The Measurement of Multidimensional Poverty

Abstract

Many authors have insisted on the necessity of defining poverty as a multidimensional concept rather than relying on income or consumption expenditures per capita. Yet, not much has actually been done to include the various dimensions of deprivation into the practical definition and measurement of poverty. Existing attempts along that direction consist of aggregating various attributes into a single index through some arbitrary function and defining a poverty line and associated poverty measures on the basis of that index. This is merely redefining more generally the concept of poverty, which then essentially remains a one-dimensional concept. The present paper suggests that an alternative way to take into account the multidimensionality of poverty is to specify a poverty line for each dimension of poverty and to consider that a person is poor if he/she falls below at least one of these various lines. The paper then explores how to combine these various poverty lines and associated one-dimensional gaps into multidimensional poverty measures. An application of these measures to the rural population in Brazil is also given with poverty defined on income and education.

Keywords

It is a pleasure to acknowledge the permission granted by Springer Nature to reprint this article from Journal of Economic Inequality, Volume 1, 2003.

Appendix: Formal Statement of the Axioms Used in the Paper

Strong Focus (SF). For any $n \in N,(X,Y) \in M^{n} ,z \in Z,j \in \{ 1,2, \cdots ,m\}$, if (i) for any i such that $x_{ij} \ge z_{j} ,y_{ij} = x_{ij} + \delta$, where $\delta > 0$, (ii) $y_{tj} = x_{tj}$ for all $t \ne i$, and (iii) $y_{is} = x_{is}$, for all $s \ne j$ and for all i, then $P\,(Y;z) = P\,(X;z)$.

Weak Focus (WF). For any $n \in N,(X,Y) \in M^{n} ,z \in Z$, if for some $i,x_{ik} \ge z_{k} ,$ for all k and (i) for any $j \in \{ 1,2, \ldots ,m\}$, $y_{ij} = x_{ij} + \delta$, where $\delta > 0$, (ii) $y_{it} = x_{it}$ for all $t \ne j$ and (iii) $y_{rs} = x_{rs}$ for all $r \ne i$ and s then $P(Y;z) = P(X;z)$.

Symmetry (SM): For any $(X;z) \in M \times Z,P(X;z) = P(\Pi \,X;z)$, where $\Pi$ is any permutation matrix of appropriate order.

Monotonicity (MN). For any $n \in N,(X,Y) \in M^{n} ,z \in Z,j \in \{ 1,2, \cdots ,m\}$, if (i) for any i $y_{ij} = x_{ij} + \delta$, where $x_{ij} < z_{j}$, $\delta > 0$, (ii) $y_{tj} = x_{tj}$ for all $t \ne i$, and (iii) $y_{is} = x_{is}$, for all $s \ne j$ and for all i, then $P(Y;z) \le P(X;z)$.

Continuity (CN): For any $z \in Z,P()$ is continuous on M.

Principle of Population (PP). For any $(X;z) \in M \times Z,k \in N,P(X^{k} ;z) = P(X;z)$, where $X^{k}$ is the k-fold replication of X.

Scale Invariance (SI). For any $(X;z) \in M \times Z,k \in N,P(X;z) = P(X^{\prime};z^{\prime})$ where $X^{\prime} = X\Lambda ,z^{\prime} = z\Lambda$, $\Lambda$ being the diagonal matrix diag $(\lambda_{1} , \cdots ,\lambda_{m} ),\lambda_{i} > 0$ for all i.

Subgroup Decomposability (SD). For any $X^{1} ,X^{2} , \ldots ,X^{K} \in M$ and $z \in Z$:

$$ {P^n}(X;z) = \sum\nolimits_{i = 1}^K {\frac{{{n_i}}}{n}\,\,{P^{{n_i}}}\,({X^i};z)} , $$

where $X \in M$ is the attribute matrix $\left[{\begin{array}{*{20}c} {X^1} \\ {X^2} \\ \\ \\ {X^k} \\ \end{array} }\right]$ witn n rows and m columns, $n_{i}$ is the population size corresponding to $X^{i}$ and $n = \sum\nolimits_{i = 1}^{K} {n_{i} }$.

Definition of a Pigou–Dalton Progressive Transfer. Matrix X is said to be obtained from $Y \in M^{n}$ by a Pigou–Dalton progressive transfer of attribute j from one poor person to another if for some persons i, t: (i) $y_{tj} < y_{ij} < z_{j}$, (ii) $x_{ij} - y_{ij} = y_{ij} - x_{ij} > 0,\,x_{ij} \ge x_{tj}$, (iii) $x_{rj} = y_{rj}$ for all $r \ne i,\,t$ and (iv) $x_{rk} = y_{rk}$ for all $k \ne j$ and all r.

One-dimensional Transfer Principle (OTP). For all n ∈ N and $Y\, \in \,M^{n}$, if X is obtained from Y by a Pigou–Dalton progressive transfer of some attribute between two poor, then $P\,(X;\,z) \le P\,(Y;z)$, where z ∈ Z is arbitrary.

Multidimensional Transfer Principle (MTP). For any $(Y;z)\, \in \,M\, \times Z$,if X is obtained form Y by multiplying $Y_{p}$ by a bistochastic matrix B and $BY_{p}$ is not a permutation of the rows of $Y_{p}$, then $P\,(X;z) \le P\,(Y;z)$, given that the attributes of the non-poor remain unchanged, where $Y_{p}$ is the bundle of attributes possessed by the poor as defined with the attribute matrix Y.

Definition of a Correlation Increasing Switch. For any $X \in M^{n} ,n \ge 2$, (j, k) ∈{1,2,…,m}, suppose that for some $i,t,x_{ij} < x_{tj} < z_{j}$ and $x_{tk} < x_{ik} < z_{k}$. Y is then said to be obtained from X by a ‘correlation increasing switch’ between two poor if: (i) $y_{ij} = x_{tj}$, (ii) $y_{tj} = x_{ij}$; (iii) $y_{rj} = x_{rj}$ for all $r \ne i,t$, and (iv) $y_{rs} = x_{rs}$ for all $s \ne j$ and for all r.

Non-decreasing Poverty Under Correlation Increasing Switch (NDCIS). For any $n \in N$ and $n \ge 2,X \in M^{n}$, z ∈ Z, if Y is obtained from X by a correlation increasing switch, then $P(Y;z) \ge P(X;z)$.

The converse property is denoted by NICIS.