The Estimation of the Growth and Redistribution Components of Changes in Poverty: A Reassessment Florent Bresson∗ Cemafi, Université de Nice Sophia Antipolis version 1.1 September 1, 2008

Abstract What are the respective contributions of growth and inequality changes to observed poverty variations? Many studies have attempted to provide some empirical evidence to answer this question using case studies with decompositions of observed poverty spells. Most of them rely on two decomposition frameworks suggested by Datt and Ravallion (1992) on the one hand, and Shorrocks (1999) and Kakwani (2000) on the other hand. However, despite their properties, these techniques are not appropriate for such an accounting exercise. Here, following Muller (2006), we propose an alternative decomposition procedure that is consistent with definitions of growth and inequality effects stemming from time-integral calculus. Contrary to the aforementioned methods, the proposed technique simultaneously fits the observed pattern of income distributions changes and does not produce large residual components.

JEL classification: I32, C00. Key words: Poverty variations decomposition, growth effect, inequality effect.

∗

Contact: [email protected]. This paper is an extended version of Bresson (2008b) that includes the corrections suggested in Bresson (2008a). The author would like to thank Roland Kpodar and Sylvain Chabé-Ferret for their helpful comments.

1

1

I NTRODUCTION

What are the respective contributions of growth and inequality changes to observed poverty variations? Inspired by the theoretical debates concerning the validity of “trickle-down” mechanisms and the necessity to achieve “pro-poor” growth, many studies have tried to provide some empirical evidence to this question. Many of them are case studies with decompositions of observed poverty spells (see Bhanumurthy and Mitra, 2004, Wan and Zhang, 2006, Baye, 2006, Dhongde, 2007, for recent illustrations). Generally, these studies rely on two decompositions frameworks suggested by Datt and Ravallion (1992) on the one hand, and Shorrocks (1999) and Kakwani (2000) on the other hand. However, we argue that these techniques are not appropriate for such an accounting exercise in spite of their attractive properties. Here, following Muller (2006), we propose an alternative decomposition procedure that is consistent with definitions of growth and inequality effects stemming from time-integral calculus. Contrary to aforementioned methods, the proposed technique simultaneously fits the observed pattern of income distributions changes and does not yield large residual components. Section 2 reviews traditional decompositions techniques in single- and multi-period contexts while section 3 is dedicated to the presentation of the new decomposition method. In section 4, the differences between the techniques are illustrated using Nicaraguan income distributions for the period 1993-2001. Section 5 concludes with further comments.

2

E XISTING DECOMPOSITIONS

Let Θ be any absolute poverty measure that complies with traditional core axioms of poverty measurement (see Zheng, 1997, for a review).1 A common practice in the poverty related literature is to characterize an income distribution by its mean value and its relative distribution. The measure Θ can thus be expressed as a function of the poverty line z, the mean income µ, a set π of r parameters that fully describes the relative income distribution, and a vector α that accounts for ethical preferences of the social evaluator.2 Hence, Θ = Θ(z, µ, π, α). We suppose that the poverty line is time-invariant, an assumption that entails to consider real incomes. In the same manner, the vector α is assumed constant since there is little reason to compare the level of poverty between two dates on the basis of different ethical preferences. As a consequence, the notation of Θ can be simplified into Θ(µ, π). 1

To the list of core axioms defined by Zheng (1997), some invariance axiom should also be added so as to get an oprational set of axioms for poverty measurement. Usually, the chosen poverty measures are based on the scale invariance axiom, but our developments are also valid for measures based on rival invariance axioms (see Bresson and Labar, 2007, for a recent review of invariance axioms). 2 Generally, π is defined as a vector of parameters that fully describes the Lorenz curve. This interpretation is normativelly non-neutral since it entails focusing on the sole poverty measures that comply with the scale invariance axiom. As this choice can be debated (see Amiel and Cowell, 1992, for empirical evidence), we prefer a more general and neutral interpretation.

2

The objective of the decompositions considered in the present paper, is to assess the respective contribution of mean income growth and inequality changes in observed poverty variations. Let C denote that theoretical contribution of growth and I the one attributed to relative distribution changes. As µ and π are the sole variable determinants of Θ, our goal is to obtain the following decomposition between dates t and t + k: Θ(µt+k , πt+k ) − Θ(µt , πt ) = ∆t,t+k Θ(µ, π) = C t,t+k + It,t+k . 2.1

(2.1)

T HE SINGLE PERIOD CASE

In spite of the apparent simplicity of the procedure, the computation of C and I is not straightforward because of the intrinsic non-separability of Θ (∂2 Θ/∂µ∂πi 6= 0∀i ∈ {1 . . .r }).3 This explains the existence of rival procedures in the literature for the calculation of these growth and inequality effects. Here we focus on the decomposition procedures proposed by Datt and Ravallion (1992) at one hand (thereafter called DR decomposition), and Shorrocks (1999) and Kakwani (2000) at the other hand (thereafter noted KS decomposition).4 In the former case, the suggested decomposition is: ∆t,t+1 Θ(µ, π) = Θ(µt+1 , πt ) − Θ(µt , πt ) + Θ(µt , πt+1 ) − Θ(µt , πt ) +R t,t+1 , {z } | {z } | C t ,t +1 :=C t ,t +1

