Variations in aggregate poverty indices can be due to differences in average poverty intensity, to changes in the welfare distances between those poor of initially unequal welfare status, and/or to emerging disparities in welfare among those poor of initially similar welfare status. This note uses a general cost-of-inequality approach that decomposes the total change in poverty into a sum of indices of each of these three components. This decomposition can serve inter alia to integrate horizontal and vertical equity criteria in the poverty alleviation assessment of social and economic programs. The use of these measures in briefly illustrated using Tunisian data.Poverty, Vertical Equity, Horizontal Equity, Targeting, Tunisia

Bibi, Sami

Duclos, Jean-Yves

English

Research Papers in Economics

 
 
 
CIRPÉE 
Centre interuniversitaire sur le risque, les politiques économiques et l’emploi 
 
 
 
 
 
Cahier de recherche/Working Paper 03-12 
 
 
 
 
Decomposing Poverty Changes into Vertical and Horizontal 
Components 
 
 
 
Sami Bibi 
Jean-Yves Duclos 
 
 
 
 
 
Février/February 2003 
 
 
 
 
 
 
 
 
 
 
_______________________ 
Bibi : CIRPÉE and Faculté des Sciences Économiques et de Gestion de Tunis, Campus Universitaire, Bd. 7 Nov., 
El Manar, C.P. 2092, Tunis, Tunisia ; fax 216-71-93-06-15 
samibibi@gnet.tn 
Duclos : CIRPÉE and Département d’économique, Pavillon DeSève, Université Laval, Québec, Canada, G1K 
7P4 ; fax 1-418-656-7798 ; phone 1-418-656-7096 
jyves@ecn.ulaval.ca 
 
This research was partly funded by Canada’s SSHRC, Québec’s FQRSC, and the PEP programme of the 
International Development Research Centre. We are grateful to Jean-Michel Grandmont and Philip Merrigan for 
their useful comments. Abstract: Variations in aggregate poverty indices ca be due to differences in average 
poverty intensity, to changes in the welfare distances between those poor of initially 
unequal welfare status, and/or to emerging disparities in welfare among those poor of 
initially similar welfare status. This note uses a general cost-of-inequality approach 
that decomposes the total change in poverty into a sum of indices of each of these 
three components. This decomposition can serve inter alia to integrate horizontal and 
vertical equity criteria in the poverty alleviation assessment of social and economic 
programs. The use of these measures is briefly illustrated using Tunisian data. 
 
 
 
