1 Introduction
Henri Theil’s book on information theory (**Theil** 1967) provided a landmark in the development of the analysis of **inequality** measurement. The signif- icance of the landmark was, perhaps, not fully realised for some time, al- though his in‡uence is now recognised in standard references on the analysis of income distribution. Theil’s insight provided both a method for thinking about the meaning of **inequality** and an introduction to an important set of functional forms for modelling and analysing **inequality**. Theil’s structure laid the basis for much of the work that is done on **decomposition** by popu- lation subgroups. The purpose of this paper is to set Theil’s approach in the context of the literature that has since developed and to demonstrate that its contribution is more far-reaching than is commonly supposed.

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In reviewing the literature, we found that the aforementioned studies had focused on summery measures of income **inequality** such as Gini and **Theil** coefficient and employed the regression-based **decomposition** method. However recent development in the research of income **inequality** emphasizes estimating the entire wage distribution and decomposing the changes of the distribution. This distributional approach provides comprehensive information. It shows the earning dispersion of different income groups. It can answer questions such as whether earnings are more dispersed among the upper incomers or lower incomers, and whether the **inequality** is driven by the presence of the extreme high incomers (appeared as a long upper tail) or very low incomers (a long lower tail). Moreover, the distribution-based **decomposition** is more general than the **decomposition** of specific income **inequality** indexes. Once it is performed, the **decomposition** of a specific index follows immediately.

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Morduch and Sicular (2002) have suggested a speci…c regression-based method for decomposing **inequality**. They claim that their method possesses three main advantages: (a) it yields an exact allocation of contributions to the identi…ed variables, (b) it is general, in that it can be employed with a variety of **inequality** **indices** and **decomposition** rules, (c) it is associated with a simple procedure for deriving standard errors and con…dence intervals for estimated components of **inequality**. In this comment we argue that these points are not

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We set in this paper a coherent theory based on functional empirical processes that allows to consider both the poverty and the **inequality** **indices** in one Gaussian field in which the study of the influence of the one over the other is done. We use the General Poverty Index (GPI), that is a class of poverty **indices** gathering the most common ones and a functional class of **inequality** measures including the Entropy Measure, the Mean Logarithmic Deviation, the different **inequality** measures of Atkinson, Champernowne, Kolm and **Theil** called **Theil**-Like **Inequality** Measures (TLIM). Our results are given in a unified approach with respect to the two classes instead of their particular elements. We provide the asymp- totic laws of the variations of each class over two given periods and the ratio of the variation and derive confidence in- tervals for them. Although the variances may seem somehow complicated, we provide R codes for their computations and apply the results for the pseudo-panel data for Senegal with a simple analysis.

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One of the most relevant studies concerning the **decomposition** of income ine- quality at the European level is carried out by Beblo and Knaus ( 2001 ). Their results show a **Theil** index of around 0.185 by using income data of ten European countries for the year 1995. Moreover, they find that between-country **inequality** contributes 3.4%, for incomes before transfers and taxes, and 9.3%, for disposable income, to the aggregated income **inequality**. A more recent approach is carried out by Hoffmeister ( 2009 ). He refers to the EU-25 and detects a convergence of national income levels and within-country personal income **inequality** between 1994/1995 and 1999/2000. Furthermore, he states that **inequality** is rising in the Social-Democratic regime but decreasing in Mediterranean welfare states. Papatheodorou and Pavlopoulos ( 2014 ) analyse the development of **inequality** in the EU-15 between 1996 and 2008 by cal- culating **Theil** **indices**. They find that the contribution of between-country **inequality** decreases from 14.8% in 1996 to 4.9% in 2008. Moreover, Papatheodorou and Pav- lopoulos ( 2014 ) find that southern and liberal European countries show the highest **inequality** and relatively contribute most to aggregated **inequality**, whereas Scandi- navian countries show the opposite result. Eurofound ( 2017 ) measures aggregated **inequality** for the EU-28 between 2005 and 2013 and reports a decrease in between- country **inequality** and an increase in overall **inequality** since 2008.

