1
An evaluation of the empirical calculation methods of the Gini coefficient
2
I. Introduction, Aims and Rationale Economic inequality is a persistent and pressing issue having the potency to rouse resentment amongst a nation’s population, give rise to social and economic upheavals and provoke strong arguments about its magnitude, impacts and potential solutions. I became interested in the issue of global inequality only after witnessing varying degrees of poverty within and between the areas of residence that have dominated my life: India (various areas within) and Singapore. It is fascinating to see that such drastic inequalities could exist in the small vicinity of cities as seen in the figure below.
Figure Figure 1: India: Poverty Poverty and Affluence in the the same plot of of land
I was curious as to how such a crucial measure that defined various governmental policies was calculated, withstanding the income disparity within a geographical region and the vastness of data required for an accurate calculation in countries such as India. This led my basic research in the area from which I discovered the prevalence of mathematics in generalising formulae to represent economic inequality. Through the lessons in school, I was able to recognise the basic principles behind some of these formulae which further probed me to investigate. This is because, an additional desire of mine was to apply the deep study we did of calculus and series in school to something more tangible and real. As such, to understand the process of rendering mathematics pertaining to real life sociopolitical situations reliable and trustworthy, I decided to focus on income inequality, comparing various ways of calculating the Gini Coefficient (a global standard), particularly
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that of India. Through the investigation, I am to identify the reasons for unreliability (if any) and to understand what a perfect measure of economic inequality would be.
II. Background Information The Gini Coefficient is the most renowned and widely employed measure of inequality, and is a standard in governmental calculations. It is named after its founder, Corrado Gini, who discovered it in 1912. The value of a region’s Gini Coefficient ranges between 0 and 1 and is based on the net income of residents. Here, 0 represents perfect equality with each resident earning the same income and 1 represents perfect inequality where 1 person earns all of the income (Bourne). As such a higher Gini coefficient value would mean greater disparity between the incomes of the richest and poorest earners in a particular region. There are a number of different ways to calculate the Gini coefficient. These include graphical methods that involve the cumulation of various data points and frequencies such as the Lorenz curve and more theoretical ones such as Pareto’s distribution function. These are the 2 methods I will be analysing and comparing against each other. The reliability will be on the basis of the closeness of the values extracted from each method to the value released by the Indian government for the year 2013 which was G=0.510 in 2013 (Nair). Method 1: Using the Lorenz Curve: Trapezium Rule The most common way of viewing the GINI coefficient is through the generalised Lorenz curve. e m o c n I f o n o i t r o p o r P e v i t a l u m u C
L0 ( x )
=
x
10 x
L1 ( x )
2 =
!1
1023
Cumulative Proportion of Population
Figure 2: The line of perfect equity and an arbitrary Lorenz curve
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In reference to Figure 2, this curve depicts the percentages of a defined population arranged from the poorest to the richest on the horizontal ( x ) axis and the cumulative percentage of income enjoyed by a segment of a nation’s population. For example, Quintile 3 shows the cumulative percentage of income earned or wealth by the 1st, 2nd and 3rd quintiles combined. Since 0% of the population have 0% of the income, the curve passes through point A (0,0) and since 100% of the population enjoy all the income, the curve passes through point B (1,1) as seen in the diagram. As such a Lorenz curve runs from one corner of the unit square to the diagonally opposite corner. This serves as the benchmark for a perfectly equal distribution of income indicated by the curve L0 ( x ) . 10 x
Figure 2 displays an arbitrary yet possible Lorenz curve
L1 ( x )
2 =
!1
1023
. The degree of
income inequality is defined by the deviation of the Lorenz curve from the line of perfect inequality. This deviation (Gini coefficient) is measured by the area underneath the Lorenz curve, as we will observe. With a Lorenz curve plot such as the one above, we can measure the Gini coefficient. The general formula to be used in the investigation is represented by the following integral:
!
=
%
" " # # $ ! "# # $$#
!
"
"
This calculates the area between the curve of perfect inequality and a Lorenz curve divided by the area under the perfect inequality curve. In Figure 1 for example, the Gini coefficient of L1 ( x ) is measured as the area LA (Lorenz area) between the curve and L0 ( x )
divided by the area under
L0 ( x )
as highlight in magenta and orange respectively.
