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Müller, Adrian
Conference Paper
Clarifying Poverty Decomposition
Proceedings of the German Development Economics Conference, Zürich 2008, No. 30
Provided in Cooperation with:
Research Committee on Development Economics (AEL), German
Economic Association
Suggested Citation: Müller, Adrian (2008) : Clarifying Poverty Decomposition, Proceedings of
the German Development Economics Conference, Zürich 2008, No. 30
This Version is available at:
http://hdl.handle.net/10419/39900
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Clarifying Poverty Decomposition
Adrian Muller∗†
Abstract: I discuss how poverty decomposition methods relate to integral
approximation, which ultimately is the foundation of every decomposition of
the temporal change of a quantity into key drivers. This offers a common
framework for the different decomposition methods used in the literature,
clarifies their often somewhat unclear theoretical underpinning and identi-
fies the methods’ shortcomings. In light of integral approximation, many
methods actually lack a sound theoretical basis and they usually have an
ad-hoc character in assigning the residual terms to the different key effects. I
illustrate these claims for the Shapley-value decomposition and methods re-
lated to the Datt-Ravaillon approach and point out difficulties in axiomatic
approaches to poverty decomposition. Recent developments in energy and
pollutant decomposition offer some improved methods, but ultimately, a fur-
ther development of poverty decomposition should account for the basis in
integral approximation.
∗Socio-economic Institute, University of Z¨urich, Bl ¨mlisalpstrasse 10, 8006 Z¨urich,
Switzerland; phone: 0041-44-634 06 06; e-mail: adrian.mueller@soi.uzh.ch
†Many thanks to Uma Rani and ˚
Asa L¨ofgren for helpful remarks and inspiring discus-
sions, and to Erika Meins for the correction of my English. The usual disclaimer applies.
1
NOTE: this is a working paper; for a the next version and the presentation,
I will in particular give additional illustration and motivation for equation
(1) and its interpretation as the fundamental equation for decomposition.
Keywords: poverty analysis, poverty measures, decomposition, Shapley-
value, inequality
JEL: I32, C43
1 Introduction
Decomposing some key variable in components assigned to different driving
forces is a common exercise in many areas. Classical areas are the evolution of
energy use and pollutant emissions (e.g. Ang 1995, 2004; Bruvoll and Larsen
2004) or poverty and inequality measures (e.g. Shorrocks 1999; Kakwani
2000; for a recent review, see Heshmati 2004).
Such decomposition is related to the general index number theory as de-
veloped for price and quantity indices. The calculation of such indices is
based on integral approximations and the price and quantity index literature
is aware of this (Trivedi 1981). The awareness of this basis in integral ap-
proximation has, however, been lost in most of the literature on energy and
pollutant decomposition (Muller 2006), and in poverty decomposition in par-
ticular. The lack of this connection to the underlying basic formalism makes
current efforts to develop optimal decomposition approaches somewhat ar-
bitrary and often difficult to understand. This is the case for the recently
developed Logarithmic Mean Divisia Index (LMDI) approach in the energy
and pollutant context (Ang 2004, Muller 2006), for example, and for the
discussion of the residual and zero or negative values in this same context.
2
Decomposition of poverty measures is even more arbitrary in that the
motivations for the choice of a certain method usually lie even further from
the underlying integral approximations. This is especially confusing in the
context of the Shapley-value decomposition (Shorrocks 1999; Baye 2005)
where reference is made to game theoretic concepts. In addition, critique
emerges regarding the performance of these methods and claims arise that
they are only suitable in special cases (Sastre and Trannoy 2000; Fiorio 2006).
Explicit reference to integral approximation as the underlying formalism
of decomposition would not only clarify issues and make the methods easier
to access, it would also add to the understanding of the problem of zero
and negative values virulent in energy and pollution decomposition and it
would shed a new light on the discussion of the residual. This has been
done for energy and pollutant decomposition in Muller (2006). Zero and
negative values are no topic in poverty decomposition because they either do
not occur or they do not pose any problem in the methods currently applied.
The residual, on the other hand, is present in some classical approaches to
poverty decomposition and usually given the somewhat vague interpretation
of interaction effects (e.g. Datt and Ravaillon 1992). This interpretation is
often criticized and the absence of a residual term in the newer approaches
related to the Shapley-value is seen as an advantage (Baye 2005).
