A New Extension of Bourguignon and Chakravarty Index to Measure Educational Poverty and Its Application to the OECD Countries

Abstract

The consequences that educational underperformance has on both individuals and society as a whole lead policy makers and planners to focus on how to measure it properly. The aim of this paper is to propose an index to measure educational poverty which, taking as a starting point the economic literature on multidimensional poverty measurement, turns out to be appropriate in the educational context. With this purpose, the following two features are demanded: (1) an individual should be identified as poor whenever they do not reach the basic level of knowledge in at least one of the relevant subjects; (2) the degree of poverty of individuals who present the same level of insufficiency in some subjects but have different scores in others should be different. Based on these premises, we introduce a multidimensional adjusted poverty index, called BC^a index, which is an extension of Bourguignon and Chakravarty index, and we apply it to measure educational poverty in the OECD countries by using data from PISA 2012 and 2015 reports.

Appendices

Appendix 1: Calculation of the Proportion of the People Who Are Poor and Deprived in Any Set of Attributes

Let $k \in N, J = \left\{ {1, \ldots ,k} \right\}$ and $S \in P\left( J \right){ \setminus }\emptyset ,$ where $P\left( J \right)$ denotes the set of all the subsets of J. We define the exclusive identification function, ${\text{ID}}^{E} :{\mathbb{R}}_{ + }^{k} \times P\left( J \right){ \setminus }\emptyset \to \left\{ {0,1} \right\}$, such that for any $x \in {\mathbb{R}}_{ + }^{k}$ and any $S \in P\left( J \right){ \setminus }\emptyset$,

$$ ID^{E} \left( x, S \right) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {\quad if\;x_{j} \ne 0 \,\forall j \in S,\;x_{t} = 0 \,\forall t \in J{ \setminus }S } \hfill \\ 0 \hfill & {\quad otherwise} \hfill \\ \end{array} } \right. $$

Therefore, for any $\left( {{\mathbb{X}},z} \right) \in {\mathcal{M}} \times {\mathbb{R}}_{ + }^{k}$, $J = \left\{ {1, \ldots ,k} \right\}$, $S \in P\left( J \right){ \setminus }\emptyset ,$ and by denoting the i-th row of matrix G $\left( {{\mathbb{X}},z} \right)$ by $g_{i} \left( {{\mathbb{X}},z} \right)$, then

$$ \mathop \sum \limits_{i = 1}^{n} ID^{E} \left( {g_{i} \left( {{\mathbb{X}},z} \right),S} \right) $$

provides the number of poor who are exclusively deprived in the set of attributes S.

Therefore, for any $\left( {{\mathbb{X}},z} \right) \in {\mathcal{M}} \times {\mathbb{R}}_{ + }^{k}$, $j \in J = \left\{ {1, \ldots ,k} \right\}$,

$$ \mathop \sum \limits_{i = 1}^{n} ID^{E} \left( {g_{i} \left( {{\mathbb{X}},z} \right),\left\{ j \right\}} \right) $$

provides the number of poor people who are exclusively deprived in attribute $j$. Moreover, $\sum\nolimits_{i = 1}^{n} {ID^{E} \left( {g_{i} \left( {{\mathbb{X}},z} \right),J} \right)}$ provides the number of people deprived in all the attributes.

We denote, for all $x \in {\mathbb{R}}$, by

$$ ID\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}c} 1 & { \quad if\;x \ne 0 } \\ 0 & {\quad otherwise} \\ \end{array} } \right. $$

Moreover, for any $\left( {{\mathbb{X}},z} \right) \in {\mathcal{M}} \times {\mathbb{R}}_{ + }^{k}$, $j \in J = \left\{ {1, \ldots ,k} \right\}$, $i \in N = \left\{ {1, \ldots ,n} \right\}\;{\text{and}}\;S \in P\left( J \right){ \setminus }\emptyset ,$

$$ \mathop \sum \limits_{i = 1}^{n} ID\left( {\mathop \sum \limits_{{S \in P\left( J \right){ \setminus }\emptyset }} ID^{E} \left( {g_{i} \left( {{\mathbb{X}},z} \right),S} \right)} \right) $$

provides the number of poor in the population;

$$ \frac{{\mathop \sum \nolimits_{i = 1}^{n} ID^{E} \left( {g_{i} \left( {{\mathbb{X}},z} \right),S} \right)}}{{\mathop \sum \nolimits_{i = 1}^{n} ID\left( {\mathop \sum \nolimits_{{S \in P\left( J \right){ \setminus }\emptyset }} ID^{E} \left( {g_{i} \left( {{\mathbb{X}},z} \right),S} \right)} \right)}} $$

is the proportion of poor who are exclusively deprived in all the attributes belonging to S;

$$ \frac{{\mathop \sum \nolimits_{i = 1}^{n} ID^{E} \left( {g_{i} \left( {{\mathbb{X}},z} \right),S} \right)}}{n} $$

is the proportion of the population that is exclusively deprived in all the attributes belonging to S.

