Poverty- From Early Views to Fuzzy

Approach Shreyo Mallik

Abstract

It has been a long term practice to define & explain poverty in many ways

right from the primitive times. This dates back to the ages of Aristotle, Mill,

Bentham, Hume when poverty came to be defined in a way that was much

formal than how it was thought to be. However, poverty has had always

been a vague concept due to of it being the degree of deprivation of social

& economic necessities. Marx had spoken of the deprivation of the working

class. In "Das Capital", Marx sympathizes for the "working class" &

condemns "capitalism" for their suffering. Marx speaks of "poverty" or

deprivation within capitalism, & strictly demands the eradication of

capitalism itself. In later times, however, came socialism, & much later

came the "mixed economy"- a mixture of capitalism & socialism. In such a

structure that evolved, a new class called the "middle class" came into

being. Prof. A. K. Sen calls this middle class the "non-rich non-poor limbo".

While in the Aristotelian era, Aristotle had considered only two sections in

the society- the rich & the poor, &, that the non-rich included only the poor,

Sen (1976) considers within the non-rich section both the poor & the non-

rich non-poor segment of the population. Prof. Sen mathematically

analyzes this problem- though in the similar lines of Aristotelian logic. He

uses Gini's coefficient in the poverty measure that he suggested. Giorgi &

Crescenzi (2001) modifies it replacing Gini's coefficient by Bonferroni's

index. Further modifications were provided by Chakravarty (2006). Sen's

thesis brings the focus on the non-rich non-poor limbo. His measure shows

empirically that it is this section of the society that matters the most in the

distribution of wealth, given a socio-economic-politico structure. The

burgeoning "non-rich non-poor limbo" ceases the downward movement of

the macroeconomic quantities trickling down through the classes. This gets

coupled up with a major political stability in an economy where the "non-

rich non-poor limbo" sustains with vigorous propensity. Such political

stability in turn stops poverty from being eradicated. This paper aims at

analyzing Sen's works, his motives, and his outlook of how poverty can be

handled in a quantitative yet analytical way. The paper computes estimates

of poverty using various standard indices and concludes by estimating

poverty using the relatively new concept of fuzzy approach. The data set

has been obtained from the National Sample Survey Organization (NSSO).

Data for the state of West Bengal for the period 2004-5 have been used in

this paper.

1. Introduction

Poverty is the degree of deprivation that any community or society at large faces.

However, the concept of poverty is shrouded in vagueness. In the 21st century,

the standard meaning of poverty stands as ‘the state of material & social

deprivation’.

From the 18th century onwards, there has been a radical shift in the history of

poverty from being an ideal sate to an execrable condition that societies seek to

ameliorate. Poverty is one of the real threats to the social order & barriers the

path of the societal development. The course of development of the notion of

poverty followed the following path:

Contemporary debate: Hayek points out that the rich with higher income

differential can pay higher tax & in turn benefit the poor. Smith speaks of

an ‘Egalitarian society’ (every individual having the sustainable minimum).

Socialist Townshend condemns the possession of excessive wealth.

The history of poverty attained a new dimension with the introduction of

the Marxian thought in it. The basis of Marx’s thought on poverty is rooted

deeply into Aristotle’s thought on society & economy. So, to have a deeper

insight on poverty, we shall throw some light briefly on Aristotle & Marx’s

conceptions on social & economic thought.

Aristotle attributes ‘techne’ or technical reason to production. Marx also

does the same & following Aristotle, he too removes moral consideration

from production. Both evaluate other economic institutions such as wealth

acquisition, social division of labor, etc. more politically.

Basic Needs Approach (BNA):

It was developed by Paul Streeten et al (1981) & Frances Stewart (1985).

The definition of basic needs changed with the progress of the human

civilization. For instance, having an x-ray clinic in certain proximity of an

urban household has today become a necessity, but it was not so a few

decades before. Modern theorists include that the deprived should be

provided with the needs to survive.