As in energy decomposition, though, the zero residual is not a good crite-
rion to identify optimal decomposition methods. The methods are much bet-
ter understood if tied to the underlying integral approximations, where the
presence of some residual due to approximation errors is natural. Referring
to this basis offers a common framework for the decomposition approaches
3
mainly applied in the poverty and inequality context, i.e. the Shapley-value
based decomposition and the decomposition methods similar to the one pre-
sented in Datt and Ravaillon (1992). I will show that these methods are
special and not entirely consistent approaches to approximate the underly-
ing integrals. In general, decomposition would gain, irrespective of where it
is applied, if this common ground in integral approximation would be appre-
ciated. Ultimately, it could be promising to develop improved methods on
this basis.
Section 2 introduces the general formalism of decomposition, illustrates
how it is linked to integral approximation and how decomposition is dis-
cussed in the energy and pollution context. Section 3 presents some of the
main methods of poverty decomposition currently applied and illustrates how
they relate to each other and to the general formalism based on integral ap-
proximation. Conclusions are drawn in section 4.
2 A General Formalism for Decomposition
The aim of dynamic decomposition analysis1is to identify and assess the
(relative) magnitude of different variables driving the time development of a
key quantity. One example is the total industrial energy use and how much
the changes in sector-wise energy efficiency, in the relative size of the different
1In a static decomposition analysis, a key variable is decomposed for one period to
investigate the differences between several groups, such as states or castes. As often done
in energy analysis, a dynamic approach can also be differentiated to account for the effects
of group structures. A static decomposition is usually of only restricted interest, as the
information on time development is missing. I therefore focus on the dynamic approach,
but add some more remarks on the static approach in section 3.
4
sectors, and in the size of the total industry contribute to its development.
Another example is how much changes in mean income and inequality con-
tribute to changes in total poverty within a country.
I develop the following general formalism. I will show in section 3 how
the methods commonly used for poverty decomposition can be seen as spe-
cial cases of this general formalism. The key quantity of interest shall be
P(t) =P(x1(t), ..., xm(t)), depending on mtime-dependent drivers xi(t), i =
1, ...m, t ∈[T0, Tn].2The change in Pis given by its total derivative dP
dtand
the change from T0to Tncan be written as
∆PT0,Tn:= P(Tn)−P(T0) = ZTn
T0
dP
dtdt=
=ZTn
T0∂P
∂x1
∂x1
∂t +∂P
∂x2
∂x2
∂t +... +∂P
∂xm
∂xm
∂t dt= (1)
=ZTn
T0
∂P
∂x1
∂x1
∂t dt+ZTn
T0
∂P
∂x2
∂x2
∂t dt+... +ZTn
T0
∂P
∂xm
∂xm
∂t dt.
The part containing the derivative with respect to xiis then interpreted
as the contribution of changes in xito the total change in P. I denote this
by ∆Pxi
T0,Tn. Usually, the functions involved are not known for all points t∈
[T0, Tn], but only for some discrete points of time, most often equally spaced
(e.g. annually): T0, T1, T2, ..., Tn−1, Tn. The integrals then have the following
structure and the integrands are basically only known at the endpoints (i=
1, ..., m):
2Pcan further be differentiated according to some group-structure of interest, i.e.
P=PG
g=1 Pg, where Pgis the value for Preferring to group g, but this does not change
the general argument and I use the simpler notation without this additional structure.
5
∆Pxi
T,T +1 =ZT+1
T
∂P (x1, ..., xm)
∂xi
∂xi
∂t dt. (2)
Decomposing Pthus boils down to solving such integrals. Because of
the lack of information, though, i.e. the lack of knowledge of the underlying
functions besides for the boundary values Tand T+ 1, this is essentially an
approximation problem. The integral has to be approximated by the values
of the integrand at the endpoints of the integration range. In addition, the
presence of derivatives may cause a problem as usually only the functions
but not their derivatives are known for the endpoints. In this case, some
approximation of the derivatives is necessary as well. The integral can thus
be written as a function Jor ˜
Jof the values at the end-points:3
∆Pxi
T,T +1 ≈
≈JP(T), xi(T),∂P
∂xi
(T),∂xi
∂t (T),
P(T+ 1), xi(T+ 1),∂P
∂xi
(T+ 1),∂xi
∂t (T+ 1)≈
≈˜
JP(T), xi(T), P (T+ 1), xi(T+ 1),
P(T−1), xi(T−1), P (T+ 2), xi(T+ 2),(3)
As the decomposition problem in energy and pollutant analysis is framed,
∂P
∂xiis usually known due to the particular structure of Pbeing a product of
3Jincludes the derivatives directly, while they are approximated in ˜
J. For ˜
J, I chose
the general formulation including P(T−1), xi(T−1), P (T+2) and xi(T+ 2), as they may
enter the formula dependent on how the derivatives are approximated. An approximation
of the derivative at T+ 1 from the right side, for example, usually depends on the value
at T+ 2.
