Measuring poverty. Araar Abdelkrim and Duclos Jean-Yves. PEP-PMMA training workshop Addis Ababa, June overty and. olitiques.

Transcription

1 E Measuring poverty Araar Abdelkrim and Duclos Jean-Yves E-MMA training workshop Addis Ababa, June 2006 E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 1/43

2 roperties of overty Indices Main properties of poverty indices: A- Focus axiom overty is invariant to income changes of the non-poor if these non-poor do not become poor E E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 2/43

3 Axioms of overty Indices B- Anonymity overty does not depend on information other living standards or incomes. C- opulation rinciple overty should be invariant to replications of the population: merging two identical distributions should not alter poverty. D- Scale Invariance overty is homogeneous of degree zero with respect to incomes Y and the poverty line z such as: (λy, λz) = (Y, z) E E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 3/43

4 roperties of overty Indices E- Monotonicity overty declines, or does not rise, following an increase in the incomes of the poor. F- The igou-dalton Transfer rinciple overty should be sensitive to inequality within the poor: transfers from poorer to richer increases poverty. G- Subgroup Decomposability Total poverty is a weighted mean of poverty levels within each group. E E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 4/43

5 overty Indices and Functional Forms A- THE EDE AROACH One can identify a set of poverty indices that are linked by their functional form to the EDE income principe. A general form of these indices is: E (z; ρ, ǫ) = z ξ (z; ρ, ǫ) (1) where: ξ (z; ρ, ǫ) is the EDE income of the distribution of censored income and censored income: Q (p; z) = min(z, Q(p)). If we note the average censored income by µ (z), we can write what follows: (z; ρ, ǫ) = z µ (z) + c (z; ρ, ǫ) }{{}}{{} Average Gap Cost of Inequality (2) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 5/43

6 overty Indices and Functional Forms Example: Suppose that: arameters of aversion to inequality: ǫ = 0 and ρ = 2. With this, the EDE income is defined as follows: ξ(ρ = 2, ǫ = 0) = where y 1 y 2...y i y i+1... y n. Let the poverty line be z = 400. E 2(n i) + 1 n 2 y i (3) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 6/43

7 overty Indices and Functional Forms E Table 1: Illustrative Example Rank i Q(p i ) = y i Q (p i, z) = yi w i = 2(n i)+1 n 2 w i yi Average ξ (z; ρ, ǫ) = X i w i y i = (4) c (z; ρ, ǫ) = µ (z) ξ (z; ρ, ǫ) = (5) (z; ρ, ǫ) = {z} {z} {z } z µ (z) c (z;ρ,ǫ) = (6) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 7/43

8 overty Indices and Functional Forms E opular EDE overty Indices The Clark-Hemming-Ulph s overty Index : (z; ρ = 1, ǫ) = The Watts Index : ) 1 1 ǫ ( 1 z 0 Q (p; z) (1 ǫ) dp, when ǫ 1, ( ) 1 z exp 0 ln(q (p; z))dp, when ǫ = 1. W(z) = 1 0 ( ln z Q (p; z) (7) ) dp. (8) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 8/43

9 overty Indices and Functional Forms The Chakravarty Index : C(z; ǫ) = 1 E opular EDE overty Indices 1 0 ( Q ) 1 ǫ (p; z) dp, 0 ǫ < 1. (9) The S-Gini overty Index : When ǫ = 0, we obtain the class of S-Gini indices of poverty: (z; ρ) (z; ρ, ǫ = 0) = z z 1 0 Q (p; z)ω(p; ρ)dp (10) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 9/43

10 overty Indices and Functional Forms E opular Gap overty Indices overty gap indices are simply functions of poverty gaps, g i = max(z y i, 0). Denoting the weight on income y i as ω i, a general form for this class of indices is: (z) = n ω i g i (11) i=1 If the weight is simply the proportion of individuals that have income y i, then the index equals the average poverty gap. E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 10/43

11 overty Indices and Functional Forms The most popular class of poverty gap indices is the Foster-Greer-Thorbecke (FGT) poverty indices: (z; α) = n w i gi α = i=1 When α > 1, the weight ω i = w i g (α 1) i E n i=1 w ii(g i > 0) when α = 0 n i=1 w ig i when α = 1, n i=1 w ig (α 1) i g i when α > 1, increases with gaps. When α > 1, this poverty index respects the igou-dalton transfer axiom. (12) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 11/43

