These cases suggest that the introduction of substitutes may or may not affect the possibility that we see a non-monotonicity in the relationship between inequality and changes in v. If the cost of obtaining the investment privately is very high, we would still expect to see this type of relationship; however, if the cost is lower than the benefits, even if it is only lower than the benefits to the advantaged group, the result will break down. It is worth noting that Case 2 is unlikely to come up when thinking about investments that an outside entity like the government might provide: if everyone already gets some investment privately, there is little incentive to provide another version of it.
In any given situation, even with substitutes, the applicability of this non-monotonicity then comes down to how costly a private investment is. There are some circumstances in which Case 3 is clearly applicable. One clear example is boiling water and the typhoid vaccine. Boiling water is expensive, but it is probably not prohibitively expensive, and it could well be the case that the cost is low enough that it makes sense to do it for the advantaged group, but too high for the disadvantaged group. In other situations, we are more clearly in Case 1; for example, when public education was initially introduced, private education was extremely expensive, and accessible only to a very small number of people. For most of the population – advantaged or disadvantaged – this alterative was not an option. Of course, there are a variety of situations in which there simply are no substitutes for some investment – for example, food – in which case this issue is moot.
A second key assumption in the model is that the investment is discrete. Obviously, many investments are continuous, and it is interesting to consider to what extent the results above go through in that case.
Again, we focus on a very simple model that incorporates this feature. We assume that the social planner chooses a level of investment n, and the value of this is ΛX f(n), where X ∈ A, D and f 0 ( n ) ≥ 0 . E a c h u n i t o f n c o s t s v , a n d , a s i s i m p l i c i t i n t h e d i s c r e t e m o d e l , w e a s s u m e t h a t t h e s o c i a planner has an additive utility function over consumption of some other good c, which has a price of 1 and ΛX f(n); the total budget is Y . The social planner solves the below maximization. l
maxnc + ΛX f(n)
s.t. c + nv = Y
T h e s o l u t i o n i s s i m p l e : a t t h e o p t i m u m n ∗ , Λ X f 0 ( n ∗ X ) = v . T h e q u e s t i o n o f i n t e r e s t i s h o w t h e