Keywords: Poverty, Vertical Equity, Horizontal Equity, Targeting, Tunisia 
 
JEL Classification: D12, D63, H53, I32, I38 1 Introduction
This note presents a method that decomposes aggregate poverty differences into
movements in average poverty intensity and into changes in the vertical and hor-
izontal locations of the poor. Such aggregate poverty differences can be due, for
instance, to the effect migration, socio-economic mobility or growth. They can
also arise from the impact of a policy or when comparing the impact of two poli-
cies. In decomposing such poverty differences, we will account for the role of
three summary measures:
1. A measure of the differences in average poverty intensity, which captures
by how much the average poverty gap is affected by a distributional change.
This measure is distribution-insensitive across the poor. In a policy context,
it can be linked to popular targeting-accuracy indicators and related to rates
of beneﬁt leakages.
2. A measure of the vertical impact of the change. This captures the extent to
which vertical inequality in the distribution of poverty gaps is reduced by a
distributional change. For policy purposes, it can serve to assess the respect
of the vertical equity (VE) criterion, which demands a search for a reduction
in the welfare gaps that separate unequal individuals.
3. A measure of the horizontal impact of the distributional change. In a policy
context, this can be linked to the horizontal equity (HE) of the program. The
”classical” deﬁnition of HE indeed deﬁnes HE as ”the equal treatment of
equals” (see Musgrave (1959)), and there is horizontal inequity (HI) when
HE is violated.
The paper mainly shows how a simple combination of the above measures
can capture the trade-offs as well as the differences between indicators of average
poverty intensity and indicators of vertical and horizontal distances. This can be
useful for descriptive as well as for policy design purposes.
Section 2 outlines the basic methodology, Section 3 shows how to decompose
the total cost of inequality into vertical and horizontal contributions, and Section
4 illustrates brieﬂy the methodology using 1990 Tunisian household data.
22 The basic methodology
2.1 Poverty and inequality
Consider a vector y = (y1;y2;:::;yN;n1;n2;:::;nN) of living standards yh (in-
comes, for short) for a population of n =
PN
h=1 nh individuals. Let the poverty
line be denoted as z. Many of the common poverty measures can be expressed
in terms of poverty gaps, gh(z) = max(z ¡ yh;0), with g(z) the vector of these
poverty gaps.1 An important subset of these measures is the class of the FGT
(Foster, Greer and Thorbecke (1984)) additively decomposable indices, which are
deﬁned as:
P®(g(z)) = n
¡1
N X
h=1
nhgh(z)
® (1)
where ® may be considered as a measure of aversion to inequality of poverty gaps.
In the manner of Atkinson (1970) for the measurement of social welfare and
inequality, letΓ®(g(z))bethe”equally-distributedequivalent(EDE)povertygap”,
viz, that poverty gap which, if assigned equally to all individuals, would produce
the same poverty measure as that generated by the actual distribution of poverty
gaps. Using (1), Γ®(g(z)) is given implicitly for ® > 0 as
Γ®(g(z)) = P®(g(z))
1
® for ® > 0:
Note that Γ1(g(z)) is the average poverty gap. For ® > 1, the more important the
difference between Γ®(g(z)) and Γ1(g(z)), the more unequal is the distribution
of poverty gaps. A natural measure of the cost of inequality is then given by:
C®(z) = Γ®(g(z)) ¡ Γ1(g(z)) for ® ¸ 1: (2)
Because C®(z) is given in per capita money-metric terms, it can be compared
directly to Γ1(g(z)). By (2), total poverty can be expressed as:
Γ®(g(z)) = Γ1(g(z)) + C®(z); ® ¸ 1: (3)
Note that it is only when the poverty gaps are equally distributed across the total
population that the cost of inequality becomes zero.
1 On this, see for instance Jenkins and Lambert (1997).
32.2 Poverty and targeting
Now consider a distributional change i which leads to an income distribution yi
with respective yi
h, gi
h(z), gi(z), and Ci
®(z). Assume that the per capita change
in income is given by ½i. The leakage of that change away from the poor is then
given for a change i by
L
i(z) = ½
i ¡ (Γ1(g(z)) ¡ Γ1(g
i(z)): (4)
The overall poverty impact is given by:
E
i
®(z) = Γ®(g(z)) ¡ Γ®(g
i(z)): (5)
Ei
®(z) can be thought of as a poverty-effectiveness measure of the change i. Using
(2 ), we can rewrite (5) as:
E
i
®(z) = ½ ¡ L
i(z) + C®(z) ¡ C
i
®(z): (6)
