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1
Generalized Stochastic Dominance with Respect to a Function
Lecture XX
I.
Meyer, Jack
“Choice among Distributions.”
Journal of Economic Theory
14(1977):
32636.
A.
The general idea of the manuscript is to restrict the risk aversion coefficient
for stochastic dominance to those risk aversion coefficients in a given interval
() () ()
12
rx rx rx
<<
.
B.
This problem will be solved by finding the utility function
()
ux
which
satisfies:
rx
x
01
''( )
'( )
[,]
≤−
≤
∀ ∈
and minimizes
[]
Gx
Fx u xd
x
() '
−
∫
0
1
1.
Given that this integral yields the expected value of
Fx
minus the
expected value of
( )
, the minimum will be greater than zero if
is preferred to
( )
by all agents who prefer
to
( )
.
2.
If the minimum is less than zero, then the preference is not unanimous
for all agents whose risk aversion coefficients are in the state range.
3.
Another problem is that utility is invariant to a linear tranposition.
Thus, we must stipulate that
( )
u
′
=
.
C.
The problem is then to use the control variable
( )
′′
−
′
to maximize the
objective function
−−
∫
x
0
1
subject to the equation of motion
'( ) '
=−
and the control constraints
≥
+≥
1
2
0
0
with the initial condition
( )
u
′
=
.
1.
Rewriting the problem, substituting
( )
zx
′
yields
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View Full Document AEB 6182–Agricultural Risk Analysis and Decision Making
Fall 2004
Professor Charles Moss
Lecture 20
2
[]
()
max
( )
( ) '( )
'( ) '
'( ) ( )
−−
=
−+
≤
−≤
∫
Gx
Fx u xd
x
st
u x
u x z x
zx
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This note was uploaded on 07/15/2011 for the course AEB 6182 taught by Professor Weldon during the Fall '08 term at University of Florida.
 Fall '08
 Weldon

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