Gini Multi-Decomposition by Subpopulation in Fuzzy Poverty: Evidence of Chad

This study aims to analyze the configuration of inequalities in Chad through a multidimensional approach. The multidimensional approach adopted and based on the fuzzy set is also called the theory of fuzzy set. The multidimensional poverty indices are calculated by the formulations of Cerioli and Zani. These indices are broken down following the method of Camilo Dagum by groups. The data used come from the Household Budget and Consumption Surveys of 2003 of 6695 households and 2011 of 9259 households. The result shows that the Gini indices indicate less unequal situations which are around 0.263 in 2003 to 0.278 in 2011. Social policies for the reduction of multi-inequality must be directed towards the most disadvantaged households.


Introduction
The continuous widening of inequalities in Africa is sometimes accompanied by an increase in poverty. The World Bank also recognizes that inequality can contribute decisively to poverty. Chad, like other African nations, has experienced high and increasing inequality over the years. However, in Chad, social policies for poverty reduction seem to result in a decline in income poverty: 54.8% in 2003 against 46.7% in 2011 (Gadom et al., 2019), but they are often not sufficient to significantly modify the structure of income distribution (Gini index: 39.82 in 2003(Gini index: 39.82 in against 43.32 in 2011. They must now direct resources towards the underprivileged and target groups at risk. This involves analyzing the multi-inequality of different sub-groups of households on the basis of certain criteria such as place of residence, sex of the head of household. The construction of a profile of How to cite this paper: Demsou, T. (2022). Gini Multi-Decomposition by Subpopulation in Fuzzy Poverty: Evidence of Chad. poverty inequality facilitates the formulation of targeted measures in the context of the fight against poverty and inequality. Section 2 reviews previous works. In Section 3, we present the data sources and methodology. Section 4 presents the empirical results and Section 5 the conclusion.

Previous Works
Earlier empirical works on income inequality have tried to decompose using the additive decomposition method (Rao, 1969). The authors (Fei et al., 1978) applied in Taiwan (China) during the period 1964-1972and Fields (1979 in urban Colombia (1967-68) the Rao's method.
Unlike previous approaches, Shorrocks (1982Shorrocks ( , 1984 proposed a method to decompose the inequality, as measured by the squared coefficient of variation. This method offers rules for decomposing measures of income inequality either by subgroup (subpopulation) or by source of income. The first method is limited to breaking down the structure of the index into an intra-group measure and an inter-group measure. When a population is divided into two groups (for example, men and women), the intragroup coefficient represents the intensity of income inequality that prevails within the groups. The between-group measure symbolizes the inequalities that exist between groups of the population. Shorrocks (1982) used this method in the USA (1968)(1969)(1970)(1971)(1972)(1973)(1974)(1975)(1976)(1977). Jätti (1997) used Shorrocks (1982's decomposition of the squared coefficient of variation in Canada, the Netherlands, Sweden, the UK, and the USA in 1980. Garcıa-Penalosa & Orgiazzi (2013) extended Jätti (1997's in Canada, Germany, Norway, Sweden, the UK, and the USA for three decades. The author introduced a method that can be simultaneously decomposed into subgroups and attributes (Chakravarty, 1988).
We use for the decomposition the Gini index proposed by Dagum (1997), which allows the disaggregation of the total inequality into three components in fuzzy poverty. The first represents the contribution to the total inequality of inequalities within each group of a population; the second, contribution to the total inequality of the net inequalities between each pair of groups that prevails within the population. And last, the contribution of the intensity of the transvariation between groups, in other words, the contribution to the total inequality of the inequalities between groups derived from the overlap between the distributions.

Data
The data sources come from harmonized surveys on household living conditions, an initiative of the WAEMU Commission. They are carried out by the National Statistical Institutes (INSEED). Samples of households surveyed: 9259 households in 2011 and 6695 in 2003. The samples are representative at the national level with stratification by place of residence (urban/rural) and by region. In addition, the household questionnaire is composed of nine main topics classified as follows: identification of household members, composition, education of different members, employment, housing characteristics, household assets, health of members, retrospective expenses as well as self-consumption.

