Robust comparisons of poverty and inequality

Robustness analysis PEP and UNDP June 2010 – 1 / 29

Robust comparisons of poverty andinequality

Abdelkrim Araar, Sami Bibi and Jean-Yves Duclos

Workshop on poverty and social impact analysisDakar, Senegal, 08-12 June 2010

Outline

Outline

Objectives

Introduction

Absolute poverty

Relative poverty andinequality

Conclusion


Introduction

Absolute poverty

Relative poverty and inequality

Conclusion

Objectives


� Understand why different choices of indices can affect inequality andpoverty results significantly;

� See how we can construct and think about “classes” of distributiveindices;

� Define dominance curves;� Distinguish first- and second-order dominance;� Use both primal and dual dominance tests, and understand their

respective usefulness;� Understand the distinction between poverty and inequalitydominance.

Introduction

Outline

Objectives

Introduction

Key issue

Ordinal sensitivity

Absolute poverty


Conclusion


Key issue


� Quoting from Kolm (1976):

“I can take any country and prove that in some period (whateverit is) inequality has increased or decreased in it, or any twocountries and prove that inequality is higher in the one or intheother, by choosing different inequality measures, all of whichwould probably seem good and valuable at first sight.”

� The same issue matters for comparing poverty, by choosing differentpoverty lines and/or poverty indices.

Ordinal sensitivity


Table 1: Sensitivity of poverty comparisons to the choice ofz andP (z)

Rank DistributionA DistributionB percentilepii = 1 350 370 0.2i = 2 400 380 0.4i = 3 450 450 0.6i = 4 500 500 0.8i = 5 550 550 1.0

(α; z) PA(α; z) PB(α; z) ∆P (α; z)(0; 360) 0.20 0.00 -0.2(0; 390) 0.20 0.40 0.2(1; 390) 0.02 0.015 -0.005

Ordinal sensitivity


� Ideally then, distributional judgments should not be too dependent uponsome particular measures of poverty or inequality.

� It is often possible to make poverty and inequality comparisons “ethicallyrobust” by drawing on methods from the theory of stochastic dominance.

� These use classes of “acceptable” poverty (or inequality) indices of someethical order,s, and check whether it is possible to rank distributionsaccording to all members of those classes.

� As the ethical order increases, the criteria put increasingly strongconstraints on how poverty indices should rank distributions of livingstandards.

Absolute poverty

Outline

Objectives

Introduction

Absolute poverty

Assumption andnotation

A general povertyindex

First-order absolutepoverty dominance

Second-orderabsolute povertydominance

Primals-orderabsolute povertydominance

Comparing first- andsecond-orderdominance

Rawlsian judgment,Central - WesternUganda 1999(α = 30)


Conclusion


Assumption and notation


� Consider a vectory of individuals standard (income, for short) ranked inincreasing order such thaty1 < y2 < ... < yn (adjusted for differences inneeds and prices).

� The inflation factorνi ∀i and household size serve to weight observationsappropriately. For simplicity, we assume that the relativeindividualweightφi =

νi∗siN

= 1

n∀ i.

� Let pi be the proportion of individuals in the population that enjoy a levelof income that is less than or equal toyi:

pi =in

� DefineQ(pi) as the income of that individual whose rank — or percentile— in the distribution ispi. A proportionpi of the population has incomelower than or equal to his; a proportion1− pi is richer than him.

Assumption and notation


� Let the absolute poverty line be denoted asz.� For dual dominance analysis, we will need quantiles censored at the

poverty linez, Q(pi; z), defined as:

Q(pi; z) = min(Q(pi), z)

and generalized Lorenz curves:

GL(pj; z) =1

n

∑j

i=1Q(pi; z).