Since, at point B, the coordinates are (1,1) , this forms a right angled triangle with point A and (1, 0) being the other two vertices, which is highlighted in a light shade of orange. Hence the area under the equity curve is the area under a triangle, which is
! "
!!!! =
! "
.
As such, the Gini coefficient can be generally written as: !
"# =
! "#
=
#"# !" # "#
! =
#
where LA is the area between the two curves mentioned above and G is the Gini coefficient of L2 ( x ) , with reference to Figure 2. However, the general formula is difficult to employ in real life situations. This is because, nations collect raw data from their population in large numbers which may be difficult to formulate as a generalised graph. I
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will attempt to do this using the trapezium rule with a limited set of data acquired from the official census data of India’s income brackets as seen in the following table.
Proportion of Population: (converting % to decimals) x i
Proportion of Income (converting % to decimals yi
1
0
0
2
0.2 (first quintile)
0.061
3
0.4 (second quintile)
0.153
4
0.6 (third quintile)
0.279
5
0.8 (fourth quintile)
0.468
6
1 (fifth quintile)
1.0
Table 1: Cumulative frequency table depicting India’s income in quintiles
Figure 3: India’s quintile income proportion scatter plot
The trapezium rule refers to a rule of numerical integration that estimates the area under a curve. As such, it is a way of estimating integrals of curves by segregating the area under curve into a number of trapeziums, whose areas are then summed. To find the Gini coefficient, the data points in Table 1 can be used to formulate a number of trapeziums to represent an estimated Lorenz Curve, as seen in the figure below:
L0 ( x )
=
x
Estimated L1 ( x )
Figure 4: Area under an estimated Lorenz curve, formulated with the tra ezium rule
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Here, the summed area of trapeziums T1, T2, T3 and T4 and the triangle TR1 (in red) subtracting by the area of TR0 (in green) represents the area LA. The area of TR0, the triangle below L0 ( x ) is 1/2. Hence, in accordance to the formula stated above the Gini coefficient estimated using the trapezium rule will be: G
=
0.5 ! (0.01 + 0.02 + 0.04 + 0.07 + 0.15) 0.5
=
0.21 0.5
=
0.420
This value is an great underestimation of the governmentally stated value of the coefficient which is G=0.510. This suggests that the Trapezium rule results in the presence of a negative bias for the calculation of the Gini coefficient, rendering it a largely ineffective measure.
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Method 2: Using the Lorenz Curve: Polynomial Regression To rectify this limitation and formulate a more accurate Lorenz curve, I will attempt to formulate a polynomial graph using polynomial regression. This refers to a method of curve fitting with which a set of data is approximated using a polynomial function that takes the form f ( x ) C 0 + C 1 x1 + C 2 x 2 + ... + C n x n where C refers to a set of coefficients and n refers =
to the degree of the polynomial function. Here, the difference between the measured value of yi and the actual value of yi is referred to as the residual value R . The general model for polynomial regression can be created using the method of least squares. This method attempts to reduce the variance between the values in order to fit the data points accurately, by finding the lowest sum of residuals. Since linear and polynomial regression models are often unreliable, tending to inappropriately depict the data, residuals are used to examine their accuracy. A residual (e ) point refers to the difference between the actual value of the dependent variable (y) and the value predicted by the points on a regression curve (y 1 ) (“Finding Residuals”).