12 overty Indices and Functional Forms Define the normalised poverty gap by ḡ i (z) = g i z ; the normalised FGT poverty index is then defined as: (z, α) = E n w i ḡi α (13) This normalised poverty index decreases with the parameter α since: (z, α) α = i=1 n p i ḡi α logḡ i 0 (14) The contribution of the poorest to total poverty, after an increase in the parameter α, may seem ambiguous. i=1 E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 12/43

13 overty Indices and Functional Forms Figure 1: Contribution of poverty gaps to FGT indices 1 g(p;z)/z E (g(p;z)/z) 3 Q(p)/z (g(p;z)/z) 2 F(z) p E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 13/43

14 overty Indices and Functional Forms Note that the relative contribution of the poorest increases with an increase in the parameter α, as showed in the following figure: Figure 2: Relative Contribution of poverty gaps to FGT indices 1/F(z) 1 2 g(p) /(z; α=2) E g(p) /(z; α=1) F( µ g (z)) F(z) p E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 14/43

15 overty Indices and Functional Forms EDE poverty gaps for FGT indices FGT indices for general values of α are difficult to interpret since these indices are averages of powers of poverty gaps. They are also neither unit-free nor money-metric (except for α = 0 and 1). A simple solution to these two problems is to transform the FGT indices into EDE (Equally Distributed Equivalent) poverty gaps as follows: E ξ g (z; α) = [(z; α)] 1/α. (15) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 15/43

16 overty Indices and Functional Forms The FGT EDE poverty gaps are as follows: where E ξ g (z; α) = z µ (z) + }{{} c(z; α) }{{} Average Gap Cost of Inequality (16) c(z; α) = ξ g (z; α) ξ g (z; α = 1) (17) The normalised EDE poverty gap index is defined as follows: ξ g (z; α) = ξg (z; α) z (18) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 16/43

17 overty Indices and Functional Forms Note that the EDE poverty gap respects the igou-dalton transfer axiom. Also, the EDE poverty gap index increases with α: ξ g (z; α) α E = ξ g [ n i=1 p ig α i log(g i) α(z; α) ] log((z; α)) α 2 > 0 (19) Note that the S-Gini poverty index belongs to the class of poverty gap indices: (z; ρ) (z; ρ, ǫ = 0) = z = = Q (p; z)ω(p; ρ)dp (20) (z Q (p; z))ω(p; ρ)dp (21) g(p; z)ω(p; ρ)dp (22) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 17/43

18 Group-decomposable poverty indices Among desirable properties of poverty indices is their decomposability across population groups. This type of decomposition allows researchers and pmakers to identify the contribution of each group to total poverty. Among the most popular indices that obey the decomposability axiom across groups are FGT, the Chakravarty and the Watts indices. In general, the decomposition of these indices takes the following form: (z; α) = E K φ(k)(k; z; α) (23) where (k; z; α) and φ(k) are respectively the poverty index and the population share of group k. k E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 18/43

19 Group-decomposable poverty indices Starting from the last equation, the intensity of the contribution of each group to total poverty depends on the poverty of that group and its share in the population. E Table 2: Illustrative Example Group k φ(k) (k; z; α) Absolute Relative Contribution Contribution / /3 Total E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 19/43

20 inequality Two main factors determine the level of poverty. These are the average income and the level of inequality. When inequality equals zero, the two most basic indices of poverty (the average poverty gap and the headcount) depend on the difference between average income and the poverty line. Recall that the Lorenz curve gives a complete picture of relative income inequalities. E E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 20/43

21 inequality The link between the headcount index and poverty is: E L (H) = z µ The link between the average poverty gap index and inequality is: where (24) 1 = [z µ p ] H (25) L(H) = Hµ p µ The link between the severity index and poverty is: where (µ)l (p) = Q(p). 2 = H 0 (26) [z (µ)l (p)] 2 dp (27) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 21/43

