metadata: authors: 0: author: Essama-Nssah, B.
count: Guinea
docna: 0: docna: Assessing the distributional impact of
            public policy
docty: Policy Research Working Paper
subtopic: Taxation & Subsidies,Disability,Poverty Impact Evaluation,Poverty Monitoring & Analysis,Services & Transfers to Poor,Energy Policies & Economics,Economic Assistance,Access of Poor to Social Services,Inequality,Poverty Assessment,Poverty Diagnostics,Poverty Lines,Small Area Estimation Poverty Mapping
teratopic: Poverty Reduction
abstracts: cdata!: Economic development necessarily changes
            the welfare of socioeconomic groups to various degrees,
            depending on differences in their social arrangements. The
            challenge for policymakers is to select the changes that
            will be most socially desirable. The author demonstrates the
            usefulness of distributional analysis for social evaluation
            and, more specifically, for welfare evaluation, using data
            from the 1994 Integrated Household Survey in Guinea. Because
            the international community has declared poverty eradication
            a fundamental objective of development, the author uses a
            poverty-focused approach to social evaluation based on the
            maximum principle. This principle offers a unifying
            framework for analyzing the socioeconomic impact of public
            policy by using a wide variety of evaluation functions,
            inequality indicators (like the extended Gini coefficient),
            and poverty indices (such as Sen's index and the
            members of the Foster-Greer-Thorbecke family). The author
            also examines, within the context of commodity taxation, how
            to identify socially desirable policy options using both the
            dominance criterion and abbreviated social welfare
            functions. He includes computer routines for calculating
            various welfare indices and for plotting the relevant
            concentration curves.
geo_reg_key: 1927320,517191
repnme: repnme: Assessing the distributional impact of
            public policy
display_title: Assessing the distributional impact
            of public policy
listing_relative_url: /research/2002/09/2016298/assessing-distributional-impact-public-policy
topicv3: Poverty m1326687 695,Poverty Measurement and Analysis m1326941 707,Power and Electricity Sector m1327094 515,Social Assistance m1327201 1466,Tax Simplification m1504614 2527
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Poverty Reduction
origu: PRM
no_of_pages: 60
model_extractions: P POLICY RESFARCH WOIOKING,<sup>PAPEK2883</sup>

# Abstract

Economic developmenit necessarily<sup>chanigesthewelfare</sup> of socioCcololilic groups to various degrees,<sup>dependinig</sup> on differenctes in their social arrangements.<sup>The</sup> challenge for policvyrakers is to select<sup>the</sup><sup>**changes**that</sup> will be most socially desirable. Essama-Nssaih demonistrates the usefulness of distributional<sup>analysisfor</sup> social evaluation and, more specifically,<sup>forwelfare</sup> evaluatioll, uislig data from the 1994<sup>Integrated</sup> Houseihold Survey in Guinea. Because<sup>theinttrinationial</sup> commiiiunity has declared poverty eradication<sup>a</sup> fundamental objective of developmeit,<sup>theautIlorusesa</sup> poverty-focused approach to social<sup>evaluationibasedon</sup>

**_the_** _maximii priniciple._ This principle<sup>offersaullifying</sup> framework for analyzing the socioeconiomic<sup>impactof</sup> public policy by using a wvide variety of<sup>evaluationi</sup> functions, inequality indicators<sup>(liketheextendedGini</sup> coefficient), and poverry indices<sup>(suchasSen'sindexand</sup> the meilimbrs of the Foster-Greer-Thorbecke<sup>family).</sup> The author also examines, within the<sup>contextof</sup> commodity taxation, oiow to identify<sup>sociallydesirable</sup> policy options usinlg both the dominance<sup>criterionand</sup> abbreviated social welfare fulictions.<sup>Heincludes</sup> computer roLitinies for calculating various<sup>welfare indices</sup> and for plotting the relevant concenltrationi<sup>curves.</sup>

This paper-a product of the Poverty Reductio<sup>nGroup,PovertyReductionand EconomilicManagement Net-work-ispart</sup> of a **larger** effort in the network<sup>to</sup><sup>**ulinlerstand**thepovertxandsocialimpa;ctof publicpolicy.Copiesofthe paperare</sup> available free from the World Bank,<sup>181 8 H StreetNW, Washington,DC204.3.3.Pleasecontact OykiaoKootzeniew,room</sup> XIC4-554, telephone 202-473-5075,<sup>fax202-522-3283X</sup> , email address okootzemew\(@wvorldbanik.org.<sup>PolicyResearch</sup> Working Papers are also posted<sup>onthe Web athttp://econ.worldbank.org.Theauthor</sup><sup>**may**becontactedat</sup> bessamilanissah@(- w\vorldbank.org. September<sup>2002.(50pages)</sup>

> _The lPolic Research Workinzg Paper_<sup>_Seriesdisseminates thefindinigsof wuorkinpro)gressto) enicou ragetheexcanigeof ideasabouit_</sup> _developmient issues-An obiective oo_<sup>_the seriesis to get the findinissoit quickl),_</sup> **_vien_** _if tbe presentations are less_<sup>_than_</sup><sup>**_fullid_**</sup><sup>_polished.The_</sup> _papers_ **_carry_** _the_ **_names_** _of the_ **_aothors_**<sup>_a1ndshooild be cited according/v.Thefindings,interpretations,_</sup><sup>**_and_**</sup><sup>_conclusionsexpressed_</sup><sup>**_in_**</sup><sup>_this_</sup> _ps/Pcr are entirely those o/ the authors._<sup>_ITheydonzotnecessar)ilrepresent theviewvof theWorldBank,itsExecitivel)irectors,orthe_</sup> _coitiitr-ics they represenit._

P'roducedi by the Research Advisorv Staff
# _ASSESSING THE DISTRIBUTIONAL IMPA CT OF_<sup>_PUBLIC POLICY_</sup>

B. Essaina-Nssah Poverty Reduction Group The World Bank Washington, D.C.
# _Table_ **_of_** _Contents_

|1.Introduction........................................................<br>1|
|---|
|2.MaximinApproachtoPoverty-Focused<sup>Evaluation.................................................. 3</sup><br>TheLogicofSocialEvaluation.................<br>3.<br> <br>|
|DistributionofSocialWeights........................................................<br>4<br> <br>|
|DecompositionbyFactorComponents<sup>.......................................................</sup><br>12|
|3.TruncationbyaPovertyLine.......................................................<sup>16</sup><br> <br>|
|DistributionofSocial WeightsbyPoverty<sup>Status................................................</sup><br>17<br> <br>|
|DecomposabilityoftheGiniFamilyof<sup>IndicesbyPopulationSubgroups..........</sup><br>17<br>|
|The UseofPovertyGaps.......................................................<sup>21</sup><br>|
|HowDesirableisTruncation?.................<br>......................................27|
|4.ImpactAnalysis.......................................................<br>29<br>|
|SocialDesirabilityofFeasible Policy<sup>Options.....................................................29</sup><br>|
|UnambiguousWelfareComparisons<sup>.......................................................</sup><br>32<br>|
|UseofAbbreviatedSocial Welfare<sup>Functions......................................................38</sup><br>|
|ChangesintheAbbreviatedSocialWelfare<sup>Function..............................39</sup><br>|
|PovertyImpact.......................................................<br>40|
|5.Conclusion.......................................................<br>43|
|References.......................................................<br>45|
# _1._ **_Introduction_**

The purpose of this paper is to fit distributional<sup>impact analysiswithin</sup><sup>_the logic of_</sup> _social evaluation_ and to illustrate its implementation<sup>usingdata fromthe1994 Integrated</sup> Household Survey in Guinea'. The perspective of development as empowerment, advocated by Sen(1999) and outlined<sup>intheWorldDevelopmentReport(WDR)</sup> 2000/2001, entails essentially the expansion<sup>andthedistributionofsocioeconomic</sup> opportunities with implications for individual<sup>andcollective well-being.Distributional</sup> analysis, a key input in policy formulation<sup>andevaluation, involvesacomparisonof</sup> alternative distributions of some indicator of<sup>the livingstandard.</sup> There is an intimate relationship between<sup>evaluationanddevelopmentto the</sup> extent that the very definition of development<sup>involvesanevaluativejudgment.The</sup> domain of development hinges on the<sup>notionof thethings thatareworthpromoting(Sen</sup> 1989). The concept of living standard<sup>thereforeplaysafundamentalrolebothinthe</sup> formulation of development objectives<sup>andintheassessmentofdevelopment</sup> effectiveness. However, this multidimensional concept<sup>isnoteasy toimplement</sup> empirically. The identification of the<sup>valuable dimensionsof the living standarddepends</sup> essentially on the underlying view about<sup>personal characteristicsandsocialarrangements</sup> that are deemed important in the realization of<sup>any life plan.</sup> The Millennium Development Goals<sup>(MDGs)setbythe UnitedNationswith</sup> respect to development and poverty eradication<sup>areconsistentwiththe</sup><sup>_empowerment_</sup> approach to development. They identify _income, health, education,_<sup>_shelter_and</sup> _governance_ as critical components of<sup>the livingstandard.Ingeneralthesegoalsareset</sup> for the year 2015 relative to 1990 and<sup>aremeantto provide guidanceto both nationaland</sup> international policies and programs. For our<sup>current purposes, we selectconsumption per</sup> _capita_ as an indicator of opportunities<sup>for well-beingat thehouseholdlevel.It will be</sup> obvious that the methodology discussed<sup>inthispapercaneasilybe extendedtoother</sup> dimensions of living that may be represented<sup>by a quantitativevariabledistributed overa</sup> population.

Fundamentally, evaluation involves the examination<sup>andweighingofa</sup> phenomenon according to some explicit<sup>orimplicityardstick(Weiss21998).Wecan</sup> leam a great deal about an evaluative<sup>approachbydistinguishing</sup><sup>_theinformation_</sup> _required for passing judgments_ from that which<sup>has no directevaluativerole (Sen1999).</sup> The information pertains to the _valuable_<sup>_aspects_of the objectof evaluation,and to the</sup> rule for _combining_ these valuable elements<sup>into anaggregative judgment.Forinstance,</sup> the utilitarian approach to social evaluation<sup>is basedonthe utilitysumtotal in thesocial</sup> states under consideration. In this framework,<sup>attentionisfocusedonindividualwell-</sup>

> ' Enquete Integrale avec Module Budget-Consommation<sup>(EIBC).The survey was carriedout by the</sup> National Statistical Office within the Ministry<sup>of Plan and Cooperation.</sup>

> 2 More explicitly, this author defines evaluation<sup>as"a</sup><sup>_systematic_assessmentof the</sup><sup>_operation_and/orthe</sup> _outcomes_ of a program or policy, compared<sup>toasetofexplicitorimplicit</sup><sup>_standards,_asameansof</sup> contributing to the _improvement_ of the program<sup>or policy".</sup>

1
being as represented by the concept of utility, and the goodness of a social state depends on the sum total of utilities associated with that state [regardless of how these utilities are distributed among individuals (Sen 1999)].

Widespread poverty in the developing world remains a serious challenge for the development community and may have prompted the inclusion of poverty eradication among the MDGs. The World Bank has declared poverty reduction its overarching objective and a benchmark measure of its performance as a development institution. In view of these considerations, we emphasize poverty-focused evaluation in this paper. If development is about _empowering people_ to take charge of their destinies (Wolfensohn 1998), then _poverty_ must be seen as the _deprivation of basic capabilities_ to lead the kind of life one has reason to value (Sen 1999). In this perspective, the identification of poverty must go beyond lowness of income. This invitation however does not deny the fact that inadequate income can lead to capability deprivation. The notion of relative deprivation plays an important role in shaping the evaluative framework discussed here.

In the context of applied policy analysis, it is not enough to have an analytical or evaluative framework, available data must be processed according to that framework in order to draw relevant conclusions. Invariably, one needs a computing platform to do the job. We propose to show that the syntax of EViews can easily be exploited to do the job at hand. EViews stands for _Econometric Views,_ a Windows version of a software designed by Quantitative Micro Software (QMS) for the processing of time series data and the conduct of econometric analysis. We find that many people who are already using EViews for other purposes and interested in distributional issues may not necessarily be familiar with the fundamental concepts and techniques of distributional analysis. In addition, depending on their level of mastery of the software, they may not even think of EViews as appropriate for handling household survey data. Yet the current version of the software (Version 4.1) can handle 4 million observations per series (variable) and the total number of observations (i.e. number of variables times the number of observations per variable) is limited only by the available Random Access Memory (RAM). In addition the syntax is sufficiently straitforward to allow the user to plot common curves such as cumulative distributions, concentration or Lorenz curves. It is also easy to compute indices of social welfare, inequality and poverty.

The outline of the paper is as follows. Section 2 presents a poverty-focused evaluation framework based on the _maximin principle._ The concept of relative deprivation underpins the distribution of social weights in the formulation of aggregative judgments. Section 3 focuses on the implications of the traditional approach to poverty analysis based on the use of a poverty line. In particular, it is shown that when the distribution of welfare is truncated on the basis of a poverty line, the Gini coefficient can be additively decomposed into two components, one measuring the within group inequality and the other the inter-group inequality. Impact analysis is reviewed in Section 4. In that Section, we show how the ranking of socioeconomic situations can vary with the specification of social weights. The basic EViews routines used to produce illustrations are provided in boxes. Concluding remarks are presented in the last Section of the paper.

