Northwestern University
Marciano Siniscalchi
Fall 2009
Econ 3310
HOMEWORK 2
DUE MONDAY, OCTOBER 19, IN CLASS.
1.
Forecasting and Random Variables
Consider a finite state space
S
, a probability
P
on
S
, and a random variable
X
:
S
→
R
. Suppose
that the following penalty scheme is offered to a decisionmaker: she is asked to report a number
ξ
, and then she will be subject to the penalty [
X
(
s
)

ξ
]
2
in each state
s
∈
S
.
Show that, if the individual minimizes her expected loss from this penalty scheme, using the
probability
P
, then her optimal forecast is
ξ
= E[
X
] =
∑
s
X
(
s
)
P
(
{
s
}
), the expectation of
X
with
respect to the probability
P
.
Observe that the simple penalty scheme for a single event we talked about in class as a prelude
to our theorem about admissible forecasts is actually a special case of this result: just take
X
= 1
E
,
the indicator function of
E
.
2.
Forecasting and Expected Utility
This question asks you to verify something we claimed to be true in class:
expectedutility
decisionmakers provide admissible forecasts, even though the forecast systems they provide will
not, in general, correspond to the subjective probability they use to evaluate bets.
Consider a finite state space
S
, and consider an individual who must provide a forecast system
π
:
P
(
S
)
→
R
, under the penalty scheme we described in class: that is, if the realized state is
s
,
the individual must pay
∑
E
∈P
(
S
)
[1
E
(
s
)

π
(
E
)]
2
. As we did in class, denote the latter quantity by
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 Spring '09
 Marciano
 Economics

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