Poverty indices. P(k;z; α ) = P(k;z; α ) /(z) α. If you wish to compute the FGT index of poverty, follow these steps:

Transcription

1 Poverty dces DAD offers four possbltes for fxg the poverty le: - A determstc poverty le set by the user. 2- A poverty le equal to a proporto l of the mea. 3- A poverty le equal to a proporto m of a quatle Q(p). 4- A estmated poverty le that s asymptotcally ormally dstrbuted wth a stadard devato specfed by the user. For the frst possblty, just dcate the value of the determstc poverty le frot of "Poverty le". For the other three possbltes, proceed as follows: Clc o the butto "Compute le". Choose oe of the followg three optos: a) Proporto of mea: the proporto l should be etered. b) Proporto of quatle: eter the proporto m ad the the quatle Q(p) by specfyg the desred percetle p of the populato. c) Estmated le: eter the estmate of the poverty le z ad ts stadard devato stdz. THE FGT INDEX The uormalzed Foster-Greer-Thorbece poverty dex FGT P(; z; α) for the populato subgroup s as follows: α P(;z; α ) = sw (z y ) sw = = where z s the poverty le ad x = max(x,0). The ormalzed dex s defed by: P(;z; α ) = P(;z; α ) /(z) α If you wsh to compute the FGT dex of poverty, follow these steps: - From the ma meu, choose "Poverty FGT dex". - Choose the dfferet vectors ad values of parameters. Amog the buttos, you wll fd the followg commads: COMPUTE: to compute the FGT dex. GRAPH: to draw the value of the dex accordg to the poverty le z. GRAPH2: to draw the value of FGT α as a fucto of a rage of parameter α

2 - To compute the ormalzed dex, choose that opto the wdow of puts. THE BOUNDED INCOME AND OVERLOAD INDICES Gap dex: The Gap dex GI(; z; z2; α) for the populato subgroup s as follows: α sw (z2 y ) I(z y z2) = GI(; z, z2; α ) = sw If the dex s relatve to the group of those wth Surplus dex: The Surplus dex SI(; z; z2; = z y z2, we have: α sw (z2 y ) I(z y z2) = GGI(; z, z2; α ) = sw I(z y z2) = α ) for the populato subgroup s as follows: α sw (y z) I(z y z2) = SI(;z,z2; α ) = sw If the dex s relatve to the group of those wth = z y z2, we have: α sw (y z) I(z y z2) = GSI(; z, z2; α ) = sw I(z y z2) = Overload dex: The Overload Idex OLI(,z, α) for group s as follows: GI(, z = 0, z2 = z, α) OLI(, z, α ) = SI(, z = z, z =, α) 2 If you wsh to compute these dces of poverty, follow these steps: 2

3 - From the ma meu, choose "Poverty Bouded come dex ". - Choose the dfferet vectors ad values of parameters. z Lower boud Compulsory z2 Upper boud Compulsory z Poverty le Compulsory for OLI α alpha Compulsory Amog the buttos, you wll fd the followg commads: COMPUTE: to compute the selected dex. GRAPH: to draw the value of the overload dex as a fucto of a rage of poverty les z. THE WATTS POVERTY INDEX The Watts poverty dex PW(; z) for the populato subgroup s defed as: ( ) sw log(y / z) PW(;z) = = sw where z s the poverty le ad x = max(x,0). = If you wsh to compute the Watts dex of poverty, follow these steps: - From the ma meu, choose "Poverty Watts dex". - Choose the dfferet vectors ad values of parameters. Amog the buttos, you wll fd the followg commads: COMPUTE: to compute the Watts dex. GRAPH: to draw the value of the dex accordg to the poverty le z. THE S-GINI POVERTY INDEX The S-G poverty dex P(;z; ρ ) for the populato subgroup s defed as: ρ ρ (V ) (V ) P(;z; ρ ) = z (z y ) ad V = sw ρ h = [ V ] h= where z s the poverty le ad x = max(x,0). 3

