothing
– they are pleased to transfer consumption from period 1 to period 2-, and in addition by
d) Remember aggregate uncertainty means that the total quantity available at date 2 is not
the same for all the states. If one agent is risk-neutral, he will however be willing to bear
9.2. 1. Because of the variance term diminishing utility, consumption should be equated
across states for each agent.
2. There are many Pareto optima. For example, the allocations below are both Pareto
optimal :
t=0
t=1
1
2
Allocation 1
Agent 1
4
3
Agent 2
4
3
Allocation 2
Agent 1
5
4
Agent 2
3
2
The set of Pareto optima satisfies:
8cc;6cc),cc thus(and cc:)c,c,c(),c,c,c( 2
0
1
0
1
1
2
1
2
2
1
2
1
2
1
1
2
2
2
1
2
0
1
2
1
1
1
0
3. Yes. Given E(c) in the second period, var (c) is minimized.
b.
1. The Pareto optima satisfy
)c6ln(
4
3
)c6ln(
4
1
c8cln
4
3
cln
4
1
cmax 1
2
1
1
1
0
1
2
1
1
1
0
c,c,c 1
2
1
1
1
0
.2Q;2Q
Q5
1
Q1
12
1
1
1
2
1
1
1
.2Q;2Q
Q1
1
Q5
12
2
1
2
2
2
1
2
Thus,
12
1
3
1
4
1
Q1
1
4
1
P1
1
1
12
3
3
1
4
3
Q5
1
4
3
P1
2
2
Allocations at Equilibrium:
t=0
t=1
1
2
Agent 1:
3
1
4)2(
12
3
)2(
12
1
4
3
3
Agent 2:
3
2
3)2(
12
3
)2(
12
1
4
3
3
This is a Pareto Optima, consumption is stabilized in t=1. However, since agent 1 had
3. Now only (1,0) is traded. The C.E. will not be Pareto optimal as the market is
incomplete. The C.E. is as follows:
3
1
1
4
4
1
11
Q2
1
The F.O.C.’s are :
1
1
1
1Q5
4
Thus
.2Q;2Q
Q5
1
Q1
12
1
1
1
2
1
1
1
12
1
3
1
4
1
P1
.
Allocation
t=0
t=1
1
2
Agent 1:
6
5
3
6
1
4
3
5
Agent 2:
6
1
4
6
1
4
3
1
Consumption is stabilized in state
1
: effectively agent 1 buys consumption insurance
from agent 2.
9.3. The Pareto optima satisfy:
)c6ln(
2
1
)c6ln(
2
1
c6cln
2
1
cln
2
1
5.c25.max 1
2
1
1
1
0
1
2
1
1
1
0
c,c,c 1
2
1
1
1
0
The F.O.C.’s are
025.:c1
0
0)1(
c6
1
)
2
1
(
c
1
)
2
1
(5.:c 1
1
1
1
1
1
0)1(
c6
1
)
2
1
(
c
1
)
2
1
(5.:c 1
2
1
2
1
2
21
6
c ,c2c6
c6
1
c
1
2
11
1
1
1
1
1
1
1
1
1
.
21
6
c ,c2c6
c6
1
c
1
2
11
2
1
2
1
2
1
2
1
2
Thus
1
2
1
1cc
, and therefore
2
2
2
1cc
;
If there is no aggregate risk and the agents preferences are the same state by state, then a
t=0
t=1
1
2
Agent 1:
.25 if 0
.25 if 6
21
6
21
6
Agent 2:
.25 if 6
.25 if 0
21
1
16
21
1
16
25.
, indeterminate
In the second case (state 2 endowment = 5 for agent 1, 3 for agent 2), there will be a
b. The agents’ problems are:
)R4ln(
1
1
1
1
1
2
Agent 1:
5.1
2
1
2
4
4
Agent 2:
5.4
2
1
4
2
2
c. Let us assume the firm can introduce 1 unit of either security. Either way, the problems
of the agents and their F.O.C.’s are not affected. What is affected are the market clearing
conditions:
If 1 unit of Q is introduced
If 1 unit of R is introduced
0RR
1QQ
21
21
1RR
0QQ
21
21
Let’s value the securities in either case.
If one unit of Q is introduced:
The F.O.C.’s become
Agent 1:
1
R
1
Q
R4
1
P
Q2
1
P
Agent 2:
12
R
212
Q
R2
1
2
1
R2
1
2
1
P
)Q5(2
1
Q14
1
2
1
2
1
Q4
1
P
The equations involving R are unchanged. Thus
4
1
PR
,
0R,0R 21
.
t=0
t=1
1
2
Agent 1:
14
8
2
14
3
3
8
2
4+2/3
4
Agent 2:
14
5
4
14
3
3
5
4
3-2/3
2
If one unit of R is introduced:
The first order conditions become, with market clearing imposed:
Agent 1
1
R
1
Q
R4
1
P
Q2
1
P
Agent 2
12
R
12
Q
R3
1
2
1
R2
1
2
1
P
)Q4(2
1
Q4
1
2
1
P
So,
Q
P
is unchanged, and
4
1
Q
P
,
2Q,2Q 21
.
t=0
t=1
1
2
Agent 1:
14
9
2
14
3
3
2
2
1
2
4
4+2/3
Agent 2:
14
6
4
14
3
3
1
2
1
4
2
2+1/3
The firm is indifferent as to which security it sells – either way it receives the same thing.
Either way a Pareto optimum is achieved since, with no short sales constraints, the
9.4. a) At a P.O. allocation there is no waste and there are no possibilities to redistribute goods
and make everyone better off. From the viewpoint of social welfare there seems to be no
argument not to search for the realization of a Pareto Optimum. Beyond considerations of
b) The answer to a) indicates we should care since complete markets are required to guarantee that a Pareto optimal
allocation is reached. Are markets complete? certainly not! Are we far from complete markets? Would the world be