Capability Approach (CA):

It was developed by Prof. A.K. Sen. It dates back to the Lecture at Stanford

University on ‘Equality of What?’ It gradually developed through

publications in various articles.

Its inspirations are rooted in Aristotle (wealth is not evidently the good we

are seeking for it is merely useful for the sake of something else), Classical

Political Economy, Smith ,1776 (economic growth & expansion of goods &

services are necessary for human development), Marx(1844), Rawls ‘Theory

of Justice’ ,1971 (emphasis on ‘self-respect & accessibility to primary

goods), Issiah Berlin’s (1958) classic essay ‘Two Concepts of Liberty’ (fiercely

attacks the positive concepts of freedom).

CA – A Brief Overview

Sen is concerned with the rights & freedoms of individuals as well as

communities at large. In short, CA camouflages BNA. CA is an alternative

approach to standard poverty analysis. It deals at large with poverty,

inequality & human development. It has similarity with Smith (‘Analysis of

Necessities’), Aristotelian ‘eudaimonia’ (human flourishing) & also with

Marx (freedom & rights).

Conceptual Foundation of CA:

Commodity → capability (to function) → function(ing) → utility (e.g.

Happiness)

Functioning is the use a person can make of the commodities at his/her

command. Capability is the ability to achieve given functioning. Functioning n-

tuple is the combination of doings/beings constituting the state of person’s

life, eg., live long, escape morbidity, read & write, etc. Capability set is the set

of above functioning vector; neither the dimension (n) of the vector nor its

weights are fixed, they are subjective.

Sen challenges the utility approach. He doesn’t distinguish between different

pleasure & pain or different kinds of desires. He points out that

happiness/desire fulfillment is only one aspect of human existence. There are

many other things of intrinsic value (notably freedom & rights) being neglected

by Welfare Approach. For instance, the Post Famine Health Survey in India

revealed disparity between observed health of widows & their subjective

impressions of their physical state.

Criticism of CA (Williams, Nassbaum, Qizilbash):

i. Sen insists valuable intrinsic capabilities.

ii. He makes inter-personal comparisons of well-being in the presence of

potential disagreements about the valuation of capabilities including

relative weights to be assigned to capabilities.

iii. Sen’s concept of ‘deliberate democracy’ is too idealistic and lacks

proper applicability.

Sen, on the contrary, defends himself by saying that the intersection of different

peoples’ ranking is typically quite large. He suggests a range of methods including

‘dominance ranking’ & ‘intersection approach’ for extending incomplete ordering.

Sen’s CA thus gives rise to Multi-dimensional Poverty Index (MDPI), where we

may have various capabilities such as bodily health, bodily integrity; senses,

imagination & thought, emotions, practical reason, affiliation, other species, play,

practical & material control over one’s environment, etc. The traditional

estimation of poverty is based on the assumption that a person’s well-being can

be represented by the functioning ‘command over resources’ (income).

Nowadays many authors recognize that poverty is a complex phenomenon that

cannot be reduced to the sole monetary dimension and this leads to the need for

a multidimensional approach that consists in extending the analysis to a variety of

non-monetary indicators of living conditions. However, although having an

income is not itself a functioning, but many functionings, like being well-

nourished or having a decent home, depend crucially on it. As observed by Anand

and Sen, ‘in an indirect way – both as a proxy and as a causal antecedent – the

income of a person can tell us a good deal about her ability to do things that she

has reason to value. As a crucial means to a number of important ends, income

has, thus, much significance even in the accounting of human development’

(2000).

It is, therefore, important to look at poverty from the point of view of income as

‘capability’. We would be dealing with the uni-dimensional poverty index with the

only capability as income and look at poverty through Sen’s Index (Sen, 1976) and

its modifications.

The format of presentation is as follows: Section 2 describes the classical

approach to poverty measures; Section 3 explains the fuzzy logic approach;

Section 4 presents the data and results and finally, Section 5 concludes.

