Session 11

ReviewPoverty - IntroductionSpaceIdentificationAggregation

TodayPoverty measuresAxioms

Poverty – Introduction

Recall3 aspects of distributionSize, spread, poverty

NoteOnly poverty has official measures Q/ Why? Why particular concern with poverty?

Sen (1976) Two steps1. Identification2. Aggregation0. Space

0. SpaceQ/ Which variable? Poverty of what?Here – income, consumption, or a single dimensional

achievementLater – Sen contends we should examine inequality in a different

space

Cumulative Distribution Function

Income s

Cum

ula

tive p

opu

lati

on

F(s)

H

Income s

Cum

ula

tive p

opu

lati

on

1

.5

Exx = (2, 8, 4, 1)

Fx(s)

2 4 6 8

Q/ SpecificsWhich income?Among whom?Family size?Over what period of time?What about durable goods?In kind income?Rich uncles?Gvt. transfersBribes and black market income?Price differences?Inflation?Taxes? Etc. See Citro and Michael

1. IdentificationQ/ Who is poor?Historical answers

Booth in LondonRowntree in YorkOrshansky in USCitro-Michael in US

A/ Set poverty line zTypes of Poverty lines See Foster 1998

Absolute za

Relative zr

Subjective zs

Hybrid zh

Examples

Citro and Michael (National Academy)Proposed new method for USCorrected biggest problems

UpdatingSen “Poor Relatively Speaking”

Impact on policy? NothingWhy not?

2. AggregationQ/ How much poverty is there?Historical answer – counting

Sen (1976)Find P(x;z) poverty measure

Income s

Cum

ula

tive p

opu

lati

on

1

.5

μ=3.75

Exx = (2, 8, 4, 1)

Fx(s)

2 4 6 8z

ExamplesNumber of poor Q(x;z)Headcount ratio H(x;z)Aggregate poverty gap A(x;z)Income gap ratio I(x;z)Per capita poverty gap P1(x;z)

Q/ What about inequality among poor?Sen measure S(x;z) uses Gini among poor

FGT measure P2(x;z) uses sq Coeff of var among poor

FGT class Pα(x;z)

Session 11

ReviewPoverty - IntroductionSpaceIdentificationAggregation

TodayReflectionsPoverty measuresAxioms

“Poverty often deprives a man of all spirit and virtue; it is hard for an empty bag to stand upright”

- Benjamin Franklin“Loneliness and the feeling of being unwanted is

the most terrible poverty.” - Mother Teresa.

“Poverty is the worst form of violence.”- Mahatma Gandhi

“The mother of revolution and crime is poverty”- Aristotle

“It is a tragic mix-up when the United States spends $500,000 for every enemy soldier killed, and only $53 annually on the victims of poverty.”

- Martin Luther King

Reflections

“Every gun that is made, every warship launched, every rocket fired signifies, in the final sense, a theft from those who hunger and are not fed, those who are cold and are not clothed.”

- Dwight D. Eisenhower“Poverty is lack of freedom, enslaved by crushing

daily burden, by depression and fear of what the future will bring."

- A person from Georgia"If you want to do something and have no power

to do it, it is talauchi (poverty).” - A person from Nigeria

"Lack of work worries me. My children were hungry and I told them the rice is cooking, until they fell asleep from hunger.”

- An older man from Bedsa, Egypt."When one is poor, she has no say in public, she

feels inferior. She has no food, so there is famine in her house; no clothing, and no progress in her family."

- A woman from Uganda"For a poor person everything is terrible - illness,

humiliation, shame. We are cripples; we are afraid of everything; we depend on everyone. No one needs us. We are like garbage that everyone wants to get rid of.”

- A blind woman from Tiraspol, Moldova

Q/ What does poverty mean to you?

Q/ Is there one aspect of a person’s life that indicates poverty, or is it a combination and cumulation of deprivations?

A prevailing notion“….poverty must be seen as the deprivation of

basic capabilities rather than merely lowness of incomes, which is the standard criterion of identification of poverty.” - A. Sen

Note Today we investigate the standard criterion.

