Comparative statics with respect to w. In this part of the question, we will consider how household behavior may change

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Comparative statics with respect to w. In this part of the
question, we will consider
how household behavior may change if the government increases the
level of the public
input, i.e., w increases. Is the household likely to increase b,
i.e., will public inputs
“crowd in” private inputs, enhancing health effects? Or is the
household likely to
decrease b, i.e., will public inputs “crowd out” private inputs,
blunting health effects?
We will show that this depends crucially on what households believe
about ∂
2h
∂b∂w, i.e.,
5
whether households believe public and private inputs are
complements (
∂
2h
∂b∂w > 0) or
substitutes (
∂
2h
∂b∂w < 0).
(a) Describe intuitively why this (crowd-in vs. crowd-out depends
on whether public
and private inputs are complements or substitutes) is reasonable.
You don’t
need any equations, and you can answer this in 2 sentences: “If b
and w are
complements, then ..., so it’s reasonable to think that I’ll ...
“If b and w are
substitutes, then ..., so it’s reasonable to think that I’ll
...)
(b) As in (a), but semi-formally. You don’t need to do a full
derivation, but you
should refer to specific equations (especially the condition for
optimal c and b)
and specific terms in those equations.
(c) (Extra credit) Show formally that
db∗
dw = −
∂g/∂w
∂g/∂b
= −
∂
2U
∂h2
∂h
∂b
∂h
∂w +
∂U
∂h
∂
2h
∂b∂w!
/p


∂
2U
∂h2
∂h
∂b !2
+
∂U
∂h
∂
2h
∂b2

 /p
= −
∂
2U
∂h2
∂h
∂b
∂h
∂w +
∂U
∂h
∂
2h
∂b∂w
∂
2U
∂h2
∂h
∂b !2
+
∂U
∂h
∂
2h
∂b2
(3)
Hint: remember the assumptions on cross-partials and second
derivatives, especially Uch, hwwand hbb.
(d) (Extra credit) Show that (3) and an assumption ∂
2h
∂b2
≤ 0 together imply that
i. If ∂
2h
∂b∂w < 0, then db∗
dw < 0
ii. If ∂
2h
∂b∂w = 0, then db∗
dw < 0 but “only a little”
iii. If ∂
2h
∂b∂w > 0, then it’s likely but not certain that db∗
dw < 0