More on multidimensional, intertemporal and chronic poverty orderings

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1 Carnegie Mellon University Research CMU Society for Economic Measurement Annual Conference 2015 Paris Jul 23rd, 9:00 AM - 11:00 AM More on multidimensional, intertemporal and chronic poverty orderings Florent Bresson University of Auvergne, florent.bresson@udamail.fr Jean-Yves Duclos University of Laval, JYVES@ECN.ULAVAL.CA Follow this and additional works at: Part of the Economics Commons Florent Bresson and Jean-Yves Duclos, "" ( July 23, 2015). Society for Economic Measurement Annual Conference. Paper This Event is brought to you for free and open access by the Conferences and Events at Research CMU. It has been accepted for inclusion in Society for Economic Measurement Annual Conference by an authorized administrator of Research CMU. For more information, please contact research-showcase@andrew.cmu.edu.

2 More on multidimensional, intertemporal and chronic poverty orderings Florent Bresson Jean-Yves Duclos : CERDI, CNRS Université d Auvergne : CIRPÉE, Université Laval & FERDI

3 A rich man s problem

4 A rich man s problem An impressive bunch of poverty indices with sound axiomatic frameworks have been proposed for:

5 A rich man s problem An impressive bunch of poverty indices with sound axiomatic frameworks have been proposed for: Multidimensional poverty: Chakravarty, Mukherjee & Ranade (1998), Tsui (2002), Bourguignon & Chakravarty (2003), Chakravarty, Deutsch & Silber (2008), Alkire & Foster (2011), Giraud (2012), Chakravary & d Ambrosio (2013), Permanyer (2014)...

6 A rich man s problem An impressive bunch of poverty indices with sound axiomatic frameworks have been proposed for: Multidimensional poverty: Chakravarty, Mukherjee & Ranade (1998), Tsui (2002), Bourguignon & Chakravarty (2003), Chakravarty, Deutsch & Silber (2008), Alkire & Foster (2011), Giraud (2012), Chakravary & d Ambrosio (2013), Permanyer (2014)... Intertemporal poverty, Calvo & Dercon (2009), Foster (2009), Hoy & Zheng (2011), Bossert, Chakravarty d Ambrosio (2012), Busetta & Mendola (2012), Canto, Gradin & del Rio (2012), Zheng (2012), Dutta, Roope & Zank (2013)...

7 A rich man s problem An impressive bunch of poverty indices with sound axiomatic frameworks have been proposed for: Multidimensional poverty: Chakravarty, Mukherjee & Ranade (1998), Tsui (2002), Bourguignon & Chakravarty (2003), Chakravarty, Deutsch & Silber (2008), Alkire & Foster (2011), Giraud (2012), Chakravary & d Ambrosio (2013), Permanyer (2014)... Intertemporal poverty, Calvo & Dercon (2009), Foster (2009), Hoy & Zheng (2011), Bossert, Chakravarty d Ambrosio (2012), Busetta & Mendola (2012), Canto, Gradin & del Rio (2012), Zheng (2012), Dutta, Roope & Zank (2013)...

8 A rich man s problem An impressive bunch of poverty indices with sound axiomatic frameworks have been proposed for: Multidimensional poverty: Chakravarty, Mukherjee & Ranade (1998), Tsui (2002), Bourguignon & Chakravarty (2003), Chakravarty, Deutsch & Silber (2008), Alkire & Foster (2011), Giraud (2012), Chakravary & d Ambrosio (2013), Permanyer (2014)... Intertemporal poverty, Calvo & Dercon (2009), Foster (2009), Hoy & Zheng (2011), Bossert, Chakravarty d Ambrosio (2012), Busetta & Mendola (2012), Canto, Gradin & del Rio (2012), Zheng (2012), Dutta, Roope & Zank (2013)... Which measure is appropriate?

9 The problem

10 The problem Poverty is lower with distribution B when compared with distribution A if: P B (λ) P A (λ) 0 with: P: the poverty measure, λ: a function that defines the poverty frontier.

11 The problem Poverty is lower with distribution B when compared with distribution A if: P B (λ) P A (λ) 0 with: P: the poverty measure, λ: a function that defines the poverty frontier. Contingency of the result with respect to λ and P.

12 The problem Poverty is lower with distribution B when compared with distribution A if: P B (λ) P A (λ) 0 with: P: the poverty measure, λ: a function that defines the poverty frontier. Contingency of the result with respect to λ and P.