2. Classical Approach to Poverty Measures

Let X be the distribution of a monetary variable (INCOME) concerning a

set of n units. We indicate by xi the income of the ith unit i = 1(1)n, and

we assume that 0 ≤ x1 ≤ x2 ≤ · · · ≤ xn. Our primary interest relies in

assessing whether a given unit can be considered poor or not. The classical

approach to poverty mainly consists of introducing a poverty line, say xP,

and concluding that the generic ith unit is poor when xi < xP (and non-poor

or rich otherwise).

Some Basic Notations

The membership function ( ) is given by:

( )

Let

be the number of the poor, where U denotes the indicator function.

With regard to the poor units, the poverty-gap associated to the ith unit

is

gi = xP − xi , ( )

Thus, gi is the (positive) difference between the poverty threshold xP and

the income xi , ( ) .

Let

be the mean of the poverty-gaps of the poor.

The income mean over the p poor units is:

& the mean income over the i poorest units ( ) is:

Sen’s Index using Gini’s Coefficient

In order to determine the Sen poverty measure, we can now recall the

following three indices:

which are the poverty-gap ratio, the Gini coefficient among the poor and the head

count ratio, respectively.

Sen’s Index for the Classical Model

Sen suggests the following poverty measure for the classical model:

where the latter holds for large p.

S is equal to 0 when there are no poor and equal to 1 when all the units

have no income (x1 = x2· · · = xn = 0).

Modification of Sen’s Index

A modification of S has been proposed by Giorgi and Crescenzi (2001), who

suggest using the Bonferroni index in place of the Gini index. The

Bonferroni index B is given by

&, among the poor as

∑

∑

where m denotes the mean income computed over the entire distribution.

Utility of Bonferroni’s Index over Gini’s Coefficient

i. The need for the Bonferroni index instead of the Gini is that the former is

more sensitive than the latter to the poorest units belonging to the income

distribution.

ii. If an amount of income moves from a donor to a recipient, the variation of

the Gini ratio depends only on the distance between their ranks, whereas

the variation of the Bonferroni index also depends on their exact positions in

the income ordering.

Modified Sen’s Index using Bonferroni’s index

In an axiomatic framework, the Giorgi and Crescenzi (2001) measure is

given by

where the latter holds when p is large.

Limitations of Classical Approach

Suppose to deal with three units such that x1 = xP − ε, x2 = xP + ε (with ε > 0)

and x3 >> xP. Following the classical approach, we can conclude that the

first unit is poor, whereas the remaining two are rich. However, this

contradicts the common thinking. Let us start with units 1 and 2. The

former unit is considered poor and the latter rich. However, both units can

be classified as poor. In fact, also unit 2 is approximately poor because

her/his income is very close to the poverty line, even if it is slightly higher.

Also the classification of units 2 and 3 is in contrast with the human

common-sense reasoning. Both units are classified as rich. Nonetheless, it

seems to be more realistic to conclude that the former is rich to some very

limited extent and the latter is definitely rich.

3. Fuzzy Logic Approach

In the fuzzy logic approach, we assign to every unit a degree of poverty

ranging from 0 to 1 according to the corresponding income. This is done by

introducing the membership function. In particular, if indicates the

attribute ‘poor’ (the symbol ‘∼’ denotes that the fuzzy logic approach is

adopted), the membership function ( ):R+→*0,1+ allows us to express

to what extent the ith unit is poor. This is done by specifying the degree of

poverty that makes xi a member of .

Membership Function

The membership function can be defined as:

where f(xi) is a decreasing function from [xP, xR) to [0,1] such that f(xP) = 1, f (xi )

< 1 xi > xP , f (xi ) > 0 xi < xR . Further,

An Example of Membership Function

A possible choice for f is:

where β is a positive parameter tuning the decreasing trend of the membership

function.

Behavior of Membership Function for different values of 𝛽

For β = 1 such a trend is linear.

For β < 1, the decreasing trend is more rapid with respect to the linear case.

The opposite holds when β > 1.

Basic Notations & Formulae in Fuzzy Logic Approach

In place of p in the classical model, in the fuzzy logic approach, we have

It is easy to see that .

