1.3 The Aggregation Step
1.3.4 The Axiomatic Approach
Chakravarty, Mukherjee and Ranade (1998) were the first to apply the axiomatic approach in order to derive multidimensional poverty indices. The idea behind this approach is to define a list of desirable properties that a poverty index should satisfy. This list is then utilized to derive those classes of poverty indices that satisfy these properties. Thus, the axiomatic approach provides the most transparent way to take care of value judgments by explicitly defining properties that poverty indices may or may not satisfy. It is this specific property that makes the axiomatic approach so appealing for the analysis of the interplay between the two concepts of efficiency and distributive justice, the main objective of this volume.
Most of the axioms that are introduced in the following analysis are so-called core axioms that have been derived by the generalization and extension of the core axioms of the one-dimensional framework to fit the multidimensional framework (e.g. Chakravarty, Mukherjee and Ranade 1998, Bourguignon and Chakravarty 1999, Tsui 2002, Bourguignon and Chakravarty 2003, Chakravarty and Silber 2008). However, it is the main message of this volume that the axioms that have been derived so far are unable to sufficiently capture the interplay between the two concepts of efficiency and distributive justice. In fact, as will be demonstrated in the following, they either account for the one or the other but not for both.
This chapter concentrates on the axiomatic foundation for ordinal poverty measures; the second chapter will do the same for the cardinal case.
12 Note, however, that the information theory approach reaches far beyond being a simple add-on for other methods. Indeed, it opens the way to more generalized poverty measures which are more sophisticated than, for instance, the average functions of the axiomatic approach (Maasoumi and Lugo 2008, p. 8).
The Axiomatic Foundation for Ordinal Poverty Measures Non-Distributional Axioms
Anonymity (AN): For any z∈Zand X∈ Xn ,P(X;z)=P(ΠX;z) where Π is any permutation matrix of appropriate order.
AN states that any characteristic of persons apart from the attributes j are irrelevant for
MN requires poverty measures not to increase if, ceteris paribus, the condition of individual h that is poor with respect to attribute l improves.
Principle of Population (PP): If for anyz∈Z,X∈ Xn, andm∈ℕ Xm is a m-fold replication of X, thenP(Xm;z)=P(X;z).
PP ensures that poverty measures do not depend on population size, thereby allowing cross-population and cross-time comparisons.
SF demands that giving a person more of an attribute with respect to which this person is not deprived will not change the poverty measure – even if the person is deprived in (an)other attribute(s). SF provides the theoretical foundation for the component poverty approach13. Subgroup Decomposability (SD): For any X1,...,Xv∈ Xn and z∈Z, poverty levels of population subgroups according to ethnic, spatial or other criteria which makes it a valuable property for policy makers.
Factor Decomposability (FD): For anyz∈ZandX∈ Xn, ( ; ) ( ; )
FD facilitates the decomposition of poverty measures into different attribute combinations.
Joint fulfilment of FD and SD allows a twofold decomposition of overall poverty according to
13 There exists a weaker version of this property that provides the theoretical foundation for the aggregate poverty line approach (e.g. Maasoumi and Lugo 2008).
subgroup-attribute combinations that improves the targeting of poverty-alleviating policies. A rather restricting implication of FD is that it requires poverty indices to be additive.
Normalization (NM): For anyz∈ZandX∈ Xn,P(X;z)=1 if xij =0∀i, j and P(X;z)=0 if .
, j i z
xij ≥ j∀ Thus, P(X;z)∈
[ ]
0,1.NM is a technical property that simply requires poverty measures to be equal to zero in case all individuals are non-poor and equal to one in case all individuals are completely deprived.