Appendix 2: Proof of Proposition 1

Proposition 1

Given$\left( {{\mathbb{X}}, z,m} \right) \in {\mathcal{M}} \times {\mathbb{R}}_{ + }^{k} \times {\mathbb{R}}_{ + }^{k}$the Adjustment Factor,$A_{i} \left( {{\mathbb{X}},z,m} \right)$, satisfies:

• i. It is non-negative; if for any${\text{i}} \in {\text{N}}$and$j, t \in \left\{ {1, \ldots , k} \right\}$, $j \ne t, r_{ij} \left( {{\mathbb{X}}, z, m} \right) > 0$and$g_{it} \left( {{\mathbb{X}}, z} \right) > 0$, then$A_{i} \left( {{\mathbb{X}},z,m} \right)$ > 0; and$A_{i} \left( {{\mathbb{X}},z,m} \right) = 1$when the individual is deprived in all the attributes.

• ii. It is increasing with respect to $\phi_{i} \left( {G\left( {{\mathbb{X}}, z} \right)} \right)$ and decreasing with respect to $\phi_{i} \left( {R\left( {{\mathbb{X}}, z,m} \right)} \right).$

• iii. Deprivation and non-deprivation levels considered to construct it are defined in a parallel way.

• iv. For each$i \in {\text{N}},$the curves defined by the sets of pairs$\left\{ {(\phi_{i} \left( {G\left( {{\mathbb{X}}, z} \right)} \right) ,\phi_{i} \left( {R\left( {{\mathbb{X}}, z,m} \right)} \right) \in \left[ {0,1\left] \times \right[0,1} \right]\text{ / } BC_{i}^{a} \left( {{\mathbb{X}},z,m} \right) = C} \right\}$, called$BC_{i}^{a}$-isopoverty curves, are increasing and convex.

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Proof

• i. It is obviously satisfied by definition of the Adjustment Factor,$A_{i} \left( {{\mathbb{X}},z,m} \right)$.

• ii. $A_{i} \left( {{\mathbb{X}},z,m} \right)$ is increasing with respect to $\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right)$

  since

  $$ \partial A_{i} \left( {{\mathbb{X}},z,m} \right)/\partial \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) = \frac{{\phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right)}}{{\left[ {\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) + \phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right)} \right]^{2} }} > 0. $$

  $A_{i} \left( {{\mathbb{X}},z,m} \right)$ is decreasing with respect to $\phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right)$) since

  $$ \partial A_{i} \left( {{\mathbb{X}},z,m} \right)/\partial \phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right) = \frac{{ - \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right)}}{{\left[ {\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) + \phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right)} \right]^{2} }} < 0. $$

• iii. Deprivation and non-deprivation levels considered to construct the Adjustment Factor are defined in a parallel way since $A_{i} \left( {{\mathbb{X}},z,m} \right) =\upzeta(f_{1} \left( {G\left( {{\mathbb{X}},z} \right)} \right)$, $f_{2} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right)$ where $f_{1} = f_{2} = \phi_{i}$.

• iv. For each $i \in {\text{N}},$ consider the $BC_{i}^{a}$-isopoverty curve $\left\{ {(\phi_{i} \left( {G\left( {{\mathbb{X}}, z} \right)} \right) ,\phi_{i} \left( {R\left( {{\mathbb{X}}, z,m} \right)} \right) \in \left[ {0,1\left] \times \right[0,1} \right]\text{ / } BC_{i}^{a} \left( {{\mathbb{X}},z,m} \right) = C} \right\}$.

