Fuzzy Subset Theory in the Measurement of Poverty by Ronald Yacat

III

Number 34, Journal Volume of Philippine XIX, No, 1, Development First Semester 1992

III _S

FUZZY SUBSET THEORY IN THE MEASUREMENT OF POVERTY _ M.C.S. Bantilan,F.T. BantilanJr. and M.M. de Castro_

INTRODUCTION While the pervasiveimprecisioninthe measurementof incomeiswell known,researchers concerned with understandingthe phenomenonof povertyand the welfare of societycontinueto rely heavilyon measures of incomebased onhouseholdsurveys,Alternativemeasuresofwelfare, however,are continuouslybeingsoughtout. Forexample, expenditureis used to substitutefor income as a measure of welfare. The subjective assessmentof respondentsregardingtheirownlevelof welfarein the socalled perceptionsurveys hasalso gainedpopularity. Four importantcriticismsonthe traditionalmeasures,as emphasized by Cerioliand Zani (1990), are reiterated.They are: Incomedata generatedthroughinterviewresponsesare often times underestimated; Povertyisa multidimensional phenomenonsothat complementary indicatorsreflecting povertyshouldbe taken intoaccount; O There is a gradualtransitionfrom extreme povertyto wealth; El Income itselfis vague as a concept, What has not been exploredin traditionalmeasures of povertyis the extensive set of categoricalvariables,indicatingstandardof living,long availablefrom existingsurvey data. The difficultyof incorporatingthese * Paper presented at the "Consultative Workshop on the Methodologies for an Integrated Analysis of Poverty and Income Distribution and Their Uses for Policy Formulation," BEAMED. Innotech Building, Quezon City, 21-24 July 1992. Respectively, Associate Professors ofthe DepaJ-tmentof Economics, College of Economics and Management and Institute of Mathematical Sciences and Physics,University of the Philippines at Los Bathos;and Development Management Officerof the Development Planning and Research Project (DPRP), NPPB-National Economic and Development Authority (NEDA. Financial support was provided by the United Nations Development Program. The authors deeply appreciate the encouragement of Dr. Camilo Dagum and the excellent assistance provided by Joanne M. Pattugalan and Rommel Ruby, also of the DPRP, NPPS-NEDA. The authors are entirely responsible for the form and content of this paper,

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indicatorsin the traditionalframeworkhas precludedtheir use in deriving measuresof povertyand welfare. The theory of fuzzy subset providesa new approachto the use of traditionaleconomicvariables such as incomeor expenditureto derive new measuresof poverty. Moreover,the approachcan readilymake use of the extensive informationcontainedin the set of standard of living indicators. Instead of the all or nothingmembershipto the poor set, it allowsfor partialmembership,whereby one doesaway with the conventional definitionof the poverty line and takes into accountthe gradual transitionfrom povertyto the state of wealth. In the incorporationof the informationfromthe set of categoricalvariablesavailablefromhousehold surveys, it takes into account the multi-dimensionalityof the poverty phenomenon.The approach provides a formalism in the use of other variablesin paralleland incomplementationwith incomeand expenditure. The theory of fuzzy subset opensup new approachesin the analysis of incomedistribution,the refinementof the concept of poverty and its measures. THEORETICAL CONSIDERATIONS Definition of Fuzzy Subset Idea of Fuzzy Subset Much of our judgment depends on our ability to classify objects accordingto qualitiesor characteristicsof interest.Forexample, we may classifyobjectsaccordingto size:withrespectto the qualityorcharacteristicofbeingsmall.To be concrete,letus considerthe setofpositivewhole numbers-- 1,2, 3, 4, 5, 6, 7, 8, 9, 10. Let E denote the set, i.e. E = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} And let A denote the subset of E characterized by the quality of being small. According to ordinary set theory, each element of set E either belongs or does not belong to the subset A. This is expressed by the characteristic function uA(i), where UA(i) =1, if i belongs to A = 0, if i does not belong to A We may, for example, set the numbers 1, 2, 3, 4, 5 as belonging to the subset A; and the rest as not belonging to A. Ordinary set theory does not allow values between 0 and 1 for the characteristic function. One is forced

BANTILANeta/.: MEASUREMENT OF POVERTY

to set up a sharp/abruptdividinglinebetweenthequalityof beingsmalland notsmall.In ourexample,thesharpboundaryliesbetween5 and 6. In most cases,however,thissharpboundaryisnotrealistic;the ideaofbeingsmall is fuzzy. Furthermore,thetheory of fuzzy subsetalsoallowsvalues between 0 and 1. Thus, membershipto a subset is no longer an all or nothing proposition.This idea of partial membershipis more realistic-- in consonance withour valuationof whether objectsbelongto a subsetor not in accordance with the characteristicsby which we classifythem. Fuzzy subsettheory is an extensionof the ordinary set theory. Goingbackto ourexample,let usassumethat 1 isdefinitelysmall;and that 10 is definitelynot small.In the frameworkof partial membership,it makes sense to assign a gradationof values to the elements of set E. representingtheirdegreeof belongingnesstothesubsetA, decreasingas the numberincreases. In this case. the generalizedcharacteristicfunctionnowis calledthe membershipfunction,uA(i), to the subsetA, where u,(i)=x.

