4.1. Univariate vs. Bivariate Inequality

The extended concentration index has been obtained by applying a concept used for the measurement of the inequality of income – the extended Gini coefficient – to the measurement of the socioeconomic inequality of health. Basically, a one-dimensional construction is transplanted into a two-dimensional context. It cannot be taken for granted, however, that anything which works well in a univariate environment is automatically suited for a bivariate world.

Here we propose a simple test to check whether an indicator is a good measure of the degree of association between socioeconomic status and health. Imagine that we turn the world upside down: for a brief moment of time the poor and the rich switch roles (one may think of Carnival). More specifically, let us assume that the poorest person and the richest person switch their health levels, that the second poorest and the second richest person switch their health levels, etc. In formal terms, this leads to a new health function

*g*(

*p*)

*h*(1−

*p*), which is the health function

*h*(

*p*) turned upside down. Our test consists of looking at the reaction of the indicator when the health function

*h*(

*p*) is replaced by

*g*(

*p*). We say that an indicator

*I* passes the ‘upside down’ test if

*I*(

*h*) and

*I*(

*g*) are always of the opposite sign (or both equal to zero). In other terms, if the indicator states that distribution

*h*(

*p*) is pro-poor (c.q. pro-rich), then it must always state that distribution

*g*(

*p*) is pro-rich (c.q. pro-poor).

It is not difficult to verify that the extended concentration index does not pass this test, except when ν =1, a case we excluded, or when ν = 2, the standard concentration index. The reason for this lies in the asymmetric nature of the weighting function. This can be illustrated by looking at the case where the chances of having high or low health levels are symmetrically distributed over the rich and the poor. An extreme example of such a symmetric distribution is the one in which only the richest and the poorest individuals have a very high health level, and all others the minimum level. This is of course a very unequal distribution, but it may be argued that since there is no systematic bias in favour of either the rich or the poor, the index of socioeconomic inequality should therefore be equal to zero. This is exactly what we find if we use the standard concentration index, but not if we use the extended concentration index with ν different from 1 or 2.

4.2. A General Formulation

When looking for an alternative, we will try to remain as close as possible to the extended concentration index. Let us consider indices of the following type:^{2}

where

*f* (μ

_{h}, ε) is a normalization function,

*w*(

*p*, ε) a weighting function, and ε an distributional sensitivity parameter. These indices belong to the class of

Mehran (1976) measures, applied to socioeconomic inequality. It is customary to restrict the attention to indices for which the weighting function is an increasing function of

*p* and for which the weights sum up to zero, i.e.

.

^{3} In addition, we assume that the weighting function is continuous and not identically equal to zero (because then the index would always be equal to zero, as in the case ν =1 above). With regard to the normalization function we assume that it is positive-valued, which implies that there is no level of average health or of the distributional sensitivity parameter such that

*f* (μ

_{h}, ε) = 0. We call

*f* (μ

_{h}, ε)

*w*(

*p,* ε) the normalized weighting function. It is immediately clear that (

5) is a special case of (

8), with ε = ν,

*f* (μ

_{h}, ν) = 1/μ

_{h} and

*w*(

*p*, ν) = 1− ν(1−

*p*)

^{ν−1}.

4.3. The Symmetry Property

The following result specifies for which type of weighting function an indicator passes the ‘upside down’ test, i.e. is such that *I*(*h*, ε) and *I*(*g*, ε) are always of the opposite sign (or both equal to zero).

Theorem The index

*I*(

*h*, ε) passes the ‘upside down’ test if and only if the weighting function is inversely symmetric around

, i.e. if and only if we have

*w*(

*p*, ε) = −

*w*(1−

*p*, ε) for any 0≤

*p* ≤1.

