APPENDIX

A.2. On the Concavity of X_ede with respect to (X_f, X_m)
 
 
Earlier we showed that X_ede was a monotonic increasing function of its arguments (X_f, X_m), demonstrated in equation (2). This is clearly a desirable property for a measure of social achievement, even once it has been adjusted for equity. Moreover, we saw that if X_f < X_m (and p_f p_m), then a unit increase in X_f adds more to X_ede than a unit increase in X_m. But will there be "diminishing returns" to unit increases in X_f , and for that matter to unit increases in X_m ? This might seem like an appealing property for the measure of social achievement X_ede to possess.
 
To address this question, we have to differentiate equation (2) partially with respect to X_i. This gives
 
 
 
 
Hence,
 
 
 
 
 
 
In general, therefore, the behaviour of will depend on both the first and second derivatives of the function V(X) at X_i and at X_ede.
 
Let us now consider two special cases of the function V(X): the constant elasticity marginal valuation (or constant relative inequality aversion) form; and the constant absolute inequality aversion form.²¹ For the case
 
 
 
 
we have V'(X) = X^-and V"(X) = -X^--1 . Substituting into equation (3) above, we get
 
 

 
 
But we know that for this functional form for V(X), X_ede satisfies
 
 
 
Hence the expression

 
is equal either to
 

or to

depending on whether i = f or i = m. In any event, the expression
 
 
 
in equation (4) will be negative²², and therefore

For the isoelastic form for V(X), we have shown that X_ede does indeed increase at a diminishing rate in each of its arguments X_f and X_m.
 
The second special case we consider for the function V(X) is the constant absolute inequality aversion form:
 
 
 
 
up to a positive affine transformation. Here g is the (positive) parameter of absolute inequality aversion, which is defined in general as -V"(X)/V'(X). In this case,
 
 
 
 
and
 
 
 
 
Substituting into equation (3), we get
 
 
 
 
 
 
 
But we know that for this functional form for V(X), Xede satisfies
 
 
 
 
Hence the expression

is equal either to

or to

depending on whether i = f or i = m. In any event, the expression

in equation (5) will be negative²³, and therefore

Thus in both the main cases of constant relative and of constant absolute inequality aversion, we have the result that X_ede increases at a diminishing rate with respect to the individual achievement X_i, for all i. But will this result be true of any concave function V(X)? An examination of equation (3) shows that (²X_ede/X_i²) can fail to be negative if V"(X_i) is close to zero and V"(X_ede) and (X_ede/X_i) are large in absolute terms.²⁴

To construct a counterexample, we choose V"(X) = 0 in the neighbourhood of variation of X_i, and V"(X) < 0 in the range around X_ede, while noting that X_ede depends both on X_f and on X_m. Let X_f < X_m, and let us here evaluate Take X_f = 0 and let X_m lie on a linear segment of the function V(X). Thus consider the function 

Then V'(X) = 1/2 for X ³ 1.
 

Now we choose X_m ³ 1, but not so large that X_ede will also lie on the linear segment containing X_m. For simplicity, assume that p_f = p_m =

.

Then

 
so that

For V(X_ede) < 1, i.e. for X_m < 3, we have

Therefore,

and

 
which is strictly convex in X_m. Hence

for the parameters and ranges of the variables we have chosen.

It is easy to see that the result will still hold if a small curvature were added to the function V(X) along its linear segment, so that V"(X) < 0 throughout. If the function were made slightly strictly concave in this range, we would still get

Moreover, using smoother splines, we could make higher-order derivatives of the function continuous at X = 1. (In our example, the second derivative of the function V(X) is not continuous at X = 1; it is equal to (-1/4) just to the left of X = 1, and to 0 just to the right of X = 1.)

The basic intuition behind the counterexample is that if social value from increasing an individual achievement goes up (essentially) linearly, then to bring about equal unit increases in V(X) we will have to raise X_ede at an increasing rate if X_ede lies on a diminishing marginal returns segment of the function.
 

21. These two forms, named as such by Atkinson (1970) in the inequality literature, correspond in the risk literature to constant relative risk aversion and to constant absolute risk aversion, respectively (Arrow 1965, Pratt 1964).

22. When there are more than two arguments, we will have  

23. When there are more than two arguments X_f and X_m, we will have

24. (X_ede /X_i ) is always less than unity for the case of constant absolute inequality aversion. Here

But for the case of constant relative inequality aversion, we can make (X_ede /X_i ) as large as we like. For example, let (X_m /X_f ) = l, p_m = p_f = 1/2, and e = 1/2. Then

  as