(iii) Well,
φ
w
=
pu
′′
(
w
+
a
∗
(
H

r
))(
H

r
) + (1

p
)
u
′′
(
w
+
a
∗
(
L

r
))(
L

r
)
is ambiguous, since
H

r >
0 but
L

r <
0. (If
u
′′′
()
<
0, however, it turns out that if
w
↑
then
a
∗
↑
)
(iv) We use the envelope theorem. In particular, let
V
(
H, L, p
) = [
pu
(
w
+
a
(
H

r
)) + (1

p
)
u
(
w
+
a
(
L

r
))]
a
=
a
*
Then
∂V
(
H, L, p
)
∂H
=
pu
′
(
w
+
a
∗
(
H

r
))
a
∗
which has the same sign as
a
∗
.
∂V
(
H, L, p
)
∂L
=
pu
′
(
w
+
a
∗
(
L

r
))
a
∗
which has the same sign as
a
∗
,
∂V
(
H, L, p
)
∂p
=
u
(
w
+
a
∗
(
H

r
))

u
(
w
+
a
∗
(
L

r
))
Since
u
() is increasing, this is positive if
w
+
a
∗
(
H

r
)
> w
+
a
∗
(
L

r
), or
a
∗
H > a
∗
L
. So if
a
∗
>
0,
∂V/∂p >
0, but if
a
∗
<
0,
∂V/∂p <
0.
4
6. Suppose an agent maximizes
max
x
f
(
x, c
)
yielding a solution
x
∗
(
c
). Suppose the parameter
c
is perturbed to
c
′
. (i) Use a secondorder Taylor
series expansion to characterize the loss that arises from using
x
∗
(
c
) instead of the new maximizer,
x
∗
(
c
′
).
Show that for small changes in
c
, the loss to the objective function is proportional to
(
x
∗
(
c
)

x
∗
(
c
′
))
2
.
(ii) Explain the strengths and weaknesses of using this as a model of “sub
rational” decisionmaking, where agents fail to reoptimize after an unexpected shock. How would
you construct such a model?
Linearize
f
(
x, c
′
) in
x
∗
(
c
) around
x
∗
(
c
′
) to get
f
(
x, c
′
) =
f
(
x
∗
(
c
′
)
, c
′
) +
f
x
(
x
∗
(
c
′
)
, c
′
)(
x

x
∗
(
c
′
)) +
f
xx
(
ξ, c
′
)
(
x

x
∗
(
c
′
))
2
2
where
ξ
is between
x
and
x
∗
(
c
′
). Since
f
x
(
x
∗
(
c
′
)
, c
′
) = 0 from the FONCs, we can rearrange to
get
f
(
x, c
′
)

f
(
x
∗
(
c
′
)
, c
′
) =
f
xx
(
ξ, c
′
)
(
x

x
∗
(
c
′
))
2
2
Evaluate at
x
=
x
∗
(
c
), and
f
(
x
∗
(
c
)
, c
′
)

f
(
x
∗
(
c
′
)
, c
′
) =
f
xx
(
ξ, c
′
)
(
x
∗
(
c
)

x
∗
(
c
′
))
2
2
So the loss that the agent incurs by not reoptimizing is proportional to (
x
∗
(
c
)

x
∗
(
c
′
))
2
, so
the loss is “not too large” as long as
x
∗
(
c
) and
x
∗
(
c
′
) are close.
One strength is that it is simple.
We can easily see how the secondderivative is related
to the approximate loss in welfare, and the model can be applied to many situations.
One
weakness is that even though
c
and
c
′
might be close, the optimal policies may be very far
apart, or
f
xx
(
ξ, c
′
) might be very large. This makes it hard to judge whether or not the loss
is large without looking at a particular problem. I would introduce a fixed cost to changing
policies, so that whenever the agent wants to move from
x
∗
(
c
) to the new optimum
x
∗
(
c
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 Fall '12
 Johnson
 Supply And Demand, implicit function, c′, FONCs