The decomposition of inequality and poverty

Transcription

1 The decomposton of nequalty and poverty THE DECOMPOSITIO OF THE FGT IDEX The FGT poverty ndex for a populaton composed of K groups can be wrtten as follows: P(z;α) = K φ(k)p(k; z; α) k = where P(k;z; α ) s the FGT poverty ndex for subgroup k and φ (k) s the proporton of the populaton n ths subgroup. The contrbuton of group k to the poverty ndex for the whole populaton equals φ( k)p(k; z; α). To perform the decomposton of the FGT ndex: - From the man menu, choose the tem: "Decomposton FGT Decomposton". Poverty lne z Compulsory Group numbers separated by "-" k - - k Compulsory REMARK: The group numbers separated by the dash "-" should be nteger values. For example, we may have two subgroups coded by the ntegers and. In ths case, we would wrte n the feld «Group umbers» the values "-" before proceedng to the decomposton. THE DECOMPOSITIO OF THE FGT IDEX FOR TWO GROUPS To perform the decomposton of the FGT ndex for two groups: - From man menu, choose the tem: "Decomposton FGT Decomposton for two groups". parameter values as follows Poverty lne z Compulsory umbers for the subgroups separated by "-" k - k Compulsory

2 In the output wndow, you wll fnd the followng nformaton: - The FGT ndex for the whole populaton. - The FGT ndex for each of the two subgroups. - The dfference n the ndces of the two groups: P(;z; α ) P(; z; α) - The percentage dfference n the contrbuton of the two populaton subgroups, ( φ ()P(; z; α) φ()p(; z; α)) / P(z; α) To compute the standard devatons for these statstcs, choose the opton computng wth standard devaton. THE DECOMPOSITIO OF THE FGT IDEX ACROSS GROWTH AD REDISTRIBUTIO EFFECTS Accordng to Datt & Ravallon (99) approach, we can decompose varaton of the FGT Index between two perods, t and t, nto growth and redstrbuton effects as follows: var aton t t t t t t t t P P = P(, π ) P(, π ) + P(, π ) P(, π ) + R /ref = C C t t t t t t t t P P = P(, π ) P(, π ) + P(, π ) P(, π ) + R / ref = var aton C Varaton = Dfference n poverty between t and t. C = Growth Impact. C = Contrbuton of redstrbuton effect R = Resdual Ref : Indcates the perod of reference. t t P(, π ) : the FGT ndex of the frst perod when we multply all ncomes of the frst t t perod by the rato / t t t P(, π ) : the FGT ndex of the second perod when we multply all ncomes y of the t t second perod by the rato / Accordng the Shapley approach, we can decompose varaton of the FGT Index between two perods, t and t, nto growth and redstrbuton effects as follows: P + P = C C Varaton C t y

3 C C = t t t t t t t t ([ P(, π ) P(, π )] + [ P(, π ) P(, )]) π = t t t t t t t t ([ P(, π ) P(, π )] + [ P(, π ) P(, )]) π To perform the decomposton of the FGT ndex across growth and redstrbuton effects: - From the man menu, choose the tem: "Decomposton Growth and redstrbuton". Poverty lne z Compulsory We note here that the Shapley approach s equvalent to the axomatc Kakwan approach for ths decomposton. THE SECTORAL DECOMPOSITIO OF DIFFERECES I FGT IDICES We can decompose dfferences n FGT nto sub-group dfferences n poverty and populaton proportons as follows: P P = φ (k)( P (k;z; α) P (k;z; α) ) K k= Varaton C Varaton = Dfference n poverty between and. C = Intra-sectoral or ntra-group mpacts C = Impact of changes n subgroup proportons C3 = Interacton effect K + P(k;z; α) ( φ(k) φ(k) ) k= C K + ( P (k;z; α) P (k;z; α) )( φ(k) φ(k) ) k= C3 To perform ths decomposton: - From the man menu, choose: "Decomposton Sectoral". 3