2
## **_2. Maximin Approach to Poverty-Focused_**<sup>**_Evaluation_**</sup>

A poverty assessment is a key ingredient<sup>in the formulationof a poverty reduction</sup> strategy. It is a determination of<sup>thenature,extentanddeterminantsof povertyatboth</sup> the _individual_ and _societal_ level.<sup>Ultimately,such anassessmentis an exercisein social</sup> evaluation to the extent that<sup>itoffersa</sup><sup>_criterion_forcomparing alternativesocial</sup> arrangements. Before considering the implied<sup>distributionof socialweights,wefirst</sup> review the logic of evaluation.

# **_The Logic of Social Evaluation_**

The notion of _evaluation_ is usually<sup>contrastedwith that of</sup><sup>_valuation._The former</sup> relates to the assessment of the<sup>relativemeritsof actions whilethelatterappliestothe</sup> comparison of things (Dasgupta<sup>2000).</sup> Thus, evaluation applies to strategies<sup>and</sup> policies<sup>3</sup> . Essentially, evaluation entails four<sup>basicaspects:(1)the</sup><sup>_identification_of the</sup> object of evaluation along with<sup>its valuabledimensions;(2) the</sup><sup>_valuation_of thevarious</sup> components of the object; (3)<sup>the formulationof an</sup><sup>_overall judgment;_and (4) the</sup><sup>_ranking_</sup> of alternatives.

We can learn a great deal about<sup>anevaluative approachbydistinguishing</sup><sup>_the_</sup> _information required for passing judgments_<sup>fromthat which has nodirect evaluativerole</sup> (Sen 1999). The information pertains<sup>to the</sup><sup>_valuable aspects_of theobjectof evaluation,</sup> and to the rule for _combining_<sup>thesevaluable elementsintoanaggregativejudgment.</sup> When seen as an assessment of<sup>individual advantageandsocialprogress,</sup><sup>_theobject of_</sup> _social evaluation is the determination_<sup>_of theextenttowhichtheprevailing social_</sup> _arrangements maintain and improve_<sup>_thelivingstandard oftheparticipants._The</sup> distribution of the living standard<sup>within the population is thus the yardstickby whichwe</sup> judge the performance of a socioeconomic<sup>system.Theidentificationof thevaluable</sup> dimensions of the living standard<sup>correspondstothespecificationof whatSen(2000)</sup> calls _the focal space_ of evaluation.

To arrive at an overall judgment,<sup>oneneedstoaggregate individualconditions</sup> into an indicator of a social state.<sup>InSen's terminology, this aggregationrulerepresents</sup> _the focal combination._ One standard<sup>approachis to defineanadditively separablesocial</sup> welfare function as follows.

### (2.1)

<!-- Start of picture text --> W = jhxh<br>h=1<br><!-- End of picture text --> where **ph** is the social weight<sup>attachedto thelevelofthewelfareofindividual(or</sup> household) h, as indicated by<sup>**xh.**Wemayinvokethe conceptof</sup><sup>_Equally Distributed_</sup>

> **3A** strategy is a conditional action<sup>thatdependson thestateof theworldoronactionstakenby other</sup> decision makers. A policy is a definite<sup>courseof actionchosenfromafeasiblesettogovern currentand</sup> future actions. A policy statement<sup>identifies</sup><sup>_objectives_andassociated</sup><sup>_means._</sup>

3
_Equivalent_ (EDE) welfare to abbreviate the above function as follows: **W(VP)=W(x). VP** is **-he** level of welfare such that, if enjoyed equally by each member of society, collective well-being would be equivalent to the one associated with the observed distribution<sup>4</sup> . The definition of the covariance between two variables allows us to express the EDE welfare as: where both f and x are n-dimensional vectors. No generality is lost if we normalized social weights such that p=l. In that case, average social welfare could be written as:

# (2.3) _V =_ **_x +_** cov(x,, )

The covariance term would be equal to zero if either , or x were constant. A constant x means that everybody has the same level of living, and therefore there is no inequality to worry about. However, a constant 1 means everybody receives equal consideration in social evaluation regardless of her/his standard of living. In this case, social evaluation is not concerned with inequality. Thus, expressions (2.2) and (2.3) may be viewed as a decomposition of the social welfare function into the _size_ and _distribution_ components. The former is represented by the mean of the distribution of the living standard indicator, while the second is measured by the covariance between the social weights and the levels of the living standard.

The focal space and the focal combination define the _informational basis_ of social evaluation. This may be viewed as a set of value judgments underlying the specification of the objects of value both from the individual and social perspective. Once the relevant indicator of the living standard has been selected, the specification of a social welfare function boils down to the distribution of social weights through the specification of **ph** for each individual or class of individuals. How can one go about this? _One possibility is to invoke a theory of social justice to guide the distribution of social weights._ We use the maximin principle in building up a poverty-focused evaluative framework.

# **_Distribution of Social Weights_**

A focus on poverty in social evaluation requires that extra consideration be given to the worse-off relative to the better-off both analytically and politically. Sen (1997: 33) explains the ethical underpinning of the ordinary Gini coefficient in terms of the _pairwise maximin principle._ According to this criterion the welfare level of any pair of individuals must be equated to the welfare of the worse off of the two. This is consistent with the _Dalton principle_ of transfers according to which rich to poor transfers improve social welfare. This value judgment may be built in the social welfare function by first ranking

> **4** The concept is used by Atkinson (1970) in a normative approach to the measurement of income inequality.

_4_
individuals according to<sup>somecriterionof social desert,then assigning social weightsin</sup> such a way that of any<sup>two individualsthemoredeservingreceivesahigherweight.</sup> More specifically, suppose<sup>thatwerank individualsaccordingtotheirlevelof</sup> consumption **xh,** then we<sup>assignsocialweights on the basis of ranksin sucha way that if</sup> h<i, then **phg3i** . The Dalton principle is<sup>thereforeconsistent withanynonnegative,</sup> monotonic and weakly<sup>decreasing socialweighing schemeph(MaysharandYitzhaki</sup> 1995: 797).

One such weighing scheme<sup>underpinstheclassofsocial evaluationfunctions</sup> associated with the Gini<sup>family of inequalityindices.To see what is involved, consider a</sup> group of v individuals<sup>selectedatrandomfromalargepopulation.</sup> A particular individual with a level of<sup>well-beingxh will feel relativelydeprived if thereis at least one</sup> other individual with a<sup>higherlevel of livingthan</sup><sup>**xh.**The likelihood thatthe other (v-1)</sup> individuals have a level<sup>of well-beinghigher than</sup><sup>**xh**isestimatedas:(1</sup><sup>_-ph_</sup><sup>**_)V-_**where</sup> **ph** iS an estimate of the<sup>cumulative distributionfimctionof x in the population(CDF)at</sup> the point **xh.** We may<sup>selectnormalizedsocial weights 5such that:</sup>

- (2.4)

Given that the CDF<sup>ismonotonicallyincreasingbetween0and1, itisclearthat the</sup> weights defined in (2.4)<sup>willdecreasemonotonicallyaswemovefromthelowerto</sup> higher ranking individuals,<sup>in terms of living standard.</sup> Average welfare in the<sup>wholesociety is thusequal to:</sup>

# (2.5)

If there were no inequality<sup>(i.e.everyonehadthesame levelof livingstandard),the</sup> covariance in the above<sup>expressionwouldbeequal tozeroandaveragelivingstandard</sup> would be equal to pt. We therefore conclude<sup>thatthesocialcostof inequalityis equal</sup> to: _v_ cov[x, (1- p)"-<sup>1] .</sup> It<sup>is knownthat</sup><sup>_V(v) =,[1_-</sup><sup>_G(v)],_whereG(v) is the extended</sup> Gini coefficient and v<sup>isinterpretedasanindicatorof social aversiontoinequality</sup> (Lamnbert 1993b). Therefore,<sup>expression(2.5)impliesthattheextendedGini coefficient</sup> is equal to: 5 The fact that the average<sup>weightisequalto one canbe establishedmuch easilyin thecase of acontinuous</sup> distribution. In this case,<sup>we have:j (1</sup><sup>_-_p) vdp =---</sup> v(l _- p)_ **_v-_** _dp =-_ _5_
The ordinary Gini coefficient is obtained by setting the aversion parameter equal to **26.** What happens if individuals are ranked according to some variable other than x? For instance, let y stand for an indicator of needs. Then rank individuals in such a way that the neediest comes first. Let q stand for the cumulative distribution under y. The covariance expression now becomes cov[x, (1- q)V-<sup>1</sup> ]. The average welfare is: where C.y(v) is the extended concentration index of x with respect to y defined as follows: Figure 2.1 reveals the weighing scheme associated with the social evaluation criterion defined by (2.3). The scheme is a function of the aversion parameter (v). When this parameter is equal to one, the social evaluation function, V(1), assigns equal weight to everyone regardless of their level of well-being. This weight is also equal to one. For all values of the aversion parameter greater than 1, the poorest individual receives a weight equal to v. Next, all people whose rank falls below the cut-off point (where the v- level curve is equal to one) receive a weight between v and 1. Finally, the social evaluation function assigns a weight between zero and one to people whose relative rank falls between the cut-off point and 1. On the basis of figure 2.1, it is clear that as the aversion parameter increases above one, the distribution of social weights is tilted such that the weight assigned to the poorest equal v and that assigned to the richest equal zero.

The figure suggests that the cut-off rank is a solution to the following equation.

> Thus the rank at which the social weight switches from being greater or equal to one to

> less than one is given by the following formula:

> **6** In that case, we note that:

> -- cov[x, (1 - p)] = -- [cov(x,l) - cov(x, p)] =-cov(x, p) _= G._ This expression says jul jux Ax that the ordinary Gini coefficient is equal to twice the covariance between x and its relative rank, divided by the mean of x.

6
Figure 2.1. Distribution of Social Weights<sup>as aFunction of the AversionParameter</sup> <!-- Start of picture text --> 7<br>6-<br>5 4<br>4-<br>3<br>2<br>-<br>1<br>-<br>0<br>0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0<br><!-- End of picture text --> Table 2.1 shows cut-off ranks<sup>computedaccordingto(2.10).</sup> These results suggest that the cut-off rank is a<sup>decreasingfunctionof theaversionparameter.As this</sup> parameter increases, the focus is shifted<sup>tothe lowerendof thedistribution.Thussocial</sup> evaluation will attach more weight<sup>to thesituationof thepoorestgroup.For instance,</sup> when the aversion parameter is<sup>equalto100,socialevaluationattachesmore weightto</sup> the lowest 5 percent of the distribution.<sup>When theaversionlevel reaches200,thefocus</sup> shifts to the poorest 3 percent of<sup>thepopulation.</sup> According to this table, the social evaluation function based on the<sup>ordinaryGinicoefficient assignsmore weightto every</sup> body up to the median. The weights<sup>then declineat a constantrate of 0.5from2to 0.</sup> Table 2.1. Cut-off Rank as a Function<sup>of the Aversion Parameter.</sup>

|o|1.1|1.2|_1.5_|2.0|3|6<br>|10<br>|20<br>|30<br>|50<br>|100<br>|200<br>|
|---|---|---|---|---|---|---|---|---|---|---|---|---|
|p*|0.62|0.60|0.56|0.50|0.42|0.30|0.23|0.15|0.11|0.08|0.05|0.03|
|Sourc|e:Com|puteda|ccording|to(2.1|0)||||||||

The empirical implementation of<sup>theframework encapsulatedinexpression(2.5)</sup> requires an estimate of the cumulative<sup>distributionfunctionandthecomputationofa</sup> covariance. The cumulative distribution function<sup>**p=F(x)**givestheprobabilityof</sup> observing an individual with a<sup>living standardatmostequaltox.Theslopeofthis</sup> 7
fimction at a particular point gives the associated density function. There are several ways of estimating a cumulative distribution on the basis of a sample of size n, depending on the method of adjustment for the non-continuity. The ordinary approach is to rank the observations in increasing order of the variable of interest, and divide the rank of each observation by the total number of observations. Let p stand for the ordinary estimate of the CDF for observation with rank r, then we may write: (2.11) p= _r n_

# Box 2.1 Lerrnan-Yitzhaki CDF Estimator in EViews

SUBROUTINE LYCDF( SERIES X, SERIES S, SERIES W) SERIES YH=X/S SORT YH SERIES POPH=S*W SERIES WH=POPH/@SUM(POPH) SMPL @FIRST @FIRST SERIES FH=WH SMPL @FIRST+1 @LAST FH=FH(-1 )+WH SMPL @ALL SERIES F_HAT=FH-0.5*WH ENDSUB The SORT command in EViews is used to rank observations in increasing or decreasing order of a particular variable (series). The syntax is: SORT(options) SERIES_NAME. The ascending order is the default option. The other option is D (for descending order). EViews also provides a command, called CDFPLOT, that displays according to the option chosen: empirical cumulative distribution functions, survivor functions, and quantiles with standard errors. In the context of welfare analysis, the survivor function gives the probability of observing an individual with a living standard at least equal to a specified level. The survivor function is therefore equal to one minus the CDF. A _quantile,_ also known _asfractile,_ is a level of living below which lies a given fraction of the population. The median is an example of a quantile to the extent that it is the value below which lies 50 percent of the population.