(2.2)

It ,t +1 :=I t ,t +1

with R being a residual term. The DR procedure is very appealing since C and I exhibit very intuitive meanings: the growth (inequality) effect C (I ) corresponds to the variation of Θ that would have occured for the observed rate of growth (inequality change) if the realtive income distribution (the mean income) was held fixed. This means that the proposed formulas estimates the respective impacts of mean income and inequality changes since they are computed through a comparison with counterfactual situations. The residual term R appears when both µ and π varies, that is in nearly all cases. The authors interpret it as the difference between the inequality effect computed with respect to mean income in t + 1 and the one obtained with respect to mean income at date t , that is I t+1,t − I t,t+1 . However, they notice that it can equivalently be seen as the difference C t+1,t − C t,t+1 . In other words, it represents the interactive effect of the simultaneous variations of µ and π on poverty. In practice, the size of this interactive term may be nonnegligible.5 This is a rather puzzling result since it represents a clear failure in accounting for the whole respective contributions of growth and inequality changes in observed poverty 3

A noticeable exception is the class of poverty measures defined by Tsui (1996). In order to save space, the methods suggested by Jain and Tendulkar (1990) and Kakwani and Subbarao (1990) have been skipped. The main drawback of these procedures is that they cannot be used to compare the relative contribution of growth and inequality changes in terms of poverty alleviation since they are not computed on the same basis. 5 Considering the empirical illustration conducted by the authors, the residual term account for about fifty percents of total poverty variations in some cases. 4

3

variations. To get a decomposition which respects relation (2.1), it is then necessary to share the residual term R between the growth and inequality effects. The solution suggested by Shorrocks (1999) relies on the use of the Shapley (1953) value developed in game theory so as to define an efficient rule for the sharing of gains among players. The intuition behind this rule is that every participant should gain the mean value of the marginal contributions that he adds in every possible coallitions with the other players. A formal general presentation of the Shapley value is given in appendix A. Its application to poverty variation yields the following decompostion: C t ,t +1 :=C t′ ,t +1

∆t,t+1 Θ(µ, π) =

z¡

¢ ¡}| ¢{ Θ(µt+1 , πt ) − Θ(µt , πt ) + Θ(µt+1 , πt+1 ) − Θ(µt , πt+1 )

+

¡ ¢ 2¡ ¢ Θ(µt , πt+1 ) − Θ(µt , πt ) + Θ(µt+1 , πt+1 ) − Θ(µt+1 , πt ) 2 {z

|

It ,t +1 :=I t′ ,t +1

(2.3) .

}

The decomposition has also been proposed by Kakwani (2000) who followed an axiomatic approach to derive equation (2.3). The author argued that the three following conditions should be met for a decomposition procedure to be valid: i) when the growth (inequality) effect is zero, then the change in poverty must be entirely due to change in income inequality (mean income), ii) if both growth and inequality effects are negative (positive), then poverty should decline (increase), and iii) the growth (inequality) effect from the initial to the final date must be the opposite of the growth (inequality) from the final to the initial date. It can ′ easily be shown that C t,t+1 = C t,t+1 +

2.2

R t ,t +1 2

′ and I t,t+1 =I+

R t ,t +1 6 2 .

T HE MULTIPLE PERIODS CASE

If the decomposition is applied in a multiple periods context, both Datt and Ravallion (1992) and Kakwani (2000) notice that the formulas presented for the single period context are not valid any more since they do not comply with the sub-period additivity principle. Suppose we are analyzing the evolution of poverty over a period 0 to T . Let t , u and v be three dates such that 0 6 t 6 u 6 v 6 T , {t , u, v, T } ∈ N. Sub-period additivity means that the growth (ineqality) effect between t and v should be the sum of the estimated growth effects for the sub-periods t to u and u to v , i.e. in formal terms:7 C t,v = C t,u + C u,v

and It,v = It,u + Iu,v .

(2.4)

However, this property is not satisfied by any of the decomposition procedures presented in the last section. Therefore, to deal with this issue, one has to choose between 6 7

This solution has already been considered by Datt and Ravallion (1992) but judged arbitrary. In Datt and Ravallion (1992), the sub-period additivity condition is also applied to the residual term.