The poverty effectiveness of the change is thus a function of the average change
½i, the leakage to the non-poor Li(z), and the redistributive impact C®(z)¡Ci
®(z).
3 Horizontal and vertical effects
3.1 Horizontal effects
For any ﬁxed yh in pre-change y, let Ω(yh) denote the group of nh equals located
at point yh. Let °i
®(gh(z)) then be the post-change EDE poverty gap at yh,
°
i
®(gh(z)) =
0
@n
¡1
h
X
Ω(yh)
g
i
h(z)
®
1
A
1=®
: (7)
Using the cost-of-inequality approach developed in Section 2, a natural measure
of the local cost of horizontal variability at yh is then given by:
´
i
®(gh(z)) = °
i
®(gh(z)) ¡ °
i
1(gh(z)) ¸ 0: (8)
In a policy context this can be interpreted as a local cost of HI at yh, generated
by post-policy inequality within the members of Ω(yh). An obvious next step is
4to aggregate the ´i
®(gh(z)) across the yh. Using population shares to do this2, an
aggregate index of horizontal variability (and thus of HI) is obtained as:
H
i
®(z) = n
¡1
N X
h=1
nh´
i
®(gh(z)): (9)
3.2 Vertical effect
Focus now on the distribution of the local EDE poverty gaps °i
®(gh(z)). Denote
this distribution as °i
®(z) = (°i
® (g1(z));:::;°i
® (gN(z));n1;:::;nN). The cost of
inequality with °i
®(z) is then given by:
C
¤i
® (z) = Γ®(°
i
®(z)) ¡ Γ1(°
i
®(z)): (10)
C¤i
® (z) can then be interpreted as the cost of inequality of a post-change distribu-
tion in which everyone is attributed his group-equivalent poverty gap. The vertical
effectiveness (or vertical equity VE) of that change can then be assessed through a
comparison of (10) with the cost of inequality in the initial distribution of poverty
gaps:
V
i
®(z) = C®(z) ¡ C
¤i
® (z): (11)
3.3 Overall poverty effectiveness
We then have:
Theorem 1 The poverty effectiveness of a distributional change i is given by
E
i
®(z) = ½
i ¡ L
i(z) + V
i
®(z) ¡ H
i
®(z): (12)
Proof of Theorem 1. See appendix.
If we assume identical the per capita impact of two distributional changes, 1
and 2, such that ½1 = ½2, and if we denote ∆F = F 2 ¡ F 1, the difference in
poverty effectiveness between two distributional changes is given by:
∆E®(z) = ¡∆L(z) + ∆V®(z) ¡ ∆H®(z): (13)
Note that the formulation of (13) shows clearly the nature of the trade-off
that can emerge between leakage and vertical and horizontal effects. A change
2 See for instance Blackorby, Donaldson and Auersperg (1981), Musgrave (1990), Lambert
and Ramos (1997), Duclos and Lambert (2000), and Bibi (2002).
5can dominate another even with a higher leakage and a lower degree of vertical
effectiveness if it introduces less horizontal variability. When ® = 1, however,
V i
1(z) = Hi
1(z) = 0, which says that differences in poverty effectiveness depend
solely on differences in leakages away from the poor.
4 An application to Tunisia
We illustrate the use of the methodology presented above using a 1990 Tunisian
survey, ”Enquˆ ete Nationale sur le Budget et la Consommation des M´ enages 1990”
(National Household Budget and Expenditure Survey). This household survey is
multipurpose and provides information on consumption expenditures for various
items as well as extensive socio-demographic information on 7734 households.
The main anti-poverty program currently in force in Tunisia is based on the subsi-
dization of food consumption and thus on ”commodity targeting”.3 Government
expenditures on that program have been substantial throughout the 1980’s and the
1990’s, amounting to 4.1% of GDP in 1984, 2.9% in 1990, and 2% in 1995. We
compare the outcome of this program with that of an alternative one based on re-
gional targeting – involving the same overall budgetary outlay for the government
– in the manner of Kanbur (1987).4 For expositional simplicity, we ignore the
extent of deadweight losses under commodity targeting. A real per capita poverty
line z of 360 Tunisian Dinars per year (roughly equal to the often-used US$1-a-
day line) is used. As in Duclos and Lambert (2000), we identify the post-policy
distribution of pre-policy equals using a non-parametric estimation of the joint
distribution of pre-policy and post-policy incomes.
Table 1 shows the estimates of the poverty effectiveness measures following
this hypothetical reform. Brieﬂy, the impact of regional targeting of transfers
would be more variable horizontally than that of the current system of commodity
targeting, as shown here by H®(z) for ® = 2;3. But although the HE viola-
tions which would arise with this hypothetical reform would certainly reduce its