6
the various xi(Muller 2006). Besides the integral, only the derivative of xi
remains then to be approximated.
How to best approximate ∆Pxi
T,T +1 , i.e. how to optimally choose Jor ˜
Jis
implicitly driving all the different approaches to decompose energy use and
pollutant emissions. “Implicitly” only, though, as awareness of the basis in
integral approximation is largely missing in the literature. The problem is
basically seen as one of choosing the correct weights for the known values at
the two end-points to best calculate the change in Pover the whole range in
between. Choosing weights actually has its roots in integral approximation,
as the simplest method to approximate an expression such as equation (2)
consists in replacing the integral with the product of the value of the inte-
grand at the upper or lower end-point times the distance on the ordinate ∆T,
in this case equaling one: J=∂ P
∂x
∂x
∂t |T+1 resp. T. This gives a weight of one
to the upper or lower boundary and a weight zero to the other. Weighting
both boundaries equally results in the average of the two values times ∆T,
equaling one again: J= [∂P
∂x
∂x
∂t (T+ 1) + ∂ P
∂x
∂x
∂t (T)]/2. These are three ap-
proximations replacing the true function by different types of step-functions.
They are analogous to classical indices in the price/quantity context (the
Laspeyres, Paasche and Marshall-Edgeworth index) and especially the first
two were also applied in (early) energy decomposition, but they have several
disadvantages, such as a usually rather large residual term or the asymmetry
regarding the boundaries (Ang 2004).
The energy decomposition literature has then developed further approaches
based on more flexible weights (e.g. the Divisia approach J=∂P
∂x
∂x
∂t (T) +
α[∂P
∂x
∂x
∂t (T+ 1) −∂ P
∂x
∂x
∂t (T)] with α∈[0,1]; α= 0,1 or 1
2replicate the three
7
methods mentioned above).4This strategy separated decomposition even
further from its base in integral approximation, as the single most impor-
tant criterion for good weights became associated with a vanishing residual,
meaning that Jor ˜
Jnot only approximate, but exactly replicate the left-
hand side in equation (3).5This however is a misleading criterion, as the
chance to exactly approximate the unknown integral based on the boundary
values only is very small. A zero residual thus bears the danger of having
been forced to be zero by just randomly or without strong reasons appor-
tioning it to the different parts of a decomposition. A decomposition with
zero residual thus needs not at all be superior to one with some residual -
which, if it is too large, however clearly also spoils the explanatory power of
the result.6
4∂x
∂t (T+ 1) and ∂ x
∂t (T) are usually approximated by the slope of the straight line joining
the endpoints, i.e. ∂ x
∂t (T+ 1) ≈x(T+ 1) −x(T)≈∂x
∂t (T), thus giving the same value and
simplifying formulae such as the Divisia Index above. This strategy could be criticized
because of its inconsistency by taking the approximation from the right for the value at
the left boundary Tand the value from the left at T+1. This leads to potentially different
results for ∂x
∂t (T) depending on whether it is part of a term between T−1 and Tor between
Tand T+ 1. However, the strategy makes sense if seen in the context of replacing the
whole unknown function with straight lines joining the known values (as for the Divisia
with α=1
2, i.e. the Marshall-Edgworth Index), for example.
5This criticism applies for example to the LMDI currently advocated as the most ade-
quate index for energy and pollutant decomposition (Ang 2004).
6See Muller (2006) for an illustrative simulation of these issues in energy/pollution
decomposition.
8
3 Poverty Decomposition
As for energy decomposition, there is a range of methods for poverty decom-
position. The choice of a certain method is sometimes based on some formal
symmetry arguments or axioms (Shorrocks 1982; Tsui 1996; Kakwani 2000),
but in most cases rather ad-hoc. There is no awareness of the underlying
approximation problem, although the decomposition methods proposed can
be understood in this frame (see section 3.2 below). Poverty decomposition
usually refers to decomposing some kind of poverty measure P, often the
classical measure introduced in Foster et al. (1984), into parts corresponding
to the effects of temporal changes in the mean income µ, the income distri-
bution Land the poverty line z:P=P(µ, L, z). This can be normalized by
z, i.e. the function to be investigated afterward depends on only two instead
of three variables: ¯
P(µ
z,L
z). It follows from the discussion above that the
general decomposition of the poverty measure reads
∆PT,T +1 =ZT+1
T
∂P
∂µ
∂µ
∂t dt+ZT+1
T
∂P
∂L
∂L
∂t dt+ZT+1
T
∂P
∂z
∂z
∂t dt, (4)
where the integrals involved have the same structure as discussed above and
similar problems related to their approximation are encountered. This also
illustrates the formal equivalence of poverty and energy decomposition.