4

changing the formula corresponding to the multiperiod effects, modifying the equations related to the sub-period effects, or revising both expressions . In Datt and Ravallion (1992), the second solution is adopted: sub-period additivity is satisfied if sub-periods effects are computed with respect to a unique reference income distribution. As the initial distribution of the whole period (date 0) can be considered as a natural choice, equation (2.2) is modified in the following way: ¢ ¡ ¢ ¡ ∆t,u Θ(µ, π) = Θ µu , π0 − Θ(µt , π0 ) + Θ µ0 , πu − Θ(µ0 , πt ) +R t,u , {z } | {z } | C t ,u :=C t0,u

(2.5)

It ,u :=I t0,u

where the superscript indicates that year 0 has been chosen for the reference distribution. The multi-period decompostion proposed by Kakwani (2000) consists in redefining both sub-period and whole-period effects. It relies on a version of the sub-period additivity principle that is more restrictive than the one defined by Datt and Ravallion (1992) since conditions (2.4) have to hold ∀u ∈ {0, 1 . . . T }. The introduction of this condition in his axiomatic framework yields the following the decomposition: ∆t,u Θ(µ, π) =

´ ´ T ³ T ³ 1 X 1 X ′ ′ C t,j +C ′j ,u + I t,j + I ′j ,u , T + 1 j =0 T + 1 j =0 {z } | {z } | C t ,u :=C˜t0,T ,u

(2.6)

It ,u :=I˜t0,T ,u

where the superscript 0, T corresponds to the reference period on which are based the estimations. Although the multi-period context has not been investigated by Shorrocks (1999), it can easily be seen that decomposition (2.6) is a departure from the application of the Shapley value. Indeed, a literally use of this tool would require to dismiss the sequence of observed changes in the income distribution. As the Shapley value is computed using the different possible sequence of players participations, marginal contributions of growth and inequality changes would be estimated using every arrangements of the different ∆t,t+1 µ and ∆t,t+1 π that are observed over the whole period. For instance, in a two period context (dates 0, 1 and 2), using the Shapley decomposition entails considering the 24 arrangements of ∆0,1 µ, ∆0,1 π, ∆1,2 µ and ∆1,2 π. The corresponding growth effect C S for the second period

5

would then be: ¢ 1¡ ¢ 1¡ Θ(µ0 + ∆1,2 µ, π0 ) − Θ(µ0 , π0 ) + Θ(µ2 , π0 ) − Θ(µ1 , π0 ) 4 12 ¢ 1¡ ¢ 1¡ + Θ(µ0 + ∆1,2 µ, π0 + ∆1,2 π) − Θ(µ0 , π0 + ∆1,2 π) + Θ(µ2 , π1 ) − Θ(µ1 , π1 ) 12 12 ¢ 1¡ ¢ 1¡ Θ(µ2 , π0 + ∆1,2 π) − Θ(µ1 , π0 + ∆1,2 π) + Θ(µ0 + ∆1,2 µ, π2 ) − Θ(µ0 , π2 ) + 12 12 ¢ 1¡ ¢ ¢ 1¡ Θ(µ0 + ∆1,2 µ, π1 ) − Θ(µ0 , π1 ) + Θ(µ2 , π2 ) − Θ(µ1 , π2 ) + ∆1,2 µ, π1 ) − Θ(µ0 , π1 ) . + 12 4 (2.7)

S C 1,2 =

which clearly differs from the effect stemming from equation (2.6).

2.3

D RAWBACKS

As noted earlier, the DR decomposition framework has been widely criticized due to the presence of a non-negligible residual term. Indeed, it represents a failure in accounting for the whole respective contributions of growth and inequality changes in observed poverty variations as the observer is left with a black-box that cannot be easily interpreted. The crucial point with this technique is that it does not measure what it is designed for since it is based on a confusion between “contributions” and “impacts”. Indeed, DR effects can be interpreted as the impacts of growth and inequality changes on the level of poverty since they corresponds to the variation of poverty that would have occured for a given factor’s variation other things being equal. A consequence is that DR growth and inequality effects are perfectly orthogonal as they are estimated using counterfactual situations. However, C and I are simultaneously determined and our feeling is that the estimation of contributions should not skip this association. This remark does not mean that the DR decomposition is false but that it does not suit the estimation of the global contributions of growth and inequality changes. An other issue is that the solution suggested in the multiple period context is puzzling since the DR effects do not exhibit such an appealing interpretation as for the single period 0 context. For instance, we could question the interest of knowing C t,t+1 for t 6= 0 since it

corresponds to the change in inequality that would have occured if income inequality were constant for the income distribution at the reference year. Thus it is very poorly related to observed poverty in dates t and t + 1. The KR decomposition framework is not free from all criticism. Recently, Muller (2006) argued that the single-period KS decomposition procedure is not satisfying and that its theoretical background is confusing (statistical phenomenons are not players). In particular, he considers that the evaluation of any decomposition technique should not be based on the capacity to provide a null residual or to comply with any practical mathematical property. In this spirit, we emphasize that the desirability of the third axiom proposed by Kakwani (2000) 6

should be questioned. Yet, the aim of poverty variation decompositions is to assess the contribution of factors that drive the evolution of poverty through time. Fundamentally, since time development is a one-way road, it does not matter whether the effects estimated when considering a move from the income distribution t + 1 to the one in t are equal (in absolute value) to those that effectively corresponds to observed evolutions from t to t + 1. One can also notice that the multi-period decomposition defined in equation (2.6) implies that the estimated growth and inequality effects are path-dependant. From this equation, it can be seen that any effect corresponding to a given sub-period depends on income distribution that are considered before, during and after the sub-period of interest. For instance, in the case of a four-period analysis (with years 0, 1, 2, 3, 4), the estimation of C˜ 0,4 1,3

would not only be driven by the income distributions in years 1 and 3, but also with those observed for the years 0, 2 and 4. While the influence of the income distribution in year 2 may be reasonnably be accepted, it seems more difficult with the initial and final distributions.