Proof (i) Let us rewrite (

8) as

If

*w*(

*p*, ε) = −

*w*(1−

*p*, ε) for any 0≤

*p* ≤1, then obviously we have

. Hence we obtain

. Likewise we derive that

. Since

*g*(

*p*) =

*h*(1−

*p*) and μ

_{g} = μ

_{h}, it follows that

*I*(

*h*, ε) = −

*I*(

*g*, ε). This proves sufficiency. (ii) Suppose that the weighting function is not inversely symmetrical around

. Then we can always find an interval [

*a*,

*b*], where

, such that

. Since at least one of these integrals is different from zero, we can assume without loss of generality that

, so that we can write

, where θ ≠ 1. Consider a health distribution characterized by a function

*h*(

*p*) with the following properties:

*h*(

*p*) =

*c* >0 for

*a* ≤

*p* ≤

*b* and 1 −

*b* ≤

*p* ≤ 1 −

*a*, and

*h*(

*p*)=0 elsewhere. For this distribution we have

and also

. Since μ

_{g} = μ

_{h}, we obtain

*I*(

*h*, ε) =

*I*(

*g*, ε) ≠0, which means that the indicator does not pass the upside down test. This proves necessity.

The sufficiency part of the proof shows that indicators which pass the upside down test are always such that *I*(*h*, ε) = −*I*(*g*, ε). This is what we call the *symmetry* property. Although the symmetry property is at first sight a stronger requirement than passing the upside down test, Theorem 1 reveals that they are equivalent. The theorem also shows that if we want the symmetry property to hold, then we are obliged to abandon the weighting function of the extended concentration index.

4.4. The Symmetric Index

In order to construct an index with the symmetry property, we have to replace the asymmetric weighting scheme of the extended concentration index by an inversely symmetric weighting scheme. This implies that if we want to maintain relatively high negative weights for the poorest individuals, we need to give relatively high positive weights to the richest individuals. The symmetric index we propose here is defined as follows:

with β >1.

^{4} In terms of expression (

8), we have ε = β and:

One can check that for β = 2 we have *w*(*p*, 2) = 2*p*−1, which means that for this value the symmetric index coincides with the extended concentration index with ν = 2.

The weighting scheme has been devised in such a way that those with fractional ranks above the median always have positive weights, and those below the median always negative weights. As can be seen from , by taking 1< β < 2 we give relatively higher weights to those with a fractional rank close to the median, while by taking β > 2 we give relatively higher weights to those at the upper and lower end of income distribution. In the most extreme case (β → +∞) the symmetric index tends to

. The value of the index varies between −β/2 and +β/2.

^{5} Just as for the extended concentration index, the distance between the bounds is equal to the value of the distributional sensitivity parameter, β, and coincides with the distance between the weights of the richest and the poorest individual.

4.5. Comparing the Extended Concentration Index and the Symmetric Index

Because of the symmetry property, the individual who occupies the median position in the income distribution plays a pivotal role in the calculation of the symmetric index. Suppose there is a ceteris paribus increase of the health level of one person located at position *p* in the socioeconomic distribution. What would be the effect of such a change upon the value of the index measuring socioeconomic health inequalities? Let us start at *p* = 0 (the poorest individual). Obviously this is a pro-poor change, and we expect the index to become more pro-poor, i.e. to decrease in value. This implies that we always have *w*(0, ε) < 0. Next, let us increase *p* and wonder from what value of *p* the change becomes pro-rich, i.e. at which point *w*(*p*, ε) turns positive. If we think that this threshold value *p*^{*} should be lower than the median, we could opt for the extended concentration index: given *p*^{*} < 1/2, if we choose the value of the distributional sensitivity parameter ν^{*}, where ν^{*} is such that *p*^{*} =1−(1/ν^{*})^{1/(ν* −1)}, we obtain the desired result. If, however, we decide that the threshold value should always be equal to the median, the symmetric index seems a more appropriate choice.

The threshold value *p*^{*} demarcates the group of the poor from the group of the non-poor. We believe that the choice of *p*^{*} = 0.5 is a reasonable point of departure as 0.5 is the expected location of a person. In other words, the lower half of the population is considered as poor, and the upper half as rich. We do not exclude that another value, say *p*^{*} = 0.25, might be more appropriate than our *a priori* choice, but without additional information (e.g. on income levels) we think it is very hard to make a case for such an alternative boundary. By construction, rank-dependent inequality measures leave that kind of information out of consideration, and therefore naturally lead us to take *p*^{*} = 0.5, at least as a starting point.