In the following, I will introduce the most common methods for decom-
position of changes in poverty or inequality measures (the decompositions in
the spirit of Datt and Ravaillon (1992), the Shapley-value decomposition and
some further related approaches) and show how they relate to the general
framework presented above.
9
3.1 Common Approaches to Poverty Decomposition
Most poverty measure decomposition approaches assume that the contribu-
tion of one variable to total change in poverty can be separated if all other
variables are kept constant, i.e. if an unobserved “counterfactual situation”
is correctly constructed. In particular, the choice of the time period, in which
to keep the other variables constant, is crucial and various possibilities for
this differentiate the methods. This approach leads to decompositions such
as (taking the normalized form with ¯µ:= µ
zand ¯
L:= L
z)
∆¯
PT,T +1 =¯
P(¯µ(T+ 1),¯
L(T+ 1)) −¯
P(¯µ(T),¯
L(T)) = (5)
=h¯
P(¯µ(T+ 1),¯
L(T+ 1)) −¯
P(¯µ(T),¯
L(T+ 1))i+
+h¯
P(¯µ(T+ 1),¯
L(T+ 1)) −¯
P(¯µ(T+ 1),¯
L(T))i+¯
R=
= ¯µ(i.e. growth)-effect + ¯
L(i.e. inequality)-effect + ¯
R,
where ¯
Ris the residual - also referred to as the interaction effect between
growth and changes in inequality, given by ¯
R=¯
P(¯µ(T),¯
L(T+1))−¯
P(¯µ(T+
1),¯
L(T+ 1)) + ¯
P(¯µ(T+ 1),¯
L(T)) −¯
P(¯µ(T),¯
L(T)) (Datt and Ravallion 1992;
Baye 2004).
The residual thus has a similar structure as the decomposition itself and
the whole formula has a rather ad-hoc character by adding and subtracting
terms to get the effects of interest and then correcting for it by collecting
their corresponding negatives in the residual, thus guaranteeing the validity
of the formula. Datt and Ravallion (1992) also observe that this residual
can be quite large, thus invalidating the whole approach. Equation (5) also
depends on the period chosen as base period, as it is not symmetrical in Tand
10
T+ 1. This method nevertheless is applied without discussion of potential
problems, e.g. in Grootaert (1995) or Kraay (2006).
A similar approach is proposed by Jain and Tendulkar (1990),
∆¯
PT,T +1 =h¯
P(¯µ(T+ 1),¯
L(T+ 1)) −¯
P(¯µ(T),¯
L(T+ 1))i+
+h¯
P(¯µ(T),¯
L(T+ 1)) −¯
P(¯µ(T),¯
L(T))i,(6)
where the residual is zero, but the two effects are calculated with reference
to different base periods and the decomposition is again not symmetric in T
and T+ 1.
This situation led other authors (e.g. Kakwani 2000; Mazumdar and
Son 2001; Bhanumurthy and Mitra 2003; Son 2003) to suggest a symmetric
alternative of this decomposition by averaging the formulae with base periods
Tand T+ 1. Kakwani (2000) in particular motivates this by proposing a set
of axioms any poverty decomposition should fulfill (cf. footnote 12 below).
This leads to a symmetric decomposition without residual and the growth
and inequality effects have the same combination of mixed base periods:
∆¯
PT,T +1 =1
2h¯
P(¯µ(T+ 1),¯
L(T+ 1)) −¯
P(¯µ(T),¯
L(T+ 1)) +
+¯
P(¯µ(T+ 1),¯
L(T)) −¯
P(¯µ(T),¯
L(T)i+
+1
2h¯
P(¯µ(T+ 1),¯
L(T+ 1)) −¯
P(¯µ(T+ 1),¯
L(T)) +
+¯
P(¯µ(T),¯
L(T+ 1)) −¯
P(¯µ(T),¯
L(T))i.(7)
This decomposition can be applied to any numbers of variables. Given a
poverty measure Pdepending on mvariables x1, ..., xm, the contribution of
xito changes in Pcan be defined to be a combination of all terms
11
∆Pxi
T,T +1 (πs−1,m−s) = [P(..., xi(T+ 1), ...)−P(..., xi(T), ...)],(8)
where πs−1,m−sis any m−1-vector with s−1 entries T+1 and m−sentries T.
