Tutorial 6 - Asset Economy with Expected Utility
Overview
• I will be covering Q1 and Q2, please do Q3,4 in your own time
Question 1
Suppose one agent is risk averse and one agent is risk neutral.
(a) Draw the contract curve.
Recall the contract curve is all the feasible trade outcome that are Pareto efficient.
How can we arrive at this graph so quickly ? Well, suppose that that is was efficient
for the risk averse agent to carry some risk (i.e not on the 45 degree line). You can
then increase the total utility by giving that risk to the risk neutral individual who is
not affected by it. It must then be true that the risk averse agent carries no risk on
the contract curve.
If the edgeworth box was a square then the risk neutral individual would also carry no
risk. That is because there is no aggregate risk, a square edgeworth box means that
the total resources are the same in both states.
We have Wilson’s Mutuality Principle:
In complete markets all idiosyncratic risk (i.e risk purely from the allocation of endow-
ments) is insured in the competitive equilibrium
(b) How risk averse is the representative consumer?
The representative consumer behaves like the consumers on average in that they as a
single person economy reach the same outcome.
In our economy risk neutral agents don’t care about risk and the risk averse agents
don’t take any risk. The representative consumer is then risk neutral.
1
(c) Suppose now both are risk averse with CARA utility, what would you expect the
contract curve to look like?
Wilson tells us that the marginal share of risk is proportional to the risk tolerance.
Here both have constant risk tolerance and so the contract curve is somewhere between
the two dashed lines and parallel to them. If agent one on the left was less risk tolerant
the line would be closer to the left, while if the agent on the right was less risk tolerant
the line would be closer to the right.
Note that if there was no aggregate risk, the the two dashed lines would be the same
line as neither has to carry any risk.
Question 2
Suppose we have utilities v1(y) and v2(y), endowment vectors w
1 and w2 and arrow prices
α.
We assume there is no aggregate uncertainty. The total endowments must be the same for
both states, let that be w
w := w11 + w
2
1 = w
1
2 + w
2
2
(a) For this part assume they agree on the state probabilities (π, 1− π)
Claim: π
1−π =
α1
α2
We take π as given so we need to find the arrow prices to prove the claim.
The agent maximises expected utility subject to her BC, that is
max πv1(y
1
1) + (1− π)v1(y12)
s.t α1y
1
1 + α2y
1
2 = α1w
1
1 + α
1
2w
1
2
First order conditions give us
π
1− π
v′1(y
1
1)
v′1(y
1
2)
=
α1
α2
2
An identical condition for agent 2 would also hold.
If one of the agents is risk neutral then v′ is constant and so we have proven our claim.
If the agents are risk averse, first assume for a contradiction that we have π
1−π <
α1
α2
.
Then we must have
v′i(y
i
1)
v′i(y
i
2)
> 1, i = 1, 2 for the F.O.Cs to hold. We can rewrite these as
v′1(y
1
1) > v
′
1(y
1
2)
v′2(y
2
1) > v
′
1(y
2
2)
Concavity of the the utilities means we have y11 < y
1
2 and y
2
1 < y
2
2, which gives us
y11 + y
2
1 < y
1
2 + y
2
2
contradicting our assumption that there is no aggregate risk and so total consumption
in each state should be equal.
If we assume conversely that π
1−π >
α1
α2
, by a very similar process we also get a contra-
diction but all the inequalities would just be going the other direction. If we want the
details we would have
v′1(y
1
1)
v′1(y
1
2)
< 1 and
v′2(y
2
1)
v′2(y
2
2)
< 1 which gives us
v′1(y
1
1) < v
′
1(y
1
2)
v′2(y
2
1) < v
′
1(y
2
2)
which by concavity would give us a contradiction
y11 > y
1
2, y
2
1 > y
2
2 =⇒ y11 + y21 > y12 + y22
Hence proving our claim.
(b) Claim: if agents have utility given by ui = min{x1, x2} then they will insure each other
completely.
Their indifference curves are going to be right angles with the kink on the 45 degree
line. As there is no aggregate uncertainty, the edgeworth box is a square and the two
indifference curves touch on the 45 degree line - i.e both are perfectly insured.
(c) What if now we assume the two agents have different beliefs, specifically that π1 > π2.
The assumption then implies that
π1
1− π1 >
π2
1− π2
Claim: π1
1−π1 >
α1
α2
> π2
1−π2
3
We have the same F.O.C we found before
π1
1− π1
v′1(y
1
1)
v′1(y
1
2)
=
α1
α2
π2
1− π2
v′2(y
2
1)
v′2(y
2
2)
=
α1
α2
We then employ a similar strategy to before where we assume certain conditions are
true for a contraction.
So we would first suppose that α1
α2
> π1
1−π1 >
π2
1−π2
which would imply by the same logic as before that
v′1(y
1
1) > v
′
1(y
1
2)
v′2(y
2
1) > v
′
1(y
2
2)
which again by the concavity of the the utilities means we have y11 < y
1
2 and y
2
1 < y
2
2,
which gives us the contradiction
y11 + y
2
1 < y
1
2 + y
2
2
you would then suppose that π1
1−π1 >
π2
1−π2 >
α1
α2
and do a very similar process to find a
contradiction. The details are
v′1(y
1
1)
v′1(y
1
2)
< 1 and
v′2(y
2
1)
v′2(y
2
2)
< 1 which gives us
v′1(y
1
1) < v
′
1(y
1
2)
v′2(y
2
1) < v
′
1(y
2
2)
which by concavity would give us a contradiction
y11 > y
1
2, y
2
1 > y
2
2 =⇒ y11 + y21 > y12 + y22
The only scenario left not contradicted is what we want to prove, so it must be true.