HOMEWORK 2 SOLUTIONS
Northwestern University
Marciano Siniscalchi
Fall 2008
Econ 3310
1.
Forecasting and Random Variables
The individual minimizes
∑
s
P
(
{
s
}
)[
X
(
s
)

ξ
]
2
with respect to
ξ
. Differentiating with respect to
this variable yields
∑
s
P
(
{
s
}
)2[
X
(
s
)

ξ
](

1) = 0; dividing by two, recognizing that
∑
s
P
(
{
s
}
)
ξ
=
ξ
and rearranging terms yields the required result.
2.
Forecasting and Expected Utility
Suppose
π
is not admissible, so there is
ρ
such that
L
ρ
(
s
)
< L
π
(
s
) for all
s
∈
S
. Then
W

L
ρ
(
s
)
>
W

L
π
(
s
) for all
s
, so
u
(
W

L
ρ
(
s
))
> u
(
W

L
π
(
s
)) for all
s
because
u
is strictly increasing.
But then E[
u
(
W

L
ρ
)]
>
E[
u
(
W

L
π
)], so the individual strictly prefers to provide the forecast
system
ρ
rather than the forecast system
π
. It follows that, if a forecast system
π
maximizes the
individual’s expected utility, it must be admissible.
3.
Expected Utility Calculations
(i) Write
U
(
Z
) for the expected utility of the generic lottery
Z
. Then we have
U
(
X
) =
1
3
√
100 +
1
3
√
50 +
1
3
√
0
≈
5
.
69036; the certainty equivalet C[
X
] of
X
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 Spring '09
 Marciano
 Economics, Utility, Probability distribution, Lρ, Marciano Siniscalchi Econ

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