  Since

  $$ BC_{i}^{a} \left( {{\mathbb{X}},z,m} \right) = \frac{{\left( {\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) } \right)^{2} }}{{\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) + \phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right)}} = C $$

  then

  $$ \phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right) = \frac{{\left( {\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) } \right)^{2} }}{C} - \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right). $$

  Then,

  $$ d\phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right)/d\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) = \left( {2\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) - C} \right)/C > 0 $$

  since

  $C = BC_{i}^{a} \left( {{\mathbb{X}},z,m} \right) = \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) \times A_{i} \left( {{\mathbb{X}},z,m} \right) \le \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) < 2\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right)$, that is, the $BC_{i}^{a}$-isopoverty curve is increasing.

  Moreover,

  $$ d^{2} \phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right)/d\left( {\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) } \right)^{2} = 2/C > 0, $$

  so, the $BC_{i}^{a}$-isopoverty curve is convex.

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Appendix 3: Proof of Remark 1

• i. The Permanyer’s index is not consistent in the sense that it measures the deprivation and non-deprivation levels by applying different functions to surplus and poverty gaps.

• ii. The construction of Permanyer’s index requires estimating $\left[ {k\left( {k - 1} \right) + 1} \right]$ parameters more than our index (where k is the number of attributes).

• iii. Permanyer’s index is not sensitive to the deprivation levels of the individuals because it is not a function of $\phi_{i} \left( {G\left( {{\mathbb{X}}, z} \right)} \right)$.

• iv. In general, the isopoverty curves defined from Permanyer’s index are not convex.

Proof

• i. See the definition of excess gaps and poverty gaps.

• ii. By definition, Permanyer’s index (Permanyer 2014) requires estimating $\gamma$ and the values of $\lambda_{jl}$ for all $j,l \in \left\{ {1, \ldots ,k} \right\}$, such that $l \ne j$, that is, $\left[ {k\left( {k - 1} \right) + 1} \right]$ parameters more than our proposal, where k is the number of attributes. In Permanyer’s words, his proposal is over-parametrised.

In order to prove (iii) and (iv), we consider the following specification of the general Permanyer’s formulation of a multidimensional poverty index ($P^{P} )$.

$$ P^{P} \left( {{\mathbb{X}}, z,m} \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \psi \left( {\left[ {\mathop \sum \limits_{j = 1}^{k} \left( {g_{ij} \left( {{\mathbb{X}}, z} \right)\mathop \prod \limits_{l = 1}^{k} \varphi_{jl} \left( {r_{il} \left( {{\mathbb{X}}, z,m} \right)} \right)} \right)^{\theta } } \right]^{1/\theta } } \right) $$

with $\varphi_{jl} \left( {r_{il} (\left( {{\mathbb{X}}, z,m} \right)} \right) = 1 + \left( {\lambda_{jl} - 1} \right)r_{il}^{\gamma } \left( {\left( {{\mathbb{X}}, z,m} \right)} \right)$, where $\psi \left( x \right) = \left( {x/k^{1/\theta } } \right)^{\alpha }$, $\lambda_{jl} = \lambda$ for all j, l, $\lambda \in \left( {0,1} \right]$ and $\gamma > 0$. Then,

$$ P^{P} \left( {{\mathbb{X}}, z,m} \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left[ {\left( {\frac{1}{k}\left( {\mathop \sum \limits_{1 \le j \le k} g_{ij}^{\uptheta} \left( {{\mathbb{X}}, z} \right)} \right)} \right)^{1/\theta } \times \left( {A_{i}^{P} \left( {{\mathbb{X}}, z,m} \right)} \right)} \right]^{\alpha } = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {P_{i}^{P} \left( {{\mathbb{X}}, z,m} \right)} \right)^{\alpha } $$

If $1 - \lambda = \beta$, $\beta \in \left[ {0,1} \right)$, the Permanyer’s correction function for any agent i can be written as $A_{i}^{P} = \left( {1 - \beta r_{i1}^{\gamma } \left( {{\mathbb{X}}, z,m} \right)} \right)\left( {1 - \beta r_{i2}^{\gamma } \left( {{\mathbb{X}}, z,m} \right)} \right) \cdots \left( {1 - \beta r_{ik}^{\gamma } \left( {{\mathbb{X}}, z,m} \right)} \right) = \mathop \prod \limits_{l = 1}^{k} \left( {1 - \beta r_{il}^{\gamma } \left( {{\mathbb{X}}, z,m} \right)} \right).$ Thus, the individual poverty levels of any individual $i \in \left\{ {1, \ldots ,n} \right\}$ according to this index is:

$$ P_{i}^{P} \left( {\left( {{\mathbb{X}}, z,m} \right)} \right) = \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) \times A_{i}^{P} \left( {{\mathbb{X}}, z,m} \right) = \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right)\mathop \prod \limits_{l = 1}^{k} \left( {1 - \beta r_{il}^{\gamma } \left( {{\mathbb{X}}, z,m} \right)} \right). $$

