# 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:

Λ_{A }v > ε_{i }Λ_{D }v > ε i

and that the difference of interest is F (Λ_{A }

v)

F (Λ_{D }

v).

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

# This difference

can be represented

=

Z Λ Λ D

f (x

v) dx

Differentiating this with respect to v yields:

d dv

=

Z Λ Λ D

f 0 ( x

v) dx

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.

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