1. INTRODUCTION:
Human beings, however selfish they are, are also partially altruist (Smith, 1759). They, when asked to view impersonally, prefer equity and so equality of distribution of valued things like income and wealth. Inequality means absence of or deviation from equality. Inequality is a quantitative but non-specific measure and should be clearly distinguished from inequity, the qualitative and broader counterpart of it.
Measures of inequality, especially of income, are tried to be constructed under the assumption that equity demands equal distribution of income and an increase in inequality reduces social welfare. A change in income distribution leads to higher inequality if and only if it reduces welfare. Pigou (1912), Dalton (1920), Lorenz (1905) and Gini (1936) have pioneering works in this field. The works of Pigou and Dalton are mainly on the reasonableness criteria of such measures and those of Lorenz and Gini are on direct measures of inequality. Lorenz measure is completely reasonable judged by the axioms prescribed by Pigou and Dalton but it fails to give complete ordering of inequality. On the other hand, Gini measure gives complete ordering of inequality, but to make the ordering complete it relies on some assumptions which are not completely reasonable.
Inequality in a region/country normally refers to absence of equality in the distribution of economic variables like income, expenditure, wealth etc. among the individuals/households in that region/country. Inequality can be calculated from data on such variables at individual/household level. For India, National Sample Survey Organisation (NSSO) collects such household data at the national level annually and/or quinquennially since India’s independence on a sample basis and that data is used for inequality estimation in India. Inequality for India and its states can be estimated only for expenditure and not normally for any other economic variable because NSSO collects household data only for expenditure of the sample households and not for income or wealth. NSSO itself estimates inequality for rural and urban sector of India and its states through two relative measures – Gini coefficient and Lorenz curve and we all use those estimates for such purpose.
Many authors use the estimates provided by the NSSO for explaining the trends and patterns of inequality in the rural and urban sectors of the states of India and many of them have their own estimates on the basis of the NSSO data. Deaton and Dreze (2002) have estimated inequality for India and its states for the years 1993-94 and 1999-00 by using (i) difference of log AM and log GM and (ii) variance of log values and have observed that inequality in India has increased in this period. Dev (2007) have shown that inequality measured by Gini coefficient has an adverse effect on poverty in the period from 1983 to 2004-05. Sen and Himanshu (2004) have shown that inequality, measured by Gini index, in rural and urban India has sharply increased in the 1990s. Majumdar, Sarkar and Meheta (2017) have used both the Gini index and the general entropy measure to examine the pattern of change in inequality in India in the pre-reform and the post-reform period. Thus, Gini index has been used by almost all of them for measuring inequality in India.
If Y1, Y2, …, Yn are
income levels of n individuals of a region/country in non-decreasing order with
mean income m then Gini coefficient for income distribution
of this population is given by 
(1)
Some academicians prefer to express Gini coefficient as
(2)
It is a single-valued quantitative or scalar measure. It is an additive measure. It is a relative (as opposed to absolute) measure – a measure relative to mean income. It is a unit free measure. For infinite population it is an index measure – its value lying between 0 and 1. It is a relative measure in another sense – it is relative to maximum possible absolute or relative value of inequality. The first measure is actually a per person per unit of mean income inequality in the Lorenz-Gini family. It is a relative measure of inequality and satisfies the population replication criterion accurately (for both finite and infinite population) but it is an index measure of inequality only for infinitely large population. On the other hand, the second measure is an index measure of inequality accurately (for both finite and infinite population) but satisfies the population replication criterion only for infinitely large population.
On the other hand, Lorenz curve is given by the locus of percentage income appropriations by percentage cumulative populations having income in the non-decreasing order. It is a plural (as opposed to singular) measure, or a multi valued measure – a qualitative measure. It is also a relative (as opposed to absolute) and so unit free measure. It is also an index measure under the assumption of infinitely large population. Gini coefficient is equal to Lorenz ratio. They use same principle of inequality measurement. Both of them are relative measures. Both of them are unit free measures. Both of them are index measures for infinite population. Lorenz curve is actually the graphical or plural counter-part of Gini coefficient. They belong to the same family. Let us call the family the Lorenz-Gini family.