3

A N INTEGRAL - BASED DECOMPOSITION PROCEDURE

Muller’s (2006) most forceful argument against the aforementioned decomposition techniques is that they are not consistent with the theory of integration. Indeed, the observed poverty variation between t and t + 1 can be rewritten in the following manner: ∆t,t+1 Θ(µ, π) =

Zt+1

d Θ(µq , πq )

t

dq

(3.1)

d q,

where q is time. Equation (3.1) can easily be linked to the issue of poverty variations decompositions into growth and inequality components. Using a first-order Taylor expansion, we obtain: ∆t,t+1 Θ(µ, π) ≃

Zt+1 |

t

∂Θ(µq , πq ) ∂µ ∂µ {z

∂q

C t ,t +1

r Zt+1 ∂Θ(µ , π ) ∂π X q q i

dq + } |i =1

t

∂πi {z

It ,t +1

∂q

dq .

(3.2)

}

Comparing equations (3.2) and (2.2) is informative about the nature of the DR decomposition. This latter also corresponds to a first-order Taylor expansion of ∆t,t+1 Θ(µ, π), but the first-order derivatives are substituted by linear interpolations all over the time interval. An alternative expression of equation (2.2) is then: C t ,t +1

z }| { Θ(µ2 , π1 ) − Θ(µ1 , π1 ) (µ2 − µ1 ) ∆t,t+1 Θ(µ, π) = µ2 − µ1

7

+

r Θ(µ , π , π−i ) − Θ(µ , π ) X 1 i ,2 1 1 1

πi ,2 − πi ,1 {z

i =1

|

(πi ,2 − πi ,1 ) +R t,t+1 .

(3.3)

}

I t ,t +1

where π−i is the vector π minus its i -th component. The presence of the residual term in (3.3) can be explained by the non-linear relation between Θ and its different arguments. As observed variations of µ and π are generally not marginal, this approximation is unlikely to yield satisfying approximations of equation (3.2), except for small variations of Θ’s determinants. If anyone intends to realize a decomposition that is based on equation (3.2), it is thus necessary to have an estimation of the first-order derivatives of Θ with respect to µ and π all over the intervals defined by their initial and final values. A solution is the use of local first-order Taylor expansion for marginal variations of each factor. Assuming that the evolution of these parameters are linear between t and t + 1 (i.e. ∂µ/∂q and ∂πi /∂q are constant over the time interval [t , t +1]), we can divide ∆t,t+1 µ and ∆t,t+1 π into s small identical variations. Then, for each variation j = 1 . . . s, first order derivatives of the poverty measure can be approximated in the following manner: ∂ Θ µt+ j −1 , πt+ j −1 ≃ s s ∂µ ´

³

´ ∂ ³ Θ µt+ j −1 , πt+ j −1 ≃ s s ∂πi

³ ´ ³ ´ Θ µt+ j , πt+ j −1 − Θ µt+ j −1 , πt+ j −1 s

s s s , µt+ j − µt+ j −1 s s µ ¶ ´ ³ −i Θ µt+ j −1 , πi ,t+ j , π j −1 − Θ µt+ j −1 , πt+ j −1 s

t+

s

s

s

s

πi ,t+ j − πi ,t+ j −1 s

(3.4)

.

(3.5)

s

Plugging these approximations into equation (3.2) yields the following decomposition: C t ,t +1 :=Cˆt ,t +1

∆t,t+1 Θ(µ, π) ≃ +

z s X

}| { ³ ´ ³ ´ Θ µt+ j , πt+ j −1 − Θ µt+ j −1 , πt+ j −1

j =1 s X r X

j =1 i =1

s

µ

s

s

s

¶

³ ´ Θ µt+ j −1 , πi ,t+ j , π−i j −1 − Θ µt+ j −1 , πt+ j −1 . s

s

|

t+

{z

s

It ,t +1 :=Iˆt ,t +1

s

(3.6)

s

}

which tends to equal equation (3.2) as s → +∞. Thus, the suggested procedure consists in decomposing observed variations of µ and π into s small equal variations and then, for each micro-variation, to compute the growth and inequality effects stemming from the DR decomposition. Summing up these s growth (inequality) micro-effects gives the desired global growth (inequality) effect. The practical implementation of this technique is reported in appendix B. When considering poverty variations between dates t and t + k, the information related to mean income and inequality changes is more complete since the income distributions

8

in t + 1, t + 2. . . t + k − 1 can be observed. Consequently, the assumption that the evolution of µ and π was linear during the whole period is not necessary anymore and additional information should be exploited for the computation of the multi-period effects. However, the hypothesis of linear evolutions of growth and inequality during each subperiod has to be maintained. The multi-period decomposition can be then defined as: ∆t,t+k Θ(µ, π) ≃

k X

Cˆt+j −1,t+j +

Iˆt+j −1,t+j .

