Decomposition Procedures for Distributional Analysis:
A Unified Framework Based on the Shapley Value
Anthony F. Shorrocks
University of Essex
and
Institute for Fiscal Studies
First draft, June 1999
Mailing Address:
Department of Economics
University of Essex
Colchester CO4 3SQ, UK
shora@essex.ac.uk
1
1. Introduction
Decomposition techniques are used in many fields of economics to help disentangle and
quantify the impact of various causal factors. Their use is particularly widespread in studies
of poverty and inequality. In poverty analysis, most practitioners now employ
decomposable poverty measures — especially the Foster et al. (1984) family of indices —
which enable the overall level of poverty to be allocated among subgroups of the
population, such as those defined by geographical region, household composition, labour
market characteristics or education level. Recent examples include Grootaert (1995),
Szekely (1995), Thorbecke and Jung (1996). Other dynamic decomposition procedures are
used to examine how economic growth contributes to a reduction in poverty over time, and
to assess the extent to which the impact of growth is reinforced, or attenuated, by changes
in income inequality: see for example, Ravallion and Huppi (1991), Datt and Ravallion
(1992) and Tsui (1996). In the context of income inequality, decomposition techniques
enable researchers to distinguish the “between-group” effect due to differences in average
incomes across subgroups (males and females, say), from the “within-group” effect due to
inequality within the population subgroups. (See ???). Decomposition techniques have also
been developed in order to measure the importance of components of income such as
earnings or transfer payments.
Despite their widespread use, these procedures have a number of shortcomings which
have become increasingly evident as more sophisticated models and econometrics are
brought to bear on distributional questions. Four broad categories of problems can be
distinguished. First, the contribution assigned to a specific factor is not always interpretable
in an intuitively meaningful way. As Chantreuil and Trannoy (1997) and Morduch and
Sinclair (1998) point out, this is particularly true of the decomposition by income
components proposed by Shorrocks (1982). In other cases, the interpretation commonly
given to a component may not be strictly accurate. Foster and Shneyerov (1996), for
example, question the conventional interpretation of the between-group term in the
decomposition of inequality by subgroups.
The second problem with conventional procedures is that they often place constraints
on the kinds of poverty and inequality indices which can be used. Only certain forms of
indices yield a set of contributions that sum up to the amount of poverty or inequality that
X
k
, k ' 1, 2, ..., m
I ' f(X
1
, X
2
, ... , X
m
)
f(
@
)
2
requires explanation. Similar methods applied to other indices require the introduction of a
vaguely defined residual or “interaction” term in order to maintain the decomposition
identity. 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 and
Aronson (1993).
A less familiar, but potentially much more serious, problem concerns the limitations
placed on the types of contributory factors which can be considered. Subgroup
decompositions can handle situations in which the population is partitioned on the basis of a
single attribute, but have difficulty identifying the relevant contributions in multi-variate
decompositions. Nor is there any established method of dealing with mixtures of factors,
such as a simultaneous decomposition by subgroups (into, say, males and females) and
income components (say, earnings and unearned income). As more sophisticated models
are used to analyse distributional issues, these limitations have become increasingly evident.
The studies by Cowell and Jenkins (1995), Jenkins (1995), Bourguignon et al. (1998), and
Bouillon et al. (1998) illustrate the range of problems faced by those trying to apply current
techniques to complex distributional questions.
The final criticism of current decomposition methods is that the individual applications
are viewed as different problems requiring different solutions. No attempt has been made to
integrate the various techniques within a common overall framework. This is the main
reason why it is impossible at present to combine decompositions by subgroups and income
components. Yet the individual applications share certain features and objectives which
enable a common structure to be formulated. Let I represent an aggregate statistical
indicator, 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 can
write
(1.1) ,
where is a suitable aggregator function representing the underlying model. The
objective in all types of decomposition exercises is to assign contributions C to each of the
k
factors X , ideally in a manner that allows the value of I to be expressed as the sum of the
k
factor contributions.