The elements of this vector indicate at which time the variables other than
xi, i.e. x1, ..., xi−1, xi+1, ..., xm, are taken in both the terms on the right hand
side in equation (8).7For mvariables, a certain combination of svariables
taken at T+ 1 and m−sat Tthus shows up in the final expression stimes
with a positive sign, stemming from the positive part of equation (8), for
each variable at T+ 1. And correspondingly, it shows up m−stimes in the
final expression with a negative sign, stemming from the negative part, but
referring to the corresponding expression for s+1.8The condition that in the
end only the original terms remain, i.e. ∆ ¯
PT,T +1 =P(x1(T+ 1), ..., xi(T+
1), ..., xm(T+1))−P(x1(T), ..., xi(T), ..., xm(T)), requires coefficients unequal
1 for the various terms. In the simplest case, the coefficients of the positive
terms can be chosen to be 1
sand for the negative ones 1
m−s, for s6= 0 and
s6=m, and 1
m−s=1
mfor s= 0 while the positive part is absent, and 1
s=1
m
for s=m, where the negative part is absent.
A more general choice of the coefficients is then γ(m, s)1
sand γ(m, s)1
m−s
with γ(m, 0) = 1 = γ(m, m). This decomposition is symmetric and residual-
free. Choosing γ(m, s) = s!(m−s)!
m!then gives the Shapley-value coefficients
(see e.g. Baye (2005), taking s+ 1 instead of sfor the negative terms)
7This is a type of ceteris paribus reasoning employing all combinations of how the other
variables can stay constant: each at Tor at T+ 1
8πs−1,m−sgives svariables at T+ 1 in the positive parts of ∆Pxi
T,T +1 (πs−1,m−s) and
s−1 at T+ 1 in the negative ones. Correspondingly, s+ 1 gives svariables at T+ 1 in
the negative parts that combine with the corresponding terms referring to s.
12
and the decomposition coincides with the Shapley-value based poverty de-
composition as introduced in Shorrocks (1999), which is seen as one of the
best methods currently available (Baye 2004, 2005; Kolenikov and Shorrocks
2005). For two variables, this is equivalent to equation (7). It is not clear,
though, how this or any other specific choice of weights might be motivated.
The derivation of this decomposition as just described is transparent but
it lacks a sound motivation. However, in my opinion, the game-theoretic
background of the Shapley-value neither offers additional relevant motiva-
tion (i.e. motivation related to the problem of poverty decomposition, which
has no tie to game theory) on how the decomposition should best be done,
respectively on why to choose γ(m, s) in this particular way. Admittedly,
the Shapley-value has some distinct axiomatic background (symmetry, no
essential player, additivity), but I will show in the next subsection that the
Shapley-value is not optimal in the light of decomposition as integral approx-
imation, and that thus these axioms cannot be employed as a motivation for
the method’s optimality.
There are other approaches aiming at improving poverty decomposition.
Dercon (2006) bases decomposition on a micro-level assessment of single
households and their status as poor or non-poor and how this changes be-
tween periods. Another different approach is based on the linkages captured
in the Social Accounting Matrix (e.g. Thorbecke and Jung (1996) and ref-
erences therein). Thirdly, Fournier (2001) discusses an approach explicitly
taking into account changes in the different underlying variables and their
correlations separately. This is, in fact, similar to taking some terms of the
Shapley-value approach into account and explaining part of the remaining
13
residual by building counterfactuals based on the rank-correlation structure.
Usually, some residual remains. Fourthly, there are regression-based ap-
proaches to decomposition (see e.g. Juhn et al. (1993), Borooah (2005) or
Wan and Zhou (2005) and references therein). The regressions, however,
refer to the definition, choice, or identification of the variables the decom-
position is based on or the construction of the counterfactual case, while
the decomposition itself (i.e. the combination of the terms where only one
variable changes) is again made according to the common approaches as de-
scribed in this subsection. Similarly, Di Nardo et al. (1996) discuss a kernel
estimation approach to construct counterfactuals needed for the decompo-
sition into changes attributable to single variables, while the decomposition
ultimately is again a variant of the approaches discussed above. This should
not be seen as an encompassing list and I do not discuss these alternative
approaches in more detail (for a recent review of methods, see also Heshmati
(2004)).
