Poverty measurement 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 Poverty Indicators Lines Poverty Analysis
Adding up poverty: Headcount q H N q = no. 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; Adding up poverty: Headcount Adding up poverty: Poverty Gap 1 q z yi PG n i 1 z y1 ,..., yq z y q 1 ,..., yn 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: q ( Z z ) * q 1 ( Z yi ) PG Z *q n i 1 Z 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 1 z yi SPG n i 1 z q 2 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 Adding up poverty: FGT-measures q 1 z yi P ( 0) n i 1 z 0 : P0 H (Headcount) 1: 2 : P1 PG (Poverty Gap/Depth) P2 SPG (Squared Poverty Gap/Severity) Additivity: the aggregate poverty is equal to population- weighted sum of poverty level in the various sub-groups of society. Range: yi 0 P H yi z P 0
Weight to the poorest Rawls welfare function: maximize the welfare of society's worse-off member. Adding up poverty: FGT-measures Derivatives 1 P z yi yi z P0 P1 1 0; ; yi yi z 1 z
P2 z yi 2 2 yi z Adding up poverty: 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 poverty: poverty Example 2 Poverty line = (6) Initial distribution: (1,2,3,4,5,6,7,8,9,10); HC: Poverty gap: (5/6,4/6,3/6,2/6,1/6,0) SPG: (25/36,,0) Poverty Alleviation Budget $6 Case 1: (6,3,3,4,5,6,7,8,9,10); HC PG: (0,3/6,3/6,2/6,1/6,0..0) SPG: (0,9/36,9/36,4/36,1/36,0..0) Case 2: (1,2,6,6,6,6,7,8,9,10); HC PG: (5/6,4/6,0,,0) SPG: (25/36,16/36,0,,0) = 0.50 = 0.25 = 0.16
= 0.40 = 0.15 = 0.07 = 0.20 = 0.15 = 0.11 Social Welfare function Utilitarian Social Welfare Function. Social states are ranked according to linear sum of individual utilities: n W ui ( x ) i 1 We can assign weight to each individuals utility: n W ai ui ( x ) i 1 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: z F ( y ) f ( x )dx 0 Poverty deficit curve:
z z D( z ) ( z x ) f ( x )dx F ( x )dx 0 0 Poverty severity curve: z z S ( z ) ( z x ) F ( x )dx D( x )dx 0 0 Robustness: First Order Dominance Test If the poverty incidence curve for the A distribution 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: 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: 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: Third Order Dominance Test If the poverty severity curve for 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: 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. 1 1.5
2 3 F(z) A 1/3 1/3 2/3 1 B 0 2/3 1 1 D(z) A 1/3 2/3 4/3 7/3 B 0 2/3 5/3 8/3 S(z) A 1/3 1 7/3 14/3
B 0 2/3 7/3 15/3 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: Egypt, poverty changes between 1996 and 2000 The percentage of the poor for All Egypt.
The Poverty Gap Index for all Egypt. 1995/96 0.9 0.35 1999/2000 0.8 0.3 0.7 0.25 0.2 P1 0.5 0.4 0.15 0.3 0.1 0.2 0.05 0.1 0 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 0 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 Poverty Line Z as %of mean 1
1995/96 1999/2000 Severity of Poverty Index for All Egypt. 0.16 0.14 0.12 0.1 P2 P0 0.6 1995/96 0.08 1999/2000 0.06 0.04 0.02 0 0.35 0.4 0.45 0.5 0.55 0.6
0.65 0.7 0.75 Poverty Line Z as % of mean 0.8 0.85 0.9 0.95 1 Poverty profiles: Additivity How poverty varies across sub-groups of society. Useful to access how the sectoral or regional patterns of economic change are likely to affect aggregate poverty. Additive poverty measures: (e.g., FGT class). Suppose population is divided into m mutually exclusive sub-groups. The poverty profile is the list of poverty measures Pj for j=1,,m. Aggregate poverty for additive poverty measures: m nj P Pj n j / n and Pj p( z j , yij ) / n j j 1
i 1 Aggregate poverty is a population weighted mean of the sub-group poverty measures. Poverty profiles: Example Urban population (2,2,3,4) Rural population (1,1,1.5,2,4) nu 4 i 1 i 1 P0u p( z j , yiu ) / n j p (3, yiu ) / nu 0.75 5 P0 r p(2, yir ) / nr 0.80 i 1 2 P Pj n j / n (0.75* 4 0.80 *5) / 9 0.78 j 1 Zu=3,Zr=2,n=9,nu=4,nr=5, Direct way: n=9; q=7; H=q/n=0.78 Poverty profiles: Two types Two main ways to present poverty profiles: Type A: Incidence of poverty for sub-groups defined by
some characteristics (e.g., place of residence) Type B: Incidence of characteristics defined by the poverty status. Region Number of persons Poverty profile Poor Non-poor Type A: Type B: % of regional % of total population who population who are poor are poor South 100 100 50 33 North 200 600 25 66 Poverty profiles: Select the target region for poverty alleviation. Geographic targeting. If one chooses South more money will go to poor. So Type A is preferable. Minimizes the poverty gap. General rule: When making the lamp-sum transfers with the aim to minimize the aggregate value of FGT type of poverty Pa the next unit of money should go to the sub-group with the highest value of Pa-1.
5 % of poor % of population Poverty profiles: Egypt (Type A) Poverty measurements by gender of individual AREA URBAN Sex of Person Male Female Total RURAL Male Female Total Total Male Female Total Mean N Mean N Mean N
Mean N Mean N Mean N Mean N Mean N Mean N PO 29.1054 63411 27.1285 61876 28.1290 125287 52.4891 51304 49.9152 49526 51.2248 100830 39.5633 114715 37.2588 111402 38.4279 226117 P1 7.5857
3.5100 226117 Poverty profiles: Multivariate Univariate: Simple cross-tabulation of poverty measures against specific variables Multivariate: Poverty measure is modeled as a function of multiple variables: or poverty regression Model household expenditure or income first and then predict poverty measures based on this regression. Do not run probit on poverty measure when expenditure data is available. Steps: Estimate regression: Log(Ci)=+Xi+I Predict consumption: E(Ci)=Exp(Xi+2/2) Calculate poverty rates based on predicted consumption, or Calculate probability of being poor, then the national headcount index will be equal to weighted average of the predicted probability, etc. Simulations. Regression of log consumption per capita on characteristics of household and household head for seven regions of Egypt. Household characteristics Log household size Log household size2 Share of children 0-6 Share of children 7-15
Share of elderly Share of adult females Share of adult males Share of literate Share of university Share of unemployed Characteristics of the head Age Age2/100 Male Female Education Illiterate Read & Write Basic Secondary Diploma University Postgraduate degree Working status Government Public Private Foreign/JVC Unemployed Out of labor force Industry of employment Agriculture Metro Upper urban Upper rural
Metro Child born in the family Family member looses job Female headed households Head education Change from illiterate to read and write Change from illiterate to basic Change from illiterate to secondary Change from illiterate to diploma Change from illiterate to University degree Change from illiterate to postgraduate degree Sector of employment Unemployed employed in the government job employed in the public sector job employed in the private sector job employed in the foreign firm 56.22 26.93 -21.95 Upper Urban 63.93 113.06 -17.47 Upper Rural 45.02 13.26 2.92 Lower
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