Method
Fuzzy indices of poverty (multidimensional poverty) provide a framework for better understanding poverty through its multiple facets.
is the weight attached to the attribute (Cerioli, 1990) ( ) 1 2 , , , , , The fuzzy poverty index is defined as: is the contribution of household j a to the overall fuzzy poverty index And the ration ( ) k X µ reflects the degree of deprivation of attribute k X for the population of n households. Dagum & Costa (2004) introduced decomposition by attribute by demonstrating that it is possible to calculate the contribution of attribute k X to the overall fuzzy poverty index. The fuzzy poverty index is defined as The authors get the (absolute) contribution of attribute k X to the multidi- This expression makes it possible to decompose the Gini measure into a intragroup component w G and a "gross" intergroup component gb G (Dagum, 1997): where, jj G is the Gini measure measured on the group j P and where jh G is the Gini measure measuring the inequalities between the groups j P and h P .
The Gini measure between two groups ( jh G ) is constructed by calculating the sum of the difference binaries of the fuzzy poverty index between people of groups different, then normalizing this sum: The decomposed Gini measure indeed shows a third term that evaluates the inequalities coming from the overlap zone between their distributions of the different groups studied. Let jh D be the directional economic distance (Dagum, 1997) allowing us to measure the proportion binary differences of the fuzzy poverty index calculated from each of the area non-overlapping distributions of fuzzy poverty index groups j P and h P . The decomposition of the Gini coefficient into three components: where ij d is the raw directional distance which is a weighted average of the differences of fuzzy poverty index where j F and h F and are respectively the distribution functions of the fuzzy poverty index in the groups j P and h P .
And jh p is moment of order 1 of transvariation: which is the weighted average of the binary differences in fuzzy poverty index Therefore, it is possible to measure the contribution of inequalities at inside of the w G G groups, the net contribution of inequality between nb G G groups and the contribution of transvariation intensity between t G G groups.
The index measures the inequalities generated by the fuzzy poverty index of the wealthiest groups on average widening gaps with those less wealthy groups. Suppose that as soon as the groups have the same average the inequalities sharp between them, groups are null (this is possible since because if 0 j h jh D µ µ = ⇒ = ). The term t G on the contrary measures inequalities resulting from the income of the less wealthy groups on average, which create gaps in income for individuals from the wealthiest groups on average.

Choice of Indicators and Cut-Offs
The question of the choice of deprivation indicators has been discussed at length by Cheli, Ghellini, Lemmi and Pannuzi in (Cheli et al., 1994) and by Cheli and Lemmi in (Cheli & Lemmi, 1995). These authors note that the choice of deprivation indicators is of fundamental importance. Furthermore, they recommend, in the analysis, to clearly distinguish the effect variables and the cause variables of poverty. Finally, Miceli in (Miceli, 2006) points out that the choice of deprivation indicators is particularly delicate and cannot intervene without a dose of arbitrary more or less and that the fuzzy measurement obtained is ultimately conditioned by the data availability. The deprivation thresholds first identify the people experiencing deprivation in each selected indicator. Deprivations are dichotomous.
The selection of socio-economic attributes to study the state of poverty was made on the basis of multidimensional notions of poverty, information from the Ecosit3 in 2011 and Ecosit2 in 2003 surveys and the Sustainable Development Goals (SDGs). This selection is very important because each of the selected attributes explains the degree of deprivation and social exclusion of the households studied (Miceli, 2006;Mussard & Alperin, 2005;Ambapour, 2009). For questions related to the choice of dimensions and capacities, see (Sen, 1993;Atkinson, 2003;Alkire, 2011;Alkire & Foster, 2011).
The selected variables are in this Table 1 below and the lack of the income dimension would be justified by the fact that it would already act on almost all the other dimensions selected (for example having a permanent home depends on its income, energy, etc.).

Analysis of Fuzzy Poverty in Chad
The weight w k represents the intensity of deprivation linked to the dimension X k .
the weights of Cerioli & Zani (1990) defined according to an inverse relation of the average degree of deprivation relative to the indicator j. According to this weighting system, more weight is assigned to the most common indicators.
The We note from this Table 2 that we have a global view on the causes of poverty, the methods of decomposition give us more detailed and precise information on the true causes of the determination of the multidimensional phenomenon of poverty.

Fuzzy Poverty by Gender
Understanding the phenomenon of poverty through the sex of the head of the household can provide useful elements for targeting actions aimed at improving the living conditions of the poor. Indeed, to propose policies that can help reduce poverty, the authorities need to know whether the phenomenon of poverty is linked to the gender of the head of the household or not. In Table 3

Fuzzy Poverty of Residence
In

Fuzzy Inequality Index at National Level
The results in Table 5

Fuzzy Gender Inequality
In Table 6 The breakdowns of the national Gini indicators in

Fuzzy Inequality by Place of Residence
The contributions to the national Gini index in Table 7 are: 53.45% in 2003 and 49.06% in 2011 and coming from the Gini index shows that it is because of the inequality due to the rural environment and the others indicate that it is the urban environment. In 2011, the Gini index indicates an almost equal share of intra-group and inter-group inequality.

Conclusion
The