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In this article we provide an overview of the Gini **decomposition** and the generalized entropy **inequality** measures, a free access to their computation, an application on French wages, and a different way than Dagum to demonstrate that the Gini index is a more convenient measure than those issued from entropy: **Theil**, Hirschman−Herfindahl and Bourguignon.

Françoise Seyte Michel Terraza
LAMETA, University of Montpellier I LAMETA, University of Montpellier I
Abstract
In this article we provide an overview of the Gini **decomposition** and the generalized entropy **inequality** measures, a free access to their computation, an application on French wages, and a different way than Dagum to demonstrate that the Gini index is a more convenient measure than those issued from entropy: **Theil**, Hirschman−Herfindahl and Bourguignon.

Since a transfer of income from a poorer to a richer person entails a transfer of the shortfall from the latter to the former, the poverty measure is bound to decrease if the **inequality** com- ponent involved in the index is deﬁned in terms of either incomes or shortfalls. In fact, in the mentioned decompositions this third component refers to income **inequality** or to shortfall in- equality indistinctly 3 . However, as will be shown below, the choice between income and shortfall **inequality** is not innocuous and diﬀerent choices may lead to contradictory results. This diﬃculty arises not only in poverty measurement but also in diﬀerent economic ﬁelds in which bounded variables are involved. Recent papers (among them Clarke et al. [6], Erreygers [7] and Lambert and Zheng [18]) deal with this issue in health measurement. The results derived by Lambert and Zheng [18] may have a straightforward application to the measurement of the **inequality** among the poor. They introduce a property of consistency which requires that achievement and shortfall **inequality** rankings should not be reversed, and show that all relative and intermedi- ate **inequality** **indices** fail their requirement. Accordingly, whenever a relative or intermediate **inequality** index is involved in the **decomposition** of a poverty index, the **inequality** component is not consistent. We think this is a serious drawback which may distort the conclusions in the

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1 Introduction
According to recent comparative studies on OECD countries, the high- est income **inequality** is found in the US, followed by the UK and Italy, the latter two presenting similar figures using standard **inequality** measures (Atkinson et al., 1995; Smeeding, 2000). However, while the US and the UK present a roughly increasing trend of income **inequality** since the 1970s, Ital- ian household income distribution exhibits substantial fluctuations but no clear trend (Brandolini and D’Alessio, 2001; D’Alessio and Signorini, 2000). Brandolini and D’Alessio (2001) find that demographic characteristics are able to explain only a limited amount of overall Italian household **inequality** but do not investigate the issue further. They reach this conclusion by using decompo- sition of income **inequality** by population groups (Bourguignon, 1979; Cowell, 1980; Shorrocks, 1980, 1984). This **decomposition** begins with dividing the population into discrete groups and then computes and combines **inequality** **indices** within each group and between the means of different groups. Although a powerful descriptive tool, it presents various limitations, including the fact that the **decomposition** can be carried out only over discrete groups, that there is no optimal rule for partitioning non-discrete variables and **decomposition** depends on the groups considered, that handling interactions among groups or multiple factors can be overwhelming, that **decomposition** by population groups does not depict a causal relationship between variables used to explain **inequality** and **inequality** itself as no control for endogeneity is available.