3.2 Poverty Decomposition and Integral Approxima-
tion
In this subsection, I discuss the poverty decomposition approaches intro-
duced above in the light of general decomposition as integral approximation
as presented in section 2. This establishes a common basis for and a new
understanding of poverty decomposition methods.
14
3.2.1 Most Common Approaches and the Shapley-Value
Approximating the terms in equation (4) by their values at the upper bound-
ary leads to expressions such as J≈∂P
∂µ
∂µ
∂t |T+1∆T, and approximating the
derivatives by the slope of the straight line joining the end-points as discussed
in footnote 4 gives
J≈P(¯µ(T+ 1),¯
L(T+ 1)) −P(¯µ(T),¯
L(T+ 1))
¯µ(T+ 1) −¯µ(T)
¯µ(T+ 1) −¯µ(T)
∆T∆T, (9)
which is the Laspeyres index. The corresponding expression can be calculated
for the variable ¯
Land both can also be evaluated at time T, thus giving
the Paasche index. The combination of the Laspeyres for both ¯µand ¯
L
gives the Datt-Ravaillon decomposition equation (5), and the combination
of Laspeyres for ¯µand Paasche for ¯
Lgives the Jain-Tendulkar formula (6).
Taking the average of the Laspeyres and Paasche indices gives the Marshall-
Edgeworth index (equivalent to the Divisia index with α=1
2). This, finally,
is the same as the Shapley-value decomposition for two variables, equation
(7).
So far, I have shown how the basic poverty decomposition methods can
be seen as special cases of integral approximation. This is however not true
any longer for the generalised formulae used in the literature and presented
above, i.e. for the Shapley-value with more than two variables. One criticism
is that in the light of the equivalence of the Shapley-value decomposition and
the decomposition method introduced in Sun (1998) (Ang et al. 2003), the
various terms in the Shapley-value can be understood as an assignment of
the residual to the various effects based on some symmetry arguments but
without further basis in the properties of the underlying functions or integral
15
approximations. Thus, all variables are treated equally, irrespective of their
properties. I illustrate this for three variables and a total which is their
multiplication:
∆P=P(T)−P(0) = x1(T)x2(T)x3(T)−x1(0)x2(0)x3(0)
= ∆P1+ ∆P2+ ∆P3,(10)
where ∆Piis the contribution of the variable xito the decomposition of P.
Replacing xT
iwith x0
i+ ∆xi, seeing ∆xias the incremental change in xifrom
period 0 to T, and symmetrically rearranging terms, we thus have
∆P= (x0
1+ ∆x1)(x0
2+ ∆x2)(x0
3+ ∆x3)−x0
1x0
2x0
3=
= ∆x1x0
2x0
3+ ∆x2x0
1x0
3+ ∆x3x0
1x0
2+
+∆x1∆x2x0
3+ ∆x1∆x3x0
2+ ∆x2∆x3x0
1+ ∆x1∆x2∆x3=
= ∆x1x0
2x0
3+1
2[∆x1∆x2x0
3+ ∆x1∆x0
3x0
2] + 1
3∆x1∆x2∆x3+
+∆x2x0
1x0
3+1
2[∆x1∆x2x0
3+ ∆x2∆x0
3x0
1] + 1
3∆x1∆x2∆x3+
+∆x3x0
1x0
2+1
2[∆x1∆x3x0
2+ ∆x2∆x0
3x0
1] + 1
3∆x1∆x2∆x3.(11)
The three last lines are ∆P1, ∆P2and ∆P3, respectively, and equal the
contributions of the three variables as identified in Sun (1998). As shown
in Ang et al. (2003), they are equal to the Shapley-value decomposition, as
can also be seen by further rearranging terms and comparing to the formulae
for the Shapley-value given above. As already indicated, the logic behind
this formula is to equally assign all the difference-terms involving ∆xi’s to
the contributions of the variables xi, i.e. a term involving s∆-factors is
16
divided by s. A pictorial illustration for this simple example are the volumes
of two cubes with edges x0
iand x0
i+ ∆xi, respectively, and how to assign the
difference in volume between the two to each of the differences in the single
edges.