13 The problem Poverty is lower with distribution B when compared with distribution A if: P B (λ) P A (λ) 0 with: P: the poverty measure, λ: a function that defines the poverty frontier. Contingency of the result with respect to λ and P. Robustness implies using criteria that make it possible to obtain rankings that do not depend on λ and P.

14 State of the art

15 State of the art Technically speaking, intertemporal poverty is multidimensional poverty (with additional axioms), so that a common framework can be used for both intertemporal and multidimensional poverty.

16 State of the art Technically speaking, intertemporal poverty is multidimensional poverty (with additional axioms), so that a common framework can be used for both intertemporal and multidimensional poverty. Chakravarty & Bourguignon (2002), Duclos, Sahn & Younger (2006) and Bresson & Duclos (2015) propose stochastic dominance conditions that make it possible to obtain robust multidimensional/intertemporal poverty orderings for broad classes of poverty indices and sets of poverty frontiers.

17 State of the art Technically speaking, intertemporal poverty is multidimensional poverty (with additional axioms), so that a common framework can be used for both intertemporal and multidimensional poverty. Chakravarty & Bourguignon (2002), Duclos, Sahn & Younger (2006) and Bresson & Duclos (2015) propose stochastic dominance conditions that make it possible to obtain robust multidimensional/intertemporal poverty orderings for broad classes of poverty indices and sets of poverty frontiers. However, they all focus on classes of continuous poverty indices while many poverty indices are not.

18 Contribution

19 Contribution Relax the continuity assumption and propose dominance conditions for bidimensional poverty indices that comply with restricted continuity,

20 Contribution Relax the continuity assumption and propose dominance conditions for bidimensional poverty indices that comply with restricted continuity, Highlight the changes induced by relaxing the continuity assumption in comparison with the unidimensional case.

21 Contribution Relax the continuity assumption and propose dominance conditions for bidimensional poverty indices that comply with restricted continuity, Highlight the changes induced by relaxing the continuity assumption in comparison with the unidimensional case. Show how the framework can easily be adapted for the ordering of the chronic component of intertemporal poverty.

22 Outline 1. Framework 2. Main results 3. Chronic poverty 4. Concluding remarks

23 Notations x := (x 1, x 2 ) R 2 is an individual profile. λ :R 2 + R is a non-decreasing welfare function. L is the set of non-decreasing functions on R 2 +. Γ R 2 + is the poverty domain. Λ is the poverty frontier. Γ 1 (λ) Γ(λ) x 1 < x 2. F (x 1, x 2 ) is the joint cumulative distribution function. H(λ) := Γ(λ) df (x 1, x 2 ). is a bidimensional headcount index.

24 DSY The DSY approach of poverty identification

25 DSY The DSY approach of poverty identification How shall we separate the poor from the non-poor in a bidimensional context?

26 DSY The DSY approach of poverty identification How shall we separate the poor from the non-poor in a bidimensional context? Duclos, Sahn & Younger (2006) defines the poor as those such that: λ(x 1, x 2 ) 0.

27 x 2 z 2 z z 1 x 1 Figure 1: The definition of the poverty domain.

28 x 2 z 2 z z 1 x 1 Figure 1: The definition of the poverty domain.

29 x 2 z 2 z z 1 x 1 Figure 1: The definition of the poverty domain.

30 x 2 z 2 z z 1 x 1 Figure 1: The definition of the poverty domain.

31 x 2 z 2 z z 1 x 1 Figure 1: The definition of the poverty domain.

32 x 2 z 2 z z 1 x 1 Figure 1: The definition of the poverty domain.

33 DSY The DSY framework

34 DSY The DSY framework Let consider bidimensional additive poverty measures of the form: P(λ)= π(x 1, x 2 ;λ)df (x 1, x 2 ). Γ(λ)

35 DSY The DSY framework Let consider bidimensional additive poverty measures of the form: P(λ)= π(x 1, x 2 ;λ)df (x 1, x 2 ). Γ(λ) DSY define the class Π(λ + ) of bidimensional poverty indices P(λ) as: Γ(λ) Γ(λ + ) Π(λ + π(x 1, x 2 ;λ)=0, whenever λ(x 1, x 2 )=0 )= P(λ) π (1) (x 1, x 2 ;λ) 0 and π (2) (x 1, x 2 ;λ) 0 x 1, x 2 π (1,2) (x 1, x 2 ;λ) 0, x 1, x 2

36 DSY The DSY solution

37 DSY The DSY solution Theorem DSY (Duclos, Sahn & Younger, 2006) P A (λ)>p B (λ), P(λ) Π(λ + ), iff F A (x 1, x 2 )>F B (x 1, x 2 ), (x 1, x 2 ) Γ(λ + ).

