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. H A = 0.75 = H B =H C ;

Adding up poverty: Headcount

Adding up poverty : Poverty Gap 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. H A = 0.75 = H B ; PG A = 0.25 = PG B.

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.

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 SPG A = 0.14; SPG B = 0.08 H A =H B, PG A =PG B but SPG A >SPG B

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

Adding up poverty: FGT-measures Derivatives

Adding up poverty 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. 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 H A > H B yet SPG A < SPG B Which policy would you choose?

Adding up poverty 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

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

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 poverty comparison results change if we make alternative assumptions?

Robustness Robustness: 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.

Robustness Robustness: 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.

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

Robustness Robustness: Formulas Poverty incidence curve: Poverty deficit curve: Poverty severity curve:

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 z max then there is more poverty in A than B for all poverty measures and all poverty lines up to z max

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 z max 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)

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