The decomposition of inequality and poverty

Transcription

1 The decomposition of inequality and poverty The decomposition of the FGT index The FGT poverty index for a population composed of K groups can be written as follows: P(z; a) f(k)p(k; z; a) K k where P(k; z; α ) is the FGT poverty index for subgroup k and φ (k) is the proportion of the population in this subgroup. The contribution of group k to the poverty index for the whole population equals φ ( k)p(k;z; α). To perform the decomposition of the FGT index: - From the main menu, choose the item: "Decomposition FGT Decomposition". - After confirming the configuration, the application appears. Choose the different Indication Variables or Choice is: Variable of interest y Size Variable s Group Variable c Poverty line z alpha α Group numbers separated by "-" k - k - Remark: The group numbers separated by the dash "-" should be integer values. For example, we may have two subgroups coded by the integers and. In this case, we would write in the field «Group Numbers» the values "-" before proceeding to the decomposition. The decomposition of the FGT index for two groups To perform the decomposition of the FGT index for two groups: - From main menu, choose the item: "Decomposition FGT Decomposition for two groups". - After confirming the configuration, the application appears. Choose the different vectors and parameter values as follows

2 Indication Variables or Choice is: Variable of interest y Size Variable s Group Variable c Poverty line z alpha α Numbers for the subgroups separated by k - k "-" In the output window, you will find the following information: - The FGT index for the whole population. - The FGT index for each of the two subgroups. 3- The difference in the indices of the two groups: P(;z; α ) P(;z; α) 4- The percentage difference in the contribution of the two population subgroups, ( φ ()P(; z; α) φ()p(; z; α)) / P(z; α) To compute the standard deviations for these statistics, choose the option computing with standard deviation. The decomposition of the FGT index across growth and redistribution effects According to Datt & Ravallion (99) approach, we can decompose variation of the FGT Index between two periods, t and t, into growth and redistribution effects as follows: P t t t t t t t t [ P( µ, π ) P( µ, π )] + [ P( µ, π ) P( µ, π )] + R / ref P Variation P C t t t t t t t t [ P( µ, π ) P( µ, π )] + [ P( µ, π ) P( µ, π )] + R / ref P Variation C C C Variation Difference in poverty between t and t. C Growth Impact. C Contribution of redistribution effect R Residual Ref : Indicates the period of reference. t t t P( µ, π ) : the FGT index of the first period when we multiply all incomes y of the t t first period by the ratio µ /µ i

3 t t P( µ, π ) : the FGT index of the second period when we multiply all incomes t y of the second period by the ratio t t µ / µ According Kakwani (997) approach, we can decompose variation of the FGT Index between two periods, t and t, into growth and redistribution effects as follows: C C P + P C C Variation t t t t t t t t ([ P( µ, π ) P( µ, π )] + [ P( µ, π ) P( µ, )]) ([ t t t t ] [ t t t t P( µ, π ) P( µ, π ) + P( µ, π ) P( µ, )]) π π i To perform the decomposition of the FGT index across growth and redistribution effects: - From the main menu, choose the item: "Decomposition Growth and redistribution". - After confirming the configuration, the application appears. Choose the different Indication Vector or parameter Choice is: Distribution - t Variable of interest Size Variable Group Variable Distribution-t y y s s c c Index of group k k Poverty lines z alpha α To compute the standard deviation of this index, choose the option for computing with standard deviation. The sectoral decomposition of differences in FGT indices We can decompose differences in FGT into sub-group differences in poverty and population proportions as follows: 3

4 P P Variation K φ(k) k ( P (k; z; α) P (k;z; α) ) K + P (k;z; α ) k K + P (k;z; α) P (k;z; α) k ( φ (k) φ (k)) ( )( ) φ (k) φ (k) Variation Difference in poverty between and. C Intra-sectoral or intra-group impacts C Impact of changes in subgroup proportions C3 Interaction effect To perform this decomposition: - From the main menu, choose: "Decomposition Sectoral". - After confirming the configuration, the application appears. Choose the different Indication Vector or parameter Choice is: Variable of interest Size Variable Group Variable Distribution Distribution y y s s c c Poverty lines z alpha α Group numbers separated by "-" k - k - To compute the standard deviation of this index, choose the option for computing with standard deviation. The decomposition of the S-Gini index by sources (or components); Let J components y add up to y, that is: J y i y i 4

