^{1}

^{*}

^{1}

^{1}

^{2}

Although the Gini coefficient is an ideal measure of income inequality, it may be applied to measure the aging inequality in a society. In this paper, an attempt has been made to develop alternative measures of aging inequality based on the Gini index. The study uses the secondary population data of Asian countries collected from the international data base, US census Bureau. From the analysis it is observed that the Gini coefficient shows equally sensitivity at all levels. The coefficient is more concern for the country which are closed to the line of absolute equality. For example, the sensitivity level in the Gini coefficient is observed much higher in Israel than in Qatar. The logarithmic transformation of Gini coefficient does not work well because it violates the transfer principle. The Geometric measure of Gini coefficient fails to measure inequality because of violating the transfer principle. On the other hand, the logarithmic transformation of geometric equivalent of the Gini coefficient works better because it shows more sensitivity than the Gini coefficient and satisfies the transfer principle. From the analysis it is also found that the trigonometric measure of Gini coefficient works better than the logarithmic transformation of geometric equivalent of the Gini coefficient because it satisfies transfer principle as well as shows higher sensitivity. Therefore, the trigonometric measure of the Gini coefficient is the best measure of aging inequality among the measures considered in the study.

Inequality is a fundamental characteristics of all graduated social parameters and conceptualized as ‘‘the average difference in the status between any pairs relative to the average status’’ [

Although inequality has long been topic of interest to sociologists, few have bothered to carefully specify what they mean by the term. It is easy, of course, to distinguish perfect equality from a state of inequality. The lack of rigor created difficulty so long as research on inequality emphasized the determinants of individual attainments. But recent efforts to test of hypothesis explaining why some societies are less equal than others have necessitated the adoption of precise measures of inequality, such as the Gini index or the standard deviation [3-5]. In the absence of clear criteria for choosing among the numerous measures of inequality, researchers have usually based their choice on convenience, familiarity, or on vague methodological grounds. Nevertheless, the decision to rank one distribution is more unequal than another has theoretical as well as methodological implications. In fact, the choice of an inequality measures is properly regarded as a choice among alternative definition of inequality rather than a choice among alternative ways of measuring a single theoretical construct [

While studying the traditional measures, it is observed that there are basically two shortcomings of traditional measures of population aging. First one is the use of cut off point for old and young age of population. For example, the cut off point of old age is 65 for developed countries and 60 years for developing countries. Similarly, the cut off point of young age is 15 in developing country and 20 in developed country. Second, the accuracy of any measure increases as the observable range of variability increases. The traditional measures consider the change of age cohorts but ignore the total pattern of the age structure of population. It is observed that when the slope of trend line (the regression coefficient) was used to measure of aging that overcome the shortcoming of traditional measures of aging [

Nath and Nazrul [

The Gini index satisfies the basic criteria of scale invariance and the principle of transfers, but two other measures: the coefficient of variation and Theil’s measure are usually preferable. While none of these measures is strictly appropriate for interval-level data, valid comparisons can be made in special circumstances. The social welfare function is considered as an alternative approach for developing measures of inequality and methods of estimation, testing, and decomposition [

Though the Gini index of concentration appeal to most economist who rank income distribution in empirical studies, but it was also used in risk analysis and financial theory. Hence it is not surprising to see that the Gini index use as a measure of dispersion in portfolio analysis [9-12].

For variables like age, where utility is neither strictly increasing nor especially relevent, the flat sensitivity of the coefficient of variation makes it appropriate choice [

The measures, the coefficient of variation and the Gini index (G) in statistics texts are only appropriate for variables measured on a ratio scale, like income or age, which have a theoretically fixed zero point [

Optimal grouping techniques (OGT) were first used for income distribution to determine Gini index [16,17]. The OGT were also used to determine the cut off point to age distribution of population. By using OGT to the US population, it is found that the age at which one becomes an older person has dramatically increased. For example, the entry age into oldness was 48.7 years in 1930 while 57.6 years in 2004. The values of Gini index that address in the contex of age distribution of US population was 0.42 in 1950 and 0.36 in 2000 [

Dalton [

Gini coefficient is the most common statistical index of diversity or inequality in social sciences [2,23]. It is widely used in econometrics as a standard measure of inter-individual or inter-household inequality in income or wealth [9,22,24]. In some studies, Gini coefficient was used to measure variability in levels of mortality among socio-economic groups. It has also been employed for analyses of the variation in degree of people's inequality in the face of death over time and across countries. In some studies, Gini coefficient was used to measure variability in levels of mortality among socio-economic groups [

Gini coefficient is computed from distributions of deaths by age in real populations. In order to avoid a bias due to different age structures, a standard population age structure was used for weighting. This approach is the same as that in economics since age at death and population distributions are independent from each other, exactly like income and population in econometrics. The use of Gini coefficient for the analysis of inequality in health in the 1980s, stressed that the individual-based measurement of inequality in health is a way to a universal comparability of degrees of inequality over time and across countries [

If we consider a particular distribution of age with n-number of groups or individuals, for same amount of transfer of ages between any two groups, Gini coefficient shows equal sensitivity provided the transfer occurs between two successive groups or individual. Moreover, we can observe that the Gini coefficient shows more concern for countries, which are close to the line of absolute equality. In order to address some of the above mentioned issues, the other indices like variance, coefficient of variation and standard deviation have also been considered, but those have incompetent either because of their total concentration on differences around mean or beacause of violating the Pigou-Dalton condition. The Pigou-Dalton condition implies that any transfer from smaller group (poor group) to higher group (rich group), other thing remaining the same, would always increase the inequality measure. In line with the same one may also think that any transfer from higher group (rich group) to smaller group (poor group), other thing remaining the same, would always decrease the inequality measure [

The study uses the secondary population data collected from international data base, US census Bureau (www. census.gov/population/data/idb) for 2011. Alternative aging indices have been applied along with conventional aging indices: proportion of older people, proportion of persons aged less than 15 years, proportion of person aged between 15 and 59 years to the Bangladesh population as well as 50 Asian countries. To see the sensitivity level of the alternative measures, special focus is given on Bangladesh, China, India, Israel, North Korea, Nepal, Qatar, Singapore, Sri Lanka and Thailand.