Household level data are usually available in the form of a weighted sample. A household of a given size represents a certain number of households in the population at large. In such a case, Lerman and Yitzhaki (1989:44) recommend the following midinterval estimator for the CDF<sup>7</sup> .

> _7_ Deaton (1997:154) proposes a similar procedure for converting household ranks into individual ranks. Assuming that there are **Whflh** people in household h (where nh is the household size and wh its absolute weight), then starting from pi=<sup>1</sup> , the rank of the first person in household h+l is given by the recursive formula: _Ph+I = Ph + nh_ **_wh_** _._ These ranks can then be normalized relative to the overall population.

8
**A li** _Wh Fh (X) =_ E _Wk +-; wo = 0_ **_k=O_** 2

(2.12) Box 2.2 Covariance Estimator<sup>of Gini-based Measuresof Welfare</sup> <!-- Start of picture text --> SUBROUTINE WEIGHCOV( SERIES X, SERIES S, SCALAR A)<br>SERIES XH=X/S<br>SERIES XHWT=XH*POPH<br>SCALAR MUX=@SUM(XHWT)/@SUM(POPH)<br>SERIES DVXH= (XH-MUX)<br>SERIES DVXHW=(POPH*DVXH)/@SUM(POPH)<br>SERIES QH=(1-F HAT)<br>SERIES QHA=(QH)A(A-1)<br>SERIES QHAW=(POPH*QHA)/@SUM(POPH)<br>SCALAR MQHA=@SUM(QHAW)<br>SERIES DVQHA= (QHA - MQHA)<br>SERIES SWT=A*QHA<br>SERIES WCOV=DVXHW*DVQHA<br>SCALAR GLY= -(A*@SUM(WCOV))/MUX<br>SCALAR SOCOST=A*@SUM(WCOV)<br>SCALAR EDEX=MUX*(1 -GLY)<br>ENDSUB<br><!-- End of picture text --> where Wh stands for the<sup>relativeweightof eachindividualinhouseholdhin the overall</sup> population. This relative weight<sup>isequalto thehouseholdsizetimestheabsolute</sup> household weight divided by<sup>total population.</sup> The subroutines presented<sup>inBoxes2.1and2.2canbe embeddedintoasingle</sup> program to compute members<sup>of the Gini family of indices.</sup> Chotilcapanich and Giffiths<sup>(2001:543)proposean alternativealgorithm basedon</sup> a linear approximation<sup>of the Lorenzcurve.</sup> This Linear-Segment<sup>Estimatorofthe</sup> extended Gini coefficient<sup>is definedby the followingexpression.</sup> where Ok iS the proportion<sup>of welfare forgroupk and</sup><sup>**7Sk**iSthepopulationshareof that</sup> group. Note that the ratio<sup>in parenthesesintheaboveexpression isalsoequal</sup> 9
to: **_Xk_** The authors of this formula present results from a Monte Carlo experiment

**_j=1_**

showing that the linear segment approach outperforms the covariance-based formula only when applied to grouped data with 20 or fewer observations. They observe little or no difference in terms of bias and mean squared error when they test both methods on data set with 30 or more observations. The authors state their main conclusion as follows: _"The two estimators have similar properties when calculated from individual observations."_ They further claim: _"When using grouped data with 20 or fewer groups, our estimator has less bias and lower mean squared error than the_ covariance estimator. _When individual observations are used, or the number of groups is 30 or more, there is little or no difference in the performance of the two estimators."_ Box 2.3. Linear-Segment Estimator of the Extended Gini Coefficient LOAD %O SMPL @ALL SERIES xl={%1}/{%2} SORT XI SERIES PoPI={%2}*{%3} SERIES PI=POPI/@SUM(POPI) SERIES WXI=POPI*XI SERIES XSHARE=WXI/@SUM(WXI) SMPL @FIRST @FIRST SERIES CUMPI=PI SERIES CUMXSH=XSHARE SMPL @FIRST+1 @LAST CUMPI=CUMPI(-1 )+PI CUMXSH=CUMXSH(-1 )+XSHARE SMPL @ALL SERIES RATIO=XSHARE/PI SERIES QI=(1 -CUMPI) SERIES QH=(1 -CUMPI(-1)) !AvMAx=6 !B=2 !CF=!B-1 !RWS=(!B*!AVMAX)-!CF VECTOR(!RWS) GINU 'EXTENDED GINI !T=1 FOR !NU=1 TO !AVMAX STEP _1I!B_ SERIES QANU=QIA(!NU) SERIES QBNU=QHA(!NU) SERIES QCNU=QANU-QBNU SERIES QDNU=RATIO*QCNU SCALAR QDSUM=@SUM(QDNU) GINU(!T)=1 +QDSUM !T=!T+1 NEXT 'End of Program 10
We applied the two methods(covariance<sup>andlinearsegments)onthreedatasets.</sup> The first is the Guinea 1994<sup>IntegratedHousehold Surveycovering4,416 representative</sup> households. As expected,<sup>theclaim thatboth methodswouldproducealmostthesame</sup> results in this case is confirmed<sup>by theresults presentedin table 2.2.Exceptfor the case</sup> when the aversion parameter<sup>isequalto1,theestimatespresentedintable2.2are</sup> identical up to the third decimal.

Table 2.2. Alternative Estimates of<sup>the ExtendedGini Coefficient</sup>

|Aversion Parameter <br>|Covariance<sup>Method </sup><br>|<sup>LinearSegments</sup><br>|
|---|---|---|
|1.0<br>|-4.58E-29<br>|6.47E-06<br>|
|1.5<br>|0.283868<br>|0.283881<br>|
|2.0<br>|0.403372<br>|0.403385<br>|
|2.5<br>|0.472096<br>|0.472112<br>|
|3.0<br>|0.517874<br>|0.517894<br>|
|3.5<br>|0.551041<br>|0.551064<br>|
|4.0<br>|0.576415<br>|0.576442<br>|
|4.5<br>|0.596588<br>|0.596618<br>|
|5.0<br>|0.613091<br>|0.613124<br>|
|_5.5_<br>|0.626898<br>|0.626934<br>|
|6.0|0.638656|0.638696|

Data Source: Guinea 1994 Integrated<sup>Household Survey</sup> In order to test the performance<sup>ofthetwomethodswhenthenumberof</sup> observations is less that 20,<sup>we chosetheextremecaseof a populationof twoindividuals</sup> with the following levels of<sup>living(25,75).The resultsarepresentedin table2.3.Both</sup> methods produced very different<sup>results. However,contrary to the conclusion reached by</sup> Chotikapanich and Giffiths<sup>(2001),thecovariancemethod outperformedthelinear</sup> segment approach. We base<sup>thisobservationonthefollowing considerations. Inthis</sup> simple case, it can be shown<sup>eitherbygeometryorbyintegration(Essama-Nssah</sup> 2000:54) that the ordinary<sup>Gini coefficientisequalto onehalf minus theproportionof</sup> the welfare enjoyed by the<sup>worseoff of both individuals.For the distributionat hand,we</sup> therefore know that the Gini<sup>coefficientis equalto0.25.</sup> The covariance method produces exactly this result<sup>whilethelinearsegment approachyieldsa much highervalue</sup> of Gini: 0.62. We further<sup>testedboth methodson the followingtextbook8exampleof a</sup> distribution of income in a population<sup>of four individuals(10,20, 30,40), again thelinear</sup> segrnent method produced<sup>muchhighervalues comparedto thecovariancemethod.The</sup> covariance method produced<sup>aGiniof 0.25whilethelinear segmentmethodgives an</sup> estimate of 0.425.

Finally, it is important to note<sup>that, when theaversionparameterisequalto one,</sup> we expect the Gini coefficient<sup>tobezero,thecovariancemethodproduces thisresult,</sup> while the linear segment method<sup>tends toyield values thataresignificantly differentfrom</sup> zero. These results cast<sup>somedoubtontheconclusionbyChotikapanich andGiffiths</sup> 8 Lambert 1993b:260
> (2001) that the linear segment estimator outperforms the covariance estimator in small samples.

Table 2.3: Altemative Estimates of the Extended Gini Coefficient

|Aversion Parameter|CovarianceMethod|LinearSegments|
|---|---|---|
|1.00|0.00|0.25|
|1.50|0.14|0.47|
|2.00|0.25|0.62|
|2.50|0.33|0.73|
|3.00|0.38|0.81|
|3.50|0.40|0.87|
|4.00|0.41|0.91|
|4.50|0.40|0.93|
|5.00|0.39|0.95|
|5.50|0.37|0.97|
|6.00|0.35|0.98|

> Source: Computed

# **_Decomposition by Factor Components_**

Let's suppose for simplicity's sake that the welfare indicator can be decomposed into two components. For instance, household expenditure may be divided into food, xl*, and nonfood expenditure, _xh_ . Using expression (2.3), average social welfare may be written as follows.

(2.14) _Vai = (,u_ + **P 2 ) +** **_[COV(X.,_** _O) +_ Cov(x 2,)] **_=_** _V 1 + V2_ In the general case where there are m components, the above expression becomes: Suppose now that we choose social weights according to the maximin principle where individuals (or households) are ranked in increasing order of total expenditure. Using the weights defined by (2.4) leads to the following expression for average social welfare.

where **Cj(v)** is the extended concentration index of component j with respect to the overall level of expenditure.

> Given that _,u_ **_=_** _Uj_ , expression (2.16) may also be written as: **_j=1_**

12
# (2.17)

where Aj _= (uj / ,)._ Note that the second expression of<sup>(2.16)isobtainedbymultiplicationof each</sup> concentration index by a neutral element<sup>(P4L).We obtainadifferentstructureof the</sup> same information by using another<sup>neutral element[G(v)/G(v)].Inthatcase,average</sup> welfare is equal to:

# (2.18)

where _lj (v)= C(v)_ . Comparing expression (2.18) to<sup>(2.5)we concludethatthe</sup> _G(v)_ extended Gini is equal to a weighted<sup>averageof the Gini Engel elasticitiesis equalto one.</sup> That is:

# (2.19)

Indeed, the ratio of the concentration<sup>indexof acomponentto theoverallGini</sup> coefficient is interpreted as an _elasticity_<sup>(LermanandYitzhaki1994, 1985;Yitzhaki</sup> 1994a). To see this interpretation, rewrite<sup>the coefficientas:</sup> The coefficient Xj is interpreted as the<sup>averagepropensity of component</sup><sup>**j**tovary with the</sup> overall indicator x. According to Yitzhaki<sup>(1994a:457)thecoefficientbj(v)maybe</sup> viewed as a nonparametric estimator of<sup>the marginalpropensityof componentjtovary</sup> with x. This interpretation is clearest<sup>whentheaversionparameter isequal to2.Then</sup> we see that: This is the _instrumental variable estimator_<sup>of the slope parameterin theregressionof the</sup> component **xj** on the overall x, using<sup>the cumulativedistributionpasaninstrument.</sup> Since **r%(2 )** is the ratio of marginal to average propensity, it is analogous to an elasticity.

13
Henceforth, we refer to this type of elasticity as Gini Engel Elasticity<sup>9.</sup> Assume<sup>now that</sup> individuals are ranked according to income. If, iij(<sup>2</sup> ) >1, commodity j<sup>isconsidereda</sup> _luxury._ If the elasticity is between 0 and 1, the corresponding commodity<sup>is considereda</sup> _necessity._ When this elasticity is negative, the commodity is an _inferior good_ (Gamer 1993: 135).

Table 2.4. Gini Elasticities for Selected Expenditure Components in Guinea (1994)

|Aversion|Total Expenditure|Cereals|Beverages|Food|Electricity<br>|Energy<br>|Non-Food<br>|
|---|---|---|---|---|---|---|---|
|1.50|1.00|0.43|1.51|1.02<br>|1.83<br>|0.85<br>|1.29<br>|
|2.00|1.00|0.50|1.43|1.05|1.76|0.92<br>|1.24<br>|
|2.50|1.00|0.55<br>|1.39<br>|1.07<br>|1.70<br>|0.95<br>|1.21<br>|
|3.00|1.00|0.58|1.36<br>|1.07<br>|1.65<br>|0.97<br>|1.18<br>|
|3.50|1.00|0.61|1.34|1.08|1.61<br>|0.98<br>|1.17<br>|
|4.00|1.00|0.63|1.33|1.08|1.58<br>|0.99<br>|1.15<br>|
|4.50|1.00|0.65<br>|1.32<br>|1.08<br>|1.55<br>|0.99<br>|1.14<br>|
|5.00|1.00|0.66<br>|1.31<br>|1.08<br>|1.53<br>|1.00<br>|1.13<br>|
|5.50|1.00|0.68<br>|1.30<br>|1.08<br>|1.51<br>|1.00<br>|1.12<br>|
|6.00|1.00|0.69|1.30|1.08|1.49|1.00|1.12|

Source: Computed Table 2.4 presents Gini Engel elasticities for selected expenditure<sup>componentsin</sup> Guinea. This information reveals that food expenditure is distributed very<sup>much like total</sup> expenditure. The distribution of expenditure on energy is also very close<sup>tothat of total</sup> expenditure. Beverages and electricity are clearly luxury goods while cereals are<sup>a</sup> necessity. In general, it appears that inequality in the distribution of household expenditure in Guinea is due to non-food expenditure.