Measures of inequality that are popularly and conveniently used or discussed in the literature are the Lorenz curve and the Gini coefficient or the Lorenz ratio beside some other measures like, the Dalton measure, the relative mean deviation, the coefficient of variation, the standard deviation of logarithms, the Atkinson measure, the Sen measure, the Theil measure, etc. There are many other measures proposed, infrequently used and discussed and it is not easy to discuss all of them. However, the essence of all those measures is well taken up through the discussions on the above mentioned measures. These measures are judged in terms of their positive versus normative characters, their completeness in inequality ordering and their ability to satisfy the reasonableness criteria.
2. The Reasonableness Criteria of Inequality:
The reasonableness criteria that are frequently referred in the literature of inequality measurement are of two categories. These criteria are related to several types of changes in income (which is being distributed) and population (among whom income is distributed) and corresponding changes in inequality that seem reasonable under the assumptions made. In the first category we have several invariance or independence criteria:
(i) Invariance under permutation or the Symmetry Criterion (SC)
(ii) Invariance under scalar multiplication or the Mean Independence Criterion (MIC)
(iii) Invariance under Population Replication or the Population Replication Criterion (PRC).
The symmetry criterion arises from the assumption of homogeneity of income of the individuals as arguments in the social welfare function, and this arises from the assumption of homogeneity of the preference, ability, willingness, etc. of individuals.
The mean independence criterion arises from the fact that we are interested not in the absolute measure of inequality but in the relative measure of inequality, specifically in the measure of relative inequality and more specifically in the index measure of relative inequality of the distribution of social welfare, and that we assume that the welfare function is either additive or additively separable and that individual social welfare functions of all individuals are homogeneous of degree one in incomes of the individuals in the economy so that the social welfare function is also homogeneous of degree one. If we do not agree upon these assumptions the MIC criterion loses its importance.
However, invariance under population replication criterion has no strong logical base and it finds little logical support only for Dalton measure and that is again, as we shall see below, due to an incomplete definition of inequality given in the Dalton measure.
In the second category we have the income transfer criteria:
(iv) The Pigou-Dalton Income Transfer Criterion (ITC) that states that a regressive transfer from a poor person to a rich (or less poor) person raises inequality.
(v) The Diminishing Income Transfer Criterion (DITC) that states that a regressive transfer between two relatively rich persons raises inequality by a smaller amount than that between two relatively poor persons.
These two criteria are based on the assumption of additive or additively separable social welfare function, on the assumption that individual social welfare is the function mainly of the individual income, on the assumption that this function is positively sloped and concave and on the assumption that inequality is measured linearly (and inversely) along the scale of social welfare.
Under criteria SC, MIC and ITC, an increase in income of any of the existing persons, say ‘P’, by an amount, say ‘A’, can be viewed as a two-step change – in the first step, income of all persons increase in equal proportion such that total income increases by ‘A’, in the second step increased income of all persons except ‘P’ is transferred to ‘P’. Thus, if ‘P’ is the richest person, then all such transfers are regressive transfers and inequality obviously increases. Similarly, if ‘P’ is the poorest person, then a decrease in the income of that person obviously raises inequality. On the other hand, a decrease in the income of the richest person (an increase in the income of the poorest person) obviously reduces inequality if even after reduction (increment) the richest person’s status to remain richest (the poorest person’s status to remain poorest) is unaltered. If status is altered, or if income of person other than the richest or the poorest person changes, then the DITC or some such criterion is to be used to measure inequality.
Let us come to the discussion of the first research question of this paper, how good and how much sufficient the Gini coefficient and Lorenz Curve are in inequality estimation. To be good it should satisfy some basic requirements/principles of inequality measure. According to Mondal and Kayet (2018), both Gini coefficient and Lorenz Curve satisfy the ‘principle of income transfer – the Pigou-Dalton principle’ and the ‘principle of proportionate additions to incomes’. In addition, they approximately, but not exactly, satisfy the ‘principle of proportionate additions to persons’. But both the Gini coefficient and the Lorenz Curve fail to satisfy the ‘principle of equal additions to incomes’ and Gini coefficient cannot completely satisfy the ‘principle of decomposition by subgroups’ and the ‘principle of decomposition by components of incomes’. Though some academicians prefer that Gini coefficient satisfies the ‘principle of decomposition by subgroups’ and the ‘principle of decomposition by components of incomes’ (e.g. Bhattacharya and Mahalanobis, 1967; Dagum, 1997; Frick et al. 2006; Costa, 2008 etc.). However, like Mondal and Kayet (2018), some other academicians state that, Gini coefficient fails to satisfy the decomposition criteria (Foster et al., 1984; Shorrocks, 1980). In addition, Chakravarty and Kundu (2016), clearly establishes that the Gini index is not a decomposable index of inequality.
