Iii well φ w pu w a h r h r 1 p u w a l r l r is

(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 second-order 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” decision-making, where agents fail to re-optimize 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 re-optimizing 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 second-derivative 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