The area between a concentration curve and the 45 degree line is known<sup>asthe</sup> area of concentration. It is equal to - cov[xj, (1 - p)]/ _pj_ = cov(xj, p)I _,uj._<sup>Therefore,</sup> r%J(2) is equal to the area of concentration for **xj** divided by the area<sup>betweenthe45</sup> degree line and the Lorenz curve of x (Yitzhaki and Lewis 1996:549).<sup>Thisgeometric</sup> interpretation provides a necessary condition for characterizing<sup>the positionofa</sup> concentration curve relative to the 45 degree line and the Lorenz curve. Thus when<sup>the</sup> Gini Engel elasticity is negative, the concentration curve is concave and<sup>lies above the 45</sup> degree line. When the elasticity is equal to zero, the curve coincides with<sup>the 45degree</sup> line. When the elasticity is between zero and 1, the concentration curve lies<sup>between the</sup> 45 degree line and the Lorenz curve. Finally, when the elasticity is<sup>greaterthanonethe</sup> concentration curve lies below the Lorenz curve.

Finally, let qj stand for the cumulative distribution<sup>obtainedbyranking</sup> individuals by increasing order of component j. Let GJ(v) be the extended Gini coefficient of component j. Expression (2.19) also implies that the overall<sup>Giniis equal to</sup>

> **9** Yitzhaki (1994a) calls it Gini income elasticity. Furthermore, (2.19) shows<sup>that Engel-aggregationholds</sup> for these elasticities to the extent that their weighted sum is equal to one. The<sup>weightsare thesharesof the</sup> total budget spent on each commodity.

14
**_j=1_**

the weighted average of the concentration indices of the components, G(v)<sup>_=_</sup> _AjCj(v)._ If we multiply each concentration index in this expression by a<sup>neutralelement</sup> [Gj(v)/Gj(v)], we obtain the following alternative decomposition of the<sup>extendedGini</sup> coefficient: Box 2.4: Plotting Concentration Curves in EViews INCLUDELYCDF LOAD %O SMPL @ALL CALL LYCDF({%1}(1), {%1}(2), {%1}(3)) SERIES P=F_HAT SERIES L45=P GROUP LCC P L45 !M={%2}.@COUNT FOR !K=1 TO !M %sv={%2).@SERIESNAME(!K) SERIES x{!K}={%1}(3)*{%2}(!K) SERIES SHARES{!K}=X{!K}I@SUM(X{!K}) SMPL @FIRST @FIRST SERIES LP{%sv} =SHARES{!K} SMPL @FIRST+1 @LAST LP{%Sv}=LP{%Sv}(-1) + SHARES{!K} LCC.ADD LP{%Sv} SMPL @ALL NEXT FREEZE(LCCS) LCC.XY LCCS.SCALE RANGE(O, _1)_ 'End of Program The coefficient Rj(v) is known as the _"Gini correlation"_<sup>betweencomponentj</sup> and overall x. This expression thus reveals that the contribution of a<sup>componentto</sup> overall inequality is determined by three factors: (1) the proportion of the<sup>componentin</sup> the total value of x, (2) the correlation between the component<sup>and thetotal, and(3)the</sup> extent of inequality in the distribution of the component.

Figure 2.2 shows the configuration of concentration curves of<sup>two itemsfrom the</sup> 1994 Integrated Household Survey in Guinea. The concentration<sup>curveof cerealslies</sup> between the Lorenz curve of total expenditure and the 45 degree<sup>line.Wetherefore</sup> expect its Gini elasticity to be positive but less than one. Expenditure on<sup>beveragesis</sup> more unequally distributed than total expenditure. The corresponding concentration curve lies below the Lorenz curve. Thus we expect the Gini<sup>elasticity of beveragesto be</sup> grater than one. The values of Gini elasticities implied by this configuration<sup>are</sup> presented in table 2.4. This table also reveals that the distribution of<sup>food expenditureis</sup> 15
very close to that of total expenditure. Expenditures on electricity and nonfood items are more concentrated in high income households.

Figure 2.2. Guinea 1994: Concentration Curves of Cereals and Beverages with respect to Total Expenditure <!-- Start of picture text -->
-<br>1.0<br>-<br>0.9<br>0.8-<br>0.7-<br>0.6-<br>0.5-<br>0.4  Tta<br>0.3 - Cereals<br>Beverages<br>0.2-<br>0.1<br>-<br>0.0<br>0.0  0.1  0.2  0.3  0.4 0.5  0.6  0.7  0.8  0.9  1.0<br><!-- End of picture text -->

# _3. Truncation by a Poverty Line_

The standard approach to poverty analysis is to partition the population exhaustively into two mutually exclusive groups on the basis of a poverty line, and to assign a weight equal to zero to the wellbeing of anybody whose living standard is above the poverty line. Such a partition raises at least two interesting issues. The first one relates to the decomposability of the Gini family of indices, and the second to the interpretation of some well known poverty indices within the framework defined by (2.1). Before considering these issues, we first examine some implications for the aggregate distribution of the social weights defined by (2.4).

16
## **_Distribution of Social Weights by Poverty_**<sup>**_Status_**</sup>

We focus here on how the relative<sup>socialweight attachedto each of thesegroups</sup> varies with the aversion parameter.<sup>Table3.1containsthe resultsof suchan experiment</sup> using household survey data from<sup>Guinea.We find thatthe poor accountfor16 percent</sup> of the mean rank, and the non-poor<sup>for84percent.Thisstemsfromthefactthatthe</sup> overall mean rank (i.e. the mean<sup>of the cumulativedistribution)isequalto0.5.The mean</sup> rank of the poor is about 0.20 while<sup>that of the non-pooris0.70.Theweightedaverage</sup> of the two numbers is equal to<sup>0.5.The weightsusedherearetherespectivepopulation</sup> shares. _When the aversion parameter_<sup>_is equal to one,each group receives,on average, a_</sup> _weight equal to its population share._<sup>As the aversion parameterincreasesabove1, most</sup> of the social weight is shifted<sup>to thepoor.</sup> Table 3.1. Distribution of Average<sup>Social Weightin Guinea by Poverty Statusin 1994 and by</sup> Level of Aversion

||ShareofMeanRank|v=l<br>|v=2<br>|v=3<br>|v=4<br>|v=5<br>|v=6<br>|
|---|---|---|---|---|---|---|---|
|Poor|0.16<br>|0.40<br>|0.64<br>|0.79<br>|0.87<br>|0.92<br>|0.95<br>|
|Nonpoor|0.84<br>|0.60<br>|0.36<br>|0.21<br>|0.13<br>|0.08<br>|0.05<br>|
|Total|1.00|1.00|1.00|1.00|1.00|1.00|1.00|

Source: Computed

# **_Decomposability_** _of_ **_the Gini_**<sup>**_Family_**</sup><sup>_of_</sup><sup>**_Indices by Population Subgroups_**</sup>

Truncation by a poverty line<sup>impliesthat the ordinaryGinicoefficient canbe</sup> written as:

## (3.1)

where the term between parentheses<sup>representswithingroupinequality,whiletheterm</sup> outside stands for between group<sup>inequality.The term</sup><sup>**si**measures the proportionof total</sup> welfare enjoyed by grout i, and<sup>**pi**is the populationshare of thatgroup.</sup> Anand(1983:320) shows that,<sup>in this case, between group inequalityis equalto:</sup>

## (3.2)

Assuming that the first group<sup>is the poor group, then theabove expressionreduces to:</sup>

## (3.3)

17
_Inequality between the two groups, as measured by the ordinary Gini coefficient, is equal to the proportion ofpoor in the population minus their income share._ This is a particular case of a general case we encountered in Section 2 when testing the covariance and the linear segment approaches to Gini calculation. In general, when the subgroup distributions do not overlap, Gb can be computed as the Gini for the distribution where everybody has her/his group average welfare.

Furthermore, in the context of the extended Gini, between group inequality is computed as follows (Yitzhaki 2002: 77): Table 3.2 shows the results of a decomposition of the extended Gini coefficients associated with the distribution of household expenditures in Guinea in 1994. The decomposition is based on (3.1) and (3.4). The first column of the table contains levels of the aversion parameter. The second column indicates within group inequality, while the third and fourth columns represent between group inequality respectively in terms of **Gb(v)** and its share in overall inequality (presented in the last colunn). These results indicate that, given the poverty line, most of the observed inequality in the distribution of household expenditure in Guinea is accounted for by the between components. When the aversion parameter ranges from 1 to 6, the share of between group inequality increases from 0 to 84 percent. _This suggests that poverty-focused evaluation must also pay attention to inequality between the poor and the rest of the population._ The use of a poverty line to divide the population into two groups means that the corresponding distributions of welfare do not overlap. It is useful to note the decomposition of the Gini coefficient when the sub-group distributions may overlap. The resulting expression suggests an indicator of social exclusion. Yitzhaki(1994b) proposes an index of the degree of overlapping between distributions which allows a more general and transparent decomposition of the Gini coefficient. In this general case, the Gini coefficient can be decomposed into three distinct components: the first component is linked to intra-group inequality, the second is an indicator of between-group inequality, and the third is an indicator of the overlapping of the distributions. The overlapping index can also be interpreted as a measure of segmentation and stratification. Such an indicator may reveal the extent of social exclusion of some socioeconomic groups.

18
Table 3.2. Inequality between Poor and Nonpoor in Guinea in 1994

|Aversion|Within|Between|Shareof<br>between|Overall|
|---|---|---|---|---|
|1.00|0.00|0.00|0.00|0.00|
|1.50|0.15|0.13|0.46|0.28|
|2.00|0.17|0.23|0.58|0.40|
|2.50|0.16|0.31|0.66|0.47|
|3.00|0.15|0.37|0.71<br>|0.52<br>|
|3.50|0.13|0.42|0.76|0.55<br>|
|4.00|0.12|0.46|0.79|0.58<br>|
|4.50|0.11|0.48|0.80|0.60|
|5.00|0.11|0.51|0.84|0.61<br>|
|5.50|0.10|0.52|0.83<br>|0.63<br>|
|6.00|0.10|0.54|0.84|0.64|

Data Source: 1994 Integrated Household Survey Suppose that a population has been divided into m subgroups<sup>indexedbyk.</sup> Consider any two groups j and k. The overlapping of group k by<sup>group jreflectsthefact</sup> that some individuals from group k have a standard of living that<sup>falls withintherangeof</sup> the distribution of the living standard within group j. The extent<sup>of thisoverlappingcan</sup> be measured by the _covariance_ between the level of the living<sup>standardin groupk and the</sup> relative rank the members of k would receive, had they been considered<sup>asmembersof</sup> group j. In this context, relative ranks are measured by the relevant<sup>cumulative</sup> distributions.

To see more clearly what is involved, let F(x) and **Fk(x)** stand respectively<sup>forthe</sup> overall CDF and the CDF within group k. Also let **Fj(x)** be the relative rank that members of k would get had they been considered as belonging<sup>to j.By definitionof the</sup> CDF, if all observations in k have a value that is less than or<sup>equalto theminimumin the</sup> range of group j, **Fk,(x)** would be equal to zero. If all values<sup>ingroupk aregreateror</sup> equal to the maximnum in j, F,(x)=1. In either one of these extreme cases, **covk[x,** F,j(x)]=O. This is the covariance between the values of x in<sup>groupkandtheirrelative</sup> rank in j. The covariance is calculated using the distribution k<sup>(Yitzhaki1994b:149).A</sup> non-zero value of this covariance reflects some overlapping of<sup>groupkby groupj.A</sup> normalized overlapping index is given by the following expression.

The index has the following properties: (1) It is greater or equal<sup>tozeroandlessthan or</sup> equal to 2; (2) It is equal to one if the distribution of group j<sup>is</sup> identical<sup>to the distribution</sup> of group k; (3) for a given segment of distribution j that<sup>fallswithintherangeof</sup> distribution k, the closer the observations of j are to the mean of k, the higher<sup>Oki.</sup> 19
The overlapping of group k with the overall population (including group k itself) is defined as: It can be shown that (Milanovic and Yitzhaki 2002: 160): where Pk iS the population share of group k. Letting **Sk** stand for the share of group k in total welfare (as represented by income or expenditure), the ordinary Gini coefficient may be written as: where _Gb_ **_=_**<sup>**_2_**</sup> **_COVlUk'Fk]_** is an indicator of between group inequality. It is equal to **_b_** twice the covariance between the mean welfare of each group and the group's mean rank in the overall distribution, divided by the overall mean of the welfare indicator. Using (3.7), we have: The first term on the right hand side of (3.9) measures within group inequality as a weighted average of the subgroup Ginis where the weights are equal to the subgroup population share times the share of welfare (as measured by the indicator x). The share of group k in x is equal to: **_Sk = Pklk_** . The second term stands for between-group _A_ inequality. The last term, is an indicator of the extent of "cross-overlapping" [to distinguish it from global overlapping measured by (3.6)]. This term is equal to zero when subgroup distributions do not overlap.

An application of the above decomposition to the distribution of per capita household expenditure in Guinea in 1994 produced the following results.