This is shown graphically in the figure below:
e
=
y ! y1
Figure 5: Depiction of residual values
Here, the sum of squared residuals is represented by:
8 2
n
SSR ! ! !" yi " (C 0 + C 1 xi i
... + C n x ) $# n i
+
1
=
In order to minimise the polynomial we take partial derivatives of this function with respect to each of the constants (C ), where we equate the residual to 0 to find the lowest value of SR (sum of residuals). Partial derivatives refer to derivatives of a function with multiple variables, where all the variables except C are held to be fixed (Weisstein). To find the Lorenz curve of India, I will restrict the investigation to quadratic regression, where the general equation is: yi
=
2
C 2 x
C 1 x + C 0
+
2
n
where:
SSR ! ! !" yi " (C 0 i
+
C 1 xi
+
C 2 x ) $# 2 i
1
=
The partial derivatives for this quadratic function will be:
!(SSR) = " 2 ' # y " (C $ !(C 0 )
+
C 1 x + C 2 x
!(SSR) = " 2 ' # y " (C $ !(C 1 )
+
C 1 x + C 2 x
!(SSR) = " 2 ' # y " (C $ !(C 2 )
+
C 1 x + C 2 x
n
0
2
) %& = 0
2
) %&x = 0
2
) %&x
i =1 n
0
i =1 n
0
2
= 0
i =1
Dividing both sides by 2 and factoring out the constants, this leads us to the following equations: n
n
C 0 n + C 1 ! xi i
i
C 1 ! xi
+
i
1
2
+
C 1 ! xi
3
i
1
=
!y 1
=
C 2 ! xi
=
! x y i
i
1
=
C 2 ! xi
4
i
1
=
n
3
n
+
equation (a)
i
i
i
1
=
n
+
=
1
=
n
2
n
i
i n
=
C 0 ! xi
C 2 ! xi
1
=
n
C 0 ! xi
+
n
2
equation (b)
i
1
=
n
=
! x y 2
i
i
1
=
equation (c)
i
1
=
which can be expressed as the following:
(1)
" $ $ $ $ $ $ $ $#
n
!x
n
i
i n
1
=
0
n
! x ! x i
1
! x 1
=
1
2
2
3
i
1
=
1
=
!x i
3
1
=
n
i
i
2
i
i
=
n
2
1
=
i
% " $ ! xi ' i ' ! C $ $ n '# & $ ! xi ' # C & $ '# i & $ ' " C % $ n $ xi ' ! '& $# i n
4
1
=
=
% ! yi ' i ' n ' ! xi yi ' ' i ' n xi yi ' ! '& i n
1
=
1
=
2
1
=
9
The creation of the matrix and its representation of the 3 equations above can be observed by looking at the multiplication of the matrices on the right hand side of (1). To multiply two matrices we need to do the dot product of each row of the first matrix and the only column of the second matrix. This calculates the sum of all the products of matching members as seen below:
" $ $ $ $ $ $ $ $#
n
!x
n
i n
n
1
i
n
! x
1
i
=
!x
i
1
i
i
C 0 n + C 1 ! xi i
1
2
4
i
1
=
n
=
i
1
=
!x
i
=
3
n
3
% ' ' ! C $ ' # & ' ( # C & ' # & ' " C % ' '& 0
!x
n
2
1
=
n
2
i
=
2
i
i
i
i
!x
i
1
=
! x ! x i
n
1
=
n
+
C 2 ! xi
1
2
i
=
1
=
n
=
!y
i
i
1
=
As seen, finding the dot product of the first row of the first matrix and the second matrix yields equation (a). Finding the dot product of the next two rows of the first matrix will result in equation (b) and (c). Therefore, matrices can be used to represent equations (a), (b) and (c). We can determine the value of the constants by multiplying both sides of (1) by the transposed first matrix:
! C $ # & C # & # C & " % 0
1
2
=
" $ $ $ $ $ $ $ $#
n
!x
n
i
i n
1
i
i
1
=
n
! x
2
i
1
2
1
=
=
!x
3
i
i
1
=
3
1
=
n
i
i
=
n
i
2
1
=
! x ! x
% ! xi ' i ' n ' ! xi ' ' i ' n xi ' ! '& i n
4
1
=
(1
" $ $ $ $ $ $ $ $#
% ! yi ' i ' n ' ! xi yi ' ' i ' n xi yi ' ! '& i n
1
=
1
=
2
1
=
To calculate the inverse matrix of a 3 ! 3 matrix, we can use the following process.
! ! " # $ # & Suppose a general matrix in the form: !" # $ % & & , where each letter corresponds to a # ' ( ) & " % real number. The inverse matrix will be:
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M
!1
1 =
M
! e f # # h i # # # ! b c h i # # # b c # e f #"
where the arbitrary minor matrix:
d
f
g
i
a g
c i
!
a d
!
a
b
c
d
=
d g
!
c f
$ & & & & & & & & & &%
e h
a g
b h
a d
b e
ad ! bc
and what is known as the determinant:
!
=
!