(3.7)

j =1

j =1

|

k X

{z

CˆtΣ,t +k

} |

{z

IˆtΣ,t +k

}

The chosen strategy is thus at odd with the one adopted by Datt and Ravallion (1992) and Kakwani (2000) since it consists in adapting the single-period decomposition for the multiple periods context. This extension to the multi-period context is elegant since it does not impose endorsing mathematical contorsions for the respect of sub-period additivity, and can be easily justified from an informational point of view. Finally, it is worth noting that the proposed decomposition is path-dependant. Nonetheless, contrary to Kakwani’s (2000) multi-period decomposition, estimated effects do not depend of changes that have occured during the periods 0 to t and t + k to T . The results obtained over any subperiod are thus consistent with those get over the whole period.

4

A N APPLICATION TO N ICARAGUA FOR THE PERIOD 1993-2001

The differences between the methods presented in the previous sections are now illustrated using household data from the 1993, 1998 and 2001 Nicaragua Living Standard Measurement Surveys.8 Each round of these nationally representative surveys provides consumption information for about 4, 200 households. The chosen poverty line is the 1993 extreme poverty line defined by the authors of the surveys, that is C$1,215 per year and per capita (see Development Research Group, 2002, appendix B, for more details). Consumption series for 1998 and 2001 are deflated using Word Bank’s CPI data. Lastly, household consumption data are equaly shared among its members so as to get individual series. As the exercise is meant to be merely illustrative, our comments will remain quite evasive concerning the Nicaraguan economic context. Table I presents some summary statistics about the income distribution and the level of poverty in Nicaragua. It can be seen that, during the period, average real income per capita fell while income inequalities decreased. As poverty declined between 1993 and 2001, whatever poverty measure is considered, we can conclude that this evolution of the relative income distribution was clearly at the benefit of extreme poor households. 8

General informations about the Living Standard Measurement Surveys are available in Glewwe and Grosh (1998). More details concerning the specific surveys used in the current study can be obtained from the LSMS website ( www.worldbank.org/html/prdph/lsms).

9

Table I: The distribution of consumption in Nicaragua from 1993 to 2001. Year Mean consumption Gini Headcount Poverty gap Squared poverty gap

1993 3, 840 50.6 21.3 6.47 2.86

1998 3, 440 43.9 17.1 4.54 1.78

2001 3, 220 41.7 17.7 4.90 1.91

Note: Inequality and poverty measures are expressed in percent. The poverty line is fixed at C$1,245 per year and per capita.

Table II: Comparision of the different decomposition techniques: Nicaragua 1993-2001, headcount index. Period 1993-1998

∆Θ Cˆ Iˆ residual Datt & Ravallion (1992) -4.16 4.33 -8.28 -0.21 (1.06)

1998-2001 1993-2001

1993-1998 1998-2001 1993-2001

1993-1998

1993-2001

(1.14)

(0.58)

-0.523 (0.12)

0.49

2.61

-0.7

-1.42

-3.73

(0.98)

(0.96)

(0.94)

(0.73)

(22.09)

-3.67

6.94

-8.98

-1.63

-0.773

(1.04)

(1.25)

(1.06)

(0.72)

(0.13)∗∗

-4.16

Kakwani (2000) 4.16 -8.32

(1.06)

(0.99)

(1.16)

0.49

2.1

-1.61

(0.98)

(0.69)

(0.98)

-3.67

6.19

-9.86

(1.04)

(1)

(1.14)

-4.16 (1.06)

1998-2001

(1.08)

Cˆ /Iˆ

0

-0.5 (0.11)

0

-1.3 (10.8)

0

-0.628 (0.09)

New decomposition 3.92 -8.09 0.01

-0.485

(1)

(1.31)

(0.275)

(0.11)

0.49

2.17

-1.61

-0.07

(0.98)

(0.85)

(1.12)

(0.11)

(1.4 × 1013 )

-1.35

-3.67

6.09

-9.7

-0.06

-0.628

(1.04)

(1.11)

(1.35)

(0.29)

(0.09)

Note: Poverty variations, growth and inequality effects as well as residual terms are expressed in percent. The poverty line is fixed at C$1,245 per year and per capita. Standard errors in parentheses estimated with a bootstrap procedure with 1,000 replications. The symbols *, ** and *** respectively indicates that the value of Cˆ , Iˆ or Cˆ /Iˆ is significantly different from the corresponding value obtained with the technique proposed in section 3 at the 10, 5 and 1% levels.