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Income **inequality** refers to the **inequality** of the distribution of individuals, household or some per capita measure of income. Lorenz Curve is used for analysing the size distribution of income and wealth and measures of **inequality** and poverty. It plots the cumulative share of total income against the cumulative proportion of income receiving units. The divergence of a Lorenz curve for perfect equality and the Lorenz curve for a given income distribution is measured by some index of **inequality**. The most widely used index of **inequality** is the Gini coefficient (for reviews of the notion and analysis of **inequality** see Subramanian 1997 and Cowell 2000). There are two parametric approaches to estimate the Lorenz curve (Ryu and Slottje, 1999). In the first approach one assumes a hypothetical statistical distribution for income distribution and in the second approach, a specific functional form is fit to the Lorenz curve directly (Chotikapanich and Griffiths 2002). An important drawback of the traditional models of the Lorenz curve is a lack of satisfactory fit over the entire range of a given income distribution. Ogwang and Rao (2000) proposed two hybrids Lorenz curves by combination of traditional models. The estimated Lorenz curve is sensitive to errors in survey data. The robustness properties of **inequality** and poverty measures assuming contaminated data with illustrations are considered in Cowell and Victoroa-Feser (1996a and 1996b). Hasegawa and Kozumi (2003) propose using Bayesian non- parametric analysis and present a method for removing the contaminated observations. Several **inequality** **indices** can be derived from the Lorenz diagram. The Lorenz Curve construction also gives us a rough but standard measure (Gini coefficient) of the amount of **inequality** in the income distribution. The index lies in the interval 0 (perfect equality) and 1 (perfect **inequality**). Among the other notable measures of **inequality** are: the range, the variance, the squared coefficient of variation, the variance of log incomes, the absolute and relative mean deviations, and Theil’s two **inequality** **indices**.

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This paper attempts to study the gender **inequality** situation in Bangladesh during 1995-96 through examining the effect the gender of a household head and their marital status has on the per capita household expenditure. The study was conducted using the **Theil** **inequality** **decomposition** technique with household expenditure data from the 1995-96 Household Expenditure Survey (HES). The study reveals that gender in Bangladesh does not seem to play a large role in explaining total **inequality** as the between component accounted for less than 5 percent of total **inequality**. However, this conclusion seems premature when the marital status of household heads are examined for their effect on **inequality**. The results show the need to distinguish amongst the various types of female headed households themselves. In this regard, De jure female headed households form an especially vulnerable group. Hence policy makers must keep this in mind when designing policy to reduce **inequality**.

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This approach is based on decomposing **inequality** **indices** for an outcome that can be expressed as a (weighted) sum of a set of factor components (e.g., Rao, 1969; Shorrocks, 1982). This was extended to the **decomposition** of outcomes that can be expressed in terms of a linear regression model in Wagstaff et al,. (2003) and applied to health concentration **indices**. Jones and Lopez (2006) extend their approach to allow for heterogeneous regression coefficient. The Wagstaff et al. (2003) regression-based **decomposition** approach for concentration **indices** has been criticised on various grounds. First, rank ignorability, that it treats the ranks, R, as fixed and does not allow for the influence of covariates on the ranking variable (Erreygers and Kessels, 2013; Heckley et al., 2016). Second, weighting function ignorability which, for the Gini coefficient, implies treating the mean of the outcome as fixed and not allowing for the influence of covariates on the mean (van Ourti et al., 2009; Heckley et al., 2016). Third, Heckley et al. (2016) are critical that the regression- based approach does not explicitly define counterfactual values of the covariates. They propose a method that uses recentred influence functions (RIF) to estimate the partial effect of covariates on a general class of concentration **indices** but that does not provide a complete percentagewise **decomposition** of the **indices**.

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This paper clarifies the relation between the **Theil** index and the loss in aggregate productivity caused by the distortions in labor allocation. My interpretation is that the difference in the **Theil** **indices** of **inequality** between two economies approximately measures the relative loss in aggregate productivity caused by these distortions. Moreover, the **Theil** index itself can be interpreted approximately as the possible maximum loss of aggregate productivity caused by these distortions.

