Appendix A: General Theoretical Results
This appendix discusses the general form of the result in Section 2. As there, I note that the conditions for vaccinations for the advantaged and disadvantaged groups are:
ΛA v > εi ΛD v > ε i
and that the difference of interest is F (ΛA
In the text I focus on the case where
F (.) is normal and show results indicating that at high v this difference will increase with decreases
in v, and at low v the difference will decrease with decreases in v. generally be true about the distribution for this to hold.
Here, I discuss what must
can be represented
Z Λ Λ D
Differentiating this with respect to v yields:
Z Λ Λ D
f 0 ( x
In order for this to be negative at high v and positive at low v, I simply require that f (.) be increasing in the left tail, and decreasing in the right tail. Any single-peaked distribution will have this property.
In contrast to the case of the normal, which is decreasing everywhere in v, I will not be able to prove this generally. However, the intuition that the disadvantaged group become relatively worse off with decreases in v at high v and relatively better off with decreases in v at low v will be true for a wide class of distributions.