0<x<l

where x isthe value ofthe membershipfunctionforthe i'h element in the closedintervalbetween0 and 1.To completeourexample,we maysetthe values of the membershipfunctionuA(i) as follows: u,(1) = 1 u,(2) = 0.9

u,(3)= 0.6 UA(4) = 0.6 U,(5) = 0.4

u,(6) = 0.2 u_(7) = 0.1

=0 UA(9) = 0 U,(10) = 0

AS designed, the value of the membershipfunction decreases as the numberincreasesaway from1. This isto conformwiththe realityof fuzzy membershipto the set of beingsmall. Rigorous Definition due to Zadeh We nowtake uptherigorousdefinitionof a fuzzysubsetdueto Zadeh (1965). Let E be a set, where each element i is a full member, i.e. uE(i) is unity.Then a fuzzy subsetA of E is a set of orderedpairs {i, UA(i)}for all elements of E where uA(i) is the degree of membershipof i in A. If uA(i) takes itsvalues

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in a set X, calledthemembershipset,then uA(i) mapsthe set E tothe set X, or i takes its values in X through the functionuA(i). We write i_uA(i) ._X The functionuA(i) islikewisecalledthe membershipfunction.It isto be notedthatthe referentialset E isan ordinaryset, sinceeach elementin E has a full membership.The subsetA, however,is a fuzzy subset,since some of its membershave partialmembership.That iswhy the theoryis calledtheory of fuzzy subsetsand nottheoryof fuzzy sets. Notation We now define the symbolswe will use in the context of poverty studies.Letthefollowinglettersdenotethecorresponding entitiesindicated: E i C

C c AJ uA(i) x_j

the referentialset orthe setof individualsor householdsin the populationof interest; the ithelement of set E; a variableusedto measurelevels of welfare;C could be a continuousvariablesuchas income;ora categoricalvariable, either dichotomousor polytomous; the jr, variable in a set of k variables; the value of the jthvariable for the ithelement of set E; the subsetof E consistingof the poor; the membershipfunctionofthe elementito the poorsubsetA; the valueof the membershipfunctionuA(i) intheclosedinterval between0 and 1, for the jth variable and for the P element of set E.

Specification of the Membership Function Determination of Critical Limits In the study of poverty, the fuzzy subset approach requires the identificationof an upper and lowerbound,orspecifiedcharacteristicsto definethree subsetsof the population,namely: a. the Subsetof the populationwho are certainly poor accordingto society'sstandard of well-being; b. the subset of the population who are certainly non-pooraccording to society's standard of well-being; c. the subset of the population who exhibit only partial membership to the poor set.

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For a given variable C, let c1bethe lower limit and c2the upper limit that divide the population into three such subsets. In the case of income, cl may be represented by an appropriately determined subsistence income; while c2may be represented by a measure of central tendency or a proportion of it. Using the notation above, we define /JA_"f(c) where f(c) gives the grade of membership to the poor subset, in terms of the value of the variable C. With the limits chosen, we have uA= 1 when c<c_ uA=0 uA=f(c),

when c>c 2 0<f(c)<l

when c,<c<c

Given a specified membership function f(c), the critical limits, c1and c2, indicating the boundaries of the three subsets, are such that lim f(c) = 1 C _

C1+

and lim f(c) = 0 C--_C

limits c1and c2define the region of transition from the state of extreme poverty to wealth. The

Expressions of Some Membership Functions The specification of the membership function/.zA(i)or f(c) is a basic step in the application of the fuzzy subset approach. An appropriate membership function may be defined in terms of the values of continuous variables (e.g., income and consumption) or in terms of values of the set of categorical variables characterizing the household. In practical applications, the following membership functions with graphical illustrations shown in Fig. 1 to 7 may be considered.

10_

(1)

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f(c)

= 1, = 0,

c<c T c>c T 0

Fig. 1

(2)

f(c)

= 1, C<C I = C=-C, Cl<C<

1 [ _

C2"C 1 = 0,

C>__C 2

Fig. 2

(3)

f(c)

=1, c<_cr

= e'k<c'cT),c > cr and k > 0.