20
Table 3.3 Gini Decomposition of Inequality in the Distribution of Household Expenditure<sup>in</sup> Guinea by Area of Residence

||Population<br>Share|Expenditure<br>Share|Mean<br>Expenditure|Mean<br>Rank|Gini|Global<br>Overlap|Cross-<br>Overlap|
|---|---|---|---|---|---|---|---|
|Conakry|0.17|0.31|849843.00|0.13|0.38|0.58|0.41|
|<br>Other|0.16|0.20|598799.30|0.10|0.38|0.85|0.69|
|Urban||||||||
|Rural|0.67|0.49|342785.10|0.27|0.33|0.96|0.29|
|Guinea|1.00|1.00|469461.30|0.5|0.40|1.00|0.00|

Data Source: 1994 Integrated Household Survey.

The overall Gini index is estimated at 0.40. The decomposition presented<sup>intable</sup> 3.3 above implies that the within-group inequality is equal to 0.14, the between-group component is equal to 0.11 while the overlapping component is 0.15.<sup>Thusabout28</sup> percent of the observed inequality in Guinea in 1994 is accounted for by<sup>the between</sup> group component (when the population is classified by area of residence).<sup>Lookingatthe</sup> overlapping components, it can be seen that no area stands out as a<sup>perfectstratum.</sup> However, in relative terms, the rural area has the least amount of overlapping<sup>withthe</sup> rest of the distribution compared to the urban areas. This fact is also supported by<sup>the</sup> pattern of mean expenditures.

# **_The_** _Use of_ **_Poverty Gaps_**

At the individual level, poverty measurement involves two basic steps:<sup>(1)the</sup> selection of a poverty line indicating a threshold below which<sup>the personis declared poor,</sup> and (2) the computation of poverty gaps measuring the relative<sup>distancebetweenan</sup> individual level of welfare and the chosen poverty line. Kakwani(1999:605)<sup>definesa</sup> class of additively separable poverty measures starting from the notion<sup>of</sup><sup>_deprivation._</sup> Let z be the poverty line, and **xh** the level of welfare enjoyed by individual<sup>h ina society</sup> comprising n individuals. Let xV(z,xb) stand for an indicator of deprivation at the individual level. The following restrictions are imposed on the indicator:<sup>(i) it isequal to</sup> zero when the welfare level of the individual is greater or equal to<sup>thatspecifiedby the</sup> poverty line; (ii) the indicator is a decreasing convex function of welfare,<sup>giventhe</sup> poverty line.

The class of poverty measured defined by (3.10) is called _additively separable_ because the deprivation felt by an individual depends only on a fixed poverty<sup>lineand</sup> her/his level of welfare and not on the welfare of other individuals in society.

21
Furthermore, if the entire population is divided exhaustively into mutually exclusive socioeconomic groups, this class of measures allows one to compute the overall poverty as a weighted average of poverty in each group. The weights here are equal to population shares. Thus, such indices are also _additively decomposable._ We wish to interpret expression (3. 10) in the context of the general social welfare function defined by (2.1). We focus our attention on one prominent subgroup of the class of poverty indices due to Foster, Greer and Thorbecke (1984). The associated deprivation function may be written as follows (Jenkins and Lambert 1997:318).

# (3.11) _YIFGT (z, xh, a) =_ max _{(1-x / z) ,O}._

If we define the relative poverty gap of individual h as gh=max{(z_xh)/z, 0}, it becomes clear that in the context of poverty-focused evaluation defined by the FosterGreer-Thorbecke (FGT) class of indices, the individual poverty gap plays a dual role in defining both individual welfare and the social weights assigned to the individual in social evaluation. To see this point more clearly, we rewrite (3.11) as:

# (3.12) _yfFGT(z,x_ **_,a)_** _=_ max{I,8(1-xh _/Z),0}._

The social weight attached to the condition of individual h is now equal to: _h=_ (IXh / _Z)a-_<sup>_1_</sup> . The parameter a is an indicator of aversion for inequality among the poor. When a=0, the social weight assigned to individual h is equal to: 1h= ( J- _h_ This coefficient balances individual deprivation in such a way that on the basis of (3.12), each poor person counts for one in social evaluation and each non-poor counts for zero. Using expression (3.10), we find that aggregate poverty is equal to: where q is now the total number of poor people in the population. Po is a measure of poverty _incidence_ in the population. This measure does not take into consideration inequality among the poor.

When the aversion parameter is equal to one, the relevant social weights are all equal to one, regardless of individual deprivation. Aggregate level poverty is given by the poverty gap ratio defined as follows:

# (3.14)

where **1tp** is the average welfare of the poor. This index is also known as a measure of poverty _intensity._ One possible interpretation of this indicator is based on the following 22
considerations. Think of x as income, and consider a situation where x is observable and it is possible to give everybody a transfer of (z-x). Afterwards, there would be no more poverty as every individual would have at least an income equal to z. Let q stand for the number of poor. The total income transferred to the poor by this operation is **q(z-pp).** Normalizing by the size of the population, we get Po(z-p.p)=zP 1. Thus in an ideal world of _incentive-preserving_ transfers and perfect targeting, **zPI** is viewed as the minimum amount of resources that must be transferred, on average, from the non-poor to the poor in order to eradicate income poverty. The poverty gap ratio does not account for inequality among the poor. To do so we must set the aversion parameter to values higher than one. Thus setting a=2 produces the following estimator for aggregate poverty.

where **tp** and **cyp** stand respectively for average welfare of the poor and the standard deviation of the welfare indicator among the poor. If everybody had a level of living equal to the poverty line, the average welfare of the poor would be equal to z. The term **cp/z** represents the coefficient of variation in these circumstances. The second expression of P2 on the right hand side of (3.15) thus reveals clearly how this estimator takes into consideration inequality among the poor.

The structure of the FGT family of poverty indicators reveals that a poverty indicator translates the type of concerns policy makers have about aggregate poverty. Typically, three dimensions are of interest: (i) _incidence, (ii) intensity_ and _(iii) inequality_ among the poor. Incidence is the proportion of the total population living below the minimum standard, while intensity (or depth) is the extent to which the well-being of the poor falls below the minimum. Most poverty indicators are designed to capture at least one of these three dimensions. Interestingly, there is a device known as the TIP curvel° which provides a graphical summary of incidence, intensity and inequality dimensions of aggregate poverty based on the distribution of poverty gaps (absolute or relative, Jenkins and Lambert 1997:317). This curve is constructed in three steps: (1) rank individuals from poorest to richest; (2) form the cumulative sum of the poverty gaps divided by population size; and (3) plot the resulting cumulative sum of poverty gaps as a function of the cumulative population share. Assuming that the n individuals in the population are ranked from poorest to richest, then for all integers **k•n** the TIP curve may be defined as (Jenkins and Lamnbert 1997: 319): It is clear from the above expression that the computation of the TIP curve is analogous to that of the Lorenz curve. Figure 3.1 below shows a TIP curve for the distribution of per capita expenditure in Guinea using the official poverty line set at

> 0 TIP stands for " three 'i's of poverty", that is incidence, intensity and inequality.

23
Guinean francs (GNF) 293,714 for 1994. This<sup>curveis based on normalizedpoverty gaps</sup> obtained by dividing absolute gaps by the poverty<sup>line.Thecurveisan increasing</sup> concave curve such that, at any given percentile,<sup>the</sup><sup>_slope is equal to the poverty gap for_</sup> _that percentile._ In the general framework defined by<sup>(2.1)and (3.12),thesocial weight</sup> attached to a percentile is a monotonic transformation<sup>of thepovertygap.Thusthe</sup> curvature of the TIP curve reveals the underlying<sup>schemeof socialweights.</sup> In this normalized case, all weights vary from a maximum<sup>of 1 for lower rankingindividualsto</sup> a minimum of 0 for people above that poverty line.<sup>The closer thestandard of living is to</sup> zero the closer the social weight gets to one.<sup>Thecloser thestandardof livingisto the</sup> poverty line, the closer the social weight is to zero.

Figure 3.1. A TIP Representation of Poverty in Guinea<sup>in1994</sup> <!-- Start of picture text --> .14<br>-<br>.12<br>.10-<br>-<br>.08<br>-<br>.06<br>.04 -<br>.02-<br>.00- I I I I I I I<br>0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0<br><!-- End of picture text --> The curve represents simultaneously the three<sup>basicdimensionsofaggregate</sup> poverty as follows: (1) the length of the<sup>non-horizontalsectionofthecurvereveals</sup> poverty _incidence_ (in figure 3.1, incidence equal<sup>about40percent);(2)the</sup><sup>_intensity_</sup> aspect of poverty is represented by the height of<sup>thecurve; and(3)thedegreeof</sup> concavity of the non-horizontal section of the curve<sup>translatesthedegreeof</sup><sup>_inequality_</sup> among the poor. The use of relative poverty gaps<sup>allows us to read off the horizontaland</sup> vertical axes, values of members of the FGT family.<sup>For Guinea,intensityis estimatedat</sup> 13 percent about. This corresponds to the<sup>vertical interceptat p=1 in figure3.1.</sup> 24
Box 3.1 Computing and Plotting a TIP Curve in EViews INCLUDE LYCDF LOAD %0 SMPL @ALL CALL LYCDF({%1}(1), {%1)(2), {%1}(3)) SERIES GAP={%2}-YH SERIES NGAP=GAP/{%2} SERIES DUMGP=O SMPL IF NGAP>O DUMGP=1 SMPL @ALL SCALAR POP=@SUM(POPH) SERIES PH=F_HAT SERIES WNGAP=(POPH*NGAP)/POP SERIES TGAPW=DUMGP*(WNGAP) SMPL @FIRST @FIRST SERIES JPW= TGAPW SMPL @FIRST+1 @LAST JPW=JPW(-1) +TGAPW SMPL @ALL GROUPJC PH JPW FREEZE(TIPC) JC.XY 'END OF PROGRAM Jenkins and Lambert(1997) explain that if all absolute<sup>povertygapswereequal</sup> among the poor, the concave segment of the TIP curve<sup>would becomea straight line with</sup> slope equal z, the poverty line. In addition, there<sup>aremaximumandminimum poverty</sup> situations that set boundaries to the TIP curve in<sup>a manneranalogousto a Lorenzcurve.</sup> The TIP curve would coincide with the horizontal axis if<sup>there were nopoorinthe</sup> population under consideration. If all incomes (or<sup>expenditures)areequalto zerosothat</sup> the relative poverty gaps are all equal to one, the TIP<sup>curvewould bea straightlinefrom</sup> the origin with a vertical intercept equal to 1 at **p=l.** Table 3.4: Poverty in Guinea by Area of Residence

|Poverty|Conakry|Other Urban|Rural|Guinea|
|---|---|---|---|---|
|Measure|||||
|Po|0.07|0.24|0.53|0.40|
|PI|0.01|0.07|0.18|0.13|
|P2|0.00|0.03|0.08|0.06|
|Source:|Computed||||

The program in Box 3.1 was used to produce figure<sup>3.1.It requiresthe user to fillin</sup> the values of three arguments. The WORKFILE containing the relevant data, a<sup>GROUP</sup> containing three series (total expenditure, the household<sup>size and the householdweight),and</sup> a SCALAR equal to the poverty line. The program can<sup>beeasilyeditedto includeformulae</sup> 25
for the three members of the FGT family discussed above. In the case of Guinea, we obtained the following results by area of residence.

The results presented in table 3.4 reveal that, poverty in Guinea (as measured by the FGT class of indices) is essentially a rural phenomenon. About 53 percent of the rural population lived in poverty in 1994. The next poorest area consists of the urban centers outside of the capital city, where poverty incidence is estimated at 24 percent. Income poverty is lowest in the capital city of Conakry where incidence stood at 7 percent.

## **_Sen 's_** _Measure of_ **_Poverty_**

The FGT family of poverty indices relies on the coefficient of variation in order to account for inequality among the poor. This dimension of poverty may be implemented by combining truncation with the weighing scheme expressed by (2.4). If **Gp** represents the Gini coefficient of the distribution of the living standard among the poor, according to Sen (1997:173), the following expression is an indicator of welfare among the poor:

## (3.17)

# _w,_ =,u,(l-Gp)

It represents the level of _per capita_ level of living, which, if enjoyed equally by every poor person, would provide the same level of social well-being among the poor as the current distribution.

Table 3.5. Guinea 1994: Inequality by Poverty Status

|Aversion|Sen|GiniforPoor|GiniforNon-PoorOverallGini|
|---|---|---|---|
|1.00|0.13|0.00|0.00<br>0.00|
|1.50|0.16|0.10|0.23<br>0.28|
|2.00|0.18|0.16|0.32<br>0.40|
|2.50|0.19|0.21|0.37<br>0.47|
|3.00|0.20|0.25|0.40<br>0.52|
|3.50|0.21|0.27|0.42<br>0.55|
|4.00|0.21|0.30|0.44<br>0.58|
|4.50|0.22|0.32|0.45<br>0.60|
|5.00|0.22|0.34|0.46<br>0.61|
|5.50|0.23|0.35|0.47<br>0.63|
|6.00|0.23|0.36|0.48<br>0.64|

Source: Computed Sen (1997:173) proposes a poverty index that is analogous to the poverty gap indicator of the FGT family. The only difference between the two is that Sen uses **wp** in lieu of **[tp.** Hence the expression: <!-- Start of picture text --> (3.18) 5 =P Pt12 wp 6<br><!-- End of picture text --> 26
It is useful to recall the axioms underpinning Sen's poverty index: _(l) focus; (2) monotonicity; (3) weak transfer;_ (4) symmetry, (5) scale invariance. Focus requires that the poverty measure depend only on the situation of the poor. The living conditions of the non-poor are irrelevant. Monotonicity requires an increase in the poverty measure whenever the well-being of the poor decreases. The weak transfer axiom makes the measure sensitive to changes in the distribution of living conditions among the very poor. Thus, if resources are transferred from one poor person to a relatively poorer one, the weak transfer axiom implies that the poverty measure should decrease.

