# Continuous Choice with Saturation

In this case, the family chooses a number of vaccines n, where n ∈ [0, 8]. Again, without vaccination the child will live with probability p, and with n vaccines, the child lives with probability p + f(n)p_{0}, where f(0) = 0 and p + f(8)p_{0 }= pˆ, to parallel the above. Each vaccination in this case costs v, as above. In this case, the family chooses n to solve a maximization problem. Specifically, for boys,

m a x n φ b ( f ( n ) ) p 0

nv

T h e fi r s t o r d e r c o n d i t i o n i n d i c a t e s t h a t t h e o p t i m a l v a l u e o f n , n ∗ b s a t i s fi e s φ b f 0 ( n ∗ b ) p 0 = v . A s n when discussing the case of continuous investments with saturation in subsection 2.2, the object of o t e d

interest in this case is

d(n_{b }

dv

n_{g })

and the non-monotonicity exists if there is a range of v over which

this is positive, and a range over which it is negative. This maps exactly into the case with saturation above, and the discussion there applies here, as well. Intuitively, in the case with saturation there will be some non-monotonicity, because there will be a range of changes in v for which we move from both groups having no vaccines to the boys having at least one vaccine. On the other end, there will be a range of changes in v for which we move from the boys having all vaccines and the girls having less than all, to a place where both groups have all vaccines. In this sense, the non-monotonicity is retained.

As discussed, this case is more complicated because how inequality moves over the rest of the range of v is dependent on the functional form of f(n). One plausible functional form is strict concavity: the first vaccination is more valuable than the second, and so on. In this case, inequality w i l l ( w e a k l y ) i n c r e a s e w i t h d e c r e a s e s i n v u n t i l v < φ b f 0 ( 8 ) p 0 ( t h a t i s , u n t i l b o y s h a v e f u l vaccination) and then decrease after that. This gives us a strict non-monotonicity. As discussed, this will not be true with any functional form for f(n), of course, but because of the saturation in any case we will get some range over which inequality is increasing and some range over which it is decreasing. l

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# Data

The analysis here is run using individual-level microdata on child health investments in India. I use primarily the second wave of the National Family and Health Survey (NFHS), which covers approximately 90,000 women and was run in 1998-1999. Women are asked about their birth history, including children ever born, dates of birth, if the children are alive, and, if not, when they died. In

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