Fig. 3

f 4'

C>_r

and k> O.

cT Rg,4

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103 f

, c>cTand

k>0

1+k(c-c_),

I 0

Fig. 5

(6)

f(c)

== 01- CTC",, 0c <C > 1/k._a < 1/"_a 0

X Fig. 6

(7)

f(c)

= =

1 , O<c<c_ 1 1 _: C1+ C= - --Sin (x) 2 2 c=,,c_ 2

.\ 0

c1 Fig. 7

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In the equations above, f(c) defines the grade of membership to the poor set according to the value of c; cTrepresents a specified poverty threshold; and c1and c2 define the bounds of c separating the poor, the transition, and the nonpoor regions. We now briefly indicate the usefulness of the functions listed above in defining the grade of membership of economic units to the poor subset. A membership function of the type given in equation (1) is handy in representing dichotomous household categories such as employment status, absence of savings, sex of household head, availability of electricity in the house, and ownership of radio, refrigerator, television and other basic household consumer durables. In this case, uA= 1 identifies membership to the deprived subset of households; while PA= 0 corresponds to nonmembership to the deprived subset. Appropriate membership functions for continuous variables and other categorical (non-dichotomous) variables such as quality of housing, tenureship status, source of water, availability/quality of toilet and sanitation, and the level of education of household members may be handled by using functions of the types given by equations (2) to (7). Take for instance, the quality of housing materials symbolized by the variable C2. The values of C2 may be defined as follows: c2 = 1 for.salvaged/makeshift materials; = 2 for mixed but predominantly salvaged materials; = 3 for light materials (cogon, nipa, anahaw); -= 4 for mixed but predominantly light materials; = 5 for mixed but predominantly strong materials; = 6 for strong materials (galvanized iron, aluminum, tile, concrete, brick stone, asbestos). Using the membership function oftype (2), and defining c1 = 2 and c2 = 6, then f(c) = = = = = =

1 for salvaged/makeshift materials; 1 for mixed but predominantly salvaged materials; .6 for light materials (cogon, nipa, anahaw); .4 for mixed but predominantly light materials; .2 for mixed but predominantly strong materials; 0 for strong materials (galvanized iron, aluminum, tile, concrete, brick stone, asbestos).

As another example, consider the level of per capita income of the Filipino household and define c1and c2 as an appropriately measured subsistence income and meanpercapita incomeof the reference population,

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respectively.Forthe Philippinesin 1988, let ustake C 1 to representa level of subsistencethreshold,say P 3,016, and c2 equalto the mean income, say P 8,008. The correspondingmembershipfunction of type (7) is presentedinFig. 8. In effect,householdssufferingfrom severedeprivation or malnutritionare consideredcompletelypoor; those having incomes higher than the average would be non-poor;while those with income betweenc1andc2areassignedgraduallydecreasinggradesofmembership to the poorsubset. Variations of the functionsof the types (1) to (7) may of course be applied to determine appropriategrades of membership to the poor subset. Sampling Scheme Suggested by the Fuzzy Subset Approach If the purposeis to derivepovertymeasuresfora givenpopulation,a cost-effective sampling scheme is suggested by the fuzzy subset methodology.This scheme comprisestwo stages, namely: (1) a sample ofthe reference populationrepresentativeof allsectorsincludingboththe poor and the nonpoor is taken and questions relating only to basic structuralcharacteristicsare asked;and (2)thegroupof householdsat risk of beingpoorisderivedfrom thefirst-stagesampleand onwhichdetailed informationaboutincome,expenditureandcategoricalvariablesreflecting their level of welfare are obtained. This approachprovidesa less costlyway of obtainingthe required informationaboutthe sectorof interest,i.e. thosewho areat riskof being poor,and subsequentlyof computingthe grade of membershipof each member in the poorsubset.Those identifiedas definitelynon-poorin the first stage are automaticallyassigneda grade of membershipequal to zero. The fuzzy subsetapproachhas a clear advantage. The data requirementsof the fuzzy subsetapproachis less stringent thantherequirementsof themoreoftenusedincomedistributionmodelling approach.In the latterapproach,responsesonincome,expendituresand othervariables are requiredfrom all sectorsof the reference population. The undercoverageand underreporting occurring with respecttoresponses from the non-poorsector, particularlythe verywealthy households,have been identifiedassourcesof inaccuracyof measuresderivedfromincome distributionmodelingeffortswhich uses probabilitydensityfunctionsof incomeor otherobservablevariables.