" $
# %
!&
'
#
(
%
+
)
' " ( $
=
!""% ! #$# ! &"'% ! #(# + )"'$ ! "(#
A curve can then be generated for a quadratic function by solving for the coefficients in the matrix. In the case of India, we have the information of the proportion of income earned by each quintile of the population shown in Table 1. Inputting the
xi
and yi values depicted in the table into matrix equation 1, we get the
following:
! # # # "
C 0 C 1 C 2
$ & & & %
=
! # # #"
6
3
2.29
3
2.29
1.8
2.29
1.8
1.5664
$ & & &%
'1
! # # #"
1.961 1.6152 1.42668
$ & & &%
To solve for the inverse of matrix, we must first find its determinant which can be calculated by summing the product of a cofactor of the first row and their respective minor matrix:
6
2.29 1.8 3 1.8 !3 1.8 1.5664 2.29 1.5664
+
2.29
3 2.29
=
6(3.59 ! 3.24) ! 3(4.70 ! 4.12) + 2.29(5.4 ! 5.24)
=
0.60347
2.29 1.8
The reciprocal of this can be multiplied to the following matrix to give us the transposition of the matrix:
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‘
" $ 2.29 $ 1.8 $ 1 $ 3 ! 0.60347 $ 1.8 $ $ 3 $ 2.29 $#
1.8
!
1.5664
3
1.8
3
2.29
2.29
1.5664
2.29
1.8
6
3
2.29
1.8
2.29
6
2.29
1.5664
2.29
1.5664
2.29
!
1.8
6
!
2.29
6
3
1.8
3
2.29
3
" 0.347 !0.577 0.156 $ !0.577 4.154 !3.93 0.60347 $ $# 0.156 !3.93 4.74 " 0.490 % 0.220 !0.815 $ ' $ !0.815 !5.553 ' 5.869 $ ' $ 0.220 ' !5.553 6.697 $# '& 1
=
=
We can determine the value of the coefficients,
C 0 , C 1 , C 2
% ' ' ' ' ' ' ' ' '&
% ' ' '&
, by substituting this into the
original equation.
! # # # "
C 0 C 1 C 2
$ & & & %
=
! # # # #"
! # # # # #"
C 0 C 1 C 2
0.490
!0.815
0.220
!0.815
5.869
!5.553
0.220
!5.553
6.697
$ & & & &%
=
! # # # # # # # "
$ &! &# &# & #" &%
'0.04139503523151611 '0.04057415997524583 1.0179444066877004
1.961 1.6152 1.42668
$ & & &%
$ & & & & & & & %
This would give us the quadratic equation: 2
y
=
C 2 x
y
=
1.02 x
+ 2
C 1 x + C 0
! 0.041 x ! 0.041
In the equation above, the coefficients have been represented up to 3 significant figures for ease of observation. The resultant Lorenz curve (Lq) amidst the scatter points of input can be seen below:
e m o c n I f o n o i t r o p o r P e v i t a l u m u C
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L0 ( x )
=
x
Lq ( x)
=
1.02 x
2
! 0.041 x ! 0.041
Cumulative Proportion of Population
Figure 6: The resultant Lorenz curve from quadratic regression
From the nature of the curve, we can tell that it does not pass through the data points exulted in Table 1. This suggests that the prediction of the y values for all x based on a limited set of data does not accurately portray the income proportion of each segment of the population for India. From the deviations of the data points (highlighted by the red points in Figure 6) from the best fit curve, we can formulate a table to depict each residual point: x
y
y1
e
0
0.00
-0.04
0.04
0.2
0.06
0.01
0.05
0.4
0.15
0.14
0.01
0.6
0.28
0.35
-0.07
0.8
0.47
0.64
-0.17
1.0
1.0
1.102
-0.102
Table 2: Residual Plot Data for Table 1
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The residual sum of squares, as explained earlier is a measure that indicates the degree to which a statistical model is a good fit for a data set. The value of SSR in this case is SSR 0.048404 which suggests that the although the quadratic line draws a suitable best fit line, it does not perfectly represent the data. More significantly, it does not fulfil the requirements of a Lorenz curve which is that it pass through the origin and point B (1,1) . =
This was a limitation I recognised only after the computation of data, and drawing the curve out using graphing software. I realised that using quadratic regression might not be an appropriate method to sketch a Lorenz curve. To combat this issue, I decided to use polynomial regression to define a polynomial of a higher degree using the data points in Table 1. Since we have 6 data points, a polynomial equation of the fifth degree can be constructed to represent the Lorenz curve. I chose to use a fifth degree polynomial here with the general equation of, yi
=
C 5 x
5
+
C 4 x 4
+
C 3 x
3
+
C 2 x 2 + C 1 x + C 0 , since this is the maximum
order of a polynomial that can be created using 6 data points, presumably result in the most accurate Lorenz curve possible. The aforementioned equation (1) can be alternatively written as:
! # # # # # "
1
2
x
1
1
x
1 2
x
2
x
2
!
