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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
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
1. Poverty: New Dictionary of the History of Ideas, Berry, Christopher, 2005
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