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Figure 8 A SAMPLEMEMBERSHIPFUNCTION

f(c) I

P 3,016

P 8,008 Per Capita Income (In pesos)

EVALUATION OF INDIVIDUAL DEPRIVATION Alternative Data Sources Alternative data sources for weltare measures are available for evaluatingthe grade of deprivationof a unitin the referencepopulation. Among these are income, consumptionexpendituresand standard of livingindicators.Incomeand expendituredataare availablefromstandard surveysregularlyconductedbygovernmentsurveybureaus.Forexample, the Family income and Expenditure Surveys (FIES) conductedby the NationalStatisticsOffice for morethanthree decades nowprovidethese typesof information.The surveydata containsa richcollectionofvariables, bothcontinuousand categorical,whichindicatesthe household'slevel of welfare. Income, expendituresand savings are among the continuous variables available. Categoricaldata describingthe householdinclude type of dwellingunit,sourceof watersupply,presenceof electricityin the house,absence or presenceof savings,sanitationfacilities,ownershipof radioor ownershipof car, etc, The evaluation of the membership function through subjective judgmentsof the individualsor householdsthemselves (Van Praag 1971 orHagenaars1986) mayalsobe utilized,Data availablefrom theseries of

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perception surveys conducted by the Social Weather Station of the Philippines,for example, giveus the subjectivejudgmentsof individuals abouttheirownperceptionoftheirlevelofwelfare. Interviewees'responses to the question"Mahirap?o Hindimahirap?"and their answersregarding where they thinkthey are in the poor-notpoorrange are available. Measurement of Individual Deprivation A measure of individualdeprivationis the value of the membership function uA(i)•The evaluation of the membership function maps the economicuniti via the functionf(c) onto the closedinterval [0,1]. Let there be k variables C1, C2.... Ck that describethe set E of n individuals.Withthecriticallimits,c1and cz0appropriatelychosenfor each variable, define xii

= f(ci_ forc l<cii<c 2 = 1 for cii <__c 1 = 0 for Cij >__C2

where x.. isthe value of the membershipfunct=on - forthe = •thindividualand • for thejt_variable•A measureof individualdeprivationis then derivedby computingthe weightedaverage acrossthe k variables,i.e., k

(8)

PA(i)= j=l _ x_ W_/iT.lW j.=

where wjis a weight appropriatelychosenfor each variable C, reflecting itslevel of,mportance.If equalweightsareassumed,thenthemembership functionreducesto •

•

p,(i) = 1/k _ x_ j=l

Examples This section presents applicationsof the fuzzy subset approach to sample data collected in the Philippines in 1988 through the FIES. Responses from a random sample consistingof 18,922 households, representinga total of 10,533,927 households,were obtained through interviews.This data set is utilizedin the followingexamples. Using Categorical Variables Seven categoricalvariablesare chosenforthis exampleto describe howvariousaspectsof the level of welfare of householdsare taken into

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account in parallel. Philippine data from the FIES survey of 1988 is used. Another paper entitled "Application of Fuzzy Subset Theory in the Measurement of Poverty in the Philippines" considers a more comprehensive set of variables from the FIES data set. The variables chosen for this example includes: C_ C2 C3 C4 Cs Cs C7

= = = = = = =

Sanitation or type of toilet facilities (SANITATION); Walling materials of dwelling unit (WALL); Presence of electricity in the house (ELEC); Source of Water Supply (WATER); Absence of Savings (ABSAVE); Ownership of Radio (RADIO); Ownership of Car (CAR).

Table 1 defines the codes describing each variable including the specification of the limits, c1 and c2. Application of equation (2) yields estimates of the extent of deprivation for each household with respect to each variable j, j = 1..... 7. Weights, denoted by vj, equal to the logarithm of the inverse rate of households showing the corresponding symptom of deprivation is applied to each variable, that is, vj =

log [nj/Mt], j = 1..... n n

Mi =

_. x,j i=1

where MI refers to the sum over all households of the degree of deprivation with respect to the attribute or variable j; and nj is the total number of households for which the variable j is observed. In this example, all seven variables are applicable among the population of households. Thus, nj is fixed at 10,533 928. Table 2 illustrates the frequency of occurrence (M_of these indicators for possible membership to the poor subset. Presented in the last column of Table 2 are the normalized weights, denoted by wj, computed by taking the relative percentage ofvjwith respect to its total, i.e., VI Wi =

j=l It is observed that the weighting system developed attaches greatest weight to basic necessities compared to the other attributes (variables) not widely available to the members of the reference population. In this

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Table 1 CODE DESCRIPTIONS OF VARIABLES SELECTED FROM THE FIES SURVEY DATA, 1988 Variables C1= SANITATION

C2= WALL

Description Kind of toilet Facilities None Others (pail system) Open pit Close pit Water sealed Walling materials of dwelling unit Salvaged/makeshift materials Mixed but predomi* nantly salvaged materials Light materials Mixed but predominantly light materials Mixed but predominantly strong materials Strong materials