It may be assumed that, on average, the components that account for a larger portion of total income would play a larger role in pushing households closer to or over the poverty line. Although this assumption seems to make sense, the relationship is more complex. Simply taking the mean of each component as a crude measure of its relative impact on poverty alleviation is insufficient, since it fails to incorporate the different distributional effects among the components. In other words, the income share of each component in the income of different individuals is not identical for all individuals, or income components are not distributed in the same way for different individuals. Duclos & Araar (2006) developed an algorithm that overcomes this problem. Rather than attempting to gauge the direct impact of each income component on poverty alleviation, they investigate the level of poverty that would have existed in the absence of this income source in overall income. They propose the testing of the marginal effect that each component has on poverty alleviation by comparing the FGT **indices** against what they would have been in the absence of a specific source of income. Although this method appears straightforward – simply calculating the drop in the poverty index that is due to the inclusion of an additional income source – it is not. The fact that the marginal effect of each component is largely dependent on the order in which different income components are included in the model is potentially problematic. Duclos and Araar (2006) circumvent this path- dependency problem by making use of a Shapley-value. Instead of calculating the effect of a specific income component on a predetermined subset of income components, the Shapley-value calculates the average marginal effect of that component over all possible income subsets (including the empty subset). The results of applying this method are presented in table 5 below.

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A further **decomposition** analysis was performed across rural and urban areas. Table 5 shows results for the marginal impact of **inequality** in within-and between- household expenditure component on **inequality** and poverty, as well as the elasticities of the impact on overall poverty at the rural level. In general, the results show that the marginal impact of within-component **inequality** on poverty is higher than that of between-component **inequality**. However, the corresponding elasticities are higher for between-component than within component **inequality**. The signs of the elasticities of changes in component **inequality** with respect to poverty vary across poverty **indices**. The elasticities for within-and between-component **inequality** were both positive for poverty severity and poverty gab while only within-component elasticity was positive for headcount poverty. Also while the marginal impacts on overall poverty of a change in within-component **inequality** were all positive, the marginal impact for between-component was only positive for headcount poverty. It is worth noting that the marginal impact of a change in within-component **inequality** on overall **inequality** was positive while that of between- component was negative. The results suggests that, in rural Ghana, reductions in within- component **inequality** are more effective in reducing overall poverty and **inequality**, irrespective of poverty index used. However, with regards to the elasticity of impact, a reduction in between- component **inequality** is only effective in reducing poverty gab and poverty severity. Specifically, a 1% reduction in within-component **inequality** leads to a 0.26%, 1.63% and 2.96% reduction in poverty headcount, poverty gab and poverty severity, respectively. On the other hand, a 1% reduction in between-component **inequality**, all things equal, leads to a 4.29% and 11.29% reduction in poverty gap and poverty severity, respectively.

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Probability of having access to landed property. The results obtained highlight the effects of migration, the level of education and the economic variables. The study by Vellenga (1986), highlights the role of the cultural system of transmission. He opposes the patriarchal system to the matriarchal system. He shows that the patriarchal system discriminates against women vis-a-vis landed property. The studies launched since 1998 by the International Fund for agricultural Development (IFAD) in Ghana, constitutes the basis of studies related to Sub-Saharan Africa. These studies reveal the existence of constraints faced by women in trying to access landed property. In spite of their contribution to production, they remain discriminated against as concerns the land resource. In these studies, the cultural variable is the most significant. For cultural reasons, the lack of authority to which women are victim does not enable them to own land. For Ducan and Brants, the reason for discrimination against women vis-a-vis the land resource finds its roots in the family base. They show that the control of the land resource is reserved to men; the women being pushed to a subsidiary role have to leave the family house to build a new family elsewhere. The study by Benkes (2006) reveals the impact of cultural practices, particularly the rules of patriarchal succession, which discriminate against women. In the same vein, Kuubana, Kwaku and Halidu (2013), reveal in the case of Ghana, an **inequality** of landed property between men and women. They record an average **inequality** of about 30%, in favour of men. These studies attribute this patriarchal **inequality** to the system of inheritance, on the one hand and to financial constraints on women on the other hand. These results are put into question by Yamakoski and Keister (2006). They show that by taking into account the age brackets, particularly the young, there is little difference between men and women. The study by Ducan and Brants (2004) is not limited solely to the impact of cultural practices. They also address the role played by demographic pressure observed in the large cities south of the Sahara. They show that women generally do not enjoy a high economic capacity, which discriminates against them vis-a-vis the speculative marketing of land. Gray and Kevane (1992) show that even if access to land is guaranteed by family ties, this also depends on the position of the women in the marriage and their level of education. Gray and Kevane (1992) focus particularly on the rural areas. They highlight the capacity of women to get access to bank credit.

