Pratt’s Theorem on Risk Aversion and Portfolio Demand
1
For
t
= 0
,
1 let
u
t
be a
C
1
vNM utility with positive ﬁrst derivative everywhere. Consider the
following problem:
α
t
∈
arg max
α
≥
0
±Z
u
t
(
w
+
αx
)
dF
(
x
)
²
.
where
R
xdF
(
x
)
>
0,
R
x
2
dF
(
x
)
>
0 (so
F
is nondegenerate) and
F
has bounded support.
Theorem
Let
u
0
be
strictly more concave
than
u
1
:
u
0
(
z
) =
T
(
u
1
(
z
)) for all
z
∈
R
+
, where
T
:
Range
(
u
1
)
→
R
is strictly concave. Then
α
1
≥
α
0
.
Proof
: Let
u
0
be strictly more concave than
u
1
. Deﬁne
V
(
α,t
) =
Z
u
t
(
w
+
αx
)
dF
(
x
)
,
and note that wlog we may take the domain of
V
(
·
,t
) to be
R
++
. (Why?) We have
V
α
(
α,
0) =
Z
u
0
0
(
w
+
αx
)
xdF
(
x
) =
Z
T
0
(
u
1
(
w
+
αx
))
u
0
1
(
w
+
αx
)
xdF
(
x
)
±
T
0
(
u
1
(
w
))
Z
u
0
1
(
w
+
αx
)
xdF
(
x
) =
Z
h
T
0
(
u
1
(
w
+
αx
))

T
0
(
u
1
(
w
))
i
u
0
1
(
w
+
αx
)
xdF
(
x
) +
T
0
(
u
1
(
w
))
Z
u
0
1
(
w
+
αx
)
xdF
(
x
)
<
T
0
(
u
1
(
w
))
Z
u
0
1
(
w
+
αx
)
xdF
(
x
) =
T
0
(
u
1
(
w
))
V
α
(
α,
1)
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This note was uploaded on 11/19/2010 for the course ECON 202 taught by Professor Schlee during the Spring '10 term at ASU.
 Spring '10
 schlee
 Microeconomics, Utility

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