� Poverty gaps are also often used for poverty measurement, defined at apercentilepi as:

g(pi; z) = z −Q(pi; z)

A general poverty index


� Consider the following general formulation of a poverty index:

P (z) =1

n

n∑

i=1

π(yi; z) (1)

� A simple way to draw dominance curves of orders is to calculate, fordifferentz, the poverty level yielded by the indexP (α; z) for α = s− 1,i.e.,

P (α; z) =1

n

n∑

i=1

(

g(pi; z)

z

)α

. (2)

First-order absolute poverty dominance


� The first class of poverty indices is denoted byΠ1(z+), and includes all

indices that decrease whenever the income level of any poor personincreases (themonotonicity principle, also named the Pareto criterion).

� Indices that are part of that class should also obey thepopulation,anonymity, andfocus principles.

� A necessary and sufficient condition for a distributionB to first-orderpoverty dominate a distributionA for all poverty lines untilz+ is that(Atkinson (1987), Foster and Shorrocks (1988), etc.):

PB(α = 0; z) ≤ PA(α = 0; z) ∀z ∈ [0, z+]. (3)

or, alternatively (using a partial Pen parade as Duclos and Araar (2006)),

QB(pi) ≥ QA(pi) ∀pi ∈ [0, F (z+)]. (4)



Table 1: Checking for first-order dominance (α = 0)

Rank Dist. A Dist. B pi

i = 1 350 370 0.2i = 2 400 380 0.4i = 3 450 450 0.6i = 4 500 500 0.8i = 5 550 550 1.0

z PA(α; z) PB(α; z) ∆P (α; z)z ≤ 350 0.0 0.0 0.0z = 365 0.2 0.0 -0.2

365 ≤ z ≤ 370 0.2 0.0 -0.2371 ≤ z ≤ 380 0.2 0.2 0.0381 ≤ z ≤ 400 0.2 0.4 0.2



0.0

1.0

2.0

3.0

4P

(alp

ha=

0, z

)

.2 .28 .36 .44 .52 .6poverty line

central western

Central and Western Uganda 1999

Second-order absolute poverty dominance


� We can add to the earlier first-order principles thePigou-Dalton transferprinciple. This leads to second-order indices, whose class is denotedbyΠ2(z

+).� Π2(z

+) excludes those indices inΠ1(z+) which can increase with a

Pigou-Dalton inequality-reducing transfer principe. One such index isthe headcount index,P (α = 0; z).

� A necessary and sufficient condition for a distributionB to second-orderpoverty dominate a distributionA is that

PB(α = 1; z) ≤ PA(α = 1; z) ∀z ∈ [0, z+]. (3)

or, alternatively (using the generalized Lorenz curve)

GLB(pi) ≥ GLA(pi) ∀pi ∈ [0, F (z+)]. (4)



Table 1: Checking for second-order poverty dominance (α = 1)

Rank GLA(pi) GLB(pi) pi

i = 1 70 74 0.2i = 2 150 150 0.4i = 3 240 240 0.6i = 4 340 340 0.8i = 5 450 450 1.0

z PA(α; z) PB(α; z) ∆P (α; z)z ≤ 350 0.0 0.0 0.0z = 360 0.006 0.0 -0.006z = 370 0.011 0.0 -0.011z = 380 0.016 0.005 -0.011z = 390 0.021 0.015 -0.006z = 400 0.025 0.025 0.0



0.0

02.0

04.0

06P

(alp

ha=

1, z

)

.2 .28 .36 .44 .52 .6poverty line

central western


Primal s-order absolute poverty dominance


For higher-order classes of poverty indices,Πs(z+), we know from

Duclos and Makdissi (2003) that:

PB(z)− PA(z) ≤ 0 for all P (z) ∈ Πs(z+)

if and only if

PB(α = s− 1; z) < PA(α = s− 1; z) for all z ∈ [0, z+]. (3)

� When (3) is found overz ∈ [0,+∞[, there is also social welfaredominance at orders. Social welfare dominance is indeed povertydominance whenz+ extends up to the highest income levels.

� Πs(z+) amounts to Rawlsian social welfare function whens → ∞ —

social welfare is then sensitive only to the well-being of the poorest.