poverty impact, they would not be considered enough here to offset its higher
3 Details about this program can be found in Tuck and Lindert (1996).
4When the minimization of P®(g(z)) at the national level is the policymaker’s objective, the
available budget should be allocated such as to equalize the P®¡1(gj(z)) of each region j to a
common value. Our regional targeting scheme thus works as follows. Transfers are ﬁrst awarded
to everyone living in the poorest region such as to equalize the region’s P®¡1(gj(z)) to that of the
next poorest region. Transfers are then awarded to each person living in these two poorest regions
such as to equalize their P®¡1(gj(z)) to that of the third poorest region. This pattern is repeated
until the entire available budget is spent.
6vertical effect (as shown by V®(z)) and lower rate of leakage (shown by L(z))).
Overall, therefore, E®(z) is larger for regional targeting. This also serves to show
how this paper’s decomposition methodology can be useful for understanding and
optimizing the poverty impact of poverty alleviation schemes.
References
[1] Atkinson, A. B. (1970), On the Measurement of Inequality, Journal of Eco-
nomic Theory 2, 244-263.
[2] Bibi, S.(2002), HorizontalInequityandtheRedistributiveEffectoftheAnti-
Poverty Design, paper presented at the eighth annual conference of The Eco-
nomic Research Forum, Cairo, January 15 - 17, 2002, Egypt.
[3] Blackorby, C., D. Donaldson and A. Auersperg (1981), Ethical Social In-
dex Numbers and the Measurement of Effective Tax / Beneﬁt Progressivity.
Canadian Journal of Economics, vol. 17, 683-694.
[4] Duclos, J.-Y. and P. J. Lambert (2000), A Normative Approach to Measuring
Classical Horizontal Inequity, Canadian Journal of Economics, vol. 33, 87-
113.
[5] Foster, J. E., J. Greer and E. Thorbecke (1984), A Class of Decomposable
Poverty Measures, Econometrica, vol. 52, 761-765.
[6] Jenkins, S. P. and P. J. Lambert (1997), Three ’i’s of Poverty Curves, with an
Analysis of UK Poverty Trends, Oxford Economic Papers. vol. 49, 317-327.
[7] Kanbur, R. M. (1987), Measurement and Alleviation of Poverty with an Ap-
plication to the Effects of Macroeconomic Adjustment, IMF Staff Papers.
vol. 34, 60-85.
[8] Lambert, P. J. and X. Ramos (1997), Horizontal Inequity and Vertical Redis-
tribution, International Tax and Public Finance, vol. 4, 25-37.
[9] Musgrave, R.A., (1959), TheTheoryofPublicFinance, McGraw-Hill, New-
York.
[10] Musgrave, R. A., (1990), Horizontal Equity, Once More, National Tax Jour-
nal, vol.43, 113-122.
7[11] Tuck, L. and K. Lindert (1996), From Universal Food Subsidies to a Self-
Targeted Program, A Case Study in Tunisian Reform, World Bank Discus-
sion Paper, no. 351, The World Bank, Washington, DC.
5 Appendix
Proof of Theorem 1. Recall that by (10) we have
C
¤i
® (z) = Γ®(°
i
®(z)) ¡ Γ1(°
i
®(z)): (14)
Noting that Γ®(°i
®(z)) = Γ®(gi(z)) and adding and subtracting Γ1(gi(z)) on the
right-hand side of (14), we ﬁnd
C
¤i
® (z) = C
i
®(z) ¡ H
i
®(z): (15)
Since the VE of change i is given by
V
i
®(z) = C®(z) ¡ C
¤i
® (z); (16)
subtracting C®(z) from the two sides of (15) and using (16), it follows that
C®(z) ¡ C
i
®(z) = V
i
®(z) ¡ H
i
®(z): (17)
The proof of Theorem 1 follows from substituting (17) into (6).
8Table 1: Poverty effectiveness of two types of targeting in Tunisia (in 1990
Tunisian Dinars)
Benchmark Commodity targeting Regional targeting
½i 0 34.79 34.79
Γ1(gi(z)) 35.64 28.33 23.41
Li(z) - 27.48 22.56
Ei
1(z) - 7.31 12.23
Γ2(gi(z)) 75.55 65.36 55.19
Hi
2(z) 0 0.28 1.53
V i
2(z) - 3.16 9.65
Ei
2(z) - 10.19 20.35
Γ3(gi(z)) 102.37 91.47 78.13
Hi
3(z) 0 0.51 2.70
V i
3(z) - 4.10 14.71
Ei
3(z) - 10.89 24.23
9

A Normative Approach to Measuring Classical Horizontal Inequity,

Ethical Social Index Numbers and the Measurement of Effective Tax / Beneﬁt Progressivity.

Horizontal Equity, Once More,

HorizontalInequityandtheRedistributiveEffectoftheAntiPoverty Design, paper presented at the eighth annual conference of The Economic Research Forum,

Measurement and Alleviation of Poverty with an Application to the Effects of Macroeconomic Adjustment,

On the Measurement of Inequality,

Three ’i’s of Poverty Curves, with an Analysis of UK Poverty Trends, Oxford Economic Papers.

Decomposing Poverty Changes into Vertical and Horizontal Components

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Decomposing Poverty Changes into Vertical and Horizontal Components

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