The theory outlined above embeds a number of assumptions. As I will detail below, in subsection 2.3, I believe that the assumptions are a good fit to the case of vaccinations in India. This subsection discusses the degree to which we expect the basic implications above to continue to hold if we relax the most important of the assumptions above. In particular, I discuss what would happen in the case where there are substitutes for the government provided-investment, and the case where investments are continuous.
The model above assumes that the investment good can only be obtained from some outside entity, at cost v + i, and there are no substitutes. In many situations, however, there are likely to be substitutes available. For example, if we think about education, private schools are a substitute for publicly-provided education. Even in the case of vaccines, there are some for which substitutes are available – typhoid can be avoided with a vaccine, but also by boiling water before drinking it.2
To model this simply, assume that the investment can be obtained either from the outside entity at a cost of v or from another source at cost t. The outcome is still valued at ΛA for the advantaged group, and ΛD for the disadvantaged group. There are three relevant cases.
Case 1 t > ΛA > ΛD: the cost of obtaining the investment privately is larger than the value of the investment for the advantaged group (and therefore larger for the disadvantaged group as well). In this case, the theory detailed above is unchanged. Since the cost of obtaining the investment privately is so large, its availability is functionally irrelevant, and choices about investment are determined by movements in v.
Case 2 ΛA > ΛD > t: the cost of obtaining the investment privately is smaller then the value of the investment for the disadvantaged group (and, therefore, also smaller than the advantaged group). In this case, everyone gets the investment regardless of the cost of provision by the outside entity, and the theory outlined above will not hold. Movements in the outside entity cost v will have no meaning, since everyone is already getting the investment even if v is extremely high.
Case 3 ΛA > t > ΛD: the cost of obtaining the investment privately is less than the value of the investment for the advantaged group, but more than the value for the disadvantaged group. In this case, we would expect the advantaged group to get the investment even if v is very high, and only the disadvantaged group investment level would be determined by changes in v. In other words, in this case we would not see a non-monotonicity: decreases in v would only decrease inequality, since they will only affect the level of investment for the disadvantaged group.
2I am grateful to an anonymous referee for suggesting this example.