The Bourguignon-Chakravarty (B-C) Poverty Family: A Characterization

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The Bourguignon-Chakravarty (B-C) Poverty Family
A Characterization

• C. Lasso de la Vega, A. Urrutia and H. Díez,
• University of the Basque Country

F. Bourguignon and K. Chakravarty (2003). Journal
of Economic Inequality.

• Theoretical papers A. Atkinson (2003) S. Bibi
  (2003)
• Empirical papers S. Bibi (2003), C. DAmbrosio,
  J. Deutsch and J. Silber (2005), S. Escalante and
  E. Yánez (2006), R. Arim and A. Vigorito (2007)

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The B-C Poverty Family A Characterization.OUTLI
NE

• Two different procedures to construct
  multidimensional indices (P.K. Pattanaik,
  S.G. Reeddy and Y. Xu (2007)).
• Bourguignon and Chakravarty (2003) introduce a
  family of multidimensional poverty measures B-C
  family
• explore a number of properties fulfilled by the
  members of this family
• show that these properties characterize the
  family.
• Concluding remarks.

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Two different procedures to construct
multidimensional indices.

• n individuals i1,,n, each endowed with k
  attributes j 1,..,k.

• Multidimensional deprivation distribution

• Ai overall deprivation for individual i,
• Ai fi ( ai1,,aik )
• P(A) Pr(A1,..,Ai,...,An)
• row-first two stage procedure

column attribute js deprivation distribution

• Bj deprivation of the attribute j
• Bj fj ( a1j,,anj )
• b) P(A) Pc(B1,..,Bj,...,Bk)
• column-first two stage procedure

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The B-C Poverty Family.

• n individuals, each endowed with 2 attributes j
  1,2, with weights wj
• zj poverty line for attribute j
• Each individual i feels a deprivation of
  attribute j

• Aggregating Deprivations for each individual i
• Combining Individual Deprivations The same
  function proposed by Foster-Greer-Thorbecke is
  used to aggregate deprivations over individuals

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1st step Aggregating Deprivations for each
individual i
Properties fulfilled by the B-C Poverty Family.

• Symmetry the name of the attributes is
  irrelevant.
• Increasing in the attribute deprivations.
• Homogeneity (1st degree) in the attribute
  deprivations.
• Homogeneity (0th degree) in the weights.
• Increasing in the weight of the attribute whose
  deprivation is the highest.
• Reflexivity ai1 ai2 a then Ai a
• Internality if ai1 lt ai2 then ai1 lt Ai lt ai2
• Aggregativity consistency in multilevel
  aggregations.

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Characterizing the B-C Poverty Family.
1st step Aggregating Deprivations for each
individual i
Proposition These eight conditions are necessary
and sufficient for the function f to be a
continuous function of the form where ? gt0
and wi 0 such that w1w21
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Properties fulfilled by the B-C Poverty Family.
2nd step Combining Individual Deprivations
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Characterizing the B-C Poverty Family.
2nd step Combining Individual Deprivations
Proposition A multidimensional subgroup
decomposable deprivation index P derived using a
row-first two stage procedure satisfies the
increasing deprivation-consistency axiom if and
only if up to a constant
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Characterizing the B-C Poverty Family.
Proposition A multidimensional subgroup
decomposable deprivation index P derived using a
row-first two stage procedure satisfies the
increasing deprivation-consistency axiom and the
individual deprivation function f satisfies the
conditions 1 through 8 if and only if up to a
constant
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Summary.

• We have proposed
• eight properties to aggregate deprivations for
  each individual
• an axiom to combine individual deprivations in a
  multidimensional index.
• We have proved that these properties
  characterize the B-C poverty family among the
  subgroup decomposable deprivation indices based
  on deprivation matrices bounded.