• iii. The previous expression shows that Permanyer’s correction function, $A_{i}^{P} \left( {{\mathbb{X}}, z,m} \right)$ is not a function depending on $\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right)$. That is, it is not sensitive to the level of deprivation of the individual since $\partial A_{i}^{P} \left( {{\mathbb{X}}, z,m} \right)/\partial \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) = 0.$

• iv. Next counter example shows that the isopoverty curves defined from Permanyer’s index, $\left\{ {(\phi_{i} \left( {G\left( {{\mathbb{X}}, z} \right)} \right) ,\phi_{i} \left( {R\left( {{\mathbb{X}}, z,m} \right)} \right) \in \left[ {0,1\left] \times \right[0,1} \right]\text{ / } P_{i}^{P} \left( {{\mathbb{X}},z,m} \right) = C} \right\},$ are not convex.

  Suppose there are two attributes, $j = \left\{ {1,2} \right\}$, $\theta = 2$, $\gamma = 1$, $\beta = 1/2$, $g_{i} = (g_{i1} ,0)$ and $r_{i} = (0,r_{i2} )$.

  Then, $\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) = g_{i1} /\sqrt 2$, $A_{i}^{P} = 1 - (r_{i2} /2)$ and $P_{i}^{P} = \left( {g_{i1} /\sqrt 2 } \right)\left[ {1 - (r_{i2} /2)} \right]$.

  If $g_{i} = \left( {0.5,0} \right)$ and $r_{i} = \left( {0,0.71} \right)$, we have that

  $$ P_{i}^{P} = \left( {0.5/\sqrt 2 } \right)\left[ {1 - (0.71/2)} \right] = 0.3225/\sqrt 2 . $$

  Now, we want to know the non-deprivation level, $r_{i2}^{*}$, which combined with the previous deprivation level increased by 0.05, $g_{i1} + 0.05,$ provides the same poverty level, $P_{i}^{P} = 0.3225/\sqrt 2 .$

  Then, solving the equation

  $$ 0.3225/\sqrt 2 = \left[ {\left( {0.5/\sqrt 2 } \right) + 0.05} \right]\left[ {1 - \left( {r_{i2}^{*} /2} \right)} \right] $$

  we obtain $r_{i2}^{*} = 0.869830152$, so $\Phi _{i}^{R} \left[ {(r_{i1} ,r_{i2}^{*} )} \right] = 0.615062799$. Therefore, the pair $\left( {\left( {0.5/\sqrt 2 } \right) + 0.05, 0.615062799} \right)$ also belongs to the isopoverty curve of level $0.3225/\sqrt 2$.

  Analogously, we want to know the non-deprivation level which combined with the initial deprivation level increased by another 0.1, $g_{i1} + 0.1$, provides the same isopoverty level, $P_{i}^{P} = 0.3225/\sqrt 2 .$

  Then, solving the equation

  $$ 0.3225/\sqrt 2 = \left[ {\left( {0.5/\sqrt 2 } \right) + 0.1} \right]\left[ {1 - \left( {r_{i2}^{**} /2} \right)} \right] $$

  we obtain $r_{i2}^{**} = 0.994420759$, so $\Phi _{i}^{R} \left[ {(r_{i1} ,r_{i2}^{*} )} \right] = 0.703161662$. Therefore, the pair $\left( {\left( {0.5/\sqrt 2 } \right) + 0.1, 0.703161662} \right)$ also belongs to the isopoverty curve of level $0.3225/\sqrt 2$.

  Then, we have that when $\Phi _{i}^{G} = 0.5/\sqrt 2$ and it increases by 0.05, it has to be compensated by an increase of the non-deprived level that amounts to $\Phi _{i}^{R} \left[ {(r_{i1} ,r_{i2}^{*} )} \right] -\Phi _{i}^{R} \left[ {(r_{i1} ,r_{i2} )} \right] = 0.113016985$ to maintain the poverty level at $0.3225/\sqrt 2 .$ However, when $\Phi _{i}^{G} = \left( {0.5/\sqrt 2 } \right) + 0.05$ and it increases by 0.05, it has to be compensated by a smaller increase of the non-deprived level, which amounts to $\Phi _{i}^{R} \left[ {(r_{i1} ,r_{i2}^{**} )} \right] -\Phi _{i}^{R} \left[ {(r_{i1} ,r_{i2}^{*} )} \right] = 0.088098863$, to maintain the poverty level at $0.3225/\sqrt 2$, which is completely unnatural in the educational context.