!
!
1
x
x
2
n
n
$ &! &# &# &# &" %
$ & & & %
C 0 C 1 C 2
=
! # # # # #"
y1 y2
! yn
$ & & & & &%
where n refers to the number of x and y coordinates. The first matrix in the equation above is known as a Vandermonde matrix which is a type of matrix that arises in the polynomial least squares fitting (Weisstein). In the case of a polynomial of the fifth degree, using the values from Table 1 this is represented as:
! # # # # # # #" ! # # # # # # # # # "
!
"
"
"
"
"
!
"#$
"#"%
"#""&
"#""!'
"#"""($
!
"#%
"#!'
"#"'%
"#"$)'
"#"!"
%$!
"#'
"#('
"#$!'
"#!$*'
"#"++'
!
"#&
"#'%
"#)!$
"#%"*'
"#($+'&
!
!
!
"
!
!
!
$
!
(
!
%
!
)
$ & & & & & & & & & %
=
!
! # # # # # # #"
!
!
!
! $# &# &# &# &# &# &# &% # # "
!
"
!
!
!
$
!
(
!
%
!
)
$ & & & & & & & & & %
!
"
"
"
"
"
!
"#$
"#"%
"#""&
"#""!'
"#"""($
!
"#%
"#!'
"#"'%
"#"$)'
"#"!"
%$!
"#'
"#('
"#$!'
"#!$*'
"#"++'
!
"#&
"#'%
"#)!$
"#%"*'
"#($+'&
!
!
!
!
!
!
=
$ & & & & & & &%
! # # # # # # #"
" "#"'! "#!)( "#$+* "#%'&
'!
!
! # # # # # # #"
$ & & & & & & &%
" "#"'! "#!)( "#$+* "#%'& !
$ & & & & & & &%
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Performing the aforementioned steps on inversion and matrix multiplication using I.T. (a calculator), due to the magnitude of the matrix we get the following matrix for the constants: ! ! $ # ! & ! $ ! # ! " & # & # & # !'$($()#)%(!&*&)( & # ! # & # '"'$!#%%(%!$+*!+& & # & # & # ! $ & # ('&%(#)"%)$*((!( & # ! & # '"!'"%*(!&%($$$+& & # % & #" &'&%!!(*%#*$**+& &% # ! & & " % =
From these values, the equation for the Lorenz curve of India in 2013 will be: !
!"!#$ !
! %$"%#& ! # + '"!#' ! ( ! %"($) ! ) + $"('# !
seen as the Lorenz curve in the diagram below, with the various scatter points defining India’s income quintiles from Table 1. e m o c n I f o n o i t r o p o r P e v i t a l u m u C
Lq ( x) L0 ( x )
=
=
1.02 x
2
! 0.041 x ! 0.041
x
L ( x )
=
5.540 x
5
! 10.148 x 4 + 6.546 x 3 ! 1.302 x 2 + 0.364 x
Cumulative Proportion of Population
Figure 7: The resultant Lorenz curve from polynomial regression
15
In comparison to the Lorenz curve derived from quadratic regression, it is observed that using a polynomial of the 5th degree is more suitable to calculate the Lorenz curve, since it goes through both the origin and point B. The gini coefficient using the integral formula according to our curve and data is: '
" " ! "#$#%& " ! '&$'%( "
!=! =
#
%
&
+
)$#%) " * ! '$*&! " ! + &$*)% " +#"
&$%%*
As can be seen, this Lorenz curve has no deviation from the data points as it interests all 6 of them seen in Table 1. Since there are no residual points, this suggests that it is a more accurate depiction of India’s income distribution that Lq obtained with polynomial regression. According to official data, the Gini coefficient of India in 2013 was G=0.510, which is not equivalent to the Gini coefficient calculated from the predicted Lorenz Curve, L . This might be a result of limited range of data used which reduces the socio-political viability of the calculations and does not accurately estimate the Gini Coefficient. In this case, polynomial regression to sketch a Lorenz curve would be more accurate with a larger set of data. Method 3: Using the Covariance formula The calculation of the Gini Coefficient using geometrical interpretations based on the Lorenz Curve, is only one of the myriad ways the index can be calculated with. An alternative method is to represent the Gini Index in terms of the covariance between income levels (proportion of population) and the cumulative distribution of income. Knowing the general formula of the Gini Coefficient using the Lorenz curve, we can rewrite it as: !