C3= ELEC

Limits

1 1 2 3 5

c1=1 c2=5

c1=1

1 2

c==3

3 3

Presence of electricity in the house No Yes

C4= WATER

Codes

1 2

Source of water supply Spring, river, stream, etc. 1 Dug well 1 Rain 1 Peddler 1 Tubed, piped well, owned by others 3 Faucet, owned by others communitywater system 4

c1 c2

c,=1 c2=6

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Table 1 (continued) Variables

Description

Codes

Limits

Tubed,pipedwell, own use

C5= ABSAVE

C6= RADIO

C7= CAR

Faucetinsidethehouse/ yard,community watersystem

Absenceof savings Savings<= 0 Savings> 0

1 2

c1=1 c2=2

Ownershipof radio No Yes

1 2

c_=1 c2=2

Ownershipofcar No Yes

1 2

c1=1 c2=2

example, the variable C2 (type of materials of dwelling unit) is assigned greater weight than Cz (ownership of car). Applying equation (8) to each household, we obtain an estimate of the amount of individual deprivation. A summary ofthe results, shown in Table 3, indicates that the membership function ranges from 0 to 1, It also indicates that about 47 percent of the population have rates of deprivation higher than the average. Fuzzy Graph for a Two-Variable Case The fuzzy subset concept is illustrated for the two-variable case. Two variables are considered, namely: (1) type of materials used for the dwelling unit (HOUSING); and (2) source of water (WATER). Table 4 contains the weighted averages of the probability of membership in the poor set considering the two variables chosen. The gradation of shades shown in the accompanying fuzzy graph (Fig. 9) approximates the possible varying degrees of membership to the poorset ofthe reference population. Using Single Continuous Variables To evaluate the membership function to the poor set, we use equation (2) for the income and expenditure data. Fixing c_ equal to P 3,015.80 (subsistence incomefor 1988 determined by the Technical Working Group

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Table 2 FREQUENCY OF OCCURRENCE OF POVERTY SYMPTOMS CORRESPONDING TO SEVEN SELECTED INDICATORS OF DEPRIVATION AMONG FILIPINO HOUSEHOLDS

Frequency of Occurrencel(MJ) 1. 2. 3. 4, 5, 6. 7,

WALL SANITATION ELEC WATER ABSAVE RADIO CAR

% of Total Population

2,562,857 3,754,222 4,255,416 4,986,613 2,904,625 3,146,277 9,945,847

,24 ,35 .40 .47 .28 ,30 .94 Total

.61387 .44807 .39672 .32478 .55950 ,52479 .02495 2,89268

,2122 .1549 ,1372 .1123 .1934 ,1814 ,0086 1.0

1ThefrequencydistTibutJon isdrawnfromasampleof18,922households.Weightsforeach samplingunit,developedforthe FIES survey,is applied.Thus, the totalpopulationof Filipino householdsin 1988 (i,e,, n = 10,533,928)is reflectedin the abovetable,

Table 3 FREQUENCY OF THE GRADE OF DEPRIVATION BASED ON SELECTED CATEGORICAL VARIABLES PHILIPPINES, 1988 pA(I)

.875 .75 .625 .5 .375 .25 .125 0

<= <= <= <= <= <= <= <=

/JA(i) = pA(i)<= pA(i) <= /JA(i)<= /JA(i) <= /JA(i)<= /JA(i) <= pA(i)<= /JA(i) <= pA(i) =

Frequency

1 1 .875 ,75 .625 .5 .375 .25 .125 0

4466, 112621,4 300123,1 1174788,2 1196477.4 1607043,8 1583621.5 2099691.5 2260242.8 194851.1

Cumulative Percentage

0.0 1.1 2,8 11,2 11.4 15,3 15,0 19,9 21.5 1.8

0.0 1,1 4,0 15,1 26,5 41.7 56.8 76.7 98,2 100.0

Quantiles: Q,速 Q_ Q_ Qgo

= = = =

1 0.864305 0.715514 0.654975

Q_s = 0.485827 Q= = 0,276537 Q2s = 0,113519

Mean = .332172 Median= ,276537 Mode = .008534

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Table 5 FREQUENCY DISTRIBUTION OF GRADES OF MEMBERSHIP IN THE POOR SET AMONG FILIPINO FAMILIES BASED ON PER CAPITA INCOME, 1988 Cumulative pA(i)

.875 .75 .625 .5 .375 •25 .125 0

<= <= <= <= <= <= <= <=

/J,,(i) /JA(i) /.JA(i) /.JA(i) /JA(i) /.JA(i) /JA(i) /J,,(i) /JA(i) /JA(i)

= 1 <= 1 <= .875 <= ,75 <=. 625 <=.5 <= .375 <=,25 <= .125 = 0

Frequency

2109484.5 1026308.8 931581,2 803400.9 696591.6 564130.2 486930.2 421329.7 38883.5 3105286,2