The extended Gini coefficient may be used in expression (3.18) to obtain an extended version of Sen's poverty indicator written as follows.

# (3.19)

Note that the generalized Sen index of poverty is analogous to the poverty gap ratio defined by (3.14). In fact both indices are equivalent when the aversion parameter v=1. The results presented in tables 3.4 and _3.5_ confirm this fact for the case of the distribution of household expenditures in Guinea in 1994. We note that S(1)=P=0.13. This observation suggests another way of extending the Sen index of poverty. Instead of using the extended Gini<sup>to</sup> measure inequality among the poor, one could use another index such as the Atkinson index of inequality Table _3.5_ shows values of extended Sen index along with estimates for the extended Gini coefficient for values of the aversion parameter ranging from 1 to 6. All indices increase monotonically with v. The results also reveal that inequality is higher among the non-poor than among the poor.

# _How Desirable is Truncation?_

To be sure, both the FGT and Sen approaches to poverty analysis are consistent with the maximin approach discussed in Section 2. Within that framework, social weights are assigned to individuals on the basis of some notion of social desert. The more deserving the individual the higher the social weight assigned to her/his situation. The Gini family of indices implements this idea as follows. First, individuals are ranked in decreasing order of social desert. Second, social weights are assigned according to (2.4) which describes each weight as a function of the relative rank of the individual and the degree of inequality aversion. As revealed by figure 2.1, the level of aversion for inequality implies a cut-off rank such that people before it receive higher social weights than people after the cut-off " The Atkinson index of inequality is also based on the concept of equally distributed equivalent (EDE) welfare applied to a specific social welfare function. In this framework, the difference between average welfare and the EDE welfare is considered as the per capita social cost of inequality. The Atkinson index is equal to the per capita social cost divided by average welfare (See Larnbert 1993b:131-138) for analytical details.

27
point. Our interpretation of expression (3.12) indicates that the FGT method<sup>does</sup> something very similar subject to the following adjustments. First, the<sup>social weightsare</sup> now a function of the relative poverty gap. Second, _the cut-offpoint is no longer_<sup>_tied to the_</sup> _choice of the aversion parameter, it is imposed exogenously by the poverty_<sup>_line._Third,</sup> everybody after the cut-off point now receives a weight of zero, regardless of<sup>her/his welfare</sup> level. Similar considerations apply to Sen's index of poverty. We argue<sup>thattheproblem</sup> here is not so much with the use of the poverty line as it is with ignoring<sup>everyperson</sup> beyond the cut-off point.

It is evident that truncation implements Sen's interpretation of<sup>thefocus</sup> assumption. As mentioned above, this interpretation renders the living<sup>standardof the</sup> non-poor irrelevant for the purpose of social evaluation. As argued<sup>in thebeginningof</sup> Section 2 of this paper, social evaluation is a critical input into public policy<sup>formulation</sup> and implementation. Indeed any policy recommendation is constrained by such an evaluative input, the reliability of which is determined by the organizing<sup>framework</sup> (view of the world) and the quality of the data set to which the framework applies.

There are two basic approaches to policymaking that shape the<sup>informational</sup> needs of the exercise. The _normative_ approach seeks to maximize a social welfare function subject to the economy's resource, technology and institutional<sup>constraints.In</sup> the classical version, the resulting optimal program is supposed to be implemented<sup>bya</sup> set of competitive and complete markets given the prevailing ownership of<sup>resources</sup> (Dixit 1996: 4-5). In the context of the _positive_ approach, policymaking<sup>isviewedas</sup> involving strategic interactions among various socioeconomic agents<sup>subject to potential</sup> _conflict_ and _cooperation._ In this context, Aristotle is reported to have<sup>stated that</sup><sup>_"It is_</sup> _when equal have or are assigned unequal shares, or people who are not_<sup>_equal,equal_</sup> _shares, that quarrels and complaints break out."_ (Young 1994:64).

In the positive perspective, a policy may be viewed as a social contract<sup>among</sup> participants which is enforceable within a given governance structure. Governance structure involves three basic elements:(1) abstract and universal rules of the<sup>socialgame;</sup> (2) enforcement institutions; and (3) mechanisms for the resolution of<sup>conflicts over both</sup> the rules and their enforcement (Fristchtak 1994) 2. Any policy choice entails a distribution of burdens and advantages that creates winners and losers. Governance capacity is therefore the ability to coordinate the aggregation of these<sup>divergingprivate</sup> interests into an outcome that can credibly be taken to represent public interest.<sup>Thisis a</sup> key consideration in the investigation of political feasibility'<sup>3</sup> of public policy.<sup>Such an</sup> investigation pertains to the way gainers and losers form coalitions and<sup>usethe political</sup> system to their advantage. As Kanbur (1994:8) explains it, the outcome hinges<sup>crucially on</sup> the threshold at which a gain or a loss becomes so significant that an individual<sup>ora group</sup>

> 12 This author further explains that governance capacity in a modem<sup>statemaybeassessedin</sup> terms of the mere existence of such rules, institutions and mechanisms, and<sup>the degreetowhich theyare</sup> universal and predictable. Govemance capacity is also determined by the<sup>extent towhichthestateis</sup> autonomous vis-a-vis private interests in society.

> **'3** Thus Kanbur (1994:4) explains that desirability is determined by welfare<sup>economicswhilefeasibility</sup> stems from political economy.

28
feels compelled to organize and fight. _At this point it may be worth distributing real or symbolic pay-offs to the losers in order to buy social peace._ The above political economy considerations suggest that truncation may be too drastic an approach to poverty-focused evaluation. Such considerations imply that we must not ignore the non-poor totally in our social calculus. The Dalton principle and its implementation by the Gini family of indices offer a way out of drastic truncation. This framework is consistent with a milder interpretation of the focus assumption which now requires only that we assign higher weights to the welfare interest of the poor relative to that of the non-poor. The particular range of social weights selection in the evaluation is an outcome of the governing political process.

## **_4. Impact Analysis_**

We view distributional impact as a comparison of social states based on the associated distributions of pay-offs from socioeconomic interaction. In this Section, we focus on the welfare impact of public policy. One particular state, known as the _counterfactual,_ plays a crucial role in the assessment of the impact of policy. By definition, the counterfactual describes the state of the world that would have prevailed had the policy in question not been implemented. Thus impact assessment involves the comparison of two states of affairs: one _with_ and the other _without_ the policy. This comparison will be based on the welfare criterion defined by (2.1). The discussion is organized around three points:(l) social desirability of feasible policy reforms; (2) unambiguous welfare comparisons; and (3) the use of abbreviated social welfare functions.

# **_Social Desirability of Feasible Policy Options_**

Policy design and evaluation involve a determination of _desirable_ actions within a _feasible_ set. Considering the public budget as the policy instrument _par excellence,_ the set of feasible collective actions must be viewed as determined by the budgetary process in the broadest sense of the term in order to capture the political economy dimension of feasibility. To be concrete, we focus on commodity taxation and assume _revenue neutrality._ Yitzhaki and Lewis(1996:543) argue that this assumption allows one to ignore the issue of the optimal size of government activity. Social desirability of a feasible action depends on the chosen social evaluation criterion or social welfare function.

In general, the impact of a policy change on the welfare of individual h, may be analyzed within the standard model of consumer behavior. The consumer is assumed to choose the best bundle of commodity that she can afford. At the optimum, this behavior can be characterized by the _indirect utility function_ defined as follows: (4.1) _vh(q,yh )=max{uh(xh) st. q.xh =yh}_ 29
where q is a vector of consumer prices and **yh** is the income of consumer<sup>h.Onecan</sup> invoke Roy's identity (Varian 1992: 106) to compute the impact of the<sup>policyonthe</sup> welfare of individual h as: Expression (4.2) above implies that the marginal impact, in terms of<sup>income,of a</sup> unit increase in the tax on commodity j (assumed to be a private good)<sup>isgivenbythe</sup> following expression: In the case of a tax increase, expression (4.3) measures the loss imposed<sup>onconsumerh</sup> as the consumption rate of commodity j. The total impact of a change<sup>inallcommodity</sup> taxes may now be expressed as:

# (4.4)

The social impact of such a reform may be written as: where **ph** is now the marginal social weight of the income of<sup>individualh.</sup> This coefficient is defined as _8h _(aW) )_ The change in social welfare induced by the tax reform may<sup>betransformedas</sup> follows. First, we change the order of summation in (4.5), then we<sup>multiplybythe</sup> **_n_** neutral element (xj/xj) where **_xj =_** _Exi_ is the aggregate _base_ of the tax on<sup>commodityj.</sup> **_h=l_** This transformation leads to the following expression: (4.6) To further apprehend the structure of the change in social welfare,<sup>weneedto</sup> consider the impact of the tax reform on the public budget. Government<sup>revenueis equal</sup> to: 30
m _B = Ztjxj(q,y)_ **_j=I_**

# (4.7)

where **xj(q,y)** is the demand for commodity j (the tax base) as a function of prices and incomes. A small change in all taxes leads to the following change in government revenue.

# (4.8)

Where **r;** is the marginal revenue associated with a change in the jlh tax. The marginal tax reform may be characterized by a vector of tax changes **dt,** or by a vector of tax revenues 6.

Revenue-neutrality implies that **dB=O.** Given that q is the vector of consumer prices, we have dt=dq. According to (4.8), we may write _dqj =_ **_Lj_ .** This implies the _r._ following expression for the change in social welfare: <!-- Start of picture text --> (4.9) dW = -E d<br><!-- End of picture text --> Three important parameters are embedded in the above expression. The _first_ one reveals the _distributional characteristic of commodity_ j. The parameter is defined as (Mayshar and Yitzhaki 1995:795): The _second_ parameter is the marginal efficiency cost of funds collected through taxation of commodity j. It is defined as: Slemrod and Yitzhaki (2001:6) explain that this parameter is an indicator of the extent of leakage from the tax base (hence efficiency cost) associated with adjustment in behavior by the taxpayer in response to the change in tax burden. This may be thought of as an 31
incentive effect of the tax on j.<sup>Themarginalefficiencycostis equal tothe ratioof the</sup> cost of funds to the taxpayers to<sup>the value of the funds going to the public treasury.</sup> The _third_ parameter is the marginal<sup>socialcost of funds definedas follows:</sup> <!-- Start of picture text --> (4.12)<br><!-- End of picture text --> The above expression reveals<sup>that themarginalsocialcostoffundsisequalto the</sup> distributional factor times the<sup>marginal efficiencycost of raisingfundsthrougha taxon</sup> commodity j.

It is instructive to note that<sup>thedistributivecomponentmayalsobewrittenas</sup> follows: When social weights are chosen<sup>accordingto(2.4),thenweknowthat</sup> **1**<sup>**.p=l**andthe</sup> second term within the brackets<sup>isequalto minusthe extendedconcentrationcoefficient</sup> of commodity j. Therefore, the<sup>marginalsocialcostof fundsfor thiscommodity isequal</sup> to:

# (4.14)

_MSCFj = MECFj [1- Cj (v)] = MECFj_<sup>_[1-_</sup><sup>**_i7j_**</sup><sup>_(v)Gx (v)]_</sup> The impact of the tax change on<sup>social welfareis thus equal to:</sup>

# (4.15)

**_m_** The policy option is feasible if<sup>_dB = E,j_</sup> **_>_** 0 has a nontrivial solution.<sup>Itissocially</sup> **_j=1_**

desirable if **dW2O.** The final determination rests<sup>onthespecificationofthe social</sup> weights **ph.** If we impose only<sup>minimal restrictionsonsocial weights,then a wide class</sup> of social welfare functions would<sup>agreeon the characterizationof the policyoption.This</sup> situation corresponds to dominance<sup>ranking.However,theselectionof a uniqueset of</sup> weights allows one to assess all<sup>policyoptionson the basis of theimplied criterion.</sup>

# **_Unambiguous Welfare Comparisons_**

As noted above, social desirability<sup>dependsonthe value judgmentsunderpinning</sup> the social welfare function. Such<sup>judgmentscan taketheformof restrictionsplacedon</sup> 32
the social weights **ph.** Suppose that we adopt an individualistic welfare function of the type defined by (2.1) subject to the restriction that **Pbh** 2 0 for each h. A Pareto-improving tax reform would have to improve the welfare of at least one individual without hurting any. This could be characterized in terms of (4.4) using the implication of (4.8) that _dqj =- ._ A Pareto-improving marginal tax reform must satisfy the following (n+l) linear restrictions: The above expression reveals that the marginal impact of the reform on consumer h is equal to minus the weighted sum of modified consumption shares. The weights are the marginal tax receipts **i,** while the modified consumption share of each commodity are obtained by multiplying each share by the corresponding marginal efficiency cost of funds.