4 Poverty lne z Compulsory Group numbers separated by "-" k - - k Compulsory We can always perform ths decomposton by usng the Shapley approach opton such as: Contrbuton of φ K = ) [ ( φ (k)p (k; α, z) ) ( φ (k)p (k; α, z) )] + [( φ (k)p (k; α, z) ) ( φ (k)p (k; α, z) ] Contrbut on of PK = ) [ ( φ (k)p (k; α,z) ) ( φ (k)p (k; α,z) )] + [( φ (k)p (k; α,z) ) ( φ (k)p (k; α, z) ] THE DECOMPOSITIO OF THE FGT IDEX BY SOURCES (OR COMPOETS) Let components J y add up to y, that s: The Shapley approach J y = y One supposes wth the Shapley approach that the contrbuton of component towards reducng total poverty s the expected value of ts margnal contrbuton when t s added randomly to anyone of the varous subsets of components that one can choose from the set of all components. When a component s mssng from that set, we assume that the observaton values of that component are everywhere replaced by 0. = To perform that decomposton of the FGT Index by sources: - From the man menu, choose the tem: " Decomposton FGT: Decomposton by sources". Poverty lne Z Compulsory Vector(s) of nterest To be selected Compulsory 4

5 THE DECOMPOSITIO OF THE FGT IDEX BY TRASIET & CHROIC COMPOETS Ths type of decomposton decomposes total poverty, observed over some tme perods, nto transent and chronc components. The Jalan & Ravallon approach t Let y be the ncome of household n perod t and for household. Total poverty s defned as follows: be the average ncome over the T perods TP( α,z) = T sw (z y ) t= = T sw = t α + The chronc poverty component s then defned as: CPC( α, z) = sw (z ) = sw = α + The transent poverty component s fnally defned as: TPC( α,z) = TP( α,z) CPC( α,z) The Duclos, Xmng and Araar approach t Let y for household. Let such as: be the ncome of household n perod t and Γα (,z) be the average ncome over the T perods be the equally-dstrbuted-equvalent (EDE) poverty gap T / α t α sw (z y ) + / α t= = Γα (,z) = [ TP( α,z)] = T sw = 5

6 The transent poverty component s defned as follows: TPC( α,z) = sw θ ( α,z) = = sw where θ =γ ( α,z) γ (,z) and T t α γ( α,z) = (z y ) + /T = t / α The chronc poverty component s defned as follows: CPC( α,z) =Γα (,z) TPC( α,z) ote that the number of perods avalable for ths type of exercse s generally small. Because of ths, a bas-correcton s typcally useful, usng ether an analytcal/asymptotc or bootstrap approach. To perform that decomposton of the FGT Index by transent and chronc components: - From the man menu, choose the tem: " Decomposton FGT: Transent and chronc". Poverty lne Z Compulsory Vector(s) of nterest To be selected Compulsory THE TRASITIO MATRIX Ths type of decomposton gves us nformaton on the fnal dstrbuton (denoted by ) for predetermned groupng wthn the ntal dstrbuton (denoted by 0). Economc transton matrx If we suppose that the bounds of the matrx are: BR { 0, B,, B, B,, Bma } percentage of populaton: = x the - whose lvng standard s between B and B k n the ntal perod - whose lvng standard s between B l and B m n the fnal perod 6

7 s gven by: = 0 <= < k l <= < m sw *I(B y B )* I(B y B ) sw = Socal transton matrx The set-up s as above, except that the bounds are now estmated as quantles of the ntal and fnal dstrbutons. To perform such estmaton: - From the man menu, choose the tem: "Decomposton Transent matrx". THE DECOMPOSITIO OF THE S-GII IDEX BY SOURCES (OR COMPOETS) Let components J y add up to y, that s: J y = y = A natural approach One natural approach to decomposng the S-Gn ndex of nequalty s as follows: I( ρ) = J = where s the coeffcent of concentraton of the th IC ( ) component and s the mean of ρ IC ( ρ) that component. The contrbuton of the th component to nequalty n y s then: The followng results appear n the output wndow: y IC ( ρ), - The S-Gn ndex for y. - The coeffcents of concentraton for every component of y. 7