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Appendix 4: Proof of Proposition 2

Proposition 2

The $BC^{a}$ index, $BC^{a} :{\mathcal{M}} \times {\mathbb{R}}_{+}^{k} \times {\mathbb{R}}_{+}^{k}\,{\rightarrow}\,\left[{0,1} \right]$ , satisfies SFI, WF, SYM, NOM, MON, NDMON, CONT, SI, PP and SUD.

Proof

• (SFI) Let us consider $\left( {{\mathbb{X}}, z,m} \right) \;{\text{and}} \;\left( {{\mathbb{Y}},z,m} \right) \in {\mathcal{M}} \times {\mathbb{R}}_{ + }^{\text{k}} \times {\mathbb{R}}_{ + }^{\text{k}}$, $j \in \left\{ {1, \ldots ,k} \right\},$$i \in \left\{ {1, \ldots ,n} \right\}$ if ${{x}}_{{ij}} \ge {{z}}_{{j}}$, $y_{ij} = x_{ij} +\updelta, \;{\text{where}} \;\updelta > 0,\; y_{tj} = x_{tj} \;{\text{for}}\; {\text{all}}\; t \ne i \;{\text{and}}\; j \in \left\{ {1, \ldots ,k} \right\},$ then

  $$ BC_{i}^{a} \left({{\mathbb{X}},z,m} \right) > 0\,{\leftrightarrow}\,\phi_{i} \left({G\left({{\mathbb{X}},z} \right)} \right) > 0\,{\leftrightarrow}\,\phi_{i} \left({G\left({{\mathbb{Y}},z} \right)} \right) > 0\,{\leftrightarrow}\,BC_{i}^{a} \left({{\mathbb{Y}},z,m} \right) > 0. $$

• (WF) If for some $i, \,x_{ik} { \geqq }z_{k} \forall k,$${\text{then}} \;g_{ik} = 0\,\, \forall k, \;{\text{which}} \;{\text{implies}}\; \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) = 0, \;{\text{so}}\; BC_{i}^{a} \left( {{\mathbb{X}},z,m} \right) = BC_{i}^{a} \left( {{\mathbb{Y}},z,m} \right) = 0.$ Moreover, since $\forall r \ne i$$BC_{\text{r}}^{a} \left( {{\mathbb{X}},z,m} \right) = BC_{\text{r}}^{a} \left( {{\mathbb{Y}},z,m} \right), \;{\text{then}} \;BC^{a} \left( {{\mathbb{Y}},z,m} \right)$ = $BC^{a} \left( {{\mathbb{X}},z,m} \right).$

• (SYM) For any $\left( {{\mathbb{X}},z,m} \right) \in {\mathcal{M}} \times {\mathbb{R}}_{ + }^{k} \times {\mathbb{R}}_{ + }^{\text{k}}$, if $\Pi$ is any permutation of the rows, then

  $$ {\Pi {\mathbb{X}}} = ({\text{x}}_{\sigma \left( i \right)j} )_{{\begin{subarray}{*{20}c} {i = 1,2,..,n} \\ {j = 1,2, \ldots ,k} \\ \end{subarray} }}. $$

Since $g_{ij} \left( {{\mathbb{X}},z} \right) = max\left\{ {0,\frac{{z_{j} - x_{ij} }}{{z_{j} }}} \right\} = max\left\{ {0,\frac{{z_{j} - x_{\sigma \left( i \right)j} }}{{z_{j} }}} \right\} = g_{\sigma \left( i \right)j} \left( {{\varPi {\mathbb{X}}},z} \right)$,

it follows that

$$ \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) = \left[ {\frac{1}{k}\left( {\mathop \sum \limits_{1 \le j \le k} g_{ij}^{\uptheta} \left( {{\mathbb{X}},z} \right)} \right)} \right]^{{\frac{1}{\theta }}} = \left[ {\frac{1}{k}\left( {\mathop \sum \limits_{1 \le j \le k} g_{\sigma \left( i \right)j}^{\uptheta} \left( {{\varPi {\mathbb{X}}},z} \right)} \right)} \right]^{{\frac{1}{\theta }}} = \phi_{\sigma \left( i \right)} \left( {G\left( {{\varPi {\mathbb{X}}},z} \right)} \right) $$