38 DSY The issue Most existing indices do not belong to Π(λ + )!

39 Restricted continuity Relaxing continuity Restricted continuity means continuity within the poverty domain.

40 Restricted continuity Relaxing continuity Restricted continuity means continuity within the poverty domain. Allows the poverty index to show a discontinuity at the frontier of the poverty domain. example

41 Restricted continuity A class of poverty indices with restricted continuity

42 Restricted continuity A class of poverty indices with restricted continuity Let ( z 1, z 2 ) be an individual profile on the poverty frontier, i.e. λ( z 1, z 2 )=0. z 1 (x 2 ) is the continuous non-negative function so that λ ( z 1 (x 2 ), x 2 ) = 0 x2. z 2 (x 1 ) is the inverse of z 1 (x 2 ).

43 Restricted continuity A class of poverty indices with restricted continuity Let ( z 1, z 2 ) be an individual profile on the poverty frontier, i.e. λ( z 1, z 2 )=0. z 1 (x 2 ) is the continuous non-negative function so that λ ( z 1 (x 2 ), x 2 ) = 0 x2. z 2 (x 1 ) is the inverse of z 1 (x 2 ). We consider the class Π(λ + ) of bidimensional poverty indices P(λ) as: Γ(λ) Γ(λ + ), π(x 1, x 2 ;λ) 0, whenever λ(x 1, x 2 )=0, Π(λ + π (x 1) ( x 1, z 2 (x 1 ) ) 0, x 1 [0, z 1 ], )= P(λ) π (x 2) ( ) z 1 (x 2 ), x 2 0, x2 [0, z 2 ], π (1) (x 1, x 2 ;λ) 0 and π (2) (x 1, x 2 ;λ) 0 x 1, x 2, π (1,2) (x 1, x 2 ;λ) 0, x 1, x 2

44 Restricted continuity A class of poverty indices with restricted continuity Let ( z 1, z 2 ) be an individual profile on the poverty frontier, i.e. λ( z 1, z 2 )=0. z 1 (x 2 ) is the continuous non-negative function so that λ ( z 1 (x 2 ), x 2 ) = 0 x2. z 2 (x 1 ) is the inverse of z 1 (x 2 ). We consider the class Π(λ + ) of bidimensional poverty indices P(λ) as: Γ(λ) Γ(λ + ), π(x 1, x 2 ;λ) 0, whenever λ(x 1, x 2 )=0, Π(λ + π (x 1) ( x 1, z 2 (x 1 ) ) 0, x 1 [0, z 1 ], )= P(λ) π (x 2) ( ) z 1 (x 2 ), x 2 0, x2 [0, z 2 ], π (1) (x 1, x 2 ;λ) 0 and π (2) (x 1, x 2 ;λ) 0 x 1, x 2, π (1,2) (x 1, x 2 ;λ) 0, x 1, x 2

45 x 2 λ(x 1, x 2 )=0 z 2 z 1 x 1 Figure 2: A member of Π(λ + )

46 Main result Theorem 1 P A (λ) P B (λ), P(λ) Π(λ + ), iff H A (λ) H B (λ) λ L s. t. Γ(λ) Γ(λ + ).

47 Main result Theorem 1 P A (λ) P B (λ), P(λ) Π(λ + ), iff H A (λ) H B (λ) λ L s. t. Γ(λ) Γ(λ + ). More demanding that Theorem DSY!

48 Increasing the ordering power

49 Increasing the ordering power Different options:

50 Increasing the ordering power Different options: Consider higher-orders of dominance (but Zheng, 1999). Add more structure (like intertemporal symmetry). Consider specific families for λ. Assume a minimal set for the poverty domain.

51 Increasing the ordering power Different options: Consider higher-orders of dominance (but Zheng, 1999). Add more structure (like intertemporal symmetry). Consider specific families for λ. Assume a minimal set for the poverty domain.

52 x 2 λ + (x 1, x 2 )=0 Γ(λ ) λ (x 1, x 2 )=0 x 1 Figure 3: A minimal poverty domain.

