1 As with the change in H 2 discussed above assume that the change in H 1 is a

# 1 as with the change in h 2 discussed above assume

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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 .

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