1. Poverty measures: Properties and Robustness
Zurab Sajaia
DECPI
The World Bank

2. A B D C E F

3. Properties and Robustness
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 measures of “poverty”?
How robust are these measures?

4. Three components of poverty analysis
Welfare
Indicators
Poverty
Lines
Poverty
Analysis

5. Adding up poverty: Headcount
where q - number of people deemed poor
N - population size
Advantages: 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 = 3.0 HA = 0.75 = HB=HC;
Table 2.1(P0)

6. Adding up poverty: Sharp poverty Low Elasticity of poverty head count with respect to the changes in poverty line
Income
Poverty line 2
Poverty line 1
0 Ranked Households n q1 q2

7. Adding up poverty: Shallow poverty High Elasticity of poverty head count with respect to the changes in poverty line
Income
Poverty line 2
Poverty line 1
0 Ranked Households n q1 q2

8. Changes in poverty when the average consumption increases
Income
Poverty line
0 Ranked Households n q2 q1

9. Changes in poverty when the consumption distribution changes
Income
Poverty line q
0 Ranked Households n

10. Adding up poverty: Headcount
Income
Poverty line
0 Ranked Households n q

11. Adding up poverty: Poverty Gap Ratio
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 = 3.0 HA = 0.75 = HB; PGA = 0.25 = PGB
Table 2.1(P1)

12. Adding up poverty: Poverty Gap
Income
Poverty line
0 Ranked Households n q

13. Adding up poverty: Poverty Gap
The minimum cost of eliminating poverty:
(Z - z)*q -- Perfect targeting
The maximum cost of eliminating poverty:
(Z*n) -- No targeting
Ratio of the minimum cost of eliminating poverty to the maximum cost with no targeting:
Poverty gap - potential saving to the poverty alleviation budget from targeting

14. 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
Advantages of SPG:
sensitive to differences in both depth and severity of poverty
hits the point of poverty line smoothly
Disadvantages: 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

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

16. Adding up poverty: FGT-measures1 P1 P2
0 Consumption or income z
Derivatives

17. Adding up poverty: FGT-measures
Range of FGT Measures:
Rawls’s welfare function: maximize the welfare of society's worse-off member
Social and economic inequalities are to be arranged ... to the greatest benefit of the least advantaged... (Theory of Justice, pages ).

18. Social 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

19. Adding 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. 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.

20. Adding up poverty: Example 1 Effect of a change in price of domestically produced goods on welfare
Price of rice in Indonesia:
Many poor households are net rice producers
Poorest households are landless laborers and net consumers of rice
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?

21. Adding 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

22. Robustness 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 results of poverty comparisons change if we make alternative assumptions?

23. Robustness: Poverty incidence curve
Each point represents a headcount 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.

24. Robustness: Poverty depth curve
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.

25. Robustness: Poverty severity curve
Poverty severity curve = area under poverty depth curve
Each point gives the squared poverty gap.

26. 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

27. Robustness: First Order Dominance Test
What if the poverty incidence curves intersect?
Ambiguous poverty ranking
What can you do?
restrict range of poverty lines
restrict class of poverty measures

28. 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).
Higher rice prices in Indonesia: very poor lose, those near the poverty line gain
What if poverty deficit curves intersect?

29. 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
Poverty deficit

30. 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