20.0 9.8 8.8 7.6 6.6 5.4 4.6 4.0 3.7 29.5

Percentage 20.0 29.8 38.6 46.2 52.8 58.2 62,8 66.8 70.5 100.0

Mean= .501263 Median= ,502428

Table 6 FREQUENCY DISTRIBUTION OF GRADES OF MEMBERSHIP IN THE POOR SET AMONG FILIPINO FAMILIES BASED ON PER CAPITA EXPENDITURES, 1988 Cumulative /JA(i)

.875 .75 ,625 .5 .375 .25 .125 0

<= <= <= <= <= <= <= <=

/JA(i) /JA(i) /.J,,(i) /JA(i) pA(i) pA(i) /JA(i) /JA(i) /JA(i) pA(i)

Mean = .502668 Median= .487672

Frequency = 1 <= 1 <= .875 <= .75 <= .625 <= .5 <= .375 <=.25 <= .125 = 0

2531767.2 896309.3 776691.4 695332.5 603938.9 528292.1 427105.0 415760.8 352534.1 3306195.5

% 24.0 8.5 7.4 6,6 5.7 5.0 4,1 3.9 3.3 31.4

Percentage 24.0 32.5 39.9 46.5 52.3 57.3 61,3 65.3 68.6 100.0

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on Poverty of the National Statistical Coordination Board (NSCB) of the Philippinesand czequalto P 8,008.40 (mean per capitaincome),we have pA(i) = 1, 0<C<C 1 = 0,

C>C 2

For per capita incomes between P 3,015.80 and P 8,008.40, the membershipfunctiontakes on valuesin the [0,1] intervalwhere HA(i) =

1, c<c 1

{cz-c}/{c 2-cl},

cl<c<c

C >__c2

Table 5 summarizesthe frequencydistributionof PA(i)based on per capita incomewhile Table 6 containsthe results based on per capita expenditures.Fig. 10 is a fuzzy graph illustratingthe distributionof continuousvariables,e,.g.,per capita incomeand per capita expenditure.As shown,the gradualtransitionfrom the stateof beingdefinitelypoorto the state of being definitely not poor is approximatedby the gradation of shades from pure blackto pure white. The average level of deprivation derivedfrombothdata setsis50 percent.The next sectionwillshowhow the above quantificationof individualdeprivationmaybe utilizedto derive a measure of the extent of povertyof the totalpopulation, POVERTY MEASURES: EVALUATION OF POPULATION DEPRIVATION Formulation of Poverty Measures A derivation of a measure of the extent of poverty of the total population is as follows: Let n be the number of individuals in a population. Then, E = {1,2 ..... n}. A generalized formulation for measures of poverty for this population can be expressed as an aggregation of individual deprivation levels where the sum over all members of the population is taken, i.e., the cardinality of the fuzzy subset A of the poor n

O = IAI = _ /JA(i). 1=i

Fig. 10 FUZZY GRAPH FOR PER CAPITA INCOME PHILIPPINES,

1988

PDF

z Z

ITI (n C '31 n".l m Z .-I "o

o m "n

SUBSISTENCE THRESHOLD

POVERTY TRESHOLD

MEAN INCOME (FIES)

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The cardinalityIAI is a naturalextensionof the traditionalconceptof set cardinality(Duboisand Prade, 1980). In otherwords,the overallextent of the deprivationof populationof size n may be measured by taking the totality ofthe levels of deprivationof all units in the population. A coefficientreflectingsociety'saversionto the state of povertymay be desiredand incorporatedthrougha coefficientof aversion, say _, to obtain I1

D = Z [PA(i)] _ I=i Subsequently, a generalized expression ofthemeasureofpoverty incidence isobtained by taking theaverageoftheindividual deprivation levels, i.e., 13

(9)

P : 1/n _. [PA (i)](' i=1

The measure P indicates the proportion of the population belonging, in a fuzzy sense, to the poor subset, where P ranges from 0to 1. In the extreme case where allmembers belongto the non-poor subset with certainty, then pA(i)= 0 for all i = 1..... n; and P = 0 (case of absence of poverty). On the other hand, if all HA(i)= 1 for all i, then P = 1 (case of extreme poverty for all). Substitutingequation (8) for HA(i), n

(10)

P = 1/n _ [_ i=1 j=l

X_l Wj

]_' k wj j=l

where k denotes the number of variables available to describe the level of individual welfare for each household i. The generalized formulation, equation (9), takes on a specific form with the choice of a particular expression of the function HA(i). Let US consider the case K = 1. For instance, take equation (2) so that equation (9) becomes q

(10)

P = 1/n T. [ i=1

C2 - C_

]", cl<c,<_c 2 c2- cl

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where HA(i)= 0 when c,> c2;HA(i)= 1 when c,< %; and q is the number of householdsat riskof povertyorwhere c,< c2.With this choicewe derive the followingmeasures of poverty. Case 1.