10

Table III: Comparision of the different decomposition techniques: Nicaragua 1993-2001, poverty gap index. Period 1993-1998 1998-2001 1993-2001

1993-1998 1998-2001 1993-2001

1993-1998 1998-2001 1993-2001

-1.93

Cˆ Iˆ residual Datt & Ravallion (1992) 1.8 -3.17 -0.564

(0.39)

(0.47)∗∗∗

(0.40)∗∗∗

(0.20)

(0.13)∗∗∗

0.34

1.15

-0.36

-0.44

-3.12

(0.33)

(0.4)∗∗∗

(0.30)

(0.19)

(51.3)

∆Θ

Cˆ /Iˆ -0.56

-1.59

2.95

-3.53

-1.01

-0.83

(0.39)

(0.50)∗∗∗

(0.38)∗∗∗

(0.25)

(0.13)∗∗∗

0

-0.44

-1.93

Kakwani (2000) 1.56 -3.49

(0.39)

(0.37)

(0.46)

0.34

0.81

-0.46

(0.33)

(0.28)

(0.37)

-1.59

2.41

-4

(0.39)

(0.38)∗

(0.46)∗

(0.09)

0

-1.73 (10.3)

0

-0.60 (0.08)

-1.93

New decomposition 1.51 -3.44 −1.01 × 10−3

(0.39)

(0.37)

(0.46)

0.34

0.81

-0.46

(0.33)

(0.28)

(0.37)

-1.59

2.32

-3.91

(0.39)

(0.37)

(0.46)

(3.94 × 10−4 )

−1.54 × 10

−4

(1.37 × 10−4 )

−1.16 × 10

−3

(3.74 × 10−4 )

-0.43 (0.09)

-1.73 (27.3)

-0.59 (0.08)

Note: Poverty variations, growth and inequality effects as well as residual terms are expressed in percent. The poverty line is fixed at C$1,245 per year and per capita. Standard errors in parentheses estimated with a bootstrap procedure with 1,000 replications. The symbols *, ** and *** respectively indicates that the value of Cˆ , Iˆ or Cˆ /Iˆ is significantly different from the corresponding value obtained with the technique proposed in section 3 at the 10, 5 and 1% levels.

This statement is confirmed by the decompostions of the variations of the headcount, poverty gap and squared poverty gap indices reported in table II to IV. So as to use the decomposition technique suggested in section 3, the size of each income distribution has been normalized to 10,000 observations.9 In this particular case, the number of microvariations has been set to 1,000 (see appendix C for an analysis of the sensitivity of the results to that choice). Since the income distribution is available for three years, the multi-period decomposition techniques described above have been used. For practical considerations, the multi-period decomposition stemming from the strict application of the Shapley value has not been employed. It is worth noting that the three decomposition techniques yield the same general con9

More precisely, we have estimated the value the quantile function at each 0,01 percent of the population using the median-unbiased estimator suggested by Fan and Hyndman (1996).

11

Table IV: Comparision of the different decomposition techniques: Nicaragua 1993-2001, squared poverty gap index. Period 1993-1998

Cˆ Iˆ residual Datt & Ravallion (1992) -1.08 0.90 -1.61 -0.36 ∆Θ

1993-2001

1993-1998

(0.22)∗∗∗

(0.13)

0.11

0.61

-0.20

-0.29

-3.01

(0.17)

(0.21)∗∗∗

(0.15)

(0.12)

(3030)

1993-2001

1993-1998 1998-2001 1993-2001

(0.14)∗∗∗

-0.96

1.51

-1.81

-0.66

-0.83

(0.22)

(0.27)∗∗∗

(0.21)∗∗∗

(0.16)

(0.14)∗∗∗

-1.08

Kakwani (2000) 0.74 -1.83 (0.17)∗∗∗

(0.26)∗∗∗

0.11

0.36

-0.25

(0.17)

(0.12)

(0.19)

(0.22)

1998-2001

-0.56

(0.24)∗∗∗

(0.22)

1998-2001

Cˆ /Iˆ

-0.96

1.15

-2.11

(0.22)

(0.18)∗∗∗

(0.26)∗∗∗

0

-0.40 (0.08)∗∗∗

0

-1.44 (26.3)

0

-0.54 (0.07)∗∗∗

-1.08

New decomposition 0.70 -1.78 −7.3 × 10−4

(0.22)

(0.17)

(0.25)

0.11

0.36

-0.25

(0.17)

(0.12)

(0.19)

-0.96

1.07

-2.04

(0.22)

(0.17)

(0.25)

(2.6 × 10−4 )

−5.4 × 10

−5

(6.4 × 10−5 )

−7.9 × 10

−4

(2.5 × 10−4 )

-0.39 (0.08)

-1.44 (28.7)

-0.52 (0.07)