22 inequality Starting from these links, we find that the cost of inequality equals to: c(z) = 2 1 = H 0 E [ z 2 2zQ(p) + Q(p) 2 z + µ p ] dp (28) What is the link between the standard Gini index and poverty indices? The Gini index is easily decomposable across poor and non-poor groups, such that: I = φ p ψ p I p + φ np ψ np I np + Ĩ (29) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 22/43

23 inequality L(p) ercentiles (p) Between Group Inequality E Lorenz Curve & overty Contribution of the non poor group ECAM I: Cameroon (1996) Contribution of the poor group E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 23/43

24 inequality The link between headcount index and the between group inequality is as follows: ( ) 1 H = µĩ (30) µ µ p overty gap index can be expressed as follows: ( ) z µp 1 = µĩ µ µ p Finally, the link between the Gini Index and 2 cannot be established directly. This is explained by the different shapes. that the distribution can have, with the same level of inequality, measured by Gini index. E (31) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 24/43

25 overty curves overty Gap Curve g(z; p) This curve indicates simply the poverty gap at a given percentile of population p. The curve naturally decreases with the rank p in the population, and reaches zero at the value of p equal to the headcount. The integral under the curve gives the average poverty gap. E E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 25/43

26 overty curves Cumulative overty Gap Curve G(z; p) This curve is defined as follows: G(p; z) = p 0 E g(p; z)dq (32) The slope of this curve is non negative and equals to g(z; p). The level of concavity of this curve at p [0, H] expresses the level of inequality within the group of the poor. E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 26/43

27 overty curves µ g (z) Figure 3: The cumulative poverty gap curve G(p;z) E A B 0 F(z) 1 p E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 27/43

28 Normalised overty Indices The normalised FGT index is: (z; α) = 1 The derived EDE poverty gap lies between 0 and 1. 0 E ḡ(p; z) α dp (33) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 28/43

29 Normalised overty Indices This makes poverty indices independent of the monetary units. This makes the indices invariant to an equi-proportionate change in all incomes and in the poverty line. Normalised poverty indices can make poverty lines act as price indices. If real poverty lines can differ according to regions or time, absolute and normalised poverty indices can give different results concerning the level of poverty. E E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 29/43

30 Growth-redistribution Decompositions As indicated above, poverty depends on two main factors which are average income and inequality. This implies that a difference in poverty across two distributions depends on the change in average income (Growth) and the change in levels of inequality (Redistribution). Suppose that one tries to explain the difference in poverty between distributions A and B using the Growth and the Redistribution components. E E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 30/43

31 Growth-redistribution Decompositions To assess the impact of Growth, one can scale incomes of A by (µ B /µ A ) and estimate the growth effect on poverty as: ( ( ) ) zµa Growth Effect = A ; α A (z; α) (34) µ B To assess the impact of Redistribution, one can scale incomes of B by (µ A /µ B ) and estimate the redistribution effect on poverty as: Redistribution Effect = E ( ( zµb B ) ; α µ A Note that the reference period is the first one (A). ) A (z; α) (35) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 31/43

32 Growth-redistribution Decompositions If one uses the second periods (B) as the referenced period, one find what follows: Growth Effect = Redistribution Effect = E ( B (α) B ( z; µ B ( B (z; α) A ( )) ; α µ A z; µ )) A ; α µ B These types of decomposition generates a residue, such that: Variation in overty = Growth Effect + Redistribution Effect + Residue (38) (36) (37) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 32/43

33 Growth-redistribution Decompositions To remove the arbitrariness, that raises from the choice of the referenced period and causes the error term, one can adopts the Shapley approach where the Shapley-factors are the referenced periods. B (z; ( α) ( A (z; α) ) ) ( ( )) zµa zµb = 0.5 A ; α A (z; α) + B (z; α) B ; α µ B µ A }{{} Shapley growth effect E ( +0.5 ( B } (39) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 33/43

34 Growth-redistribution Decompositions Example: Suppose that: overty line: z = 400. All incomes have increased proportionally by 20% in period B. The government redistributes a part of income through income taxes, such that: The excess of incomes over the poverty line is taxed by 20%: T i = 0.2 max(0, y i z)) The poor receives an allocation of the amount A = n i=1 T i H E E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 34/43

