Poverty measures:Properties and Robustness

Michael Lokshin

DECRG-PO

The World Bank

Properties and Robustness

Questions for the analyst: How do we measure “welfare”?

Individual measures of well-being When do we say someone is "poor"?

Poverty lines. How do we aggregate data on welfare into a

measure of “poverty”? How robust are the answers?

Three components of poverty analysis

Welfare

Indicators

Poverty

Lines

Poverty

Analysis

Adding up poverty: Headcount

q = number of people deemed poor

n = population size Advantage: easily understood Disadvantages: insensitive to distribution below the

poverty line e.g., if poor person becomes poorer, nothing happens to H.

Example: A: (1, 2, 3, 4) B: (2, 2, 2, 4) C: (1,1,1,4)

Let z = 3. HA = 0.75 = HB=HC;

qH

N

Adding up poverty: Headcount

Adding up poverty: Poverty Gap

1

1 1

1

,..., ,...,

qi

i

q q n

z yPG

n z

y y z y y

Advantages of PG: reflects depth of poverty

Disadvantages: insensitive to severity of poverty

Example: A: (1, 2, 3, 4) B: (2, 2, 2, 4)Let z = 3. HA = 0.75 = HB; PGA = 0.25 = PGB.

Adding up poverty: Poverty Gap

Adding up poverty: Poverty Gap

The minimum cost of eliminating poverty:

(Z-z)*q -- Perfect targeting.

The maximum cost of eliminating poverty: Z*q -- No targeting.

Ratio of minimum cost of eliminating poverty to the maximum cost with no targeting:

Poverty gap -- potential saving to the poverty alleviation budget from targeting.

q

i

iz PGZ

yZ

nqZ

qZ

1

)(1

*

*)(

Adding up poverty:Squared Poverty Gap Week Transfer Principal: A transfer of income from any

person below the poverty line to anyone less poor, while keeping the set of poor unchanged, must raise poverty

Advantage of SPG: sensitive to differences in

both depth and severity of poverty.

Hits the point of poverty line smoothly. Disadvantage: difficult to interpret Example: A = (1, 2, 3, 4) B = (2, 2, 2, 4)

z = 3 SPGA = 0.14; SPGB = 0.08

HA=HB, PGA=PGB but SPGA>SPGB

q

i

i

z

yz

nSPG

1

21

Adding up poverty: FGT-measures

1

0

1

2

1( 0)

0 : (Headcount)

1: (Poverty Gap/Depth)

2 : (Squared Poverty Gap/Severity)

qi

i

z yP

n z

P H

P PG

P SPG

Additivity: the aggregate poverty is equal to population- weighted sum of poverty level in the various sub-groups of society.

Range:

poorestthetoWeight

Pzy

HPy

i

i

0

0

Rawls welfare function: maximize the welfare of society's worse-off member.

Adding up poverty: FGT-measures

1

0 1 22

1

10; ; 2

i

i

i

i i i

P z y

y z z

P P P z y

y y z y z

Derivatives

Adding up povertyAdding up poverty: Recommendations

Does it matter in poverty comparisons what measure to use?

Depends on whether the relative inequalities have changed across the situations being compared.

If no changes in inequality, no change in ranking.

Recommendations: Always be wary of using only H or PG; check SPG. A policy conclusion that is only valid for H may be

quite unacceptable.

Adding up poverty: Example 1

Example: Effect of the change in price of domestically produced goods on welfare.

Price of rice in Indonesia: Many poor households are net rice producers, the

poorest households are landless laborers and net consumers of rise.

Policy A Decrease in price of rice: small loss to person at poverty line, but poorest gains;

Policy B Increase in price: poorest loses, but small gain to person at poverty line.

So HA > HB yet SPGA < SPGB

Which policy would you choose?

