1 as with the change in h 2 discussed above assume

1 As with the change in H
2
discussed above, assume that the change in H
1
is a “shock”
sthat occurs with no change in the purchased or endogenous inputs in the health
production function for H
1
or H
2
(C
1
C
, M
1
, C
2
C
and M
2
) and without any change in HE
1
,
HE
2
or η . Thus this change in H
1
and H
2
does not affect W
CD
.
27

It is also instructive to examine how changes in
exogenous variables that are likely to increase child
health ultimately affect children’s academic skills.
The net effect of changes in p
M,1
, p
M,2
, HE
1
and HE
2
(prices for health inputs and the overall health
environment) on T
2
is obtained directly by
differentiating equation (15), yet more can be learned
by decomposing these effects to illuminate the
pathways by which they take place. Consider first an
improvement in the health environment in the second
time period, which can be expressed as an increase in
HE
2
. Note that such a change has no effect on W
CD
,
since it does not enter the budget constraint.
Substituting (6) and (10) into (3), and (3), (7)
and (11) into (4), and then (3) and (4) into (1), and
finally (12), (13) and (14) into (1) gives a more
detailed unconditional demand function for academic
skills (T
2
). Differentiating this expression with
respect to HE
2
gives:
2
D , 2
HE
T
∂
∂
=
2
P
H
T
∂
∂ [
2
P , 2
HE
H
∂
∂
+
2
P , 2
M
H
∂
∂
2
D , 2
HE
M
∂
∂
+
C
2
P , 2
C
H
∂
∂
2
C
D , 2
HE
C
∂
∂
(21)
+
1
P , 2
H
H
∂
∂
(
1
P , 1
M
H
∂
∂
2
D , 1
HE
M
∂
∂
+
C
1
P , 1
C
H
∂
∂
2
C
D , 1
HE
C
∂
∂
)]
+
1
CD
H
T
∂
∂ [
1
P , 1
M
H
∂
∂
2
D , 1
HE
M
∂
∂
+
C
1
P , 1
C
H
∂
∂
2
C
D , 1
HE
C
∂
∂
] +
1
P
EI
T
∂
∂
2
D , 1
HE
EI
∂
∂
28

+
2
P
EI
T
∂
∂
2
D , 2
HE
EI
∂
∂
+
YS
T
P
∂
∂
2
D
HE
YS
∂
∂
=
2
P
H
T
∂
∂ [
2
P , 2
HE
H
∂
∂
+
2
P , 2
M
H
∂
∂
2
D , 2
HE
M
∂
∂
+
C
2
P , 2
C
H
∂
∂
2
C
D , 2
HE
C
∂
∂
] +
2
P
EI
T
∂
∂
2
D , 2
HE
EI
∂
∂
+
YS
T
P
∂
∂
2
D
HE
YS
∂
∂
where the last line indicates that a change in the
health environment in time period 2 comes “too late”
for parents to reverse decisions made in time period 1
(this is relaxed below). Intuitively, a government
policy that changes the health environment changes
both H
1
and H
2
, but households who are already in
time period 2 when the government policy changes
cannot change M
1
, C
1
C
, C
1
or EI
1
, so for these
households there is only a “short-run” effect of the
policy change: H
2
changes but not H
1
. A “long-run”
effect applies only to households who are still in their
first time period when the policy is implemented, or
who enter time period 1 after the policy is
implemented; for these households both HE
1
and
HE
2
change by the same amount, and the long-run
impact of that change in the health environment
incorporates households’ decisions to change M
1
,
C
1
C
, C
1
and EI
1
.