Let r & the numbers of the rich & the non-rich respectively given by

It is easy to see that

is the number of units belonging to the non-poor non- rich limbo.

Generalized Extensions of the Notations of the Classical Approach in the Fuzzy Logic Model

Instead of the number of poor, we would now be interested with the

number of non-rich. So, we compute the non-richness gap

& proceed in similar lines as in the classical model.

We define the non-richness gap ratio as:

The mean of the non-rich units is:

The mean up to the ith non-richer unit ( ) is:

Gini coefficient among the non-rich is given by:

Bonferroni index among the non-rich is given by:

∑

∑

The generalized head-count ratio is given by:

The fuzzy logic extensions of Sen’s index using Gini coefficient & Bonferroni

index are respectively given by:

Sen (1976) and Giorgi and Crescenzi (2001) prove, respectively, that Sen’s

index using Gini coefficient & Bonferroni index can be expressed as a

normalized weighted sum of the poverty-gaps gi as:

Where ( ) is a normalizing term & ( ) is a non-negative

weight associated with the ith unit.

Axiom (Ordinal Rank Weights)

The weights being associated with the non-richness-gaps are:

Theorem 1

For a large number of the non-rich, the only measure satisfying the axioms

of Ordinal Rank Weights and Normalized Poverty Value is given by:

Theorem 2

For a large number of the non-rich, the only measure satisfying the axioms

of Ordinal Rank Weights and Normalized Poverty Value is given by:

4. Data and Results

Data on Total consumer expenditure (a proxy for income data) separately

for urban & rural West Bengal for NSS 61st round (2004−05) have been

used for the analysis. The data for each of urban & rural are given

separately in terms of per capita expenditure (PCE) for 30 & 365 days

respectively. The analysis has been done for both the classical & fuzzy logic

approaches. We have computed Sen’s indices for both. In both the classical

& fuzzy logic approaches, we have taken the state-defined poverty lines for

rural (Rs. 382.82) & urban (Rs. 449.32) as the poverty lines for the rural &

urban West Bengal respectively. In case of the fuzzy logic approach, we

have taken the 2nd quantile (Q2) & 3rd quantile (Q3) as the richness lines &

hence computed & for 𝛽 respectively. In the case of the

fuzzy logic approach, we have computed the mean, standard deviation (SD)

& finally the coefficient of variation (CV) for each of the cases concerned.

Based upon these, we shall interpret the results statistically.

Graphs

Computations

Table(1) showing calculations for Sen's Indices for the Classical Approach

Rural

State-defined PL 382.82

Rural(30)

SG 0.1725317

SB 0.1745187

Rural(365)

SG 0.1437028

SB 0.1407584

Urban

State-defined PL 449.32

Urban(30)

SG 0.1285809

SB 0.1286183

Urban(365)

SG 0.1057826

SB 0.1070784

NB :- PL : Poverty Line

Tables(2-5) showing calculations for Sen's Indices for the Fuzzy Logic Approach& their Statistics