Comparing first- and second-orderdominance


� A first-order improvement in welfare or in poverty demands that the Penparade condition (4) be obeyed. This also requires all incomes to belarger.

� For a second-order improvement, richer individuals can lose providingthat the gains of poorer individuals are high enough to offset the losses ofthe richer.

� The Rawlsian-order conditions, obtained whens tends to infinity, onlyrequire that

min yB > min yA,

no matter what happens elsewhere in the distributions.

Rawlsian judgment,Central - Western Uganda 1999 (α = 30)


0.0

0001

.000

02.0

0003

.000

04.0

0005

P(a

lpha

=15

, z)

.2 .6 1 1.4 1.8 2.2poverty line

central western



Outline

Objectives

Introduction

Absolute poverty


Notation

Properties of arelative poverty index

Dual first-orderrelative dominance

Dual first-orderrelative dominance,Central - WesternUganda, 1999

Dual second-orderrelative dominance

Dual second-orderrelative dominance,Central - WesternUganda, 1999

Primals-orderrelative dominance

Primal first-orderrelative dominance,Central - WesternUganda, 1999

Primal second-orderrelative dominance,


Notation


� Let normalized incomesyi = yi/µ be incomes as proportion of the meanincome and definingQ(pi) accordingly.

� Setting the poverty line at a proportionλ of (µ) yields

Py(α; z) =1

n

n∑

i=1

(

λµy −Q(pi)

λµy

)α

+

=1

n

n∑

i=1

(

λ−Q(pi)

λ

)α

+

= Py(α; λ)

� Defining, if need be,Q(pi; λ) andg(pi; λ) accordingly.

Properties of a relative poverty index


� Consider the following general formulation of a relative povertyevaluation function:

P (λ) =1

n

n∑

i=1

π(pi; λ) (4)

� Any class of relative poverty indices of orders, denoted byΠs(λ+),

should fulfill theprinciple of population, theanonymity principle, and thescale invariance property (but not thefocus axiom, in the sense thatchanges in the incomes of the non-poor may change the povertyline andthus the poverty level).

� Scale invariance (or homogeneity): relative poverty is invariant to anyuniform proportional change in individuals’ income. Hence, for anyscalara > 0 and any distributiony, Pay (λ(aµ)) = Py(λµ).

Dual first-order relative dominance


� The first class of relative poverty indices is denoted byΠ1(λ+), and

includes all of the indices that decrease when (as a fractionof meanincome) the income level of any poor person increases.

� A necessary and sufficient condition for a distributionyB to first-orderrelatively dominate a distributionyA is that

QB(pi) ≥ Q

A(pi) ∀pi ∈ [0, F (λ+)]. (5)

� Condition (5) cannot apply across all of the values ofλ ∈ [0, +∞[. Wecannot have first-order relative inequality dominance.

Dual first-order relative dominance,Central - Western Uganda, 1999

Outline

Objectives

Introduction

Absolute poverty


Notation










24

6in

com

e as

% o

f the

mea

n

0 20 40 60 80 100percentile

Western Central

Dual second-order relative dominance


� Adding to the first-order conditions thePigou-Dalton transfer principleleads to second-order indices,Π2(λ

+).� Π2(λ

+) excludes the indices ofΠ1(λ+) that violatethe Pigou-Dalton

principe, such asPy(α = 0; λ).� A sufficient condition for a distributionyB to second-order relatively

dominate a distributionyA

LB(pi) ≥ LA(pi) ∀pi ∈ [0, F (λ+)]. (6)

� If condition (6) holds forλ+ = ∞, then we also have second-orderrelative inequality dominance. This requires that the Lorenz curves do notintersect∀ pi ∈ [0, 1].