4 If you wsh to compute the S-G poverty dex, follow these steps: - From the ma meu, choose "Poverty S-G dex". - Choose the dfferet vectors ad values of parameters. z Poverty le Compulsory ρ Rho Compulsory Amog the buttos, you wll fd the followg commads: COMPUTE: to compute the S-G poverty dex. GRAPH: to draw the value of the dex accordg to the poverty le z. THE CLARK, HEMMING AND ULPH (CHU) POVERTY INDEX The poverty dex P(;z; ε) for the populato subgroup s defed as: /( ε) * ε sw (y ) z = f ε ad ε 0 sw P(;z, ε ) = = * sw l y z exp = f ε= sw = where z s the poverty le ad y * y = z f y otherwse z If you wsh to compute the CHU poverty dex of poverty, follow these steps: - From the ma meu, choose "Poverty CHU dex". - Choose the dfferet vectors ad values of parameters. z Poverty le Compulsory Amog the buttos, you wll fd the followg commads: COMPUTE: to compute the CHU poverty dex. GRAPH: to draw the value of the dex accordg to the poverty le z. 4

5 THE SEN INDEX The Se dex of poverty PS(;z, ρ) for the populato subgroup s defed as: * PS = H I ( I)G H = q = = = sw *I(y z) = sw = sw *I(z y ) sw G * s the G dex of equalty amog the poor, z s the poverty le ad x = max(x,0). If you wsh to compute the Se poverty dex, follow these steps: - From the ma meu, choose "Poverty Se dex". - Choose the dfferet vectors ad values of parameters. z Poverty le Compulsory Amog the buttos, you wll fd the followg commads: COMPUTE: to compute the Se poverty dex. GRAPH: to draw the value of the dex accordg to the poverty le z. THE BI-DIMENSIONAL FGT INDEX The Foster-Greer-Thorbece poverty dex, P g (; z; α ), for a dcator of well-beg g ad for a populato subgroup s as follows α sw (zg x g,) P(;z; g g α ) = = sw = 5

6 where z g th s the poverty le for good g, x g, s the g compoet of household I ad t = max(t,0). The ormalsed dex s defed by: P(;z; α ) = P(;z; α)/(z) α g g g g g Uo headcout The uo headcout, based o G dmesos or commodtes, s equal to: G sw I(zg < x g ), = g= P(;z,z 2,...) = sw = Itersecto headcout The tersecto headcout, based o G dmesos or commodtes, s equal to: G sw I(zg x g, ) = g= P(;z, z 2,...) = sw = Uo sum of gaps The uo sum of gaps, usg G dmesos or commodtes, s equal to: P(;z,z,...) sw (z x ) G g g, = g= 2 = sw = Itersecto sum of gaps The tersecto sum of gaps, usg G dmesos or commodtes, s equal to: sw (z x ) * I(z x ) G G g g, g g, = g= = ˆP(;z,z 2,...) = sw = Itersecto product of gaps The tersecto product of gaps, usg G dmesos or commodtes, s equal to: 6

7 2 P(;z,z,...; α, α,...) = sw (z x ) * I(z x ) G G αg g g, g g, = g= = 2 sw = Graphcal llustrato for two commodtes Z 2 Commodty 2 I II III Z Commodty If you wsh to compute bdmesoal poverty dces, follow these steps: - From the ma meu, choose "Poverty Bdmesoal dex". - Choose the dfferet vectors ad values of parameters. z Poverty le Compulsory z Poverty le 2 Compulsory 2 α alpha Compulsory α 2 alpha2 Compulsory Amog the buttos, you wll fd the followg commads: COMPUTE: to compute bdmesoal poverty dces. 7

8 Results of ths applcato are: - FGT dex for commodty : correspodg to areas (III) the graphcal llustrato. - FGT dex for commodty 2: correspodg to areas (IIIII) the graphcal llustrato. - FGT dex for the two commodtes (Uo approach): correspodg to areas (IIIIII) the graphcal llustrato. - FGT dex for the two commodtes (Itersecto approach): correspodg to areas (II) the graphcal llustrato. Example: Food ad o-food expedtures per day F CFA (Cameroo 996). Food poverty le evaluated at 256 F CFA ad o-food poverty le evaluated at 7 F CFA. IMPACT OF A PRICE CHANGE ON THE FGT INDEX The mpact of a good s margal prce chage (deoted IMP) o the FGT poverty dex P(,z; α) s as follows: IMP = = P(;z; α) *pc pl α CD l (;z)*pc 8