Poverty Measures

Assume Identification problem solvedPoverty line selectedAnyone with income below poverty line is poor

Q/ How to aggregate data into a single indicator of poverty?

Note This was a remarkable questionVery little discussion of issue before SenBroad acceptance of headcount measuresOrigins in social choice theory – aggregation

exercises

Q/ Why a single indicator?Will also discuss “partial indices”

Capture one aspect of poverty at a time

Notation

y = (y1,…yn) income distributionz poverty liney* censored income distribution

nonpoor incomes replaced by z

z – yi* shortfall or gap

gi = (z-yi*)/z normalized shortfall or gap

g = (g1,…gn) normalized gap distribution

mi = yi*/z normalized income

m = (m1,…mn) normalized income distribution

μ(.) , | . | mean , sum P(y;z) poverty measure

Notationy = (7,3,4,8) income distributionz = 5 poverty liney* = (5,3,4,5) censored income distribution

nonpoor incomes replaced by z

z – y2* = 2 shortfall or gapg2 = (z-y2*)/z = 2/5 normalized shortfall or gapg = (0, 2/5, 1/5, 0) normalized gap distributionm2 = y2*/z = 3/5 normalized censored incomem = (1, 3/5, 4/5, 1) normalized censored

distributionμ(y) = 11/2 |y| = 22 mean , sum P(y;z) poverty measure

Q/ What functional form for P?

Headcount

Def g0 = (g10,…gn

0) indicator distributiongi

0 = 1 if poor(yi < z)

gi0 = 0 if not (yi > z)

Q(y;z) = |g0| Headcount number of poorProperties

Symmetry, Scale invariance, FocusNot replication invariant

Less useful for comparisons over time/space

Note Useful partial indexSays a lot about absolute number, not about incidence,

depth, severity, etc.

Headcount Ratio

H(y;z) = μ(g0) = Q(y;z)/n Headcount ratio

Interpretation Incidence or percentage of the population that is poor

PropertiesSymmetry, Scale invariance, Rep. invariance,

FocusViolates Monotonicity

Graph?

ExIncomes = (7,3,4,8) poverty line z = 5Who’s poor? g0 = (0,1,1,0)Headcount H = (g0) = 2/4Example: (7,3,3,8) No change in H!Example: (7,3,3,8) No change in H!

Violates monotonicityViolates monotonicity

Note: Partial indexNote: Partial indexProvides information on one aspect of Provides information on one aspect of

povertypovertyfrequencyfrequency

Ignores other aspectsIgnores other aspectsdepth, distributiondepth, distribution

Aggregate Gap

A(y;z) = nz - |y*| Aggregate gaptotal income necessary to raise all poor incomes to z

PropertiesSymmetry, Monotonicity, FocusViolates Scale invariance, Replication invariancePartial index

Income Gap Ratio

I(y;z) = |g|/Q = A/(Qz) Income gap ratio

average normalized gap of the poor, or

I(y;z) = (z –μp)/zwhere μp is the mean poor income

Note Here I is not an inequality index!

PropertiesSymmetry, Scale invariance, Rep. invariance,

FocusViolates MonotonicityPartial index: average depth of poverty among

poor

Example Incomes = (7,3,4,8) poverty line z = 5

Normalized gaps = g = (0, 2/5, 1/5, 0)Income gap = I(y;z) = |g|/Q = (3/5)/2 =3/10Example: (7,Example: (7,44,4,8) I = 2/10 ,4,8) I = 2/10 (sensitive to some (sensitive to some

increments)increments) Example: (7,3,Example: (7,3,66,,44) I = 2/5 ) I = 2/5 (not to others)(not to others)

Note: Partial indexNote: Partial indexProvides information on one aspect of povertyProvides information on one aspect of poverty

depth of poverty among the poordepth of poverty among the poor

Ignores other aspectsIgnores other aspectsfrequency, distributionfrequency, distribution