Another issue concerns the reaction of the index of socioeconomic health inequality to health transfers at different locations in the distribution. Suppose there is a transfer of health Δ from a person located at position *p*_{j} to a person located at position *p*_{i}, with *p*_{j} =*p*_{i} +*d* and *d* > 0 (i.e. the first person is richer than the second). We can compare the effect of such a transfer for different equidistant individuals. A good measure of where the transfer takes place is given by the number *z* = *p*_{j} −*d*/2 = *p*_{i} + *d*/2, i.e. the location halfway between *p*_{i} and *p*_{j}. If we believe that the effect of such a transfer should become smaller and smaller as *z* increases, we have to opt for the extended concentration index with *v* > 2. If, however, we think that the effect should be smaller the closer *z* lies to the centre (i.e. *p*^{*} = 0.5), then the symmetric index with β > 2 seems more appropriate. The first property is that of sensitivity strictly increasing with poverty, in short ‘sensitivity to poverty’; the second that of sensitivity strictly increasing when one moves from the centre towards the extremities of the distribution, in short ‘sensitivity to extremity’.

When the issue is the measurement of one-dimensional inequality, for instance of incomes, we believe ‘sensitivity to poverty’ is the appropriate distributional concept. But it can be questioned whether the same concept is also the most appropriate one in the case of two-dimensional inequality, for instance of health in relation to socioeconomic rank. In the latter case, we are not measuring the inequality of health as such, but the degree of association between the distribution of health and the socioeconomic ranking. The measure of this degree of association should take into account the whole spectrum of possibilities, and not privilege inequality in one dimension over inequality in the other. By making the measure more sensitive to one end of the income spectrum (‘the poor’) than to the other (‘the rich’), we run the risk of reducing or even neglecting part of the existing inequality. Why should a person with a low income rank but a high health level count more than a person with a high income rank but a low health level? While the symmetric index does not address this issue directly, it expresses the idea that what is happening at the extremities of the income distribution, whether it be at the high end or the low end, should carry more weight than what is happening in the middle.

This brings us to an interesting resemblance between rank-dependent indices that satisfy the symmetry property and the range, which is probably the oldest and most frequently used measure in the field of health inequality (e.g.

Townsend and Davidson 1982). The range compares the health levels of the top and bottom income groups, and its implicit value judgment is that the difference between the best and worst-off income group is what matters for health inequality. This is very much in line with the value judgements of the symmetric index with a high ‘sensitivity to extremity’ (i.e. a high value of β). Hence, the symmetric index proposed in this paper allows to bring together the value judgements underlying rank-dependent indices, such as the concentration index, and those underlying indices focusing on ‘extremes’, such as the range.

Due to its exclusive focus on income poverty, the extended concentration index may lead to counterintuitive results. Consider a health distribution in which the poorest 10% of the population have a very high health level, say

*c* > 0, the richest 20% also, and all the rest a very low health level, say 0. Since there are twice as much rich persons in good health than poor persons, we believe few people would doubt that health is distributed rather strongly in favour of the rich, and therefore we expect a positive value of the index. Yet, the extended concentration index will be

*negative* for any value of ν (approximately) higher than 3.33 (the value of the extended concentration index is in this example equal to

)

^{6}. By contrast, the symmetric index will always be positive.

The explanation of the divergence lies in the way in which the two indices treat different combinations of ranks and health levels. Low health levels always have a small contribution to the value of the extended concentration index (positive in case of a high rank and negative for low ranks); and this also holds for the symmetric index. But things are different for high health levels. In case of the extended concentration index, these lead to a moderately positive contribution for high ranks, and a very large negative contribution for low ranks; while there is no such difference (apart from the sign of the contribution) for the symmetric index, i.e. there is a large positive contribution for high ranks, and a large negative contribution for low ranks.