53 Partial headcount

54 Partial headcount F λ,1 and F λ,2 are partial headcount indices defined as: F λ,1 (x 1 ) := F λ,2 (x 2 ) := x1 0 x2 0 F ( z 2 (y 1 ) y 1 ) f1 (y 1 ) d y 1, F ( z 1 (y 2 ) y 2 ) f2 (y 2 ) d y 2.

55 A lower bound for the poverty frontier

56 A lower bound for the poverty frontier Let Γ(λ +,λ ) := Γ(λ + ) Γ(λ ) be the domain where the poverty frontier is assumed to be located.

57 A lower bound for the poverty frontier Let Γ(λ +,λ ) := Γ(λ + ) Γ(λ ) be the domain where the poverty frontier is assumed to be located. Theorem 2 P A (λ) P B (λ), P(λ) Π(λ + ), λ L s. t. Λ(λ) Γ(λ +,λ ), iff F A (x 1, x 2 ) F B (x 1, x 2 ), (x 1, x 2 ) Γ(λ + ), and H A (λ) H B (λ) λ L s. t. Λ(λ) Γ(λ +,λ ), and F A λ,t (x) F B λ,t (x) t {1,2}, x [0, z+ t ], λ L s. t. Λ(λ) Γ(λ +,λ ).

58 A lower bound for the poverty frontier Let Γ(λ +,λ ) := Γ(λ + ) Γ(λ ) be the domain where the poverty frontier is assumed to be located. Theorem 2 P A (λ) P B (λ), P(λ) Π(λ + ), λ L s. t. Λ(λ) Γ(λ +,λ ), iff F A (x 1, x 2 ) F B (x 1, x 2 ), (x 1, x 2 ) Γ(λ + ), and H A (λ) H B (λ) λ L s. t. Λ(λ) Γ(λ +,λ ), and F A λ,t (x) F B λ,t (x) t {1,2}, x [0, z+ t ], λ L s. t. Λ(λ) Γ(λ +,λ ). Third condition vanishes if we assume π(z 1 (x 2 ), x 2 )=c 0 x 2.

59 x 2 H(λ) Λ(λ) b d F (a,b) F (c,d) Λ(λ ) a c Λ(λ + ) x 1

60 x 2 Λ(λ) z + 2 F λ,1 (a) b Λ(λ + ) a F λ,2 (b) z + 1 Λ(λ ) x 1

61 Remarks Contrary to the unidimensional case, imposing a minimal set for the poverty domain raises the ordering power. In the specific case of intersection poverty domains, conditions in Theorems 1 and 2 boil down to Theorem DSY s condition.

62 General framework

63 General framework Main assumption: Intertemporal poverty can be additively decomposed into chronic and transient components, that is: P(λ)= C ( P(λ) ) + T ( P(λ) ). where C () is the chronic component and T () the transient component.

64 Different views The spell approach: poor are either chronic poor or transient poor (Rainwater, 1981; Foster, 2009). The components approach: poor can cumulate chronic and transient forms of poverty (Jalan & Ravallion, 2000).

65 x 2 λ(x 1, x 2 )=0 Γ T (κ,λ) Γ C (κ,λ) κ(x 1, x 2 )=0 x 1 Figure 4: Chronic and transient components of poverty with the spell approach.

66 x 2 λ(x 1, x 2 )=0 Γ T (λ) Γ C (κ,λ) κ(x 1, x 2 )=0 x 1 Figure 5: Chronic and transient components of poverty with the component approach.

67 Dominance checks

68 Dominance checks In both cases, the chronic poverty domain is the bottom part of the poverty domain.

69 Dominance checks In both cases, the chronic poverty domain is the bottom part of the poverty domain. Assuming that C ( P(λ) ) Π(λ + ), Theorems 1 to 6 can be used to obtain robust orderings.

70 Further work

71 Further work To do: Higher-order of dominance (Pigou-Dalton transfers, transfer-sensitivity... ),

72 Further work To do: Higher-order of dominance (Pigou-Dalton transfers, transfer-sensitivity... ), Implementation,

73 Further work To do: Higher-order of dominance (Pigou-Dalton transfers, transfer-sensitivity... ), Implementation, Discontinuities within the poverty domain,

74 Further work To do: Higher-order of dominance (Pigou-Dalton transfers, transfer-sensitivity... ), Implementation, Discontinuities within the poverty domain, Transient poverty.

75 The end Thank you for your attention.

76 π x 2 x 1 Note: the opposite of the individual poverty index is depicted on the figure. Figure 6: A bidimensional poverty index with restricted continuity. back 0