Note: Poverty variations, growth and inequality effects as well as residual terms are expressed in percent. The poverty line is fixed at C$1,245 per year and per capita. Standard errors in parentheses estimated with a bootstrap procedure with 1,000 replications. The symbols *, ** and *** respectively indicates that the value of Cˆ , Iˆ or Cˆ /Iˆ is significantly different from the corresponding value obtained with the technique proposed in section 3 at the 10, 5 and 1% levels.

clusions regardless the chosen poverty measure, i.e. the decrease in real per-capita income raised poverty while the relative distribution changes contributed to its alleviation. However, some differences deserve some attention. First, we can observe that the magnitude of the residual component obtained with the DR decomposition may be very important. For instance it is about three times larger than the observed variation of the headcount index for the period 1998-2001. On the contrary, the size of the residual term obtained with our integral-based decomposition is economically non-significant, in particular with distribution-sensitive poverty measures like the poverty gap and squared poverty gap indices. As a consequence, it can be considered as an error term and should not deserve further attention. Because of the presence of such large residual terms, DR estimated effects differs slightly from the corresponding values computed with our method and the one proposed in Kak-

12

wani (2000). The discrepencies are notably important with the headcount index. For instance, the inequality effect is about one percentage point lower with our decomposition technique than with the DR decomposition for the period 1998-2001. We can observe that this discrepency entails a diverging assessment of the relative contribution of growth and redistribution to poverty alleviation. Differences can also be observed when comparing the results of our decomposition technique with Kakwani’s (2000) method but they are generally not meaningfull from an economic point of view. However, that result would need further inquiry before claiming that the two decomposition techniques shall be deemed substitutes. To see whether the observed differences between the rival techniques are statistically significant in our application, we performed a non-parametric test based on a bootstrap procedure with 1,000 replications. Resampling was done at the household level. For each iteration, the difference with the effects obtained with our decomposition technique has been estimated and confidence interval have been computed using the observed distribution of these differences over the whole set of replications. The results show that the results of the test are highly sensitive to the choice of the poverty measure. Whereas differences are not statistically different from zero with the headcount index, a contrasting picture can be observed with the squared poverty gap. While this result is not surprising with the DR approach, it can also be observed regarding Kakwani’s (2000) technique for the periods 19931998 and 1993-2001. As a consequence, the two techniques should not be considered as perfect substitutes, even if the output are qualitatively similar in the context of our application on Nicaraguan data.

5

C ONCLUDING REMARKS

In the following paper, we proposed a new decomposition framework to account for the respective contributions of growth and inequality changes to poverty variations. Based on first-order Taylor expansions and time-integral definitions of the growth and inequality effects, the proposed procedure defines a consistant rule for sharing the residual term produced by the Datt and Ravallion’s (1992) decomposition. Our solution also contrasts with other decomposition techniques in the way multiple period decompositions are handled. Despite the drawbacks noted in section 2.3, it is worth noting that the DR and KS decomposition frameworks should not be relegated to historical surveys of poverty analysis. For instance, as explained earlier, the single-period DR decomposition is the right choice if one intends to estimate the respective impacts (i.e. ceteris paribus effects) of growth and inequality changes on the level of poverty. Regarding the Shapley value, Shorrocks’s (1999) intention was to provide a framework that could be used for both static and dynamic general decompositions of poverty measures. As our remarks are only related to a specific dynamic decomposition, the Shapley value may still remain a useful tool for static analysis (see Sastre and Trannoy, 2002, in the context of inequality measure decompositions).

13

Finally, along these lines, we just focused on decompositions of poverty variations into growth and inequality changes components. However, many other decompositions can be handled. For instance, Son (2003) suggests a four-element decomposition such that the inequality effect is itself decomposed into i) the contribution of differences in growth rates between the groups, ii) the effect of the change in inequality within the different groups, and iii) the outcome of changes in the population shares of the various groups. The decomposition can also be extended if current income are considered instead of real income for poverty measurement. This slight modification entails the addition of a purchasing power effect to the growth and inequality effects. In the same manner, we could consider the case of multidimensional poverty measure. For instance, Chakravarty, Deutsch, and Silber (2008) recently proposed a definition of the Watts multidimensional poverty index which estimation of the contribution of its five determinants to observed changes could be realized without relying on linear interpolation as in the general context of unidimensional measures considered in this paper. These extensions of the decomposition method presented here are beyond the scope of this paper, but should obviously deserve further scrutiny.

A PPENDIX A

T HE S HAPLEY VALUE

Consider a set K of m ∈ N∗ players which mutual efforts produced a gain G ∈ R. Let a coallition S be a subset from K of s players. If g (S) is the gain (or the loss) that can get the coallition S without negociating with the remaining players, the marginal contribution of player k when joining this coallition will be g (S ∪ k) − g (S). The solution suggested by Shapley (1953) is that each player k should be rewarded with its global contribution V (k), that is the arithmetic mean of each one of its possible marginal contributions wheighted by the probability of observing each coallition. In formal terms, the “just” reward is: V (k) =

m−1 X s=0

X s!(m − s − 1)! ¡ ¢ g (S ∪ k) − g (S) . m! S⊆K −k

(A.1)

Whether the marginal contributions g depends or not on the other players’ actions, an interesting property of the Shapley value is that: m X

V (k) = G.

