Poverty comparisons with cardinal and ordinal attributes

Transcription

1 Poverty comparisos with cardial ad ordial attributes Kristof Bosmas Departmet of Ecoomics, Maastricht Uiversity Luc Lauwers Ceter for Ecoomic Studies, Katholieke Uiversiteit Leuve Erwi Ooghe Ceter for Ecoomic Studies, Katholieke Uiversiteit Leuve

2 Problem : How idetify the poor? Give a poverty budle z = (z, z 2, ), a threshold for each attribute, how do we determie who is poor ad who is ot? (i) Uio (ii) Itersectio (iii) Itermediate x 2 x 2 x 2 z 2 z z 2 z z 2 z z x z x z x Ideally, leave the choice to the practitioer (see also Duclos, Sah & Youger, 2006 ; Alkire & Foster, 2008) 2

3 Problem 2: How give priority to the worse off poor? Cosider a multidimesioal Foster-Greer-Thorbecke idex: i i j j j ΣΠ α ( z x ) j (the sum over the poor oly) where x i j is idividual i s amout of attribute j Example with poverty budle z = (0, 0) ad weights α = α 2 = Idividual has budle (4, 6) ; idividual 2 has budle (6.5, 4) Accordig to the idex idividual is worse off tha 2 Assume a extra 0.5 of the first attribute ca be give away Worse off idividual gets it poverty decreases by 2 Better off idividual 2 gets it poverty decreases by 3 Udesirable coclusio: give priority to the better off! 3

4 Problem 3: How deal with ordial data? Ordial: e.g., health, housig, educatio, Example: Would you say your health i geeral is 0 = poor, = fair, 2 = good, 3 = very good, 4 = excellet? Problem i defiig priority Cardial: Should $30 go to someoe with $00 or to someoe with $200? meaigful questio Ordial: Should poit of health go to someoe with health 2 or to someoe with health 3? ot a meaigful questio See also Alkire & Foster (2008) ad Bossert, Chakravarty & D Ambrosio (2009) 4

5 Notatio Set of attributes C O (remember problem 3) Each idividual has a attribute budle x = (x C, x O ) i B x C the vector of cardial attributes (positive real umbers) x O the vector of ordial attributes (itegers, 0,, 2, 3, ) Fixed poverty budle z i B A distributio = (x, x 2, ) is a elemet of D Poverty rakig < ( better tha relatio) o D 5

6 Axiom : Additive represetability (AR) AR: There exists a cotiuous fuctio π : B R such that, for all ad Y i D, < Y if ad oly if Σ i= π( x i ) Y Y Σ i= π( y i ) AR is a strog axiom: < has to be (i) cotiuous, (ii) aoymous, (iii) separable, (iv) replicatio ivariat AR is strog, but quite commo i the literature (see, e.g., Foster & Shorrocks, 99 ; Tsui, 2002 ; Bourguigo & Chakravarty, 2003) 6

7 Axiom 2: Focus (F) Remember problem (how to idetify the poor) We defie the set of poor i as P = { i x i z } Give AR, the poor are those with π(x) > π(z) F: for all i D, if Y is obtaied from by a chage i the budle of a o-poor while keepig her o-poor, the ~ Y Give F, the fuctio π is costat for all x such that π(x) π(z) 7

8 Axiom 3: Mootoicity (M) M: for all ad Y i D ad for all poor i, we have that if x i > y i ad x j = y j for all j i, the Y Give AR ad M, the fuctio π is strictly decreasig i each attribute for budles x such that π(x) > π(z) 8

9 Axiom 4: Priority (P = CP & OP) Remember problem 2 (how to give priority to the worse off poor) CP: for all i D, for all δ = (δ C, δ O ) i B with δ C > 0 ad δ O = 0 for all poor i ad j with x i x j we have (, x i,, x j + δ, ) (, x i + δ,, x j, ) OP: for all i D, for all δ = (δ C, δ O ) with δ C = 0 ad δ O > 0 for all poor i ad j with x i x j ad x i k = x j k for all k for which δ O,k > 0 we have (, x i,, x j + δ, ) (, x i + δ,, x j, ) 9

10 Axiom 4: Priority (P = CP & OP) Remember problem 2 (how to give priority to the worse off poor) CP: for all i D, for all δ = (δ C, δ O ) i B with δ C > 0 ad δ O = 0 for all poor i ad j with x i x j we have (, x i,, x j + δ, ) (, x i + δ,, x j, ) OP: for all i D, for all δ = (δ C, δ O ) with δ C = 0 ad δ O > 0 for all poor i ad j with x i x j ad x i k = x j k for all k for which δ O,k > 0 we have (, x i,, x j + δ, ) (, x i + δ,, x j, ) 0

11 Result A poverty rakig < satisfies AR, F, M ad P iff there exist weights w k > 0 for each cardial attribute k i C, strictly icreasig fuctios v k : N R, with v k (0) = 0, for each ordial attribute k i O, a cotiuous fuctio f : R + R, with f (r) = f (ζ) for each r ζ = Σ k C w k z k + Σ k O v k ( z k ), that is strictly covex ad strictly decreasig o [0, ζ], such that, for all ad Y i D, we have < Y iff Σ i= f ( Σ w k x i k + k C Σ v k ( x i k ) ) k O Y Y Σ i= f ( Σ w k y i k + k C Σ v k ( y i k ) ) k O

12 Special case: All attributes cardial < Y iff Σ i= f ( Σ w k x i k ) k C Y Y Σ i= f ( Σ w k y i k ) k C Satisfies: Weak uiform majorizatio priciple: if Y ad = YQ with Q a o-permutatio bistochastic matrix, the < Y Correlatio icreasig majorizatio priciple Example with z = (0, 0): ( (5, 6), (9, 2) ) ( (9, 6), (5, 2) ) 2

13 A weaker form of priority (work i progress) Cardial priority (CP) is a demadig ormative priciple It asks us to disregard efficiecy costs caused by dimiishig returs to well-beig Example: Suppose x i x j, but idividual i has oly 0 uits of attribute, while idividual j has 00 uits Accordig to CP, if a extra uit of attribute ca be give away, the it should go to idividual j 3

14 A weaker form of priority (work i progress) Cosider a weaker versio of CP: For all i D, for all δ = (δ C, δ O ) i B with δ C > 0 ad δ O = 0 for all poor i ad j with x i x j ad x i k x j k for all k for which δ C,k > 0 we have (, x i,, x j + δ, ) (, x i + δ,, x j, ) Characterizes poverty measures lookig like Σ i= f ( Σ g k ( x i k ) + k C Σ v k ( x i k ) ) k O with v k as before ad appropriate coditios o f ad g k 4

15 Coclusio Characterizatio of a class of poverty measures featurig: Priority to the worse off poor Both cardial ad ordial attributes Flexibility i the choice of idetificatio criterio Work i progress: Ivestigate weaker forms of priority Derive practical coditios to check uaimity judgmets A applicatio of the latter usig EU-SILC data 5