The Gini coefficient is usually defined in terms of the Lorenz curve [

where, is the income/age of the i-th person, is the income/age of the j-th person, is the average income/age and . Eq.1 is a measure of dispersion divided by twice the mean. It is the average absolute difference between all pairs of individuals. Specifically (1) is known as Gini coefficient of mean difference given by Kendall and Stuart [

Researchers [31-34] in the field of measurement of inequality have always been in the quest of presenting of simple and easy way to calculate Gini coefficient keeping its objective. Milanovic [

The Eq.1 can be modified as follows:

Age structure of population can be categoried into three broad groups namely children, active force and elderly.

Since the data set consists of 3 different age groups (i = j = 3), the expression (4) can be rewritten as

After some straight forward simplication, the expression (5) can be written as follows:

i.e.

Ignoring the multiplier, the expression becomes

where and represent the proportion of elderly, the proportion of children and the proportion of active population as these supports for Bangladesh as well as developing countries.

In order to make Gini coefficient more rational in terms of sensitivity, we take the natural logarithm and modify the Eq.4 and Eq.5 as follows:

i.e.

For 3 different groups, the expression becomes

Since it violates the transfer principle of an ideal measure of inequality, we should look for other measures.

Majumder [

From the

divided the rectangle into two equal triangles. For each triangle, base = height = 1, as sum of proportion equal to unity. The area of triangle is

From the figure 1, it shows that the area beyond the Lorenz curve is the sum of area of n small triangles and (n – 1) rectangles. The area of each triangle is:

and the sum of n triangles is

, since

Similarly, the sum of (n – 1) rectangles is

The total area beyond the Lorenz curve is

Now the area between the diagonal line and the Lorenz curve is

We may standardise the above expression (10) with the multiplier.

Therefore Eq.10 may considered as an alternative geometric measure of Gini’s coefficient and written as

Therefore,

For 3 different groups, the expression (11) becomes

Therefore,

Taking logarithm the expression (11) and (12) become

We have developed arithmetic and geometric derivation of Gini coefficient for measuring aging inequality. In order to find simpler and alternative derivation of Gini coefficient to measure aging inequality, it is tried to develop trigonometric measure of inequality. Mojumder [

We can standardise the expression (15) by substracting n from it and multiplying by. Thus we have

i.e.

For 3 different individuals/groups, the TMG becomes

i.e.

The developed measures of inequality of population aging have been applied to Bangladesh population as well as some other selected countries. We have chosen those countries of Asia which satisfy the rank order condition. The results of the measures of inequality of population aging have been displayed in Tables 1 and 2. For convenience of the analysis, it is assumed that, and represent lower end, middle end and higher end of the distribution.

transfer of ages between two consecutive groups G changes by 1.942 percent from

From the expression (7), it may be realised that the value of G may be higher in a country where share at the lower end are comparatively smaller than the other. Our data support this claim. For example, the G value of Israel, Bangladesh and Qatar are 0.439, 0.515 and 0.837 respectively. Their corresponding share of the lower end are 0.142, 0.071 and 0.018 respectively (

It is also observed that changes of G may be higher in a country where the values of G are comparatively smaller than others. For example, when transfer of person (1 percent) belongs to specific age interval takes place between any two consecutive groups in Israel and Singapore, G changes by 2.28 percent and 1.60 percent respectively in these two countries as shown in

It is observed that the minimum and maximum value of the logarithmic transformation of the geometric index (LTGEG) are 3.251 (in Israel) and 5.393 (in Qatar) as displayed in

When transfer of ages takes place in downward direction from to, to and so on, sensitivity level gradually decreases (

It is also observed that changes of LTGEG are higher, where share at the lower end is comparatively smaller. For example, the changes of LTGEG in Qatar and Israel are 12.60 and 1.47 respectively whereas the share at the lower end are 0.018 and 0.142 for these two countries. Therefore it can be said that share of the elderly population is higher where the sensitivity of index is higher. It is also clear that this index (LTGEG) is more sensitive than the Gini coefficient (G).

The TMG is computed by using the expression (17). The observed minimum and maximum values of trigonometric measure (TMG) are 3.359 (in Israel) and 56.014 (in Qatar) respectively (

All the measures (G, LTGEG and TMG) satisfy the rank-order condition (). An inequality index satisfies the three basic properties: rank-order condition, mean or scale independence and population size independence (Anand, 1997). The two of above properties tell that if every one’s age is changed in the same proportion and similarly if number of individual/person at each age level is changed by the same proportion, the index remain invariant. It is observed that all the measures (G, LTGEG and TMG) discussed here satisfy these properties. Therefore, this measure may be considered as an alternative measure of inequality of aging. Again, from the performance of the measures CV, G and LTGEG, it is clear that LTGEG is better one. Because it overcomes all the difficulties face by other measures CV and G. Again the trigonometric measure (TMG) works very well because it follows all the properties of an ideal measure. Also its sensitivity level is higher than other measures. Considering the criteria of measure and the degree of sensitivity, TMG is the best measure of aging inequality.

We are grateful to the reviewers of this article who have given valuable suggestions.