3. Absolute Versus Relative Measure of Inequality:
Actually inequality can be viewed both in absolute and relative senses (e.g. Dalton, 1920; Kolm, 1976). Kolm (1976) has well taken up the debate between absolute and relative inequality. He has been of the opinion that inequalities can be measured by both the ways and the researchers in this field have used both of them. He has tried to define a relative measure of inequality as a ‘rightist’ measure of inequality as the richer section of the community or the capitalist class or their union prefers to accept it when income increases (by equal amount or by equal proportion) and an absolute measure of inequality as ‘leftist’ measure of inequality as the poorer section of the community or the labour class or the labour union prefers to accept it when income increases.
In the Lorenz-Gini family the absolute measure of
inequality is 
(3)
It is the pure per capita inequality and satisfies the
‘Principle of proportionate
additions to persons’ and the ‘Principle of equal additions to incomes’. An absolute measure of inequality is not
unit free and so inequality comparison across countries using different units
of measuring income or over time in the same country with inflationary
conditions becomes inconvenient. An absolute measure of inequality of this type has a fixed lower bound at 0
but its upper bound is not fixed. It is given by 
when all income is enjoyed by a single
person. For this reason also, an absolute measure of inequality is inconvenient
for inequality comparison. It is also inconvenient for inequality comparison as
it fails to compare inequality relative to mean income.
In this family the relative measure of inequality is 
. This is the Gini coefficient of the first type. It is
the measure of inequality per capita and per rupee of mean income and satisfies
the ‘Principle of
proportionate additions to persons’ and the ‘Principle of proportionate
additions to incomes’. A
relative measure of inequality is unit free and so inequality comparison across
countries using different units of measuring income or over time in the same
country with inflationary conditions becomes convenient. It is also convenient
for inequality comparison as it compares inequality relative to mean income. A
relative measure of inequality
of this type has a fixed lower bound at 0 but its upper bound is not fixed. It
is given by 
when all income is enjoyed by a single
person.
However, viewing relative measure of inequality as ‘rightist’ and absolute measure of inequality as ‘leftist’ is not completely true, because when income falls (by equal amount or by equal proportion) the richer section of the community or the capitalist class or their union prefers to accept an absolute measure of inequality and the poorer section of the community or the labour class or the labour union prefers to accept a relative measure. Anyway, these are two well-accepted views and Kolm himself was convinced of both the views. He has preferred to develop a ‘centrist’ view of inequality in between the two.
We are also actually convinced of both the views and probably like Kolm we also want to have a centrist view in between the absolute and relative views. We do not want that inequality remains constant with equal additions to incomes; rather we want that inequality fall. Similarly, we also do not want that inequality remains constant with proportionate additions to incomes; rather we want that inequality increase. Thus to have a complete view of inequality, we should have a plural view, it should be measured in both ways – absolute and relative. This may lead to a conflicting conclusion in both inter-temporal and inter-state comparisons. If we want to avoid this conflict and try to develop a singular measure, a centrist measure, we shall be in trouble once again because it is difficult to determine the relative weights of absolute and relative inequalities. We shall not go for that, rather we shall present them separately.
4. Measurement of Inequality: SD-CV Family:
Though Gini coefficient is still used popularly in inequality estimation mainly because of convenience and tradition of its use, some prefers to use standard deviation (SD) and coefficient of Variation (CV) for the purpose. For example, Subramanian and Jayaraj (2015) have used the Krtscha (1994) measure (product of SD and CV) for measuring inequality in India in a centrist way. As we shall see below there are some added advantages in using SD and CV as measures of inequality over the Gini coefficient. Moreover, many researchers raise objection against the use of only relative measures of inequality. Many of them argue that both relative and absolute measures of inequality should be given equal importance in judging the real trends and patterns of inequality.