As noted in Section 2, the Dalton principle may be built in the social welfare function by first ranking individuals according to some criterion of social desert, then assigning social weights in such a way that of any two individuals the more deserving receives a higher weight. To see clearly what is involved, consider a society of three individuals ranked from the poorest to the richest. Under the Dalton principle, the pattern of social weights is :PI1Ž2i3P3. Given that social weights are now chosen such that each is nonnegative and adjacent ones satisfy **(p-I-Phb)>O** it is desirable to express the welfare improving condition in a way that is consistent with this restriction. Mayshar and Yitzhaki (1995:797) show that a _Dalton-improving policy_ may be characterized in tenns of cumulative marginal impact.

The cumulative marginal impact is defined as: By definition, _dvh_ = _cmvh_ - _cmvh'._ The change in social welfare due to a marginal tax reform is thus equal to: Expression (4.18) is a sum of cross products. The first term of each product is known to be nonnegative. It is therefore clear that if the second term is also nonnegative the whole sum will be nonnegative as well. The reform will be welfare improving (i.e. dW20) if each cmvh **_>0._** The condition for Dalton-improvement may thus be stated generally as: 33
(4.19) The condition for Pareto-improvement implies that for a Dalton-improvement, but<sup>not the</sup> other way around. Mayshar and Yitzhaki(1995:798) emphasize that, within this framework, _the social ranking is exogenous and need not be based on income._<sup>They</sup> further note that when the ranking is based on income, the Dalton criterion is equivalent to the second degree dominance criterion based on the generalized Lorenz curve.

To further expose the structure of the Dalton criterion, on the basis of (4.9) we write the cumulative marginal impact as follows.

# (4.20)

In the above expression, the term _(x; / xj )MECFj_ represents a point on the **i=l** concentration curve of commodity j scaled by that commodity marginal efficiency cost of funds. Thus (4.20) shows that the design of a Dalton-improving indirect tax reform<sup>is</sup> equivalent to searching for a vector 8 such that **Yj8j2O** and the 8-weighted sum of scaled concentration curves for the commodities is everywhere non-positive'<sup>.</sup> Let's now focus on the simple two-commodity case where commodity k is taxed in order to subsidize the consumption of commodity s while keeping the public budget balanced. The revenue neutrality constraint implies that (oS+8o=0. The condition for Dalton-improvement may thus be written as:

# (4.21)

Since o, is negative by assumption, the reform will be Dalton-improving if the modified concentration curve for commodity s lies nowhere below that for commodity k.<sup>Wemay</sup> say that commodity s Dalton dominates commodity k. If the two commodities happen to have identical positive MECFs<sup>1</sup> 5, then we need only compare the ordinary concentration curves. It is in fact the relative magnitudes of the MECFs that determine whether to use _14_ Mayshar and Yitzhaki(1995: 800) propose the following algorithm in the case of strict revenue- _n_ ~~~~~~~~~2 neutrality. min [maxk- cmv' _(3), 0)] st. ,_ **_j_** _= 0; 3,_ 0. A solution is achieved when the J _h=1_ ~~~~~~~~j=1 objective function is equal to zero for nontrivial values of the vector 6. In fact the constraint<sup>that 8170is</sup> meant to avoid trivial solutions. The authors further explain that one must compute two sets of solutions for positive and negative values of 6,. Since the scale of the reform can be set arbitrarily, this implies checking solutions for 51=1 and for 51=-l. Finding a solution for these two values means that there exists a Dalton-improving tax reform that does not involve a change in the tax rate of commodity 1.

_15_ Estimates of MECFs are known to be sensitive to changes in the structure of preferences (Mayshar<sup>and</sup> Yitzhaki 1995:803).

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the ordinary concentration curves or the modified ones. It is instructive<sup>to note that, if the</sup> concentration curve of commodity s lies nowhere below the<sup>Lorenz curveoftotal</sup> expenditure (or income), then a subsidy on this commodity financed<sup>bya proportional</sup> income tax would increase welfare (Yitzhaki and Slemrod 1991:<sup>486). In the same spirit,</sup> Yitzhaki and Thirsk 1990:14) explain that a tax on wages may<sup>be interpretedas</sup><sup>_a tax on_</sup> _all expenditures made by workers._ Finally, to focus only on efficiency considerations, we look at **cmvn.**<sup>Inthe case</sup> of a revenue neutral reform involving two commodities, expression<sup>(4.20) implies that:</sup>

# (4.22)

_n cmv" = Xdvh = -(MECF, -MECFk)35 20_ This expression reveals that the neutral tax reform involving a<sup>subsidyforcommoditys</sup> will reduce deadweight loss if MECFs>MECFk.

_There is a necessary condition for Dalton improvement based on Gini elasticities of the two commodities involved._ Recall that if commodity s<sup>dominatescommodityin</sup> terms of expression (4.21), then it must be true that the area of<sup>concentrationofs</sup> multiplied by the marginal efficiency cost of s must be less than<sup>the areaof concentration</sup> of commodity k times its marginal efficiency cost. This condition<sup>may be stated in terms</sup> of concentration indices as follows.

# (4.23)

# - _[C3 (v)MECF, - Ck(v)MECFk ] O 0_

Based on definition (4.11), we write the ratio of the marginal<sup>efficiencycoefficientsas</sup> **_aSk_** _= _MECFk_ . Using the definition of the Gini Engel elasticity, the necessary<sup>condition</sup> _-MECF,_ for Dalton improvement can be expressed as:

# (4.24) _-_ **_[q,_** _(v) - ask m7k(v)]GX (v) 2 0_

This condition can be used to narrow down the set of commodities<sup>onwhich to perform</sup> pairwise comparisons of modified concentration curves (Yitzhaki<sup>andThirsk1990:9).</sup> When both commodities have identical MECFs, then the necessary<sup>conditionforDalton</sup> improvement presented in (4.22) reduces to the following inequality:<sup>M</sup><sup>_(V)< 7k(V)._</sup> Condition (4.21) can be verified graphically by plotting on<sup>the horizontalaxis the</sup> cumulative proportions of individuals in the income distribution,<sup>andonthe verticalaxis</sup> the difference between the concentration curve of commodity<sup>sandtheconcentration</sup> curve of commodity k, multiplied by **ask.** Yitzhaki and Slemrod(1991)<sup>call this the</sup> **DCC(p)** curve (the "difference in concentration curves" curve).<sup>This curve starts at (0,0)</sup> and ends at (1, 1- **ao).** If it lies everywhere above the horizontal<sup>axis,commoditys</sup> dominates commodity k; if it is located entirely below<sup>thehorizontalaxis,then</sup> commodity k dominates s. If the curve crosses the horizontal<sup>line, thereis no dominance</sup> to speak of.

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The **DCC(p)** curve allows a combined presentation of both the efficiency gain from the reform and its distribution among income groups. If the curve<sup>is increasingat p,</sup> then individuals located at that percentile gain from the policy. If the<sup>curveis decreasing</sup> then they are losing from the reform. **DCC(1)=(1- ak)** provides an indication of<sup>whether</sup> the cumulative gain for the whole society is positive, zero or negative.<sup>Theoutcome</sup> hinges on how **ask** compares to 1. When this ratio is equal to one, the policy<sup>reformis</sup> neutral with respect to efficiency (no efficiency gain, nor loss). However, if<sup>the</sup><sup>**DCC(p)**</sup> curve lies above the horizontal axis for all other values of p, we conclude<sup>that the policy</sup> is welfare improving (in the sense of Dalton) on account of distribution.<sup>When</sup><sup>**DCC(1)<**</sup> 0 (implying that ask>l) condition (4.21) cannot hold for all h, since it<sup>failstoholdfor</sup> h=n.

It is useful to note that the same methodology can help analyze the<sup>case wherea</sup> _subsidy is financed by a proportional income tax._ In this case, we have to<sup>comparethe</sup> concentration curve for commodity s to the Lorenz curve of the income distribution.

Figure 4.1. Difference in Concentration Curves for Cereals and Beverages<sup>inGuinea1994</sup> <!-- Start of picture text --> .30-<br>.2 5-<br>.20-<br>.1 5 -<br>.00<br>0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0<br><!-- End of picture text --> To illustrate the point, we compare the distributional characteristics<sup>oftwo</sup> commodities, cereals and beverages, using data from the 1994 integrated<sup>household</sup> survey from Guinea. Figure 2.2 shows the concentration curves of the two<sup>commodities</sup> along with the Lorenz distribution of total household expenditure. It is<sup>evidentfrom that</sup> figure that the consumption of cereals in Guinea is more equally distributed<sup>thantotal</sup> consumption and the consumption of beverages. Not having estimates for relative efficiency costs, we proceed as if **ask** were equal to one (the case of neutrality<sup>vis-a-vis</sup> efficiency). Under this assumption, we would expect welfare dominance<sup>totheextent</sup> that the concentration curve for cereals lies totally above the concentration<sup>curvefor</sup> 36
beverages over the entire range of the cumulative distribution of per capita expenditure. Figure 4.1 confirms this view. It plots the difference between the concentration of the consumption of cereals and that of beverages. The **DCC** curve lies entirely above the horizontal axis, thus it would be socially desirable to implement a commodity tax reform that would shift the tax burden away from cereals and towards beverages while keeping the public budget balanced. In addition, figure 4.1 reveals that at least 85 percent of the population would benefit from such a move.

It is useful to note the concept of TIP dominance in the context of truncated analysis. Given a poverty line z, if a TIP curve a lies entirely over another TIP curve b, we say that a TIP-dominates b. In other words, there is more poverty in situation a than in situation b, regardless of the dimension we choose to focus on: _incidence, intensity_ or _inequality_ among the poor, and for all poverty lines less than or equal to z.

Figure 4.2. A TIP Comparison of Poverty in Guinea by Area of Residence (Data Source: 1994 Integrated Household Survey) <!-- Start of picture text --> .20<br>Rural<br>-<br>.16<br>.1 2-<br>.08-<br>Other U rban<br>.04-<br>Conakry<br>.00- , I , I I I<br>0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0<br><!-- End of picture text --> Figure 4.2 illustrates the point with three TIP curves for three population subgroups in Guinea based on the area of residence of the household in 1994: (1) Conakry, the capital city; (2) other urban centers and (3) the rural area. The dominance relation among these curves reveals that, regardless of the dimension considered, poverty in Guinea (as measured by the FGT class of indices) is essentially a rural phenomenon. About 53 percent of the rural population lived in poverty in 1994. The next poorest area consists of the urban centers outside of the capital city, where poverty incidence is estimated at 24 percent. Income poverty is lowest in the capital city of Conakry where incidence stood at 7 percent.

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Our discussion of TIP curves focuses<sup>ontheFGTclassofpovertyindices.</sup> However it is worth noting that Jenkins<sup>andLambert's(1997)resultsaremoregeneral.</sup> Consider the class of Generalized Poverty<sup>Gap(GPG)povertyindices.Themembersof</sup> this class are increasing Schur-convex' functions of absolute poverty gaps given<sup>the</sup> poverty line. In addition, these indices<sup>arereplicationinvariant.Asub-classof these</sup> indices, GNPG, is defined on normalized<sup>povertygaps(e.g.FGT).Asit turnsout,TIP</sup> dominance based on absolute poverty gaps<sup>is equivalenttoa unanimous povertyordering</sup> by all members of the GPG class, and<sup>for all povertylinesthat are at mostequal to the</sup> chosen one<sup>17</sup> .

# **_Use of Abbreviated Social Welfare Functions_**

Dominance criteria provide a general<sup>frameworkforidentifying unambiguous</sup> rankings of social states in terms of<sup>the distributionof thelivingstandard.However,the</sup> relation of dominance is a _partial ordering_<sup>inthesensethatitcouldfailtorank two</sup> situations. For instance when two concentration<sup>(orLorenz)curvesintersect,weloose the</sup> ability to draw unambiguous conclusions<sup>about theinequalityorwelfarecontentof the</sup> distributions under comparison.