Dual second-order relative dominance,Central - Western Uganda, 1999


.2.4

.6.8

1cu

mul

ativ

e in

com

e sh

are

0 20 40 60 80 100percentile

Western Central

Primal s-order relative dominance


Duclos and Makdissi (2003) show that:

PBy (λ)− PA

y (λ) ≤ 0 for all P (λ) ∈ Πs(λ+)

if and only if

PBy (α = s− 1; λ) < PA

y (α = s− 1; λ) for all λ ∈ [0, λ+]. (7)

When condition (7) holds for allλ ∈ [0,+∞[, there iss-order relativeinequality dominance.

Primal first-order relative dominance,Central - Western Uganda, 1999

Outline

Objectives

Introduction

Absolute poverty


Notation










0.2

.4.6

.8P

(alp

ha=

0, z

)

.1 .4 .7 1 1.3 1.6poverty line

central western


Primal second-order relative dominance,Central - Western Uganda, 1999

Outline

Objectives

Introduction

Absolute poverty


Notation










0.2

.4.6

.8P

(alp

ha=

1, z

)

.1 .98 1.86 2.74 3.62 4.5poverty line

central western


Conclusion

Outline

Objectives

Introduction

Absolute poverty


Conclusion

Summary

STATA andDASPcommands

Exercises with Stataand DASP

References


Summary


� Distributional judgments can sometimes depend on the choice of someparticular measures of poverty or inequality.

� It is possible to check for the robustness of poverty and inequalitycomparisons by performing tests of poverty and inequality dominance.

� These tests use classes of indices that obey some properties, which areincreasingly demanding as the order of dominance increases.

� As the order of dominance tests increases, the classes of poverty andinequality indices put increasing importance on the lowestincomes.

Summary


� First-order improvements in welfare or in poverty demands incomes beuniformly larger across the distribution.

� Second-order improvements allow for losses of richer individuals to beoffset by gains for the less rich.

� Relative poverty and relative inequality comparisons use incomes asproportion of the mean income.

Summary


� Primal dominance tests use “income” curves, that is curves that show thedistribution of people according to different income thresholds. Awell-known example is the cumulative distribution function.

� Dual dominance tests use “people” curves, that is curves that show thedistribution of incomes according to different people thresholds. Awell-known dual curve is the Lorenz curve.

STATA and DASP commands


� Poverty dominance and critical poverty lines (dompov)� Inequality dominance and critical percentiles (domineq)� FGT curves (cfgt)� FGT curves with confidence intervals (cfgts)� Differences between FGT curves with C.I. (cfgts2d)� Lorenz and concentration curves (clorenz)� Lorenz/concentration curves with confidence intervals (clorenzs)� Differences between Lorenz curves with C.I. (clorenzs2d)

http://132.203.59.36/DASP/help/DASP_MANUAL_V2.1.htm#_Toc247593980








Exercises with Stata and DASP


� Part I: Exercises 5.1, 5.2, 5.3� Part I: Exercises 9.1, 9.2, 9.3, 9.4� Part I: Exercises 10.1, 10.2, 10.3, 10.4, 10.5, 10.6, 10.7� Part I: Exercises 11.1, 11.2, 11.3, 11.4, 11.5, 11.6� Part II: Exercises 6.1, 6.2

http://dasp.ecn.ulaval.ca/exercises/exercises.htm#_Toc255461631





References


ATKINSON, A. (1987): “On the Measurement of Poverty,”Econometrica,55, 749 –764.

DUCLOS, J.-Y. AND A. A RAAR (2006):Poverty and Equity Measurement,Policy, and Estimation with DAD, Berlin and Ottawa: Springer and IDRC.

DUCLOS, J.-Y. AND P. MAKDISSI (2003): “Restricted and UnrestrictedDominance for Welfare, Inequality and Poverty Orderings,”Journal ofPublic Economic Theory, 6, 145–164.

FOSTER, J. AND A. SHORROCKS(1988): “Poverty Orderings,”Econometrica, 56, 173–177.

KOLM , S.-C. (1976): “Unequal Inequalities, II,”Journal of EconomicTheory, 13, 82–111.

Robust comparisons of poverty and inequality

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