and

$$ \phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right) = \left[ {\frac{1}{k}\left( {\mathop \sum \limits_{1 \le j \le k} r_{ij}^{\uptheta} \left( {{\mathbb{X}},z,m} \right)} \right)} \right]^{{\frac{1}{\theta }}} = \phi_{\sigma \left( i \right)} \left( {R\left( {{\varPi {\mathbb{X}}},z,m} \right)} \right) $$

then,

$$ BC_{i}^{a} \left( {{\mathbb{X}},z,m} \right) = \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right)A_{i} \left( {{\mathbb{X}},z,m} \right) = \phi_{\sigma \left( i \right)} \left( {G\left( {{\varPi {\mathbb{X}}},z} \right)} \right)A_{\sigma \left( i \right)} \left( {{\varPi {\mathbb{X}}},z,m} \right) = BC_{\sigma \left( i \right)}^{a} \left( {{\varPi {\mathbb{X}}},z,m} \right) $$

$$ {\text{and}}\;BC^{a} \left( {{\mathbb{X}},z,m} \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {BC_{i}^{a} \left( {{\mathbb{X}},z,m} \right)} \right)^{\alpha } = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {BC_{\sigma \left( i \right)}^{a} \left( {{\varPi {\mathbb{X}}},z} \right)} \right)^{\alpha } = BC^{a} \left( {{\varPi {\mathbb{X}}},z} \right) $$

• (NOM)$For\;all\; z \in {\mathbb{R}}_{ + }^{k} , {\mathbb{X}} \in {\mathcal{M}}, if\,x_{ij} { \geqq }z_{j} \forall i \in \left\{ {1, 2, \ldots , n} \right\}, \;{\text{and}}\; \forall j \in \left\{ {1, 2, \ldots , k} \right\},$ then $\phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) = 0,$ so $BC_{i}^{a} \left( {{\mathbb{X}},z,m} \right) = 0$, therefore $BC^{a} \left( {{\mathbb{X}},z,m} \right) = 0.$

• (MON) Since $y_{ij} = x_{ij} + \delta$,

  (a) if $y_{ij} < z_{j} ,$ then $g_{ij} \left( {{\mathbb{Y}},{\text{z}}} \right) = g_{ij} \left( {{\mathbb{X}},{\text{z}}} \right) - \frac{\delta }{{z_{j} }}$ and $r_{ij} \left( {{\mathbb{Y}},{\text{z}},{\text{m}}} \right) = 0,$ therefore $g_{ij} \left( {{\mathbb{Y}},{\text{z}}} \right) < g_{ij} \left( {{\mathbb{X}},{\text{z}}} \right)$ and $\phi_{i} \left( {G\left( {{\mathbb{Y}},z} \right)} \right) < \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right)$. Moreover $\phi_{i} \left( {R\left( {{\mathbb{Y}},z,m} \right)} \right) = \phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right)$. By applying (ii) of Proposition 1, $A_{i} \left( {{\mathbb{Y}},z,m} \right) < A_{i} \left( {{\mathbb{X}},z,m} \right)$, so $BC_{\text{i}}^{a} \left( {{\mathbb{Y}},z,m} \right) < BC_{\text{i}}^{a} \left( {{\mathbb{X}},z,m} \right),$ then $BC^{a} \left( {{\mathbb{Y}},z,m} \right) < BC^{a} \left( {{\mathbb{X}},z,m} \right).$

  (b) If $y_{ij} >$$z_{j} , g_{ij} \left( {{\mathbb{Y}},{\text{z}}} \right) = 0, g_{ij} \left( {{\mathbb{X}},{\text{z}}} \right) > 0$ and $r_{ij} \left( {{\mathbb{Y}},{\text{z}},{\text{m}}} \right) > 0$, then $\phi_{i} \left( {G\left( {{\mathbb{Y}},z} \right)} \right) < \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right)$ and $\phi_{i} \left( {R\left( {{\mathbb{Y}},z,m} \right)} \right) > \phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right)$. Analogously by applying (ii) in Proposition 1, $A_{i} \left( {{\mathbb{Y}},z,m} \right) < A_{i} \left( {{\mathbb{X}},z,m} \right)$, so $BC_{\text{i}}^{a} \left( {{\mathbb{Y}},{\text{z}},{\text{m}}} \right) < BC_{\text{i}}^{a} \left( {{\mathbb{X}},{\text{z}},{\text{m}}} \right),$ then $BC^{a} \left( {{\mathbb{Y}},z,m} \right) < BC^{a} \left( {{\mathbb{X}},z,m} \right)$.