A decomposition rule is based on the goal to decompose a general func-
tion of some mvariables into madditive parts, each one corresponding to
the contribution of one of these variables. This is thus some type of linearisa-
tion, and basing a decomposition procedure on some symmetries on the level
of these linearised summands, as it is done in the Shapley-value approach,
treating all variables symmetrically, need not be correct. This is so as we are
not primarily interested in ∆xiitself, but rather in ∆xi=xi(t+ ∆t)−xi(t)
as a function of ∆t, which, in general, will not be linear.
I illustrate this criticism of the Shapley-value with a simulation based
on some concrete choice of the variables xias functions of t: let’s choose
x1=t, x2=t2, x3=t
4. Inserting this in equation (1), where again P=
x1x2x3, and solving the integrals gives ∆P= ∆P1+ ∆P2+ ∆P3=T4
4and
the following (exact) decomposition
∆P1=ZT
0
∂x1
∂t x2x3dt=ZT
0
t3
4dt=T4
16 ,∆P2=T4
8,∆P3=T4
16 .(12)
Using the Shapley-value equation (11), the result is different (but also exact),
which shows that the Shapley-value does not necessarily lead to the correct
decomposition9:
9Most terms are equal zero in this simple example, as x0
i= 0 for i= 1,2,3, but this
special property is not crucial for the general argument.
17
∆P1= ∆P2= ∆P3=T4
12 .(13)
For further illustration, I also state the condition for the Shapley-value
for three variables to be exact. It is, for the contribution of the first variable
(xt
i=xi(t)), the requirement that
ZT
0
∂x1(t)
∂t x2(t)x3(t)dt!
=
!
= (xT
1−x0
1)x0
2x0
3+1
2[(xT
1−x0
1)(xT
2−x0
2)x0
3+ (xT
1−x0
1)(xT
3−x0
3)x0
2] +
+1
3(xT
1−x0
1)(xT
2−x0
2)(xT
3−x0
3).(14)
Comparing this to integral approximation as discussed above shows also that
the Shapley-value contains too many terms mixing values referring to the two
different boundaries. In correct integral approximation, such mixture only
occurs via the derivative-term, i.e. for one variable only, while all the others
are evaluated either at the upper or lower boundary only.
3.2.2 Static and Axiomatic Decomposition
A somewhat different approach to decomposition is taken by authors that
address the static decomposition of differences between various groups in
the society such as spatial groups, e.g. states in a nation (Dhongde 2003;
Kolenikov and Shorrocks 2005), or different castes (Borooah 2005) rather
than changes between time periods. Formally this could be seen as the same
problem as temporal decomposition, and the same methods could be applied.
This however would assume some continuous range of parameters between
spatial groups, states, castes etc., which clearly is not the case for most
18
group-variables in reality - although approximation formulae based on the
values at the endpoints (i.e. for two groups, for example) can be applied
due to formal equivalence. Thus, the framework of integral approximation is
not adequate for such static analysis as the notion of a path connecting the
groups generally does not make sense. Postulating such a path makes the
formulae from integral approximation applicable but it will likely lack a sound
interpretation. The case is different as soon as some temporal information
is available. Then, the decomposition can be undertaken as discussed above,
employing separate group-wise analysis (i.e. separately for each state, caste,
etc.), or it could be done by directly incorporating the different group effects
as they are usually incorporated in energy decomposition (for groupings such
as by fuel type or industry sector, see e.g. Muller 2006).10
Finally, I link the poverty decomposition method based on integral ap-
proximation as described above to some axiomatic approaches in the liter-
ature. Most recent is Kakwani (2000), who sets up a system of 5 simple
rather intuitive axioms any poverty decomposition should fulfil (mainly sym-
metry and consistency properties, see the formalism below and footnote 12),
discusses and criticises existing decomposition methods in the light of these
axioms and proposes a new method that fulfils all 5 axioms. His discussion
is framed in a two-variable setting and the method he finally recommends is
just the Shapley-value for two variables.11 Adopting his notation, we consider
10A separate analysis is udnertaken for each group and then aggregated for all groups.
This leads to results quantifying the relative effects of changes in fuel-composition or
sectoral structure without further specification of how changes in the single fuels or sectors
contribute.
11He does not mention this, though - but the Shapley-value decomposition was also
introduced after this paper was originally written in 1997.
19
a poverty measure in period idepending on a measure for inequality (like the
Lorenz curve L) and a measure of the average income level µ: Θ(µi, Li). Em-
ploying the integral approximation approach to decomposition, the change
in this measure between two periods can then be written as
Θij =Zj
i
∂Θ
∂µt
∂µt
∂t dt+Zj
i
∂Θ
∂Lt
∂Lt
∂t dt=: Gij +Iij ,(15)
where Gis the growth and Ithe inequality component. Due to the properties
of integration, the decomposition based on integral approximation thus fulfills
these 5 axioms set up by Kakwani (2000)12.