Poverty Gap

P1(y;z) = μ(g) (Per capita) poverty gapaverage normalized gap across the entire population, or

P1(y;z) = (z –μ(y*))/z = HI = |g|/n

PropertiesSymmetry, Scale invariance, Rep. invariance,

Focus,Monotonicity

Poverty Gap

Example Incomes = (7,3,4,8) poverty line z = 5

Normalized gaps = g = (0, 2/5, 1/5, 0)Poverty gap = P1(y;z) = μ(g) = (3/5)/4 = 3/20

Example: (7,Example: (7,44,4,8) ,4,8) P1 = 4/20 = 4/20 (sensitive to (sensitive to

increments)increments) Example: (7,3,Example: (7,3,66,8) ,8) P1 = 2/20 = 2/20 (also to others)(also to others)

Note: Useful poverty indexNote: Useful poverty indexProvides information onProvides information on

depth and frequency of poverty among the poordepth and frequency of poverty among the poorignores distribution (violates a ignores distribution (violates a transfer principletransfer principle))

Before: (7,3,3,8) Before: (7,3,3,8) P1 = 4/20 = 4/20

After: (7,2,4,8) After: (7,2,4,8) P1 = 4/20 = 4/20 (insensitive to transfers (insensitive to transfers among among

poor)poor)

FGT (Foster Greer Thorbecke, 1984)

gi2 squared normalized gapgi

2 = (gi)2 if poor

gi2 = 0 if not

g2 = (g12,…gn

2) squared gap distribution

P2(y;z) = μ(g2) FGT indexaverage squared normalized gap across the entire population, or

P2(y;z) = H(I2 + (1-I2)Cp2) = |g2|/n

where Cp2 is squared coeff of var

among poor

Ex Incomes = (7,3,3,8) poverty line z = 5

Squared Normalized gaps g2 = (0, 4/25, 4/25, 0)FGT = P2 = (g2) = 8/100 Example: (7,2,4,8) Example: (7,2,4,8) Squared Normalized gaps = gSquared Normalized gaps = g22 = (0, 9/25, 1/25, = (0, 9/25, 1/25, 0)0)

PP11 = 10/100 (sensitive to inequality) = 10/100 (sensitive to inequality)

Note: Useful poverty indexNote: Useful poverty indexProvides information onProvides information on

distribution, depth and frequency of poverty among the distribution, depth and frequency of poverty among the poor; emphasizes situation of poorest of poor.poor; emphasizes situation of poorest of poor.

FGT family

giα normalized gap raised to α > 0= (gi)α if poor

giα = 0 if not

gα = (g1α,…gn

α) distribution

Pα(y;z) = μ(gα) FGT familyaverage α power of normalized gap across entire

populationNote

P0 is the headcount ratio, P1 is the poverty gap

AxiomsFocus: If x is obtained from y by an increment

among the nonpoor, then P(x;z) = P(y;z)Symmetry: If x is obtained from y by a permutation,

then P(x;z) = P(y;z)Replication Invariance: If x is obtained from y by a

replication, then P(x;z) = P(y;z)Scale Invariance: If (x;z') is obtained from (y;z) by a

scalar multiple, then P(x;z') = P(y;z)Monotonicity: If x is obtained from y by a simple

increment among the poor, then P(x;z) < P(y;z)Transfer: If x is obtained from y by a progressive

transfer among the poor, then P(x;z) < P(y;z)Note: Pα(y;z) satisfies: Focus, Symmetry, Replication

invariance, and Scale invariance for all α > 0; Monotonicity for α > 0; and Transfer for α > 1.

Subroup Consistency: Let x, x’, y, and y’ be distributions satisfying nx = nx’ and ny = ny’. If P(x;z) > P(x';z) and P(y;z) = P(y';z) then P(x,y;z) > P(x',y';z).

Decomposability: For any distributions x and y, we have P(x,y;z) = (nx/n) P(x;z) + (ny/n) P(y;z).

Q/ Does FGT satisfy?