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Instructions. You are encouraged to work in groups, but everybody must write
their own solution to the problem that is for grade. Good Luck! (i) (For Grade)
There are n individuals. Each individual i has constant absolute risk aversion ai >
0 and an asset that pays Xi where (X1+,...,Xn) „ N (µ1,...,µn ) Σ).
(a) What are the optimal risk sharing contracts? What is the vector of payoffs from
an optimal risk-sharing contract? Characterize the set of the vectors of
certainty equivalents from optimal risk sharing contracts.
(b) (b) Answer (a) for a1 = ………+an, μ1 = ………….= μn
How much the society as a whole are willing to pay all of these assets? Assuming
that t ` hey wr ˘ ite a symmetric contract, what is the preference relation of an
individual on
(σ2, ρ) pairs. Discuss Briefly.

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Instructions. You are encouraged to work in groups, but everybody must write
their own solution to the problem that is for grade. Good Luck! 1. (For Grade)
There are Önitely many states s 2 S. The set of outcomes is [0; 1), the amount
of consumption. Consider an expected utility maximizer with utility function u (c)
= p c. Suppose that for each state s 2 S, there is an asset As that pays 1 unit of
consumption if the state is s and 0 otherwise (these are called Arrow-Debreu
securities). Suppose also that we know the preference of the decision maker
among these assets and the constant consumtion levels; e.g., we know how he
compares an asset As to consuming c at every state.
(a) Derive the decision makerís preference relation among all acts from the
above information.
(b) Assume that the decision maker has a Öxed amount of money M,

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which he cannot consume unless he invests in the Arrow-Debreu secuirites
above, assuming that these secrities are perfectly divisible, and the price of a
unit of As is some ps > 0. Derive the demand of the decision maker for these
securities as a function of the price vector p = (ps) s2S :
Question 1 Part (a)
We know that the DM is an expected utility maximizer. Therefore her
preferences Á over acts are completely described by her utility function u :
: r0, 8q Ñ R and probability P P
U(f) := _ P(s)u(f(s))> > P(s)u(g(s)) =: U(g)
We already know what is u: u(c) = vc for ce [0, co). So we only need to find P.
Write f, for
the Arrow-Debreu security corresponding to state s: for all state s'
1 its = s
f . (s')
10 else.
Observe that U(fs) = P(s), so to find P(s) it is enough to measure U(f,). We know
how the
DM ranks fs against any ce [0, co), that is, we can say whether the DM prefers f,
to a constant
act.

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This means in particular that for every s we have enough data to find c, e [0, co)
such that
is ~ Cs. But fs ~ Cs implies that
P(s) = U(f.) = U(cs) = Vcs.
So we have found P, and we can conclude that for an arbitrary act f
U() = EVcaf(s).
BES
Remark. This exercise has a "Anscombe-Aumann flavour." In the Savage
approach, to elicit
and P, first we estimate the probability (chopping the state space using P6), and
then we use
the probability to measure utility (applying mixture space theorem). The
Anscombe-Aumann

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methodology goes the other way round. First utility is measured using obejctive
lotteries (ap-
plying mixture space theorem). Then certainty equivalents are used to measure
the probabil-
ity, which is what we do here. Even if objective lotteries are slightly misterious
objects, the
Anscombe-Aumann method is way more flexible than Savage's, and nowadays
most of the works
in decision theory adopts it (e.g., ambiguity).
Part (b)
Note that any act can be written as a non-negative linear combination of Arrow-
Debreu securities,
and any non-negative linear combination of Arrow-Debreu securities is an act.
Therefore the
portfolio problem for the DM is:
maximize U(f) over f E F subject to _ f(s)p. < M.
The solution to this problem is: for all