8 - The rato / for every component of y. - The contrbuton for every component. The Shapley approach One supposes wth the Shapley approach that the contrbuton of component to total nequalty s the expected value of ts margnal contrbuton when t s added randomly to anyone of the varous subsets of components that one can choose from the set of all components. When a component s mssng from that set, we assume that the observaton values of that component are everywhere replaced by ts average. The followng results appear n the output wndow: To perform that decomposton of the S-Gn ndex of nequalty: - From the man menu, choose the tem: "Welfare and nequalty Decomposton S-Gn decomposton". - Select the desred decomposton approach. Rho ρ Compulsory Vector(s) of nterest Index-ndex Compulsory THE DECOMPOSITIO OF THE S-GII IDEX BY POPULATIO GROUPS Let there be G populaton subgroups. We wsh to determne the contrbuton of every one of those subgroups to total populaton nequalty. atural approach We rewrte the S-Gn ndex as: where φ : the populaton share of group g; I ρ g I I I G g ρ = φ g g, ρ + ρ g= : the contrbuton of nter group nequalty to total nequalty; g : the average revenue of those n group g. : average revenue of total populaton. Ig, ρ : S-Gn of group g The Shapley approach 8

9 Ths decomposton has two steps. The frst one s to decompose total nequalty nto nter-group and ntra-group contrbutons. The second step s to express the total ntra-group contrbuton as a sum of the contrbutons of each of the groups. In the frst step, we suppose that the two Shapley factors are nter-group and ntra-group nequalty. The rules followed to compute nequalty n the presence of one or two factors are: to elmnate ntra-group nequalty and to calculate nter-group nequalty, we use a vector of ncomes where each observaton has the average ncome of ts group; to elmnate nter-group nequalty and to calculate ntra-group negalty, we use a vector of ncomes where each observaton has ts ncome multpled by the rato /. The second step conssts n decomposng total ntra-group nequalty as a sum of group nequalty. To do ths, we proceed systematcally smply by replacng the revenues of those n a group by the average ncome of that group, such as to elmnate the ntra-group contrbuton of a gven group. To perform the decomposton of the S-Gn ndex by groups: - From the man menu, choose the tem: "Welfare and nequalty Decomposton S-Gn decomposton by groups". - Select the desred decomposton approach. Rho ρ Compulsory Vector(s) of nterest Index-ndex Compulsory g THE DECOMPOSITIO OF THE GEERALISED ETROPY IDEX OF IEQUALITY The Generalsed Entropy ndex of nequalty can be decomposed as follows: K θ ˆ ˆ ˆ (k) I( ) (k).i(k; ˆ θ = φ θ ) + I( θ) k= ˆ where: φ (k) s the proporton of the populaton found n subgroup k. (k) s the mean ncome of group k. I ( k;θ) s the nequalty wthn group k. ( θ) I s populaton nequalty f each ndvdual n subgroup k s gven the mean ncome of subgroup k, (k). To perform the decomposton of the entropy ndex: 9

10 - From the man menu, choose the tem: "Welfare and nequalty Decomposton Entropy decomposton". Theta θ Compulsory Group numbers separated by "-" k - - k Compulsory The followng nformaton appears n the output wndow: - The entropy ndex for the whole populaton. - The entropy ndex for between-group nequalty I ( θ). - The entropy ndex wthn every subgroup I(k; θ ). - The rato ( (k) / ) ormalsed mean for every subgroup. - The absolute contrbuton to total nequalty of nequalty wthn every subgroup, that s, θ ( (k) / ). φ(k).i(k; θ) - The relatve contrbuton to total nequalty of nequalty wthn every subgroup. DECOMPOSITIO OF VARIATIO OF SOCIAL WELFARE IDEX BETWEE TWO PERIODS We can decompose the dfference n socal welfare (as measured by the EDE Atknson ndex) between two populatons, and, as follows: where: ξ ( ε) ξ( ε) = (I I)* + ( )*( I ) + ( )*(I I ) C C: Impact of change n nequalty. C: Impact of change n mean. C3: Interacton mpact. C C3 To perform ths decomposton: - From the man menu, choose: "Decomposton Decomposton of Socal Welfare". - Choose the dfferent vectors and epslon ε ε Compulsory 0