APPENDICES

A.1. Properties of the Gender-Equity-Sensitive Indicator X_ede

We begin with the general definition of X_ede with respect to a concave increasing social valuation function V(X). We leave the functional form of V(X) unspecified and require only that V'(X) > 0 and V"(X)  0. Then X_ede is defined through the equation (n_f + n_m)V(X_ede) = n_fV(X_f) + n_mV(X_m).

Henceforth, we denote p_f and p_m as the proportions of females and males in the total population, which is of size (n_f + n_m). By definition, p_f = n_f/(n_f + n_m) and p_m = n_m/(n_f + n_m).

It follows that (p_f + p_m) = 1, and V(X_ede) = p_fV(X_f) + p_mV(X_m). (1)

Hence 

X_ede = V^-1(p_fV(X_f) + p_mV(X_m))

with V^-1(.) a convex increasing function. As X_ede is a monotonic increasing function of a concave function, it will be at least quasi-concave in (X_f, X_m). But will it be concave? The answer to this question is of central interest in understanding the gender-equity-sensitive indicator (GESI).

The first property of X_ede that we wish to derive is simply that X_ede is monotonic increasing in both X_f and X_m. This follows directly from partial differentiation with respect to X_i of equation (1) above, for i = f, m:

V'(X_ede)×X_ede/X_i = p_iV'(X_i).

Hence

X_ede/X_i = p_iV'(X_i)/V'(X_ede) (2)

> 0 for i = f,m

because V'(X) > 0. Thus X_ede is monotonic increasing in both female and male achievements.¹⁸

Furthermore, if female achievement X_f is less than male achievement X_m, and the female population proportion p_f is greater than or equal to the male population proportion p_m , then a unit increase in female achievement will be more valuable socially than a unit increase in male achievement. This follows easily using equation (2). By assumption, X_f < X_m and V"(X)  0; hence V'(X_f) V'(X_m). Since also p_f  p_m, we have

X_ede/X_f = p_f V'(X_f)/V'(X_ede)

p_mV'(X_m)/V'(X_ede)

because V'(X_f) V'(X_m)

= X_ede/X_m.

Note finally the property of X_ede relating to changes in the population proportions of the two subgroups, p_f and p_m. A rise in the population proportion of the subgroup with a higher level of achievement will result in a higher value of X_ede. Thus if X_m > X_f, and since p_f = 1 - p_m, we have from equation (1) by partial differentiation with respect to p_m

X_ede/p_m = [V(X_m) - V(X_f)]/V'(X_ede)

> 0 because V'(X) > 0.

The results obtained hitherto are valid for any concave increasing social valuation function V(X). But how precisely does X_ede depend on the "degree of concavity" of V(X)? We know that if V(X) is linear then we have
 

where is the simple arithmetic average of the individual achievements X_f and X_m. If V(X) is strictly concave, then we will have

This is because, by definition of X_ede,
 

Hence

because V(X) is monotonic increasing in X. The analysis suggests that a "more concave" valuation function than V(X) will imply a still lower value of X_ede. We define one monotonic increasing function to be more concave than another if the former can be expressed as a concave monotonic increasing transform of the latter.

It is indeed the case that a more concave valuation function will yield a lower value of X_ede. Thus, if an increasing concave transform (.) is applied to the concave function V(X), then the equally distributed equivalent achievement corresponding to(V(X)) will be smaller than that corresponding to V(X). This is demonstrated by applying the function (.) to (p_fV(X_f) + p_mV(X_m)), and using concavity of (.) to prove the result.

Let and be the equally distributed equivalent achievements corresponding to the functions V(X) and (V(X)), respectively. Then
 

Applying (.) to both sides of this equation gives
 

 
because (.) is concave

by defenition of 
 

Hence,

because (V(X)) is monotonic increasing in X.

This inference is analogous to the Arrow (1965)-Pratt (1964) result in the theory of uncertainty that a more risk-averse individual has a lower certainty equivalent income for any given risk.¹⁹ It was also established as an inequality concerning convex functions in Hardy, Littlewood and Pólya (1952: 75-6).

The general proposition proven above on "more concave" functions can be applied to demonstrate result (2) in the text. This relates to isoelastic social valuation functions and states that the larger is the elasticity Î of the marginal valuation function V'(X), the smaller will be X_ede. We let

and prove the result for  0,   1. Thus consider the function

To apply the general proposition we must be able to write

so that (.) is an increasing concave function. We have
 

and hence
 
 

say, which provides a definition of the required function (V). It turns out that (V) is indeed an increasing concave function of V provided that v > . To show this, differentiate (V) with respect to V twice:
 

and
 

< 0 since v > .
 

Note that whether  < 1 or > 1, the quantity (1-)V is always positive (and equal to X^{1 -} ): when 0   < 1 both (1-) and V are positive, and when > 1 both (1-) and V are negative. Our definition of V(X) as X^{1 -} divided by (1-) thus circumvents the need to prove separately results for positive and negative powers of X, i.e. separately for the cases 0  < 1 and  > 1.²⁰
 
 
18. Note that this property does not necessarily obtain with arbitrarily specified measures of gender equity, such as (X_f /X_m)(p_f X_f + p_mX_m). The latter measure is equal to [(p_f X_f ²/X_m) + p_mX_f ], which is a strictly decreasing function of X_m.

19. It is equivalent to saying that a more risk-averse individual is willing to pay more to eliminate any given risk — one of several characterizations of "greater risk aversion". See, inter alia, Arrow (1965), Pratt (1964), Rothschild and Stiglitz (1970), Diamond and Rothschild (1989).

20. For the case V(X) = log X, which corresponds to an elasticity of marginal valuation  = 1, we have X = ^V. Now consider a more concave function, i.e. one with elasticity of marginal valuation v > 1. Then

 
This is an increasing concave function for all values (positive and negative) of V, because (1-v) < 0. Hence we can apply the general proposition on "more concave" functions.