Deutsches Institut fr Entwicklungspolitik (DIE)

Integrating Inter-Personal Inequality in Counting Poverty Indices:The Correlation Sensitive Poverty Index Nicole Rippin24 June 2014 Deutsches Institut fr Entwicklungspolitik (DIE)2OutlineIntroductionThe identification of the poorThe aggregation of the individual characteristics of the poor in the ordinal frameworkIII.I The Multidimensional Poverty Index (MPI)III.II The Correlation Sensitive Poverty Index (CSPI)Empirical applicationConclusionI. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion Deutsches Institut fr Entwicklungspolitik (DIE)3Insufficient income has for a long time been considered to be a good proxy for poverty in all its various facets.The income approach, however, relies on critical assumptions:Over time, serious concerns have been raised regarding the appropriateness of these simplifying assumptions (e.g. Rawls, 1971; Sen 1985, 1992; Drze and Sen, 1989; UNDP, 1997).Economic Resources Assumption: equal individual conversion factorsIgnoring in particular:Personal heterogeneitiesVariations in physical environmentDifferences in social climateUtilityGoodsAssumption: perfect and complete marketsIgnoring in particular:The role of public goodsLimited accessAsymmetric informationChoiceConversionIntroductionI. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion Deutsches Institut fr Entwicklungspolitik (DIE)I. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion It was Amartya Sen, who developed a new approach to measure poverty and welfare: the capability approach (1979, 1985, 1992, 1999, 2009).Thus, the capability approach implies a multidimensional approach to poverty measurement.Economic Resources Assumption: equal individual conversion factorsIgnoring in particular:Personal heterogeneitiesVariations in physical environmentDifferences in social climateUtilityGoodsAssumption: perfect and complete marketsIgnoring in particular:The role of public goodsLimited accessAsymmetric informationChoiceConversionCapability SetFunctioning BundleChoiceChoiceIntroduction4 Deutsches Institut fr Entwicklungspolitik (DIE)I. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion Empirical evidence demonstrates that considerable population shares might be multidimensional poor but not income poor, and vice versa (e.g. Klasen, 2000).Already a strong trend in the last decade, multidimensional poverty measurement has been given a further boost through the introduction of the first internationally comparable Multidimensional Poverty Index MPI (Alkire and Santos, 2010).However, in the multidimensional framework inequality does not only exist within, but also across dimensions; consequently there exists a tension between the two concepts of distributive justice and efficiency that does not exist in the one-dimensional framework:[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).Introduction5 Deutsches Institut fr Entwicklungspolitik (DIE)I. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion In the ordinal context, inequality across dimensions is usually considered as the spread of simultaneous deprivations across the population, thus only accounting for distributive justice.This work suggests to define inequality across dimensions as the correlation-sensitive spread of simultaneous deprivations across the population.This rigour definition accounts for the tension between the two concepts of distributive justice and efficiency that Sen mentioned and has strong implications on the identification of the poor and the aggregation of individual poverty characteristics.Introduction6 Deutsches Institut fr Entwicklungspolitik (DIE)I. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion 7Theoretical Background represents a set of n persons

represents a set of k poverty attributes

K represents the respective vector of threshold levels

+K represents a vector of weights such that

K represents the achievement vector of person i

Person i is deprived with respect to attribute j if

represents the deprivation vector of person i suchthat if and if

For any , the deprivation matrix is denoted by +NK

A poverty index is defined by

Society A has higher poverty than society B if and only if P(XA) P(XB) is the weighted sum of deprivations of person i

Deutsches Institut fr Entwicklungspolitik (DIE)I. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion 8Let be an identification function so that person i is poor if and not poor if

Three specifications for have been suggested so far:

According to the union method, deprivation in one attribute is deprivation in all attributes (perfect complements):

According to the intersection method, poverty only occurs when there is deprivation in all attributes (perfect substitutes):

Union and Intersection Method Deutsches Institut fr Entwicklungspolitik (DIE)I. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion 9Intermediate Method (Dual Cut-Off)In response to the limited practicability of union and intersection method, the idea of an intermediate approach was brought up by Mack and Lindsay (1985) and formally introduced by Foster (2007) and Alkire and Foster (2007, 2011).According to the intermediate method, individual i is poor if the weighted sum of deprivations is higher than a predetermined minimum level:

The intermediate method provides a practicable solution, the theoretical justification is, however, questionable: up to the cut-off, attributes are considered to be perfect substitutes, from the cut-off onwards, however, the very same attributes are considered to be perfect complements.There is another way to identify the poor that can be derived directly from the aggregation step by fully accounting for the two concepts of distributive justice and efficiency. Deutsches Institut fr Entwicklungspolitik (DIE)I. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion 10The Equality-Promoting Change10For any and X , is obtained from X by an equality-promoting change, if for some individuals g and h, and A distributional change is said to be equality-promoting whenever the difference in the number of simultaneously suffered deprivations between two individuals is reduced

Based on Chakravarty and DAmbrosio (2006), Jayaraj and Subramanian (2010) introduced the equality-promoting change in order to capture inequality across dimensions:Jayaraj and Subramanian (2010) then formulated the axiom Nonincreasingness under Equality-Promoting Change: For any and X , if is obtained from X by an equality-promoting change, then

The axiom captures distributive justice, yet it neglects efficiency by disregarding possible correlations between attributes. Deutsches Institut fr Entwicklungspolitik (DIE)I. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion 11The Inequality Increasing SwitchDepending on the nature as well as the strength of the correlations between attributes, poverty might very well increase under an equality-promoting change.Thus, I introduce the concept of an inequality increasing switch:Define Then, for two individuals g and h such that , matrix X is said to be obtained from matrix by an inequality increasing switch of attribute l if and 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

Duclos, Sahn and Younger (2006) for instance argue that complementarities exist between the two poverty dimensions education and nutrition as better nourished children learn better. If the degree of complementarity is strong enough, poverty decreases with increasing inequality. Deutsches Institut fr Entwicklungspolitik (DIE)I. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion 12A New AxiomBased on this concept I formulate the axiom Sensitivity to Inequality Increasing Switches: For any and X , if is obtained from X by an inequality increasing switch of non-complementary attributes, then Further, if is obtained from X by an inequality increasing switch of complement attributes, then

Example: i = 2, j = 5, z = (1 1 1 1 1)

Deutsches Institut fr Entwicklungspolitik (DIE)I. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion 13A New Class of Ordinal Poverty IndicesThe new axiom directly implies a new multiple step identification function that is nondecreasing in the number of deprivations and has a nondecreasing (nonincreasing) marginal in case attributes are considered to be substitutes (complements).The former accounts for distributive justice, the latter for efficiency.Property 1A multidimensional poverty measure P satisfies AN, NM, MN, SF, PP, FD, SD and SIIS if and only if for all and X :

with non-decreasing in and a nondecreasing (nonincreasing) marginal in case attributes are considered to be substitutes (complements).

Deutsches Institut fr Entwicklungspolitik (DIE)I. Introduction II. The Identification StepIII. The Aggregation Step IV. Empirical Application V. Conclusion 14A New Identification FunctionConsider the following multiple step identification function: 1>a1