difference in investment levels across the two groups varies with v. In particular, what is the sign of
dv . It is clear that this will vary depending on the characteristics of f(n). Below I consider three cases: the case with a possibility of saturation of n, the case where f(n) is linear, and a general case where f(n) is a differentiable, non-linear function.
Saturation Allow f(n) to have a general functional form but assume that n ∈ [0, ¯n] – that is, it is not possible to purchase more than ¯n of the investment. As before, non-monotonicity requires that there is some region with high v over which decreases in v generate more inequality, and some region with lower v in which decreases in v generate less inequality.
– Case 1: Increase in Inequality Begin with a sufficiently high value of v such that v > Λ A f 0 ( 0 ) > Λ D f 0 ( 0 ) , s o t h e r e i s n o i n v e s t m e n t f o r e i t h e r g r o u p , i m p l y i n g e q u a l i t y . A s v d e c r e a s e s , t h e r e w i l l b e s o m e p o i n t a t w h i c h Λ A f 0 ( 0 ) > v > Λ D f 0 ( 0 ) , i n w h i c h c a s e n ∗ A > 0 b u t n ∗ D = 0 , g e n e r a t i n g i n e q u a l i t y .
– Case 2: Decrease in Inequality Begin with a lower value of v such that Λ A f 0 ( ¯ n ) > v > Λ D f 0 ( ¯ n ) . W i t h t h i s v a l u e o f v , n ∗ A = ¯ n b u t n ∗ D < ¯ n . S i n c e n increase above ¯n, at this point further decreases in v will not affect the investment for the c a n n o t
advantaged group, but will increase investment for the disadvantaged group, until n over this range, inequality will be decreasing.
= ¯n ;
This indicates that there will be some non-monotonicity in the relationship between cost and inequality. In contrast to the simple normally-distributed discrete case above, exactly the shape we would expect this relationship to take outside of these two end points is ambiguous, and depends on the functional form of f(n). This is similar to the version of the model outlined in Appendix A for the discrete case with a general functional form for the distribution of F (): we know there will be some non-monotonicity, but the shape of the function between the endpoints is not defined without knowing the functional form of f(n).
f ( n ) l i n e a r I f f ( n ) i s l i n e a r , w e h a v e a c o r n e r s o l u t i o n . I n t h i s c a s e , f 0 ( n ) w i l l s i m p l y b e e q u a l to some constant; denote this φ. If φΛX > v then the social planner will choose to devote all of
the available resources to this investment, and we will have n planner chooses to spend nothing on this investment and n ∗ X
∗ X = = 0.
Y v . If φΛX < v the social Again, non-monotonicity
requires that there is some region with high v over which decreases in v generate more inequality, and some region with lower v in which decreases in v generate less inequality.
Case 1: Increase in Inequality Begin with a sufficiently high value of v such that v > φ Λ A > φ Λ D , a n d n ∗ A = n ∗ D = 0 , w h i c h i m p l i e s e q u a l i t y – n o i n v e s t m e n t f o r e i t h e r
n ∗ type. A =
Y v As v ,n ∗ D decreases, since ΛA > ΛD, there = 0, generating inequality.
v > φΛD,
Case 2: Decrease in Inequality Begin with a lower value of v such
v > φΛD. As v decreases further, eventually it will reach levels
v, where n
, and equality will be restored.
As in the case of saturation, in this model the basic intuition behind the non-monotonicity is retained in the case of a linear f(n) function. There is at least some region in which inequality will get worse with decreases in v and some region in which it will get better.