Let _ = 1 to obtain q

(11)

cz - CI

P = 1/n _. /J (i) = 1/n Y. ( _ ),c,<c,<c i=1 i=1 c2- cl

The measure givenby equation11 reducesto the traditional headcount ratio when c2 is evaluated at some specified thresholdlevel.The traditionalratio is one where each individualisidentifiedaspoorwithcertainty(/J= 1) ornonpoorwith certainty 0J = 0) depending on its location relative to an identifiedpovertythreshold.Inthe caseindicatedby equation (11), the limitof/JA(i)is equalto 1 asc_approaches%;and the limitof/JAis equalto 0 as c_approachescz. Table 7 illustrates the observationthat asthe two limits,% and c2,approachthe referencethresholdlevel,saycT,thenthetraditionalheadcount ratio is obtained. Case 2. For ct= 2, then q

P = 1/n ,_ i=1

C2- Ci

(

)', c,<c,<c

c2- c_

where/JA(i)= 0 when c=> c2;HA(i)= 1 when c_< %. For c_= 2 and the lowerboundis equal to 0, then q

C2- Ci

P = 1In _: (_)' i=1 cz The above expression is equivalent to another common measure of poverty developed by Forster, Greer and Thornbecke (FGT). Several otherspecialcases of equation(9) may be specified by takingotherfunctionalforms for/JA(i) andvariouslevels of the coefficientof aversioncÂ˘.

116

JOURNAL OF PHILIPPINE DEVELOPMENT Table 7 ESTIMATES OF POVERTY INCIDENCE BASED ON THE FUZZY SUBSET APPROACH CONSIDERING +/-,STANDARD DEVIATION BANDS ABOUT THE POVERTY THRESHOLD HIGH

5961.125 (+.0001)

PT 5888,874

6231.256 (+.001)

7476.292 (+.oo5)

8098.808 (+.007)

8876.954 (+.0095)

9032.584 (+.01)

0.53939

(-.oooi) 5608.741 (-0.001)

0.53929

4363.708

0.53094

(-0.005) 3741.191 (-0.007)

0.52312

2963.045

0.51027

(-,oo95) 2807.416

0,50737

(-O.Ol) ST = 3015.8 PT = 5920 Mean (Dagum)= 9257 SD = 311258.4 HR = 54.1

Some []

two

Results Case

Using where

Philippine

FIES Data

k = 1: Fuzzy Subset Measures of Poverty Based on Income and Expenditure, 1988,

This section presents the results of poverty measurement continuous variables, namely income and expenditure.

based on Table 8

presents results based on per capita income and expenditure data. The values contained in these tables are computed using equation 1 1, i.e,,

BANTILAN et al.: MEASUREMENT OF POVERTY

119

Table 8 ESTIMATES OF POVERTY INCIDENCE BASED ON THE FUZZY SUBSET MEASUREMENT USING FIES PER CAPITA INCOME AND EXPENDITURE DATA, 1988 c= = Upper Limit INCOME:

Statistics

EXPENDITURE

1988

Mean Per Capita Income (P 8603)

Mean PCE

(P 9516)

Mean Per Capita Expenditure (P 6554)

.53488

.48558

.60466

.54204

.48566

.60472

.57152

.46579

.62535

.60614

.55016

.65666

.64124

.58930

.68801

(P 9516)

Â˘1 = Lower Limit Subsistence .50126 Level (TWG) (P 3016) Mode .50875 (Dagum) (P 3156/a, P 3017/b) 3rd Decile .53961 (Dagum) (P 3763/a, (P 3400/b) 4th Decile .57594 (Dagum) (P 4559/a, P 4045/b) Median (Dagum) .61288 (P 5490/a, P 6407/b) a/Value of income..... b/Value of expenditure.

P = 1/n

q _. HA(i) i=1

= 1/n

q C2- CI _. ( -) i=1 c2 - c 1

As shown, alternative critical bounds (c 1and c2) are considered in the analysis. The lower limit, cl, is given by 5 levels, namely: (a) subsistence threshold determined by the TVVG of NSCB; (b) the mode of the income distribution; (c) the third decile; (d) fourth decile; and (e) the median based on the estimated Dagum model (Bantilan et al, 1991). The upper bounds took either the mean per capita income (based on the FIES data) and the

120

JOURNAL OF PHILIPPINE.DEVELOPMENT

Table 9 POVERTYMEASURESBASEDONA SET OF SELECTED CATEGORICALVARIABLES,AND PER CAPITAINCOME AND EXPENDITURESDATA=PHILIPPINES,1988 Approach Data