Distributional Axioms
I will now turn to the axioms that deal specifically with inter-personal inequality. In the context of ordinal poverty measures, inter-personal inequality is captured by a majorization criterion proposed by Chakravarty and D’Ambrosio (2006) in the context of social exclusion and formally introduced in the context of multidimensional poverty measurement by Jayaraj and Subramanian (2010):
Equality-Promoting Change (EPC): For any z∈Zand X,X′∈ Xn, X′is obtained from Xby an equality-promoting change, if for some individuals g and h,
h g h h g
g c c c c c
c′ = +1, ′ = −1, ′ ≤ ′ and c′i =ci∀i≠g,h.
Nonincreasingness under Equality-Promoting Change (NEPC): For any z∈Zand
′∈ X
X, Xn if X′is obtained from Xby an equality-promoting change, then )
; ( )
;
(X z ≥P X′ z
P .
A change is equality-promoting whenever the difference in the number of simultaneously suffered deprivations between two individuals is reduced. Jayaraj and Subramanian (2010) claim that such an equality-promoting change should not increase poverty.
However, the equality-promoting change does not distinguish between the attributes that are affected by the change, i.e. is insensitive to any kind of association among attributes and is therefore unable to account for the concept of efficiency. However, ‘[a]n attempt to achieve equality of capabilities – without taking note of aggregative considerations – can lead to severe curtailment of the capabilities that people can altogether have’ (Sen, 1992, pp. 7-8).
In fact, an increase in inter-personal inequality can increase or decrease poverty, depending on the assumed relationship among attributes. More precisely, poverty will increase as long as there is no complementary relationship among attributes since the two concepts of distributive justice and efficiency work in the same direction. However, this is no longer true in the case of complements. In this case, distributive justice considerations suggest an increase in poverty while efficiency considerations suggest a decrease in poverty. It
depends on the importance that is attributed to distributive justice considerations as well as on the degree of complementarity which of the effects will predominate in the end.
A rather vivid example is provided by Duclos, Sahn and Younger (2006, p. 950) who observe that complementarities exist between the two poverty dimensions education and nutrition as better nourished children learn better. If the degree of this complementarity is strong enough, the authors argue, it might even overcome the ‘usual ethical judgement’
(Duclos, Sahn and Younger, 2006, p. 950) that favours those deprived in more dimensions so that overall poverty would actually decrease in case education would be transferred from the poorly to the better nourished.
In order to capture this idea, I introduce the concept of an Inequality Increasing Switch:
Inequality Increasing Switch (IIS): Define di =#{cijcij =1}. Then, for two individuals g and h such that dg >dh >1, Matrix X is said to be obtained from matrix X′by an inequality increasing switch of attribute l if xhl′ <zl < xgl′ , xgl =xhl′ ,xhl = xgl′ and
i l j x x h g i x
xil = il′∀ ≠ , ; ij = ij′∀ ≠ ,∀ .
An inequality increasing switch is a switch of attributes that increases (reduces) the number of deprivations suffered by the person with higher (lower) initial deprivation.
Based on the Inequality Increasing Switch, I introduce the following property that captures the interplay between the two concepts of distributive justice and efficiency:
Sensitivity to Inequality Increasing Switch (SIIS): For any z∈Z and X,X′,X′′∈ Xn if X′
is obtained from X by an inequality increasing switch of non-complementary attributes, then ).
; ( )
;
(X z ≤P X′ z
P Further, if X′′ is obtained from X by an inequality increasing switch of complement attributes, then P(X;z)≤≥P(X′′;z)≤P(X′;z).
As mentioned before, in the case of a non-complementary relationship between attributes there is no conflict between the concepts of distributive justice and efficiency. Thus, SIIS is equivalent to NEPC, i.e. an inequality increasing switch should not decrease. The situation changes, however, in the case of complements. In this case, the concepts of distributive justice and efficiency work in different directions and it depends on the importance attributed to distributive justice considerations as well as on the degree of complementarity whether poverty increases or decreases through an inequality increasing switch.
In the following section I will utilise the axioms developed here in order to derive a new class of poverty indices and compare it to the main other indices that have been developed in an ordinal context.