Rural(30) Indices SG SB

RL Q2 Q3 Q2 Q3

Beta

0.5 0.306912 0.413589 0.306993 0.413651

1 0.35698 0.511869 0.357078 0.511948

2 0.406233 0.603603 0.406348 0.603701

Mean 0.356708 0.509687 0.356807 0.509767

SD 0.049661 0.095026 0.049678 0.095044

CV 7.182824 5.363685 7.182372 5.363478

Rural(365) Indices SG SB

RL Q2 Q3 Q2 Q3

Beta

0.5 0.280495 0.392687 0.280569 0.392744

1 0.336785 0.49946 0.336878 0.499537

2 0.393751 0.599073 0.393863 0.599169

Mean 0.33701 0.497073 0.337103 0.49715

SD 0.056628 0.103214 0.056647 0.103233

CV 5.951253 4.815965 5.950908 4.815802

Urban(30) Indices SG SB

RL Q2 Q3 Q2 Q3

Beta

0.5 0.292362 0.416851 0.292583 0.416927

1 0.354018 0.52584 0.354159 0.525943

2 0.410308 0.618554 0.410479 0.618683

Mean 0.352229 0.520415 0.352407 0.520518

SD 0.058994 0.100961 0.058967 0.100987

CV 5.970628 5.154632 5.976289 5.15428

Urban(365) Indices SG SB

RL Q2 Q3 Q2 Q3

Beta

0.5 0.266198 0.398194 0.266301 0.398268

1 0.333724 0.514384 0.333859 0.514487

2 0.396803 0.611659 0.396971 0.611788

Mean 0.332241 0.508079 0.332377 0.508181

SD 0.065315 0.106872 0.065348 0.1069

CV 5.086743 4.754087 5.086266 4.753807

Interpretations based on the computations

The table 1 corresponds to the classical approach while the tables 2-5

correspond to the fuzzy logic approach.

The value of the Sen’s index increases drastically from the classical to the

fuzzy logic approach. The values of the Sen’s indices for both the rural &

urban West Bengal have been pretty low in case of the Classical Approach.

However, they increase significantly in case of the fuzzy logic approach. This

indicates that the non-rich non-poor limbo persists with a considerable high

propensity in both the rural & urban West-Bengal.

In the classical approach, the values of the indices have been considerably

low suggesting that there are not many people in the society who are poor.

But the society that we are considering is not the Aristotelian society where

there are only rich or only poor people in the society. It’s the Marxian

society in which the society is comprised of the poor, the non-poor non-rich

limbo & the rich respectively from the bottom to the top of the economic

strata. So, computations based on the fuzzy logic approach where the non-

poor non-rich limbo has been duly considered seems to yield sensible

results.

In the fuzzy logic approach, considerable high means, low SD’s & high CV’s

suggest that there are a large number of people who have income below

the sustainable threshold. In the rural sector, this might be due to the fact

that there are a large number of poor people having income below a

certain threshold- a threshold that is required for a sustainable

consumption. This may be due to the more dependence on agriculture to

industry & continuous division of land over generations. In the urban

sector, however, this might be the result of the lack of proper technical

education amongst the urban mass ceasing them from being employed.

Further, a considerable number of people may have moved from the rural

to the urban sector in search of employment but have remained

unemployed or received considerably low-waged jobs.

This is interpretative of the fact that a considerable number of the people in

both the sectors have income below or around the sustainable threshold

which is consequent upon the intense propensity of the non-rich non-poor

limbo in the society.

5. Conclusion

The fuzzy logic approach gives the picture of the scenario where most

of the people in the society are below ‘poverty line’. This is due to the

presence of the ‘non-rich non-poor limbo’ or the ‘middle class’

which debars the downward trickling of the macroeconomic

quantities from the rich to the poor.

The macroeconomic units get stuck in the non rich-non poor-belt

where they perish being consumed.

Thus, the ‘non-rich non-poor limbo’ persists with intense propensity

in West Bengal.

Soft wares used

R

MS EXCEL

Sources

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2. On Aristotle and Marx: A Critique of Aristotelian Themes in Marxist Labor

Theory, Annie Chau, Department of Economics, Stanford University;

Advisor: Professor Takeshi Amemiya, May 2003

3. The Capability Approach: Its Development, Critiques and Recent Advances

(GPRG-WPS-032): David A. Clark, Global Poverty Research Group, Website:

http://www.gprg.org/

4. "Human Development and Economic Sustainability," World Development,

Elsevier, vol. 28(12), pages 2029-2049, December, Anand, Sudhir & Sen,

Amartya, 2000

5. A fuzzy logic approach to poverty analysis based on the Gini and Bonferroni

inequality indices: Paolo Giordani · Giovanni Maria Giorgi

6. POVERTY ESTIMATES FOR 2004-05, New Delhi, March, 2007: GOVERNMENT

OF INDIA PRESS INFORMATION BUREAU