Problem Set 1 - Solutions
Tommaso Denti
Question (i)
Part (a)
Set X̄ “
∞n
i“1
Xi , aggregate endowment. From the lecture notes, an optimal risk sharing contract
pY1 , . . . , Yn q is given by
Yi “
1{↵i
X̄ ` ⌧i
1{↵1 ` . . . ` 1{↵n
@i “ 1, . . . , n,
where ⌧1 , . . . , ⌧n P R are state-independent transfers such that
∞n
i“1 ⌧i
“ 0. Agent i gets
´
¯
”
ı
1{↵i
1{↵i
1
¯
´↵i 1{↵ `...`1{↵
E rX̄ s´ 2p1{↵ `...`
V ar pX̄ q`⌧i
´
X
1{↵n q2
n
1
1
e´↵i ⌧i “ ´e
E rui pYi qs “ ´E e 1{↵1 `...`1{↵n
.
Hence
Yi „
1{↵i
1{↵i
¯ ` ⌧i ,
ErX̄ s ´
V arpXq
2p1{↵1 ` . . . ` 1{↵n q2
1{↵1 ` . . . ` 1{↵n
where the right-hand side is the certainty equivalent of Yi for agent i.
Part (b)
Notice that ErX̄s “ nµ and
V arpX̄q “
n ÿ
n
ÿ
i“1 j “1
Cov pXi , Xj q “
n
ÿ
i“1
V arpXi q `
n ÿ
ÿ
i“1 j‰i
CovpXi , Xj q “ n
2
p1 ` pn ´ 1q⇢q.
The results in Part (a) become
1
X̄ ` ⌧i ,
n
1
↵
↵
Erui pYi qs “ ´e´↵p n ErX̄ s´ 2n2 V arpX̄q`⌧i q “ ´e´↵pµ´ 2n
↵ 2
p1 ` pn ´ 1q⇢q ` ⌧i .
Yi „ µ ´
2n
Yi “
1
2
p1`pn´1q⇢q`⌧i q
,
Summing up the agents’ certainty equivalent, the society as a whole is willing to pay
nµ ´
↵
2
2
p1 ` pn ´ 1q⇢q
for the assets. If we focus on symmetric contracts, ⌧1 “ . . . “ ⌧n “ 0. Therefore
p
2
, ⇢q Á p˜ 2 , ⇢˜q
ô
2
p1 ` pn ´ 1q⇢q § ˜ 2 p1 ` pn ´ 1q⇢˜q,
which is the agent’s preference over the assets. Comments: (i) fixing ⇢, the agent prefers lower
, and (ii) fixing 2 , the agents prefers lower ⇢.1 Intuition: V arpX̄q (and therefore V arpYi q) is
decreasing in 2 and ⇢, and the agent is risk averse.
2
Question (ii)
See solution to question 2, pset 1 2010.
Question (iii)
See solution to question 2, pset 1 2014.
Question (iv)
See solution to question 2, final 2011.
1 Notice that p1 ` pn ´ 1q⇢q • 0 by assumption, otherwise the assets are not jointly normal, that is, ⌃ is not
positive semi-definite. In fact, 1 ` pn ´ 1q⇢ is an eigenvalue of ⌃.
2
MIT OpenCourseWare
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14.123 Microeconomic Theory III
Spring 2015
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