Other axiomatic systems are presented in Shorrocks (1982) and Tsui
(1996), for example. The axiomatisation in Tsui, however, mainly refers
to the poverty measure itself and less to its decomposition, which is basically
the same as finally derived in Kakwani (2000).
Decomposition based on integral approximation does however not fulfil
the axioms of Shorrocks (1982). The assumption on symmetric treatment of
factors (his assumption 2b) is not fulfilled in his sense, where it refers to the
functional dependence of the contribution to inequality of one factor being
the same for all factors. Using his notation, in my approach, the contribution
of a factor Ykto the general inequality measure I(Y), where Y=PK
k=0 Yk
is total income built of several types of income Yk, is Sk(Y1, ..., Y K;K) :=
RT
0
∂I (Y)
∂Y k
∂Y k
∂t dt. As a functional description, this is symmetric in the different
factors, but not necessarily on the level of Sas a function of Yk. Furthermore,
Shorrocks’ approach is also criticised by several authors, e.g. by Paul (2004)
12The axioms are 1) If Iij = 0 then Θij =Gij and if Gij = 0 then Θij =Iij ; 2) if
Gij ≤0 and Iij ≤0 then Θij ≤0 and if Gij ≥0 and Iij ≥0 then Θij ≥0; 3) Gij =−Gji
and Iij =−Iji ; 4) Gij =Gik +Gkj ; 5) Iij =Iik +Ikj for all periods i, j, k;
20
for the lack of motivation for some of his conditions13 and by Fournier (2001)
as being too restrictive and, being static, as not being of primary interest —
although static decomposition is applied frequently.
It may be concluded from this discussion that generally, as in energy
and pollutant decomposition, formulating axioms for the decomposition of
changes should not be given too much weight to, especially if decomposition
is seen in the light of integral approximation. Furthermore, although the
general formulation of decomposition based on integrals may fulfil the axioms,
due to the unavoidable errors, they may not be fulfilled when it comes to
concrete approximations (cf. also Muller 2006). Investigating axioms for
poverty measures themselves, however, may clearly make sense, but this is
not the topic of this paper.
4 Conclusions
A wide range of methods for poverty or general inequality measure decom-
position is currently being applied. None of these methods, however, has a
sound basis, as none refers to integral approximation, which is the ultimate
starting point of any dynamic decomposition analysis. Muller (2006) recently
analysed these issues in the context of energy and pollutant decomposition,
where similar problems are encountered. The methods used in energy de-
composition perform somewhat more satisfactorily from a theoretical point
of view than the common poverty decomposition methods. To assess the ad-
equacy of the methods most often applied in poverty decomposition, such as
the Shapley-value, comparison with methods more directly related to integral
13Paul (2004) however retains the symmetry axiom I criticise here.
21
approximation is necessary. For energy decomposition, such an assessment
has been done for the LMDI (Ang 2004; Muller 2006), which is seen as one
of the best methods in this context. Although lacking a sound theoretical
basis, this method performs reasonably well also in relation to integral ap-
proximation, and the LMDI may be used as a reasonably reliable option for
most cases.
Thus I suggest to apply the LMDI also for the decomposition of changes
in poverty or general inequality measures. This method is more appropriate
than the Shapley-value, which has desirable features, but assigns the resid-
ual term in an inadequate manner to the different drivers behind changes in
poverty. This does not mean that results based on the Shapley-value neces-
sarily are wrong - but it is difficult to assess when it is adequate and how
large potential errors may be. Admittedly, an assessment of the performance
of the LMDI may not be easy and the best practice would be to solely rely
on integral approximation. This would work best if it is possible to collect
or access additional data for the case at hand, thus gaining additional infor-
mation on the functions to be approximated and improving the reliability of
the result.
Finally, I emphasize that, as in energy and pollutant decomposition, in-
creased reliance on axiomatic approaches is no solution to identify optimal
methods. In the light of integral approximation, desirable properties only
need to be fulfilled approximately, thus spoiling assessments of methods based
on a system of axioms. The prime example for this may be the desirability
of a zero residual, i.e. of a complete decomposition, which does not need
to hold for an approach based on approximations. It is only natural to en-
22
counter some errors when approximating - which simply lies in the nature of
an approximation in comparison to an exact solution.
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