9 where z s the poverty le, s the populato subgroup for whch we wsh to assess the mpact of the prce chage, ad pc s the percetage prce chage for good l. α α z y sw x f α ad Normalsed α z sw z = = α α IMP = sw ( z y) x f α ad Not Normalsed sw = = sw K h(z y )*x E x y z *f(z) = = = f α= 0 sw = where x l s expedture o commodty l by dvdual, ad f = max( f,0 ). Note that f the FGT dex s ormalzed: IMP = CD α l (;z) * pc If you wsh to compute these statstcs, follow these steps: - From the ma meu, choose "Poverty Impact of prce chage". - Choose the dfferet vectors ad values of parameters. z Poverty le Compulsory α Alpha Compulsory pc Prce chage % Compulsory Amog the buttos, you wll fd the followg commads: COMPUTE: to compute the mpact of the prce chage. GRAPH: to draw the value of the mpact as a fucto of a rage of poverty les z. IMPACT OF A TAX REFORM ON THE FGT INDICES A tax reform cossts of a varato the prces of two commodtes ad 2, uder the costrat that t leaves uchaged total govermet reveue. The effect of ths costrat s gve by a effcecy parameter, gamma ( γ ), whch s the rato of the margal cost of publc fuds (MCPF) from a tax o 2 over the MCPF from a tax o. 9

10 The mpact of ths tax reform (deoted IMTR) o the FGT poverty dex P(; z; α) s gve by: α X α IMTR = CD (; z) γ CD 2 (; z) * pc X2 where z s the poverty le, CD α (;z) ad CD α 2 (;z) are the cosumpto domace curves for commodtes ad 2, ad pc s the percetage prce chage of commodty. Uder the govermet reveue costrat, the percetage prce chage of commodty s gve by: X γ pc. X2 To compute the mpact of the tax reform: - From the ma meu, choose "Poverty Impact of tax reform". - Choose the dfferet vectors ad values of parameters. Vectors x Commodty Compulsory x 2 Commodty 2 Compulsory z Poverty le Compulsory α alpha Compulsory γ gamma Compulsory pc s % prce chage Compulsory Amog the buttos, you wll fd the followg commads: COMPUTE : to compute the mpact of the tax reform. CRITICAL γ : to compute the gamma at whch the tax reform wll have zero mpact o poverty. The value of ths crtcal gamma equals: CD (;z) / CD 2 (;z). GRAPH z : to draw the mpact of the tax reform as a fucto of a rage of poverty les z. GRAPH : Graph γ : to draw the mpact as a fucto of a rage of MCPF ratos. δ 2, α α LUMP-SUM TARGETING The per-capta dollar mpact of a margal addto of a costat amout of come to everyoe wth a group called Lump-Sum Targetg (LST) o the FGT poverty dex P(; z; α), s as follows: αp(, z; α ) f α ad Not Normalsed α LST = P(, z; α ) f α ad Normalsed z f(,z) f α= 0 0

11 where z s the poverty le, s the populato subgroup for whch we wsh to assess the mpact of the come chage, ad f(,z) s the desty fucto of the group at level of come z. To compute that mpact: - From the ma meu, choose "Poverty Lump-sum Targetg". - Choose the dfferet vectors ad values of parameters. z Poverty le Compulsory α alpha Compulsory Amog the buttos, you wll fd the followg commads: COMPUTE: to compute the mpact of the come chage at a partcular value of z GRAPH: to draw the mpact as a fucto of a rage of poverty les z. INEQUALITY-NEUTRAL TARGETING The per-capta dollar mpact of a proportoal margal varato of come wth a group, called Iequalty Neutral Targetg, o the FGT poverty dex P(; z; α) s as follows: P(,z; α) zp(,z; α ) α f α ad FGT s ot ormalsed µ () P(,z; α) zp(,z; α ) INT = α f α ad FGT s ormalsed µ () zf (, z) f α= 0 µ () where z s the poverty le, s the populato subgroup for whch we wsh to assess the mpact of the come chage, ad f(,z) s the desty fucto of the group at level of come z. To compute that mpact: - From the ma meu, choose "Poverty Iequalty-eutral Targetg ". - Choose the dfferet vectors ad values of parameters. z Poverty le Compulsory α alpha Compulsory Amog the buttos, you wll fd the followg commads: COMPUTE: to compute the mpact of the come chage. GRAPH: to draw the mpact as a fucto of a rage of poverty les z.