• (NDMON) Since $x_{ij} > z_{j}$, then $y_{ij} = x_{ij} + \delta > x_{ij} > z_{j}$ implies that $g_{ij} \left( {{\mathbb{Y}},{\text{z}}} \right) = g_{ij} \left( {{\mathbb{X}},{\text{z}}} \right) = 0$, therefore $\phi_{i} \left( {G\left( {{\mathbb{Y}},z} \right)} \right) = \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right)$. Moreover,

  $$ r_{ij} \left( {{\mathbb{Y}},z,m} \right) = \frac{{x_{ij} + \delta - z_{j} }}{{m_{j} - z_{j} }} > \frac{{x_{ij} - z_{j} }}{{m_{j} - z_{j} }} = r_{ij} \left( {{\mathbb{X}},z,m} \right),$$

  then

  $$ \phi_{i} \left( {R\left( {{\mathbb{Y}},z,m} \right)} \right) > \phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right). $$

  So, by applying Proposition 1, $A_{i} \left( {{\mathbb{Y}},z,m} \right) < A_{i} \left( {{\mathbb{X}},z,m} \right)$, and

  $$ BC_{i}^{a} \left( {{\mathbb{Y}},z,m} \right) \leq BC_{i}^{a} \left( {{\mathbb{X}},z,m} \right) = \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right) \times A_{i} \left( {{\mathbb{X}},z,m} \right).\quad Therefore\quad BC^{a} \left( {{\mathbb{Y}},z,m} \right) \leq BC^{a} \left( {{\mathbb{X}},z,m} \right). $$

• $\left( {CONT} \right)\,BC^{a} \left( { {\mathbb{X}} , z,m} \right)$ is a composition of continuous functions.

• (SI) Since $x_{ij}^{'} = \lambda_{j} x_{ij},\,z_{j}^{'} = \lambda_{j} z_{j} , m_{j}^{'} = \lambda_{j} m_{j}$, it is obtained that

  $$ g_{ij} \left( {{{\mathbb{X}}^{\prime}},z'} \right) = max\left\{ {0,\frac{{z_{j}^{'} - x_{ij}^{'} }}{{z_{j}^{'} }}} \right\} = max\left\{ {0,\frac{{\lambda_{j} z_{j} - \lambda_{j} x_{ij} }}{{\lambda_{j} z_{j} }}} \right\} = g_{ij} \left( {{\mathbb{X}},{\text{z}}} \right) $$

  and

  $$ r_{ij} \left( {{{\mathbb{X}}^{\prime}},z',m'} \right) = max\left\{ {0,\frac{{x_{ij}^{'} - z_{j}^{'} }}{{m_{j}^{'} - z_{j}^{'} }}} \right\} = r_{ij} \left( {\mathbb{X}}, z, m \right) $$

  Then,

  $\phi_{i} \left( {G\left( {{{\mathbb{X}}^{\prime}},z'} \right)} \right) = \phi_{i} \left( {G\left( {{\mathbb{X}},z} \right)} \right)\, {\text{and}}\, \phi_{i} \left( {R\left( {{{\mathbb{X}}^{\prime}},z',m'} \right)} \right) = \phi_{i} \left( {R\left( {{\mathbb{X}},z,m} \right)} \right),$ therefore

  $$ BC^{a} \left( {{\mathbb{X}},z,m} \right) = BC^{a} \left( {{{\mathbb{X}}^{\prime}},z^{\prime},m'} \right). $$

• (PP) For any $\left( {{\mathbb{X}},z,m} \right) \in {\mathcal{M}} \times {\mathbb{R}}_{ + }^{k} \times {\mathbb{R}}_{ + }^{k} , r \in {\mathbb{N}}$, finite, if ${\mathbb{X}}^{r}$ is the r-fold of ${\mathbb{X}}$,

  $$ \begin{aligned} BC^{a} \left( {{\mathbb{X}}^{r} ,z,m} \right) & = \frac{1}{nr} \mathop \sum \limits_{i = 1}^{nr} \left( {BC_{i}^{a} \left( {{\mathbb{X}}^{r} ,z,m} \right)} \right)^{\alpha } = \frac{1}{nr}\mathop \sum \limits_{i = 1}^{n} \mathop \sum \limits_{h = 1}^{r} \left( {BC_{{i_{h} }}^{a} \left( {{\mathbb{X}}^{r} ,z,m} \right)} \right)^{\alpha } \\ & = \frac{1}{nr}\mathop \sum \limits_{i = 1}^{n} r\left( {BC_{i}^{a} \left( {{\mathbb{X}}^{r} ,z,m} \right)} \right)^{\alpha } = BC^{a} \left( {{\mathbb{X}},z,m} \right) \\ \end{aligned} $$

