Session 11 Review Poverty - Introduction Space Identification Aggregation Today Poverty measures Axioms

Poverty – Introduction Recall 3 aspects of distribution Size, spread, poverty Note Only poverty has official measures Q/ Why? Why particular concern with poverty?

Sen (1976) Two steps 1.Identification 2.Aggregation 0. Space Q/ Which variable? Poverty of what? Here – income, consumption, or a single dimensional achievement Later – Sen contends we should examine inequality in a different space

Cumulative Distribution Function Income s Cumulative population F(s) H

Income s Cumulative population 1.5 Ex x = (2, 8, 4, 1) F x (s) 246 8

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

1.Identification Q/ Who is poor? Historical answers Booth in London Rowntree in York Orshansky in US Citro-Michael in US A/ Set poverty line z Types of Poverty lines See Foster 1998 Absolute z a Relative z r Subjective z s Hybrid z h Examples

Citro and Michael (National Academy) Proposed new method for US Corrected biggest problems Updating Sen “Poor Relatively Speaking” Impact on policy? Nothing Why not?

2.Aggregation Q/ How much poverty is there? Historical answer – counting Sen (1976) Find P(x;z) poverty measure

Income s Cumulative population 1.5 μ =3.75 Ex x = (2, 8, 4, 1) F x (s) z

Examples Number 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 P 1 (x;z) Q/ What about inequality among poor? Sen measure S(x;z) uses Gini among poor FGT measure P 2 (x;z) uses sq Coeff of var among poor FGT class P α (x;z)

Session 11 Review Poverty - Introduction Space Identification Aggregation Today Reflections Poverty measures Axioms

“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 solved Poverty line selected Anyone with income below poverty line is poor Q/ How to aggregate data into a single indicator of poverty? Note This was a remarkable question Very little discussion of issue before Sen Broad acceptance of headcount measures Origins 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 = (y 1,…y n ) income distribution zpoverty line y * censored income distribution nonpoor incomes replaced by z z – y i * shortfall or gap g i = (z-y i * )/znormalized shortfall or gap g = (g 1,…g n )normalized gap distribution m i = y i * /znormalized income m = (m 1,…m n )normalized income distribution μ(.), |. |mean, sum P(y;z)poverty measure

Notation y = (7,3,4,8) income distribution z= 5poverty line y * = (5,3,4,5) censored income distribution nonpoor incomes replaced by z z – y 2 * = 2shortfall or gap g 2 = (z-y 2 * )/z = 2/5 normalized shortfall or gap g = (0, 2/5, 1/5, 0) normalized gap distribution m 2 = y 2 * /z =3/5 normalized censored income m = (1, 3/5, 4/5, 1) normalized censored distribution μ(y) = 11/2 |y| = 22 mean, sum P(y;z)poverty measure Q/ What functional form for P?

Headcount Def g 0 = (g 1 0,…g n 0 ) indicator distribution g i 0 = 1 if poor(y i < z) g i 0 = 0 if not(y i > z) Q(y;z) = |g 0 | Headcount number of poor Properties Symmetry, Scale invariance, Focus Not replication invariant Less useful for comparisons over time/space Note Useful partial index Says a lot about absolute number, not about incidence, depth, severity, etc.

Headcount Ratio H(y;z) = μ(g 0 ) = Q(y;z)/n Headcount ratio Interpretation Incidence or percentage of the population that is poor Properties Symmetry, Scale invariance, Rep. invariance, Focus Violates Monotonicity Graph?

Ex Incomes = (7,3,4,8) poverty line z = 5 Who’s poor? g 0 = (0,1,1,0) Headcount H =  (g 0 ) = 2/4 Example: (7,3,3,8) No change in H! Violates monotonicity Note: Partial index Provides information on one aspect of poverty frequency Ignores other aspects depth, distribution

Aggregate Gap A(y;z) = nz - |y*|Aggregate gap total income necessary to raise all poor incomes to z Properties Symmetry, Monotonicity, Focus Violates Scale invariance, Replication invariance Partial 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 –μ p )/z where μ p is the mean poor income Note Here I is not an inequality index! Properties Symmetry, Scale invariance, Rep. invariance, Focus Violates Monotonicity Partial 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 =3/10 Example: (7,4,4,8) I = 2/10 (sensitive to some increments) Example: (7,3,6,4) I = 2/5 (not to others) Note: Partial index Provides information on one aspect of poverty depth of poverty among the poor Ignores other aspects frequency, distribution

Poverty Gap P 1 (y;z) = μ(g)(Per capita) poverty gap average normalized gap across the entire population, or P 1 (y;z) = (z –μ(y * ))/z = HI = |g|/n Properties Symmetry, 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 = P 1 (y;z) = μ(g) = (3/5)/4 = 3/20 Example: (7,4,4,8) = 4/20 (sensitive to increments) Example: (7,4,4,8) P 1 = 4/20 (sensitive to increments) Example: (7,3,6,8) = 2/20 (also to others) Example: (7,3,6,8) P 1 = 2/20 (also to others) Note: Useful poverty index Provides information on depth and frequency of poverty among the poor ignores distribution (violates a transfer principle) Before: (7,3,3,8) = 4/20 Before: (7,3,3,8) P 1 = 4/20 After: (7,2,4,8) = 4/20 (insensitive to transfers among poor) After: (7,2,4,8) P 1 = 4/20 (insensitive to transfers among poor)

FGT ( Foster Greer Thorbecke, 1984) g i 2 squared normalized gap g i 2 = (g i ) 2 if poor g i 2 = 0 if not g 2 = (g 1 2,…g n 2 )squared gap distribution P 2 (y;z) = μ(g 2 )FGT index average squared normalized gap across the entire population, or P 2 (y;z) = H(I 2 + (1-I 2 )C p 2 ) = |g 2 |/n where C p 2 is squared coeff of var among poor

Ex Incomes = (7,3,3,8) poverty line z = 5 Squared Normalized gaps g 2 = (0, 4/25, 4/25, 0) FGT = P 2 =  (g 2 ) = 8/100 Example: (7,2,4,8) Squared Normalized gaps = g 2 = (0, 9/25, 1/25, 0) P 1 = 10/100 (sensitive to inequality) Note: Useful poverty index Provides information on distribution, depth and frequency of poverty among the poor; emphasizes situation of poorest of poor.

FGT family g i α normalized gap raised to α > 0 = (g i ) α if poor g i α = 0 if not g α = (g 1 α,…g n α )distribution P α (y;z) = μ(g α )FGT family average α power of normalized gap across entire population Note P 0 is the headcount ratio, P 1 is the poverty gap

Axioms Focus: 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: P α (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 n x = n x’ and n y = n y’. 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) = (n x /n) P(x;z) + (n y /n) P(y;z). Q/ Does FGT satisfy?