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CHAPTER 17.
17.1. a) These utility functions are well known. Agent 1 is risk-neutral, agent 2 is risk-averse.
b) A PO allocation is one such that agent 2 gets smooth consumption.
c) Given that agent 2 is risk-averse, he buys A-D1 and sells AD2, and gets a smooth
consumption; Agent 1 is risk-neutral and is willing to buy or sell any quantity of A-D
1
2
q
q1
The price of AD securities depends only on the probability of each state.
Agent 2’s optimal consumption levels are
1/112ccc 2
2
1
22
which is 1 if = 0.5.
d) Note: it is not possible to transfer units of consumption across states. Price of the bond
is . Allocation is not PO.
Available Security
t = 0
t= 1
-pb
θ1
θ2
1
2
17.2 When markets are incomplete :
i) MM does not hold: the value of the firm may be affected by the financial structure
of the firm.
financial instruments
17.3 a. Write the problem of a risk neutral agent :
This is generic: risk neutrality implies no curvature in the utility function. If the equilibrium
change, except that the equilibrium price becomes:
PQ = ¼
b. pQ = 1/6, Q1 = 0 (the former FOC is not affected), pR = 1/3
So agent 1 sells 4 units of asset R. He reduces his t = 1 risk. At date 1, he consumes 1 unit
Post trade allocation:
t = 0
t = 1
t = 2
Agent 1
20 4/3
1
1
Agent 2
28 2/3
15
19
17.4
t=0
t=1
1
2
Agent 1
4
6
1
Agent 2
6
3
4
6.Prob4.Prob
ncEc
2
1
c
~
,cU
ncEnc
2
1
c
~
,cU
21
2
1
2
010
2
1
1
1
010
1
a. Initial utilities –
409.1716.693.
79.14.386.1
2
1
1n6.6n4.4n
2
1
c
~
,cU:1Agent 10
1
201 1
Agent 2: U c , c 6 .4 n 3 .6 n 4
23 .439 .832
4.271
%ll
b. Firm’s output
t=1
1
2
-p
2
3
6.3
6.1
p
The reverse is true for agent 1 especially on the issue of consumption smoothing across
t=1 states: he has very little endowment in the more likely state. Furthermore the security pays
If the two states were of equal probability agent 1 would have a bit less need to smooth,
and thus his demand would be relatively smaller. We would expect p to be smaller in this case.
c. The Arrow-Debreu securities would offer greater opportunity for risk sharing among the
d. Let the foreign government issue 1 unit of the bond paying (2.2); let its price be p.
Agent Problems:
1
222
QQ24n6.Q23n4.pQ4
2
2
Where, in equilibrium
1QQ 21
F.O.C’s:
26.
24.
p
1
22 Q24
Q23
2
Substituting
12 Q1Q
, these equations become:
111 Q21
4.2
Q26
6.1
pQ4
p
11 Q26
4.2
Q25
6.1
p
Solving these equations using matlab yields
4215.1Q
502.1p
1
and thus
4215.Q2
1
22
1
11
1
22
1
11 eqeqxqxq
Agent 2 : max
2
2
2
1
2
1
2
1xlnxln
s.t.
2
22
2
11
2
22
2
11 eqeqxqxq
Considering that the two Arrow-Debreu prices necessarily remain equal, agent 2 solves
2
2
2
1
2
2
2
2
2
1
2
1
2
1
x,x xln)xeex2ln(max
2
2
It is clear that he wants x to be as high as possible, that is, x =2.
The FOC wrt
2
2
x
solves for
2
eex2
xx 2
2
2
1
2
1
2
1
2
2
.
Again there is perfect consumption insurance for the risk averse agent (subject to
t=1
1
2
A bond
1
1
Technology 2
1+x
1-
x
2
1
These two securities can replicate (1,0) and (0,1). To replicate (1,0), for instance, invest a
in the bond and b in the firm where a and b are such that
b)
1e
p8
1
e
p8
3
2
2
1
2
2
8
3
1
8
1
2eep
If
12 ee
, then
12 pp
;
If
12 ee
, then
21 pp
b. There is now only one security
t=1
t=2
b
p
1
2
1
1
Agent Endowments
t=1
t=2
1
2
Agent 1
1
e
1
a
1
a
Agent 2
2
e
2
a
2
a
Now, there can be trade here even with this one asset, if, say
1
e
= 0,
1
a
> 0,
2
e
> 0,
2
a
= 0
to take an extreme case.
c. Suppose we introduce a risky asset
t=0
t=1
1
2
-p
1
z
2
z
where
0z,z,zz 2121
.
Combinations of this security and the riskless one can be used to construct the state
claims. This will be welfare improving relative to the case where only the riskless asset is
d. The firm can convert x units of (1,1) into x units of {(1,0), (0,1)}. These agents
(relative to having only the riskless asset) would avail themselves of this technology, and
inputs
firm
(inventor)
agent
outputs
a{(1,0), (0,1)}
(x-a){(1,0), (0,1)}
x(1,1)
Clearly, if a = x, the agents would ignore the inventor. However, each agent would be willing to
Suppose the inventor could choose to convert x, 2x, 3x, …, nx securities (x understood to be
small). The additional increment he could charge would decline as n increased.
