f(n) is Differentiable In this case, if we are not in a corner solution, we can totally
differentiate to explicitly solve for
. We find
n ∗ D )
Λ A f 0 0 ( n ∗ A )
Λ D f 0 0 ( n ∗ D )
The sign of this is ambiguous, as is the way the sign might vary with v. There are situations in which we would expect to see a non-monotonicity (depending on the shape of the function, including the higher-order derivatives nd the magnitude of ΛA and ΛD), but there are other functional forms in which we would expect, for example, the movements in inequality to be linear.
The model above is obviously not the only way to incorporate continuous investments. This does give a general sense, however, of how those results would vary. In cases in which we have saturation, in particular, the non-monotonicity seems fairly robust. With continuous investments without saturation it is somewhat ambiguous what we would expect, although there are specific values and functional forms which will yield non-monotonicities even without saturation. In the next subsection I turn specifically to the example of vaccinations.
An Application to Vaccination by Gender in India
The empirical work in this paper focuses on the case of vaccination in India.3 In this case, the decision maker is the family, and I assume that each family i is endowed with one child, either a boy or a girl.4 The value of a living boy child, φb is greater than the value of a living girl child, φg. This could be due to simple preference, or it could reflect the fact that the monetary return to daughters is lower. In either case, we can summarize the existence of son preference with this simple difference in values, and this paper is agnostic about the source of the difference.
Parents have the opportunity to invest in the health of their child through vaccination. In general, with the vaccinations used here, there are unlikely to be very strong substitutes. Other than quarantine, it is extremely difficult for parents to avoid exposing their children to the diseases against which they are vaccinated, so in nearly all of these cases vaccination is the only reasonable source of protection. In principle, it is reasonable to think of vaccination in two ways – either as a
3In this particular context, there are interesting parallels between this theoretical result and an older literature on wealth and intrahousehold inequality. Kanbur and Haddad (1994), for example, argue that an intrahousehold bargaining framework can predict this type of non-monotonic relationship between wealth and inequality within the household.
4For simplicity, I ignore any existing children. Obviously, the number and gender of children that the family already has may influence their utility for each new child. However, the implications described below only require that, on average, the utility from a male child is higher than the utility from a female child. Abstracting away from existing children retains the simplicity of the model. However, an earlier version of this paper discusses a framework in which existing children are explicitly modeled. The central result – the non-monotonicity – holds in that model as well (that version of the paper available from the author).