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ECON501 Advanced Microeconomic Theory 1
Fall Semester 2007
Problem Set 7
The due date for this problem set is Monday November 19
1. Assume a
f
nite state space
S
=
{
s
1
,...,s
N
}
and a onedimensional
consequence space
X
=
R
+
Suppose a person’s preference relation over the set of statecontingent
vectors
c
=(
c
1
,...,c
N
)
∈
R
N
+
, admits a Subjective Expected
Utility Representation. That is, there exists a probability vector
(
π
1
,...,π
N
) and utility index
u
:
R
+
→
R
such that for any pair
of acts
c, c
0
:
c
<
c
0
⇔
N
X
n
=1
π
n
u
(
c
n
)
≥
N
X
n
=1
π
n
u
(
c
0
n
)
Furthermore, assume that
u
(
·
) is a strictly concave and strictly
increasing function and that
π
n
>
0foreach
n
=1
,...,N
.S
in
c
e
X
=
R
+
,w
ecande
f
ne a convex combination of two contingent
consumption vectors as follows: for any pair of contingent con
sumption vectors
c
,
c
0
and any
α
in (0
,
1),
c
00
=
α
c
+(1
−
α
)
c
00
is
the contingent consumption vector for which
c
00
n
=
αc
n
+(1
−
α
)
c
0
n
for every
n
=1
,...,N.
(a) Show that if for any pair of contingent consumption vectors
c
and
c
0
c
∼
c
0
and
c
6
=
c
0
then
c
00
=
1
2
c
+
1
2
c
0
Â
c
(b) Explain why is this not a contradiction of the independence
axiom.
Notation:
for any pair of contingent consumption vectors
c
and
c
0
and any event
E
⊂
S
,let
c
E
c
0
denote the contingent consumption
vector
c
00
for which
c
00
n
=
½
c
n
if
s
n
∈
E
c
0
n
if
s
n
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View Full Document (c) Show that a preference relation that admits a subjective ex
pected utility representation satis
f
es the following property
called the
surething principle.
SureThing Principle:
Fo
ranytwocon
t
ingen
tcon
sump
t
ion
vectors
c
and
c
0
,andanyevent
E
⊂
S
c
%
c
0
E
c
⇔
c
E
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This note was uploaded on 09/04/2010 for the course ECON 501 taught by Professor Grant during the Spring '10 term at Rice.
 Spring '10
 GRANT

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