5 A natural approach One natural approach to decomposing the S-Gini index of inequality is as follows: I( ρ) J µ µ IC ( ρ) where IC ( ρ ) is the coefficient of concentration of the th component and µ is the mean of that component. The contribution of the th component to inequality in y is then: µ µ y IC ( ρ), The following results appear in the output window: - The S-Gini index for y. - The coefficients of concentration for every component of y. 3- The ratio µ / µ for every component of y. 4- The contribution for every component. The Shapley approach One supposes with the Shapley approach that the contribution of component to total inequality is the expected value of its marginal contribution when it is added randomly to anyone of the various subsets of components that one can choose from the set of all components. When a component is missing from that set, we assume that the observation values of that component are everywhere replaced by its average. The following results appear in the output window: To perform that decomposition of the S-Gini index of inequality: - From the main menu, choose the item: "Welfare and inequality Decomposition S-Gini decomposition". - Select the desired decomposition approach. 3- After confirming the configuration, the application appears. Choose the different 5

6 Indication Variables or Choice is: Size Variable s Rho ρ Vector(s) of interest Index-index The decomposition of the S-Gini index by population groups; Let there be G population subgroups. We wish to determine the contribution of every one of those subgroups to total population inequality. Natural approach We rewrite the S-Gini index as: µ I I I G g ρ φ g g, ρ+ ρ g µ where φ g : the population share of group g; I ρ : the contribution of inter group inequality to total inequality; µ g : the average revenue of those in group g. µ : average revenue of total population. Ig, ρ : S-Gini of group g The Shapley approach This decomposition has two steps. The first one is to decompose total inequality into inter-group and intra group contributions. The second step is to espress the total intra group contribution as a sum of contributions of each of the groups. In the first step, we suppose that the two Shapley factors are inter-group and intra group inequality. The rules followed to compute inequality in the presence of one or two factors are: to eliminate intra-group inequality and to calculate inter-group inequality, we use a vector of incomes where each observation has the average income of its group; to eliminate inter-group inequality and to calculate intra-group inegality, we use a vector of incomes where each observation has its income multiplied by the ratio µ / µ. g The second step consists in decomposing total intra-group inequality as a sum of group inequality. To do this, we proceed systematically simply by replacing the revenues of 6

7 those in a group by the average income of that group, such as to eliminate the intra group contribution of a given group. To perform the decomposition of the S-Gini index by groups: - From the main menu, choose the item: "Welfare and inequality Decomposition S-Gini decomposition by groups". - Select the desired decomposition approach. 3- After confirming the configuration, the application appears. Choose the different Indication Variables or Choice is: Size Variable s rho ρ Group numbers separated by "-" k -k - The decomposition of the Generalised Entropy index of inequality The Generalised Entropy index of inequality can be decomposed as follows: θ K (k) I( ) (k) µ θ φ.i(k; θ) + I( θ) k µ y where: φ (k) is the proportion of the population found in subgroup k. µ (k) is the mean income of group k. I ( k;θ) is the inequality within group k. ( θ) I is population inequality if each individual in subgroup k is given the mean income of subgroup k, µ(k). To perform the decomposition of the entropy index: - From the main menu, choose the item : "Welfare and inequality Decomposition Entropy decomposition". - After confirming the configuration, the application appears. Choose the different 7

8 Indication Variables or Choice is: Variable of interest y Size Variable s Group Variable c theta θ Group numbers separated by "-" k -k - The following information appears in the output window: - The entropy index for the whole population. - The entropy index for between-group inequality I ( θ). 3- The entropy index within every subgroup I(k; θ ). 4- The ratio ( µ (k)/ µ ) Normalised mean for every subgroup. 5- The absolute contribution to total inequality of inequality within every subgroup, that is, θ ( µ (k) / µ ). φ(k).i(k; θ) 6- The relative contribution to total inequality of inequality within every subgroup. To compute the standard deviations for these statistics, choose the option computing with standard deviation. Decomposition of variation of social welfare index between two periods We can decompose the difference in social welfare (as measured by the EDE Atkinson index) between two populations, and, as follows: where: ξ ( ε) ξ( ε) (I I )* µ + ( µ µ )* ( I ) + ( µ µ )*(I I ) C C C3 C: Impact of change in inequality. C: Impact of change in mean. C3: Interaction impact. To perform this decomposition: - From the main menu, choose: "Decomposition Decomposition of Social Welfare". - Choose the different 8

9 Indication Vector or parameter Choice is: Variable of interest y Size Variable Group Variable Group number epsilon Distribution Distribution y s s c c k k ε ε To compute the standard deviation, choose the option for computing with standard deviation. 9