(A.2)

k=1

B

T HE INTEGRAL - BASED DECOMPOSITION IN PRACTICE

Let X and Y be the income vectors that respectively corresponds to the initial and final income distribution for the country of interest. These vectors are such that individual income 14

are ordered in increasing order. For the sake of simplicity, we suppose that the size of X and Y is identical and equal to n.10 When the chosen poverty measure Θ complies with the scale invariance axiom — the value of the poverty measure is then insensitive to any change in the monetary unit —, the growth and inequality effects that ensues from Datt and Ravallion’s (1992) approach are: C X ,Y I X ,Y where

µY µX

¶ µY X − Θ (z, X ) , = Θ z, µX ¶ µ µX = Θ z, Y − Θ (z, X ) . µY µ

(B.1) (B.2)

X is the income distribution characterized by the final value of income per capita

but the Lorenz curve of the initial distribution. Conversely,

µX µY

Y is the income distribution

that would exhibit the final degree of inequality but the initial mean income. Now, consider the vector κ of the differences between Y and X . This vector is then divided into s equal vectors in order to obtain s marginal variations in mean income and inequality degree. For each j -th addition of s −1 κ to the initial income vector, the equations (B.1) and (B.2) are applied so as to get the micro growth and inequality effects that correspond to a move from distribution X +

j −1 s κ

j

to X + s κ, with j ∈ {1, 2 . . . s}. Finally, the

total growth and inequality effects are obtained by summing up of these micro-effects over the set {1, 2 . . . s}. The empirical implementation of the general decomposition procedure defined in equation (3.6) is thus:

IˆX ,Y

µ

µ

¶¶ µ ¶ j −1 j −1 Θ z, X+ κ − Θ z, X + κ , µ j −1 s s j =1 µ ¶¶ µ ¶ µ s X µ j −1 j j −1 X + κ − Θ z, X + κ , = Θ z, µj s s j =1

Cˆ X ,Y =

s X

µj

(B.3) (B.4)

´ ³ j where µ j = E X + s κ . C

S ENSITITY OF THE INTEGRAL - BASED DECOMPOSITION TO THE NUMBER OF MICRO - VARIATIONS

To assess the robustness of the results of the decomposition technique proposed in section 3, it is necessary to check wether estimated effects are sensitive to the value of s, that is the number of micro-variations used to compute the marginal effects. As the estimation may be computer-intensive, this investigation is also necessary to find an appropriate value for s so as to conciliate efficiency and precision requirements. To that purpose, we have used 10

Generally, the income series are of different sizes. It is then necessary to transform them so as to get equally-sized vectors. The most common procedure consist in estimating the whole income distribution for each year using a kernel estimator. With the help of the predicted smoothed distributions, generating income vectors of the desired length becomes straightforward.

15

the Nicaraguan data for the period 1992-1998 and have tested values of s from 100 to 5,000. The results of the estimations, reported on figure 1 show that the estimated effects behave differently regarding the headcount index on the one hand, and the poverty gap and squared poverty gap indices at the other hand. With the headcount index, it can be seen the estimated values of the growth and inequality effects are relatively unstable. For the period of interest, estimated growth effects ranges from 3.75 to 4.03, and inequality effects from 8.1 to -7.73. This variability is due to the fact that the headcount index is a step function; as a consequences, discontinuities make the decomposition technique sensitive to marginal changes of the size of each micro-variation of the income distribution. Concerning the poverty gap and squared poverty gap indices, the picture is slightly different since marginal changes of s do not yield the same variability of the poverty measure as the headcount index. For instance figures 1b and 1c show that estimated values converge to a fixed value as s increases and that low values of s may not bias the estimation of the growth and inequality effects. Our results suggest that setting the value of s in the range [500, 1000] is a fair compromise between efficiency and precision.

16

(a) Headcount index.

inequality effect

−8.1

3.7

−8.0

3.8

−7.9

3.9

−7.8

4.0

growth effect

0

1000

2000

3000

4000

5000

0

1000

2000

s

3000

4000

5000

s

(b) Poverty gap index.

inequality effect

−3.4435

1.5105

−3.4425

1.5115

−3.4415

1.5125

growth effect

0

1000

2000

3000

4000

5000

0

1000

2000

s

3000

4000

5000

s

(c) Squared poverty gap index.

inequality effect

−1.7830

−1.7820

0.7045 0.7050 0.7055

growth effect

0

1000

2000

3000

4000

5000

0

s

1000

2000

3000

4000

5000

s

Note: Growth and inequality effects are expressed in percent. The poverty line is fixed at C$1,245 per year and per capita.

Figure 1: Sensitivity of the estimated effects to the number of micro-variations: Nicaragua 1993-1998.

17

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