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In Bangladesh, while the number of diarrhoea-related childhood deaths has dropped sub- stantially, under-five morbidity rate has remained persistently high [8,9]. Globally, under-five deaths due to ARI has reduced from 1.7 million in 2000 to 0.94 million in 2013 [10]. However, the burden of ARI remains persistently high in Bangladesh (21% ARI-related mortality) [11], with approximately 50,000 children dying annually from ARI with a high pediatric hospitalisa- tion rate of 40% [12–14]. Furthermore, ARI alone was accountable for 39.8 per 1000 children in 2013 [15]. Similarly, malaria (10.3%) [16], dengue fever (12.5%) [17], chikungunya (19%) [18], and typhoid are also common causes of childhood illness [12–15,19,20]. Bangladesh has already made improvements in the child health-related parameters of its people [21]; however, the prevalence of childhood morbidity is still distressingly high and widespread [22]. Poor health is often associated with low socioeconomic status and an inability to access health ser- vices, especially in rural communities [23]. One of the major challenges for potential improve- ment is to address the high levels of **inequality** in the distribution of healthcare across regions and socioeconomic groups [24,25]. The United Nations General Assembly’s recently adopted Sustainable Development Goals (SDGs) are aimed at fighting against **inequality** so that every- body gets similar access to, and benefits of, any health policy [26]. However, to achieve the health-related goals of SDG-3 (good health and well-bring) and SDG-10 (reduced inequalities especially access to health services) [26], it is essential to measure and address **inequality** so that all strata of society can potentially benefit.

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Our main findings are as follows. The increase in post—tax income **inequality** was slower than that of pre—tax **inequality** indicating that the redistributive role of the tax system has increased over time. However, our **decomposition** reveals that most of this increase in redistribution was not due to the policy effect but a mechanical consequence of the rising **inequality** in pre—tax income. Indeed, the effects of policy changes more or less canceled out over the period as a whole — which is a direct consequence of partisan politics. Our findings are in line with popular perceptions regarding the political cycle, with disequalizing (equalizing) effects observed for policy changes implemented during Republican (Democrat) administrations (see Bartels 2008). The results for some sub— periods show large effects for actual policy changes — sometimes accounting for more than 50 percent of the increase in post—tax **inequality** (Tax Reform Act of 1986). There are also significant differences between results for the lower and upper parts of the distribution. Policy reforms enacted in the early and mid 1990s reduced income gaps at the bottom to below their 1978 values. The equalizing effect of tax policy on **inequality** at the lower half of the distribution is maintained until the end of the observation period and even enforced by provisions enacted through the American Recovery and Reinvestment Act. By contrast, no equalizing effects of policies can be discerned for the upper part of the distribution. Instead, for the period as a whole, tax policy changes affecting top—income earners appear to have slightly exacerbated trends towards widening income gaps at the top.

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Equal access to healthcare services such as dental care is defined as one of the main goals in all healthcare systems [1, 2]. Nevertheless, utilization of healthcare has been influ- enced by a variety of sociodemographic and economic fac- tors. Previous studies reported socioeconomic inequalities in dental care utilization in high-income countries [1, 4]. A limited number of studies [7–9] also suggested the effect of socioeconomic status in the utilization of dental care in some provinces in Iran. This study, for the first time, mea- sures the extent of socioeconomic inequalities in Iran and across 31 provinces. Further, the observed socioeconomic **inequality** in dental care utilization was decomposed to identify the main determinates such **inequality** among Iranian households.

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