Adding up povertyAdding up poverty: Example 2

Poverty line = (6) Initial distribution: (1,2,3,4,5,6,7,8,9,10); HC: = 0.50 Poverty gap: (5/6,4/6,3/6,2/6,1/6,0) = 0.25 SPG: (25/36,…,0) = 0.16 Poverty Alleviation Budget $6 Case 1: (6,3,3,4,5,6,7,8,9,10); HC = 0.40 PG: (0,3/6,3/6,2/6,1/6,0..0) = 0.15 SPG: (0,9/36,9/36,4/36,1/36,0..0) = 0.07 Case 2: (1,2,6,6,6,6,7,8,9,10); HC = 0.20 PG: (5/6,4/6,0,…,0) = 0.15 SPG: (25/36,16/36,0,…,0) = 0.11

Social Welfare functionSocial Welfare function

Utilitarian Social Welfare Function. Social states are ranked according to linear sum of individual utilities:

We can assign weight to each individual’s utility:

Inclusive and Exclusive Social Welfare Functions

1

( )n

ii

W u x

1

( )n

i ii

W a u x

Robustness of poverty comparisonsRobustness of poverty comparisons

Why should we worry? Errors in living standard data Uncertainty and arbitrariness of the poverty line Uncertainty about how precise is the poverty measure Unknown differences in need for the households with

similar consumption level. Different poverty lines that are completely reasonable and

defensible.

How robust are our poverty comparisons? Would the poverty comparison results change if we

make alternative assumptions?

RobustnessRobustness: Poverty incidence curve

1. The poverty incidence curve Each point represents a headcont for each possible

poverty line Each point gives the % of the population deemed

poor if the point on the horizontal axis is the poverty line.

RobustnessRobustness: Poverty depth curve The poverty depth curve = area under poverty incidence curve Each point on this curve gives aggregate poverty gap – the poverty

gap index times the poverty line z.

RobustnessRobustness: Poverty severity curve

The poverty severity curve = area under poverty depth curve Each point gives the squared poverty gap.

RobustnessRobustness: Formulas

Poverty incidence curve:

Poverty deficit curve:

Poverty severity curve:

z

dxxfyF0

)()(

zz

dxxFdxxfxzzD00

)()()()(

zz

dxxDdxxFxzzS00

)()()()(

Robustness:Robustness: First Order Dominance Test

If the poverty incidence curve for distribution A is above that for B for all poverty lines up to zmax then there is more poverty in A than B for all poverty measures and all poverty lines up to zmax

Robustness:Robustness: First Order Dominance Test

What if the poverty incidence curves intersect? -- Ambiguous poverty ranking.

You can either:i) restrict range of poverty lines ii) restrict class of poverty measures

Robustness:Robustness: Second Order Dominance Test

If the poverty deficit curve for A is above that for B up to zmax then there is more poverty in A for all poverty measures which are strictly decreasing and weakly convex in consumptions of the poor (e.g. PG and SPG; not H).

e.g., Higher rice prices in Indonesia: very poor lose, those near the poverty line gain.

What if poverty deficit curves intersect?

Robustness:Robustness: Third Order Dominance Test

If the poverty severity curve for distribution A is above that for distribution B then there is more poverty in A, if one restricts attention to distribution sensitive (strictly convex) measures such as SPG.

Formal test for the First Order Dominance –

Kolmogorov-Smirnov test

Robustness:Robustness: Examples

Initial state (1,2,3) (2,2,3) (1,2,4) – unambiguously lower poverty (2,2,2) poverty incidence curves cross. compare z=1.9 and z=2.1 poverty deficit curves do not cross Thus poverty has fallen for all distribution sensitive measures.

Example 2:Initial State A: (1,2,3) Final State B: (1.5,1.5,2)

C. F(z) D(z) S(z) A B A B A B 1 1/3 0 1/3 0 1/3 0 1.5 1/3 2/3 2/3 2/3 1 2/3 2 2/3 1 4/3 5/3 7/3 7/3 3 1 1 7/3 8/3 14/3 15/3

Robustness:Robustness: Recommendations

First construct the poverty incidence curves up to highest admissible poverty line for each distribution.

If they do not intersect, then your comparison is

unambiguous.

If they cross each other then do poverty deficit curves and restrict range of measures accordingly.

If they intersect, then do poverty severity curves. If they intersect then claims about which has more

poverty are contentious

Robustness:Robustness: Egypt, poverty changes between 1996 and 2000

The percentage of the poor for All Egypt.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Poverty Line Z as %of mean

P0

1995/96

1999/2000

The Poverty Gap Index for all Egypt.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Poverty Line Z as % of mean

P1

1995/96

1999/2000

Severity of Poverty Index for All Egypt.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Poverty Line Z as % of mean

P2

1995/96

1999/2000