3
The aim of this paper is to offer a solution to this general decomposition problem and to
compare the results with the specific procedures currently applied to a number of
distributional questions. In broad terms, the proposed solution considers the marginal effect
on I of eliminating each of the contributory factors in sequence, and then assigns to each
factor the average of its marginal contributions in all possible elimination sequences. This
procedure yields an exact additive decomposition of I into m contributions.
Posing the decomposition issue in the general way indicated by (1.1) highlights formal
similarities with problems encountered in other areas of economics and econometrics. Of
particular relevance to this paper is the classic question of cooperative game theory, which
asks how a certain amount of output (or costs) should be allocated among a set of
contributors (or beneficiaries). The Shapley value (Shapley, 1953) provides a popular
answer to this question. The proposed solution to the general decomposition problem turns
out to formally equivalent to the Shapley value, and is therefore referred to as the Shapley
decomposition. Rongve (1995) and Chantreuil and Trannoy (1997) have both applied the
Shapley value to the decomposition of inequality by income components, but fail to realise
that a similar procedure can be used in all forms of distributional analysis, regardless of the
complexity of the model, or the number and types of factors considered. Indeed, the
procedure can be employed in all areas of applied economics whenever one wishes to
assess the relative importance of the explanatory variables.
The paper begins with a description of the general decomposition problem and the
proposed solution based on the Shapley value. Section 3 shows how the procedure may be
applied to three issues concerned with poverty: the effects of growth and redistribution on
changes in poverty; the conventional application of decomposable poverty indices; and the
impact of population shifts and changes in within-group poverty on the level of poverty
over time.
Section 4 looks in more detail at the features of the Shapley decomposition in the
context of a hierarchical model in which groups of factors may be treated as single units.
This leads to a discussion of the two-stage Shapley procedure associated with the Owen
value (Owen, 1977). A number of results in this section establish the conditions under
which the Shapley and Owen decompositions coincide, and indicate several ways of
simplifying the calculation of the factor contributions. These results are then used to
k 0 K ' {1, 2, ..., m}
I ' f(X
1
, X
2
, ... , X
m
)
f(@)
X
k
X
k
, k ó S
4
generate the Shapley solution to the multi-variate decomposition of poverty by subgroups, a
problem which has not been solved before.
In Sections 5 and 6, attention turns to inequality analysis, beginning with decomposition
by subgroups using the Entropy and Gini measures of inequality. This is followed by a
discussion of the application of the Shapley rule to decomposition by source of income.
The main purpose of these applications is to see how the Shapley procedure compares
with existing techniques in the context of a variety of standard decomposition problems.
The overall results are encouraging. In all cases, the Shapley decomposition either replicates
current practice or (arguably) provides a more satisfactory method of assigning
contributions to the explanatory factors. However, the greatest attraction of the procedure
proposed in this paper is that it overcomes all four of the categories of problems associated
with present techniques. As a consequence, it offers a unified framework capable of
handling any type of decomposition exercise. After summarising the principal findings of
this paper, Section 8 briefly discusses the wide range of potential applications to issues
which have not previously been considered candidates for decomposition analysis.
2. A General Framework for Decomposition Analysis
Consider a statistical indicator I whose value is completely determined by a set of m
contributory factors, X , indexed by , so that we may write
k
(2.1) ,
where describes the underlying model. In the applications examined later, the indicator
I will represent the overall level of poverty or inequality in the population, or the change in
poverty over time. The factor may refer to a conventional scalar or vector variable, but
other interpretations are possible and often desirable; for the moment it is best regarded as a
loose descriptive label capturing influences like “uncertain returns to investments”,
“differences in household composition” or “supply-side effects”.
In what follows, we imagine scenarios in which some or all of the factors are eliminated,
and use F(S) to signify the value that I takes when the factors , have been
dropped. As each of the factors is either present or absent, it is convenient to characterise