• (SUD) For any $\left( {{\mathbb{X}},z,m} \right) \in {\mathcal{M}} \times {\mathbb{R}}_{ + }^{\text{k}} \times {\mathbb{R}}_{ + }^{\text{k}}$ and any partition of the population into s subgroups, s≥ 2:

  $$ BC^{a} \left( {{\mathbb{X}},z,m} \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {BC_{i}^{a} \left( {{\mathbb{X}},z,m} \right)} \right)^{\alpha } = \frac{{n_{l} }}{n}\mathop \sum \limits_{l = 1}^{S} \frac{1}{{n_{l} }}\mathop \sum \limits_{i = 1}^{{n_{l} }} \left( {BC_{i}^{a} \left( {{\mathbb{X}},z,m} \right)} \right)^{\alpha } = \frac{{n_{l} }}{n}\mathop \sum \limits_{l = 1}^{S} BC^{a} \left( {{\mathbb{X}}_{l} ,z,m} \right) $$

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Appendix 5: Proof of Proposition 3

Proposition 3

The $BC^{a}$ index, $BC^{a} :{\mathcal{M}} \times {\mathbb{R}}_{+}^{k} \times {\mathbb{R}}_{+}^{k}\,{\rightarrow}\,\left[{0,1} \right]$, satisfies FD if $\alpha = \theta$.

Proof

For any $\left( {{\mathbb{X}},z,m} \right) \in {\mathcal{M}} \times {\mathbb{R}}_{ + }^{k} \times {\mathbb{R}}_{ + }^{k}$, let $x_{j} = \left( {x_{ij} } \right)_{{1{ \leqq }i{ \leqq }n}}$ denote the j-th column of ${\mathbb{X}}$, which represents the achievement vector in attribute j of all the individuals $i = 1, ..,n$. Then, the poverty level due to attribute or dimension j, according to our poverty index $BC^{a}$, denoted by $BC_{dj}^{a}$, is defined for $\alpha = \theta$, as:

$$ BC_{dj}^{a} \left( {{\mathbb{X}},z,m} \right) = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left[ {\left( {\left( {g_{ij} \left( {{\mathbb{X}},z} \right)} \right)^{\theta } } \right)^{{\frac{1}{\theta }}} A_{i} \left( {{\mathbb{X}},z,m} \right)} \right]^{\theta } . $$

Then,

$$ \begin{aligned} BC^{a} \left( {{\mathbb{X}},z,m} \right) & = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left[ {\left( {\frac{1}{k}\mathop \sum \limits_{1 \le j \le k} g_{ij}^{\theta } \left( {{\mathbb{X}},z} \right)} \right)^{{\frac{1}{\theta }}} A_{i} \left( {{\mathbb{X}},z,m} \right)} \right]^{\theta } \\ & = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \frac{1}{k}\mathop \sum \limits_{j = 1}^{k} g_{ij}^{\theta } \left( {{\mathbb{X}},z} \right)A_{i} \left( {{\mathbb{X}},z,m} \right)^{\theta } \\ & = \frac{1}{k}\mathop \sum \limits_{j = 1}^{k} \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} g_{ij}^{\theta } \left( {{\mathbb{X}},z} \right)A_{i} \left( {{\mathbb{X}},z,m} \right)^{\theta } \\ & = \frac{1}{k}\mathop \sum \limits_{j = 1}^{k} \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left[ {\left( {\left( {g_{ij} \left( {{\mathbb{X}},z} \right)} \right)^{\theta } } \right)^{{\frac{1}{\theta }}} A_{i} \left( {{\mathbb{X}},z,m} \right)} \right]^{\theta } \\ & = \frac{1}{k}\mathop \sum \limits_{j = 1}^{k} BC_{dj}^{a} \left( {{\mathbb{X}},z,m} \right). \\ \end{aligned} $$

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