Decomposition Procedures for Distributional Analysis: A ...

1 Decomposition Procedures for Distributional analysis : A Unified Framework Based on the Shapley ValueAnthony F. ShorrocksUniversity of EssexandInstitute for Fiscal StudiesFirst draft, June 1999 Mailing Address:Department of EconomicsUniversity of EssexColchester CO4 3SQ, IntroductionDecomposition techniques are used in many fields of economics to help disentangle andquantify the impact of various causal factors. Their use is particularly widespread in studiesof poverty and inequality. In poverty analysis , most practitioners now employdecomposable poverty measures especially the Foster et al.

2 (1984) family of indices which enable the overall level of poverty to be allocated among subgroups of thepopulation, such as those defined by geographical region, household composition, labourmarket characteristics or education level. Recent examples include Grootaert (1995),Szekely (1995), Thorbecke and Jung (1996). Other dynamic Decomposition Procedures areused to examine how economic growth contributes to a reduction in poverty over time, andto assess the extent to which the impact of growth is reinforced, or attenuated, by changesin income inequality: see for example, Ravallion and Huppi (1991), Datt and Ravallion(1992) and Tsui (1996).

3 In the context of income inequality, Decomposition techniquesenable researchers to distinguish the between-group effect due to differences in averageincomes across subgroups (males and females, say), from the within-group effect due toinequality within the population subgroups. (See ???). Decomposition techniques have alsobeen developed in order to measure the importance of components of income such asearnings or transfer their widespread use, these Procedures have a number of shortcomings whichhave become increasingly evident as more sophisticated models and econometrics arebrought to bear on Distributional questions.

4 Four broad categories of problems can bedistinguished. First, the contribution assigned to a specific factor is not always interpretablein an intuitively meaningful way. As Chantreuil and Trannoy (1997) and Morduch andSinclair (1998) point out, this is particularly true of the Decomposition by incomecomponents proposed by Shorrocks (1982). In other cases, the interpretation commonlygiven to a component may not be strictly accurate. Foster and Shneyerov (1996), forexample, question the conventional interpretation of the between-group term in thedecomposition of inequality by second problem with conventional Procedures is that they often place constraintson the kinds of poverty and inequality indices which can be used.

5 Only certain forms ofindices yield a set of contributions that sum up to the amount of poverty or inequality thatXk,k'1,2,..,mI'f(X1,X2,..,Xm)f(@)2re quires explanation. Similar methods applied to other indices require the introduction of avaguely defined residual or interaction term in order to maintain the decompositionidentity. The best known example is the subgroup Decomposition of the Gini coefficient,which has exercised the minds of many authors including Pyatt (1976) and Lambert andAronson (1993).A less familiar, but potentially much more serious, problem concerns the limitationsplaced on the types of contributory factors which can be considered.

6 Subgroupdecompositions can handle situations in which the population is partitioned on the basis of asingle attribute, but have difficulty identifying the relevant contributions in multi-variatedecompositions. Nor is there any established method of dealing with mixtures of factors,such as a simultaneous Decomposition by subgroups (into, say, males and females) andincome components (say, earnings and unearned income). As more sophisticated modelsare used to analyse Distributional issues, these limitations have become increasingly studies by Cowell and Jenkins (1995), Jenkins (1995), Bourguignon et al.

7 (1998), andBouillon et al. (1998) illustrate the range of problems faced by those trying to apply currenttechniques to complex Distributional final criticism of current Decomposition methods is that the individual applicationsare viewed as different problems requiring different solutions. No attempt has been made tointegrate the various techniques within a common overall framework. This is the mainreason why it is impossible at present to combine decompositions by subgroups and incomecomponents. Yet the individual applications share certain features and objectives whichenable a common structure to be formulated.

8 Let I represent an aggregate statisticalindicator, such as the overall level of poverty or inequality, and let ,denote a set of contributory factors which together account for the value of I. Then we canwrite( ),where is a suitable aggregator function representing the underlying model. Theobjective in all types of Decomposition exercises is to assign contributions C to each of thekfactors X, ideally in a manner that allows the value of I to be expressed as the sum of thekfactor aim of this paper is to offer a solution to this general Decomposition problem and tocompare the results with the specific Procedures currently applied to a number ofdistributional questions.

9 In broad terms, the proposed solution considers the marginal effecton I of eliminating each of the contributory factors in sequence, and then assigns to eachfactor the average of its marginal contributions in all possible elimination sequences. Thisprocedure yields an exact additive Decomposition of I into m the Decomposition issue in the general way indicated by ( ) highlights formalsimilarities with problems encountered in other areas of economics and econometrics. Ofparticular relevance to this paper is the classic question of cooperative game theory, whichasks how a certain amount of output (or costs) should be allocated among a set ofcontributors (or beneficiaries).

10 The Shapley value (Shapley, 1953) provides a popularanswer to this question. The proposed solution to the general Decomposition problem turnsout to formally equivalent to the Shapley value, and is therefore referred to as the Shapleydecomposition. Rongve (1995) and Chantreuil and Trannoy (1997) have both applied theShapley value to the Decomposition of inequality by income components, but fail to realisethat a similar procedure can be used in all forms of Distributional analysis , regardless of thecomplexity of the model, or the number and types of factors considered. Indeed, theprocedure can be employed in all areas of applied economics whenever one wishes toassess the relative importance of the explanatory paper begins with a description of the general Decomposition problem and theproposed solution based on the Shapley value.