In the discussion about different measures of inequality, both Sen (1973) and Kolm have found that standard deviation and coefficient of variation satisfy the basic properties of absolute and relative measures of inequality respectively. Kolm has observed that these measures though satisfy the ‘income transfer principle’; they fail to satisfy the ‘principle of diminishing income transfer’. Sen rejects these measures on three grounds one of which is their failure to satisfy the ‘principle of diminishing income transfer’. The second reason is the way the deviations are taking in the formula of standard deviation and coefficient of variation. According to Sen deviations of income from the mean is less reasonable than deviations of one income from the other. The third reason lies in the squaring principle applied in the formula of standard deviation and coefficient of variation. He finds no justice in applying this principle; rather he finds that this squaring principle is making the increase in inequality from regressive transfer invariant to the levels of income of the two individuals hence dissatisfying the ‘principle of diminishing income transfer’. However, the second reason shown by Sen is not tenable because standard deviation can also be
expressed as 
(4)
The squaring principle in standard deviation and coefficient of variation that makes them not to satisfy the ‘principle of diminishing income transfer’ can be said to be unreasonable if we are convinced of the principle of diminishing income transfer but cannot be said to be more unreasonable than the measures in the Gini family. In the Gini family a regressive transfer of an amount between two poor persons with an income difference may lead to a less increment in inequality than a regressive transfer of the same amount between two rich persons with same income difference if the number of persons between the two poor persons is less than that between the two rich persons. This property makes them more unreasonable than those in the SD-CV family. The ‘principle of diminishing income transfer’ is put forward on the ground of diminishing marginal utility of income. Standard deviation of the logarithm of income is tried as a solution; but has not succeeded to be reasonable for other reasons. Diminishing marginal utility of income leads to a higher welfare loss from a regressive transfer of an amount between two poor persons with an income difference than a regressive transfer of the same amount between two rich persons with same income difference. This also leads to a larger increase in the inequality in the welfare distribution in the former case. But if we are interested in the income distribution as such and not in the implied welfare distribution then we may not bother about it. Moreover, if marginal utility diminishes at a constant rate the ‘principle of diminishing income transfer’ no longer remains reasonable, rather the inequality measures in the SD-CV family become fully reasonable.
As explained by Kolm and as is seen from
the formula, SD is a per person inequality measure and thus an absolute measure
of inequality. CV is a per person per rupee of mean income/expenditure inequality
measure and so a relative measure of inequality. The upper bound of this
relative measure is 
and so CV has to be divided by to
have an index measure of inequality in the SD-CV family. For Gini measure the
relative measure of inequality is very close to the index measure of inequality
for large population. But for the measures in the SD-CV family relative measure
of inequality (measured by CV) is widely different from the index measure of
inequality (measured by CV-index). This is the main advantage of using the absolute,
relative and index measures of inequality in the SD-CV family over the
Lorenz-Gini family. In other words, the main disadvantage of this
relative/index inequality measure in the Lorenz-Gini family is its failure to
distinguish between a relative measure and an index measure. Actually, in the
relative/index inequality measure in the Lorenz-Gini family the weights of
different incomes are so set that inequality increases at high rates for all
transfers starting from a situation of perfect equality and increases at low
rates for all transfers approaching the situation of perfect inequality making
the inequality measure converging to one.
5. Empirical Analysis:
Now we enter into the discussion of the second part of this work which is related for searching some measurable factors responsible for the occurrence of different types of inequality in India. In India, NSSO collects household expenditure data for rural and urban sector separately and estimates inequality by using such data for rural and urban sector separately. In this paper we have used the overall inequality(The overall inequality, absolute or relative, in Lorenz-Gini family or in SD-CV family are estimated simply by combining the raw data of consumption expenditure for rural and urban sectors together with their corresponding population.) (better named as combined inequality), proposed by Mondal and Kayet (2018).
5.1 Database and Methodology for Empirical Analysis:
We have collected the raw data of different
expenditure classes of consumption expenditure from seven major surveys by NSSO
from 1983 to 2011-12 to have our own estimation. The Gini coefficient and the
coefficient of variation are used as relative measure of inequality in Lorenz-Gini
family and in SD-CV family respectively. On the other hand, the absolute Gini
and standard deviation are used as absolute measure of inequality in these two
families. Beside absolute and relative measure we should consider the index
measure of inequality to get the complete picture of inequality (Mondal and
Kayet, 2018). As mentioned earlier, for Gini measure the relative measure of
inequality is very close to the index measure of inequality for infinitely
large population like India because Gini coefficient has to be divided by 
to have the Gini index and for India the
ratio of (n-1) to n is very close 1. On the other hand, for the measures in the
SD-CV family relative measure of inequality (measured by CV) is widely
different from the index measure of inequality (measured by CV-index) because
CV has to be divided by to
have an index measure of inequality in the SD-CV family. And for this reason
the CV index is used as a separate measure in this work.