I. II. III.

Categorical Variables (Example1) Per Capita Income Per Capita Expenditure

FST Approach

Modelling Approach

Published Measures (NSO)

.5013

.541

.552

.5027

.621

.650

.3322

mean per capita personal consumption expenditure based on published National Income Account estimates. These limits are given in pesos and indicated in parenthesis. Taking the subsistence threshold determined by the TVVGof NSCB as lower limit (cl) and the mean per capita income as upper limit (c2),poverty incidence is measured at 50.1 percent. The absolute magnitudes of the poverty incidence rates based on per capita expenditure is estimated at 48.6 percent. Applying alternative lower limits i.e., mode, 3rd, 4th decile and the median of the income distribution) gave poverty incidence rates of 51-61 percent which is consistently a little higher than the estimates based on corresponding expenditure data (i.e., 49-59 percent) in 1988. An interesting observation is the consistency of the results irrespective of the limits used. cI Case where k = _ Fuzzy Subset Measures Based on a Set of Selected Categorical Variables: Philippines Table 9 presents a measure of population deprivation in 1988 based on the Fuzzy Subset Theory (FST) Approach using the seven selected attributes of the previous example. A summary is presented comparing the results derived with those obtained using income and expenditure data. A comparison with other traditional measures is also provided. The higher level of estimates of poverty incidence drawn from the use of income data compared to those obtained by utilizing categorical variables may be indicative of possible underreporting of income in the FIES Survey.

BANTILANet aL: MEASUREMENT OF POVERTY

121

SUMMARY AND CONCLUSION The applicationof the conceptof fuzzy subsetsin the measurement of povertyprovidesgreatpotentialin improvingthe currentmethodologies in measuringlevels of individualdeprivationaswell as the overalllevelof deprivationof society. Thetraditionalmeasures,e.g.,theheadcountratio,requiresa specified cutoff income (e.g., poverty threshold) to separate the poor from the nonpoor, The improvement over current procedures relates to three aspects. First, the gradualtransitionfrom extreme povertyto wealth is accountedfor.The measurementof individualdeprivationtakes different grades of membershipto the poor sector depending on the observed attributeslike incomeor expenditures.Second, the use of fuzzy subsets and correspondingmembershipfunctionsfora set of attributesallowsus to synthesizealternativefuzzy relationsand takes intoaccounttheoverall levelofwelfareofthe household.The multidimentionality natureofpoverty which incomeand expenditurealone may not capture is accounted for. Third, thetraditionalincome-basedindicesmayresultin incorrectfindings as respondentsusuallyprovideimpreciseinformationabouttheirincome, An evaluationof the level of deprivationof the householdand societyis enhanced by consideringotherobservedcharacteristicswhichmay more accuratelydescribethe households'state of welfare.

122

JOURNALOF PHILIPPINEDEVELOPMENT REFERENCES

Cerioli, A. and Zani, S. "A Fuzzy Approach to the Measurement of Poverty." In Income and Wealth Distribution, Inequality and Poverty (Studies in ContemporaryEconomics). C. Dagum and M. Zenga (eds) Berlin, Heidelberg: Sprlnger-Verlag, 1990. Dubois, D. and Prade, H. FuzzySets and Systems: Theory and Applications. New York: Academic Press, 1980. French, S. " Fuzzy Decision Analysis: Some Criticism." in Fuzzy Sets and Decision Analysis H.J.L. Zimmermann, L.A. Zadeh and B.R. Gaines (eds). Amsterdam: North-Holland, 1984. Gaines, B.R. "Foundations of Fuzzy Reasoning" In Fuzzy Automata and DeclsiOt7Processes eds. M.M. Gupta, G.N. Saridis and B.R. Gaines (eds.) New York: North-Holland, 1977. C-._les,R. "The Concept of Grade of Membership" Fuzzy Sets and Systems 25 (1988)-:297-323. Hage_0a__ars, A.J.M. The Perception of Poverty. Amsterdam: North-Holland, 1986. Hisdal, Ei ".AreGrades of Membership Probabilities?" Fuzzy Sets and Systems 25 (1988): 325-348. Ka_ufman,A. Introduction to the Theory of Fuzzy Subsetsl Vol. 1. Fundamental Theoretical Elements; Vol. 2. Application (Information and Decision Processes in Humans and Machines). New York: Academic Press, 1975. Van Praag, B.M.S. "The Welfare Function of Income in Belgium: An Empirical Investigation." European Economic Review 2 (1971): 337-369. Zadeh, L.A. "Fuzzy Sets, Information and Control" Journal of Mathematical Analysis and Applications 8 (1965): 338-353. . "Probability Measures of Fuzzy Events" Journal of Mathematical Analysis and Applications 23 (1968): 421-427.