To see the graphical intuition behind the result, consider Figure 1. This figure graphs two possible cost distributions with different levels of v; the dotted line represents a distribution with better access to the health investment (lower v). The cutoffs D1, A1 and D2, A2 represent two sets of investment cutoffs (Dx is the cutoff for the disadvantaged group, Ax for the advantaged group). The mass of the distribution under the cutoff receives the investment, so the D1, A1 cutoffs represent a world with overall higher investment levels. Consider what happens to the difference in investments when we move from the solid to the dotted distribution, which represents a decrease in v. For the case of D2, A2, this movement causes a greater increase in the share receiving the investment for the advantaged group than for the disadvantaged group because both lines are on the increasing part of the distribution. In contrast for the case of D1, A1, the increase causes a greater improvement for the disadvantaged group because both lines are on the decreasing part of the distribution. It is this intuition that is central to the result.
In Appendix A, I discuss the specific generalizability of this result to other cost functions. Although it will not be true for all cost distributions, the intuition in Figure 1 is robust. In
particular, the fact that
is negative at high values of v and positive at low values will be true in
general for any single peaked distributions (although it will hold for other distributions, as well).
In addition to this basic prediction of a non-monotonicity, the model also makes a prediction about how this relationship will vary as the magnitude of the difference in values across groups varies, summarized in Proposition 2.
Proposition 2. When the relative value of the advantaged group is higher, the amplitude of the function will be larger, so we expect a larger (in magnitude) relationship between access and inequality.
Proof. The amplitude is determined by the maximum value (in absolute value terms) attained by
d dv the function increase in Λ . Note that an increase in the value of the advantage group can be represented by an d dv , holding ΛD attains its maximum absolute value twice: when ΛA = v constant. A D and when Λ = v, and this value will be the same in absolute value. At this point,
2(Λ Λ D 2σ2 ) Λ D ) 2 2 . The derivative of this with respect to ΛA is σ 2 , which is positive. So as the relative value of the advantaged
group increases, the amplitude of this function increases, and we expect the magnitude of the relationship between access and inequality to be larger.
The intuition here is, again, relatively straightforward. Referring again to Figure 1, if the cutoffs for the two groups are very close to each other, then when the curve shifts left the increase in area to the left of each cutoff will be similar. If the cutoffs are very far apart, the same curve shift causes very different magnitude increases in the area to the left.