Sen (1989: 18) explains that the dorminance<sup>approach to social evaluation providesa</sup> _minimal partial order_ on the focal<sup>space.The Paretoimprovementtestdiscussedabove</sup> (based on the social welfare function<sup>definedby2.1)isanexampleof suchaminimal</sup> ordering to the extent that it requires<sup>onlynon-negativesocialweights.Sen(1989)also</sup> explains that it is possible to go beyond<sup>the scopeof thisminimal order by further restricting</sup> social weights to lie within a particular<sup>range.The Daltoncriterion,forinstancefurther</sup> restricts the admissible range associated<sup>withthe Pareto criterion.Thenarrowerthe range,</sup> the more extensive the overall ranking<sup>willbeincomparisontominimaldominance.</sup> A social evaluation function based on<sup>a uniquesetof weights(e.g.2.5)wouldinducea</sup> complete ordering of socioeconomic states.<sup>Theweightsinquestionarea reflectionof</sup> the underlying _value judgments._ These<sup>value judgmentsdeterminethe structureof</sup><sup>_index_</sup> _numbers_ used to assess the inequality and<sup>social welfare effects of a policy based on induced</sup> distributional changes. In the end, the selection of evaluative<sup>weightsboilsdownto</sup> _priority_ setting among the different dimensions<sup>of thequality of life(e.g.componentsof</sup> consumption) and among diverse individuals<sup>orgroups(withinresource andinstitutional</sup> _constraints)._ In this subsection we focus<sup>on index numbersbased on the</sup><sup>_maximin_principle,</sup> and on _normalized poverty gaps._

**16** By analogy to Schur-concavity in the<sup>caseof inequality andwelfarewhichensuresthat inequalityfalls</sup> and welfare increased when the distribution<sup>ofresourcesis"smoothed"byequalizing transfers,Schur-</sup> convexity require that poverty fall when poverty<sup>gaps are smoothed (Jenkins and Lambert1997).</sup> 17 Jenkins and Lambert(1997) also reveal<sup>acloselink betweenthepovertyorderingsbasedonTIP</sup> dominance and those associated with<sup>generalizedLorenzdominance.</sup> They show that distribution a dominates distribution b on the basis of<sup>the generalizedLorenz curve if and only if the TIPcurvefor b lies</sup> nowhere below that of a, for all common<sup>povertylinesz.The generalizedLorenzcurveisequal tothe</sup> ordinary Lorenz curve times the mean of<sup>thecorresponding distribution.</sup>

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_Changes in the Abbreviated Social Welfare Function_ When an economy undergoes a shock (exogenous or policy), its structure and the implied distribution of welfare may change significantly. In the particular case of policy reform, the desirability and feasibility of a policy proposal depend essentially on the implications of the proposal for efficiency and equity. We may used the abbreviated social welfare function defined by (2.5) to track the effects of a policy. To simplify matters, we focus the analysis at the margin. Consider therefore the enactment of a policy that leads to marginal changes (dxn) in the living standard of the population involved. The logic behind expression (2.5) leads us to the following assessment of the induced welfare impact.

where a stands for the distribution of marginal benefits induced by the reform. The above expression is equivalent to the following'<sup>8</sup> : where _17a (v) =_ G ( ) is the ratio of the concentration of marginal benefits to the overall _Gx (v)_ Gini index of inequality in the distribution of the living standard prior to the reform.

Expression (4.26) shows that a first-order assessment of the welfare impact of a policy reform within the maximin framework can be based on three parameters: (1) the average marginal benefit of the reform **_(ta),_** (2) the overall Gini index of inequality Gl(v) and (3) the Gini Engel elasticity of the marginal benefit of the reform **la(v).** A Gini Engel elasticity greater than one means that the distribution of marginal benefits is more unequal than the initial distribution of the living standard. Therefore the reform is inducing more inequality. This distributional effect will lower welfare. The overall impact depends on the sign and size of the average benefit.

When the living standard is composed of different components, we may be interested in welfare changes due to a change in one component. Kakwani(1995:5) provides a way of computing the elasticity of V(v) with respect to the mean of the jlh component.

According to Kakwani(1995:5) this elasticity, which measures the effect of a change in the jlh income component on total welfare and may be interpreted as the sum of the "income" and the inequality effect. The income effect is equal to the share of the component in total expenditure (or income) **Xj,** while the inequality effect is equal to this share times a

> _18_ The expression is obtained by factoring out _pa_ and multiplying the resulting concentration coefficient by the neutral element [GX(v)/Gx(v)]. Also note the analogy between this expression and that of the marginal social cost of funds (4.14).

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progressivity index denoted by<sup>**_PK_**</sup><sup>_(v)._Apositivevalueof thisindex means thatthe</sup> change in the **jth** component is progressive<sup>inthe sense that it favorsthe poor morethan the</sup> rich. A negative value implies regressivity.<sup>If thechangein the</sup><sup>**jlh**incomecomponentis</sup> distributed in proportion to total income,<sup>itsconcentration indexwillbeequaltotheGini</sup> index and the progressivity index<sup>will bezero.In thiscaseneitherthe rich nor thepoorare</sup> favored. This progressivity index may<sup>guide the designof anoptimum tax (or expenditure)</sup> policy. The methodology described<sup>heremayalsobe usedtoevaluatethe effectof price</sup> changes on total welfare

# _Poverty Impact_

One may be interested in analyzing<sup>thepovertyimpactof policiesandshocks</sup> based on poverty indices such as those<sup>of the FGTfamily.Asa generalrule, the poverty</sup> impact of a policy depends on<sup>thedegreeof targeting.</sup> Options range from perfect targeting to no targeting at all.<sup>Intheextremecaseof notargeting,anyeffectof policy</sup> on the poor is linked to the "trickle<sup>downmechanism". However,policymakers may</sup> resort to a coarser approach to<sup>targetingbased onabroad definitionofpopulation</sup> subgroups (Kanbur 1987a:72). Such<sup>policieswillaffectboththepoorand the non-poor</sup> within the targeted socioeconomic<sup>group.For instance, price supportfor a particularcrop</sup> will affect both poor and non-poor<sup>farmersengagedinthecultivationof the crop.A</sup> subsidy on a particular food item<sup>willaffect allthosewhoareconsumingit.Atthe</sup> margin, such an analysis relies on<sup>the derivativesof the poverty indicesused with respect</sup> to changes in the welfare indicator.<sup>For theFGTfamilyof indices, thefirst derivativeof</sup> the poverty gap with respect to<sup>the welfareindicator is equal to:</sup> and the second derivative is: For positive values of the aversion<sup>parameter,thesederivatives indicatethat membersof</sup> the FGT family of indicators<sup>are</sup><sup>_non-decreasing convex_functionsofthewelfare</sup> indicator.

Kanbur(1987a:73) notes that<sup>apolicy mayhave eitheranadditiveora</sup> multiplicative effect on the level of<sup>individualwelfare.Considerfirsttheadditivecase.</sup> Policy implementation changes<sup>alllevelsof welfareto(xh+t),sothatindividual poverty</sup>

> **_19_** See Kakwani (1995:5-7) for<sup>details</sup>

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gaps are now equal to: _yIFGT (Z, X, a) =_ max _{(1 - (xh + t) / z)a,0} ._ This implies the derivative of the poverty gap with respect to t is equal to: This suggests that the derivative of the average poverty gap with respect to t is equal to: Kanbur interprets this expression as the shadow price of budgetary expenditure when one is seeking to minimize **Pa** subject to a budgetary constraint. When a=1, the amount by which poverty intensity changes when the welfare indicator increases marginally is proportional to the head count ratio, i.e. **Po.** This expression makes the shadow price of budgetary expenditure proportional to the difference between Pa..l and **Pa.** Another interesting aspect of poverty impact analysis relates to the responsiveness of poverty to growth and changes in inequality. Given that poverty indices are computed on the basis of a distribution of living standards, which is fully characterized by the _mean_ and the degree of _inequality,_ it is reasonable to think of a poverty index as a function of these two factors. Indeed, procedures have been developed for the decomposition of changes in poverty into _growth_ and _inequality_ components (Ravallion and Datt, 1992).

For small changes and under the assumption that the Lorenz curve shifts proportionately over the whole range of income distribution, Kakwani (1990) shows that the total percentage change in a poverty index is equal to the _growth elasticity of the index_ times the percentage change in the mean income plus the _elasticity of the index with respect to the Gini_ times the percentage change in the Gini coefficient. He further shows that the growth elasticity of the head-count index is equal to: Where **f(z)** is the density function of the welfare indicator evaluated at the poverty line. The elasticity with respect to Gini is: 41
(4.33) For the general class of additively separable<sup>poverty measures,the growth</sup> elasticity is given by the following expression.

The elasticity with respect to Gini is: These elasticities may be used to evaluate<sup>the</sup><sup>_potential for povertyreduction._</sup> Indeed, a higher growth elasticity<sup>meansa greaterpotential forpovertyreduction.</sup> Focusing in particular on poverty incidence,<sup>itcanbeseen from(4.32)thatthe</sup> responsiveness of the head-count index<sup>to growthdepends essentiallyon the</sup><sup>_initial value_</sup> of the index and the slope of the distribution<sup>functionatthe povertyline.The higherthe</sup> initial head-count, the lower the elasticity<sup>andthehighertheactualrateofgrowth</sup> required to further reduce poverty (other<sup>things being equal).</sup> In the particular case of the FGT family of<sup>indices,these elasticitiesareequal to:</sup> On the basis of expressions (4.35) and<sup>(4.36)the total impact of growthand redistribution</sup> on income poverty may be written as: Ravallion (1994b) argues that the above<sup>decompositionmay lead to large errorsin</sup> the case of big discrete changes. A different<sup>approachis therefore required.Insteadof</sup> summarizing inequality by the Gini index,<sup>onemayusea parameterizedLorenzcurve</sup> along with the mean income to decompose<sup>changes in poverty into growthand inequality</sup> components, and a residual".

> 20 Ravallion and Datt (1992) note that the existence<sup>of thisresidualdependsonwhetheror not the poverty</sup> index is additively separable between the mean<sup>andtheLorenzcurve.Theresidualwouldvanishif the</sup> mean income or the Lorenz curve remained<sup>constantover the decompositionperiod.</sup>

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> The _growth-inequality_ relationship thus seems fundamental in analyzing the

> dynamics of poverty. If growth is distributionally neutral, poverty is expected to be reduced on such a path<sup>21</sup> . If one drops the neutrality assumption, then the outcome becomes ambiguous. In this case the distributional impact and the poverty implications

> depend essentially on the initial structure of the economy and the. profile of the adjustment process.

# _5. Conclusion_

This paper sought to demonstrate how one might perform distributional impact analysis within _a poverty-focused_ evaluative framework. The approach is illustrated with household level data from Guinea, using EViews as a computing platform. The focus on poverty is important for at least two reasons. Widespread poverty in the developing world remains a serious challenge for the development community. Furthermore, the ethics of _empowerment_ which underpins current development thought entails the expansion and the distribution of socioeconomic opportunities with implications for individual and collective well-being. These considerations have prompted the development community to make poverty reduction a fundamental objective of socioeconomic development, and a benchmark measure of the performance of various social arrangements. In this context, poverty is seen as the deprivation of basic capabilities to lead the type of life one has legitimate reasons to value.

> Any evaluation is characterized by the underlying _criterion._ The object of social evaluation is the determination of the extent to which prevailing social arrangements promote individual and collective well-being. This requires identification of the objects of value, and an aggregation rule for the comparison of social states. The need for aggregation raises the fundamental issue of weights to attach to each constituent of the

> aggregate. The choice of social weights is based on _value judgments._ If one believes that an incentive-preserving transfer of resources from rich to poor would increase collective

> well-being, then the _maximin principle_ provides an adequate foundation to a poverty-

> focused evaluation. When implemented in the context of additively separable social welfare functions, the principle leads to social evaluation functions that are decomposable

> into two components revealing both the average living standard and its distribution

> among the population. The distribution component is equal to the covariance between the living standard and the social weights. Thus, the maximin approach to social evaluation offers a way of combining both efficiency and equity considerations in social evaluation. In addition, the linearity of the covariance operator allows a factor

> decomposition of the inequality components on the basis of the constituents of the living standard.

Further specification of social weights is needed to determine the extent of poverty focus of the evaluation. The Gini family of indices provides a weighing scheme based on relative ranks of individuals (according to some criterion of social desert), and on a focal parameter indicating the degree of aversion to inequality. This aversion

> 21 The time frame is important in this context as we should distinguish between short and long-run impacts associated respectively with fluctuations in output, and steady-state growth in capacity output.

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parameter also determines a cut-off<sup>ranksuch that people belowthecut-off point receives</sup> higher weights than people above<sup>it.In thatsense,thestandard approachtopoverty</sup> analysis based on the truncation of<sup>the distributionof the living standard by a povertyline</sup> is quite consistent with the maximin<sup>approach.</sup> However, this truncation appears undesirable on political economy grounds.

The criterion presented in this<sup>papercan guidetheidentificationof desirable</sup> policy options. This entails a<sup>ranking of socialstateson thebasisof thevalues assumed</sup> by the criterion in each state. The<sup>scopeof therankingdependson the acceptablerange</sup> assigned to social weights. This range is a reflection of the<sup>extentof therestrictions</sup> placed on the social weights.<sup>Themildertherestrictions,themore</sup><sup>_robust_the. implied</sup> comparisons. This is the essence of<sup>thedominance approachtoevaluationbased on the</sup> comparison of distribution functions. When dominance fails, one can<sup>resortto</sup> abbreviated social welfare functions. Such functions usually translate<sup>the conceptof</sup> equally distributed equivalent (EDE) welfare. Gini-based evaluation functions incorporate a focal parameter that<sup>summarizes theunderlyingvalue judgments.</sup> This parameter can be used in sensitivity<sup>analysis.</sup> In the end, the selection of evaluative<sup>weights boilsdowntoprioritysetting</sup> among the different dimensions of<sup>thequalityof lifeandamong diverseindividualsor</sup> groups (within resource and institutional<sup>constraints). The maximinapproach illustrates</sup> the link between social weights<sup>andtheunderlying valuejudgments.</sup><sup>_It isimportant to_</sup> _note that in the selection of evaluative_<sup>_weights,one has a choice betweena technocratic_</sup> _and a democratic approach_ (Sen<sup>1999).</sup> The ethics of empowerment imply<sup>a</sup> determination of explicit evaluative<sup>weightsfroma participatoryapproach basedon open</sup> public debate involving all concerned.<sup>Thisseems to beoneof thesure waystodetect</sup> the value system that commands<sup>respectamongthe participantsin therelevantsocial</sup> arrangement.

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