35 Growth-redistribution Decompositions E Table 3: Illustrative example Rank i Y A g A i Y B (G) g B i (G) Y B (G, R) g B i (G, R) Average B 1 A {z } 40 = Growth Effect {z } 20 + Redistribution Effect {z } 20 (40) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 35/43

36 Growth-redistribution Decompositions E Table 4: Illustrative Example Rank i Y A Y B Y (µ B, I A ) Y (µ A, I B ) Average Y (µ B, I A ) = Y A µ B (41) µ A Y (µ A, I B ) = Y B µ A µ B (42) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 36/43

37 Growth-redistribution Decompositions Reference period is A: Reference period is B: GE = (Y (µ B, I A )) (Y A ) = = RE = (Y (µ A, I B )) (Y A ) = = GE = (Y B ) (Y (µ A, I B )) = = RE = (Y B ) (Y (µ B, I A )) = = Shapley decomposition: E GE = RE = E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 37/43

38 Sectoral Decompositions Recall that the absolute contribution of group k, noted by E(k), to total poverty is defined as follows: E E(k) = φ(k)(k; z, α) (43) Between two periods or two distributions A and B, the change in total poverty equals the sum of changes in group contributions, such that: B (z; α) A (z; α) = K (E B (k; z, α) E A (k; z, α)) (44) k=1 The change in the contribution of group k is defined as follows: E B (k; z, α) E A (k; z, α) = φ B (k) B (k; z, α) φ A (k) A (k; z, α) (45) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 38/43

39 Sectoral Decompositions For a given group k, one can use the reference period A to estimate the impact of the change in poverty or in the proportion of this group on the change of its contribution, such that: E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 39/43 E overty Impact(k) = φ A (k) ( B (k, z, α) A (k, z, α) ) (46) Demographic Impact(k) = A (k, z, α)(φ B (k) φ A (k)) (47) Starting from this, differences in poverty can be expressed as follows: B (z; α) A (z; α) = K k φ A(k) ( B (k; z; α) A (k; z; α) ) }{{} within-group poverty effects + K k + A (k; z; α) (φ B (k) φ A (k)) }{{} demographic or sectoral effects K ( B (k; z; α) A (k; z; α)(φ B (k) φ A (k)) ). k } {{ } interaction or error term (48)

40 Sectoral Decompositions To remove the arbitrariness in the choice of the reference period as well as the residual, one can apply the Shapley approach. With this approach the sectoral decomposition is as follows: E B (z; α) A (z; α) = K k φ(k) ( B (k; z; α) A (k; z; α) ) }{{} within-group poverty effects + K k (k; z; α) (φ B (k) φ A (k)) }{{} demographic or sectoral effects (49) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 40/43

41 Impact of demographic changes One can simulate the impact of a proportional increase in the population share of group t on poverty. Specifically, suppose that: The population share of group t increases proportionally by λ. The population share of other groups falls proportionally to keep the condition K φ(k)φ(t) k=1 = 1φ(k) satisfied (the share falls by The impact of this on poverty is then: (z; α) = (t; z; α) k t E φ(k) 1 φ(t) 1 φ(t) ). (k; z; α) φ(t)λ (50) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 41/43

42 Impact of demographic changes One can also simulate the impact of an absolute increase in the population share of group t on poverty. Specifically, suppose that: The population share of group t increases by λ. The population share of other groups falls proportionally to keep the condition K λφ(k) k=1 = 1φ(k) satisfied (the share falls by The impact of this on poverty is then: (z; α) = (t; z; α) k t E φ(k) 1 φ(t) 1 φ(t) ). (k; z; α) λ (51) E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 42/43

43 Decomposing by income sources By supposing that total income equals the sum of C income sources, the question is: How does each component contribute to the alleviation of poverty? It is clear that in the absences of all of these sources, the gap of each individu equals the poverty line. For the non poor, the sum of component exceeds the poverty line. How then can one estimate the contribution of any component? Again here, one can use the Shapley approach to overpass the arbitrariness of the order of the choice of components. To do this, we suppose that the marginal contribution of component c in reducing poverty, is simply the reduction in poverty after adding this component to an initial set of components. E E-MMA training workshop Addis Ababa, June 2006 Distributive Analysis and overty - p. 43/43