Several measurable factors are responsible for the determination of the variability of inequality. Though in most of the works, authors are used only Gini coefficient as a measure of inequality, we have considered the above mentioned five measures for the purpose.
Public education expenditure is often analysed as a factor of income inequality. Inequality will reduce if government gives more emphasis to increase funding for public education. If there is a mixed education system – private and public, Glomm and Ravi Kumar (1992) state that inequality may not decrease under private education system but it obviously decreases under public education system. Saint Paul and Verdier (1992), Eckstein and Zilcha (1994) and Zhang (1996) develop models where they argue that government should give gradual help to education for decreasing inequality over time. The expenditure on education by public authority can reduce inequality if poorer people have access to public education. If their income is too low, they cannot benefit from public education and thereby income inequality even increases (Sylwester, 2002). Kayet and Mondal (2016) argue that public expenditure for education is one of the important tools for reducing food inequality in India. We have collected the data of public expenditure on education at different levels from the Analysis of Budget Expenditure on education in different years published by the Ministry of Human Resource Development (MHRD, Govt. of India). Per capita public expenditure on education at different levels can be obtained from total public expenditure on education at different levels divided by total population and are measured by rupees.
Kuznets (1955) proposes by the inverted-U
hypothesis that, with growth in per capita income, inequality rises initially
but after sometimes, it reaches the maximum point and then returns to the
original level. We have collected the data of net state domestic product (NSDP)
from the Central Statistical Organisation (CSO) and are measured by percent.
The annual compound growth rate of output is calculated by,
(GrNSDP)it=(lnNSDP)it -(ln NSDPi(t-k)) (5)
(Corresponding years)
i denotes the cross section of states, t denotes the current period, k denotes the previous period.
Population can play a vital role (positive
role) in explaining inequality. However, it is inversely related with index of
inequality in SD-CV family measured by CV-index as mentioned earlier, CV has to
be divided by
to have CV-index. We have collected the raw data of population from different census reports from 1981 to
2011 and have adjusted by Lagrangian non-linear interpolation method to have
our own estimation and measured by person.
Monthly per capita consumption expenditure is an important macroeconomic variable for explaining inequality especially for those countries like India where only expenditure inequality is estimated and not normally any other types of inequality. We have collected the data of monthly per capita consumption expenditure (MPCE) for rural and urban sector separately from different major survey by NSSO for consumer expenditure for the referred period and measured by rupees. The overall MPCE can be obtained from the weighted average of rural and urban MPCE and is calculated by,
(OMPCE)it=
(RMPCEconst.price) it*(RPOP)it + (UMPCEconst.price)it*(UPOP)it)
(TPOP)it (6)
Where, RMPCE, UMPCE and OMPCE denote rural, urban and overall MPCE respectively and RPOP, UPOP and TPOP denote rural population, urban population and total population respectively, i denote the cross section of states and t denotes current period.
As inequality is directly related to employment work participation rate (WPR) and share of non-agricultural employment (SNAE) may have the important role for determination of the variability of inequality. We have collected the data of rural and urban work participation rate (WPR) and share of non-agricultural employment (SNAE) from different major survey for employment-unemployment by NSSO from the referred period and measured by percent. Like the overall MPCE, the overall WPR and also the overall SNAE can be simply calculated. The overall WPR is obtained from the weighted average of rural and urban WPR and is calculated by,
(OWPR)it=(RWPR)it*(RPOP)it + (UWPR)it*(UPOP)it (7)
(TPOP)it
Where, RWPR, UWPR and OWPR denote rural, urban and overall work participation rate respectively and RPOP, UPOP and TPOP denote rural population, urban population and total population respectively, i denote the cross section of states and t denotes current period.
The overall SNAE is obtained from the weighted average of rural and urban SNAE and is calculated by,
(OSNAE)it=(RSNAE)it*(RPOP)it + (USNAE)it*(UPOP)it (8)
(TPOP)it
Where, RSNAE, USNAE and OSNAE denote rural, urban and overall share of non-agricultural employment respectively and RPOP, UPOP and TPOP denote rural population, urban population and total population respectively, i denote the cross section of states and t denotes current period.