12 FGT ELASTICITY Growth The overall growth elastcty (GREL) of poverty, whe growth comes exclusvely from growth wth a group (amely, wth that group, equalty eutral), s gve by: P(,z; α) zp(,z; α ) α f α P(z, α) GREL = zf (, z) f α= 0 F(z) where z s the poverty le, s the populato subgroup whch growth taes place, f(z) s the desty fucto at level of come z, ad F(z) s the headcout. G The overall G elastcty (GEL) of FGT poverty, s gve by: P(z; α ) µ α f α P(z, α) z GEL = f(z)( µ z) f α = 0 F(z) To compute that growth elastcty: - From the ma meu, choose "Poverty Growth Elastcty". - Choose the dfferet vectors ad values of parameters. z Poverty le Compulsory α Alpha Compulsory Amog the buttos, you wll fd the followg commads: COMPUTE: to compute the growth elastcty. GRAPH: to draw the mpact as a fucto of a rage of poverty les z. 2

13 INCOME-COMPONENT PROPORTIONAL GROWTH Chage per 00% of compoet C Assume that total come Y s the sum of C come compoets, wth Y = λ c yc ad where c c= s a factor that multples come compoet y c ad that ca be subject to growth. The dervatve of the ormalzed FGT dex wth respect to λ c s gve by P(;z, α) λ c λ c =,c= C where CD c s the C-domace curve of compoet c. Chage per $ of compoet = CD (;z, α) c The per-capta dollar mpact of growth the th group s as follows: th j compoet o the ormalzed FGT dex of the P(;z, α) j y µ () y j j = CD (;z, α) where CD s the ormalzed C-domace curve of the compoet j. Elastcty wth respect to compoet The th j compoet elastcty of poverty (as measured by the FGT dex) s: µ () j CD (;z, α) P(;z, α) j where CD s the ormalzed C-domace curve of the compoet j. - If you wsh to compute ths elastcty, choose "Poverty Compoet Elastcty". - If you wsh to compute the above mpacts, choose "Poverty Icome-Compoet Proportoal Growth", ad select oe of the three optos. - Choose the dfferet vectors ad values of parameters. 3

14 z Poverty le Compulsory α Alpha Compulsory Amog the buttos, you wll fd the followg commads: COMPUTE: to compute the statstcs. THE IMPACT OF DEMOGRAPHIC CHANGES Ths applcato computes the mpact of a chage (by a gve percetage) the populato proporto of a group t. That chage s accompaed by a exactly offsettg chage the populato proporto of the other groups. If the populato proporto of group t creases by 00 pc percet, such that φ ( t) ( φ(t)( pc) ), the total estmated mpact o poverty s as follows: K φ(t) P = φ(t)*p(t;z, α) * φ()*p(;z, α) *pc φ(t) s If the populato proporto of group s creases by a absolute 00 pc percet of the total, the total estmated mpact o poverty s as follows: populato, such that φ ( t) ( φ(t) pc) K φ() P = P( = t;z, α) *P(;z, α) *pc s φ(t) where P(;z; α ) s the FGT poverty dex for subgroup ad φ () s the proporto of the populato foud that subgroup. To perform ths estmato: - From the ma meu, choose: "Decomposto Impact of Demographc Chage". - After settg the cofgurato, the applcato appears. Choose the dfferet vectors ad parameter values as follows: t Chaged group Compulsory z Poverty le Compulsory α Alpha Compulsory Group umbers separated by "-" Compulsory REMARK: The group umbers separated by the dash "-" should be teger values. For example, we may have two subgroups coded by the tegers ad 2. I ths case, we would wrte the feld «Group Numbers» the values "-2" before proceedg to the decomposto. 4