%
" " # # $ ! "# # $$# % ! !" "# # $$# =
!
"
"
%
=
"
In this case, lets assume that the cumulative distribution function F ( x ) gives the proportion of the population having an income level below or equal to x . This is a non-decreasing function that represents the percentage of individuals with an income below x . Let’s call this proportion p . Additionally, lets assume that F ( x ) is continuously differentiable such that the following density exists: F !( x )
=
f ( x )
where for a given value of x the proportion p can be alternatively defined as: x
p
=
! f ( x ) 0
=
F ( x )
16
Using the geometrical representation of the aforementioned general formula for the Gini Coefficient, we can represent it in terms of the covariance between income levels and the cumulative distribution of income (Lubrano).
1
G
=
1! 2
! L( p) dx 0
where
is the covariance
Cov
cumulative distribution of the same income F ( y) and
=
2
Cov( x, F ( x )) between income levels
y
and the
µ µ
is average income.
The table below represents household incomes for each of India’s quintiles, as an extension of Table 1:
Proportion of Population: (converting % to decimals) x i
Proportion of Income (converting % to decimals
Household Income (Rs/Annum) yi
1
0.2 (first quintile)
0.061
19,041
2
0.4 (second quintile)
0.153
29,353
3
0.6 (third quintile)
0.279
41,220
4
0.8 (fourth quintile)
0.468
65,235
5
1 (fifth quintile)
1.0
153,872
Table 3: Table of mean income levels corresponding to each population quintile in India
Using this, the cumulative distribution of income refers to x coordinates while income levels refer to the average personal income corresponding to x segment of the population. This suggests that the Gini coefficient is proportional to the covariance between a variable and its rank. The covariance of two variables indicates how they change together. As such, it provides a measure of the degree of correlation between sets of random variables, with a positive covariance value suggesting a positively relation and a negative value, an inverse relation. Understanding the idea of covariance was especially challenging for me, since statistics was one topic that was not visited in any of my math lessons. As such, as opposed to a formulaic one, I attempted to diagrammatically understanding and explain the concept. Using the paired data in Table 3, a scatter plot is seen below:
17
Figure 8: Diagrammatical representation of Covariance
In the diagram I drew all possible rectangles that could exist between the 5 data points, colouring them red. Here, the covariance is represented net amount of red in the plot (reflecting the average covariation between the variables), which would be roughly around the middle due to darker shades of red there. Mathematically, this is shown with the formula: n
! ( x ! x )( y ! y ) i
Cov( x, y)
i
i 1 =
=
n
where:
!1
x
=
independent variable
y
=
dependent variable
n
=
number of data points
x
=
mean of independent variable, x
y
=
mean of dependent variable, y
Using the values in Table 3, we can calculate x and y first. 5
! x
i
x
=
5
! y
i
y
i 1 =
=
=
5
3 =
5
308,721 =
5
i 1
=
5
=
0.6
61, 744.2
=
µ
Substituting these values into the aforementioned covariance formula we get: 5
! ( x ! x )( y ! y ) i
Cov( x, y) = =
=
i
i =1
4 17081.28 + 6478.24 + 0 + 698.16 + 36851.12 4
15277.2
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Dividing this value by
2
we can calculate the value of the Gini Coefficient using the
µ
covariance formula: G =
2 =
61744.2
! 15277.2
0.495
As can be seen, the value of G=0.495 is not equivalent to the officially stated value for the Gini Coefficient of India in 2013 of G=0.510, calculated and published by the Indian government using their complete data. With only 5 generalised income levels used to determine the covariance between income levels and proportions of population in India, this is inevitable. By using a limited number of data points, I realised I am ignoring various idiosyncrasies that may be present in the income distribution of each individual segment. This has led to an underestimation of the Gini coefficient of India. As with the result from the first method, the reason for the discrepancy most likely lies primary in the limited access a civilian has to national income data. This creates challenges in observing the effectiveness of distinct methods to calculate the Gini Coefficient. Discussion and Analysis In this investigation, I attempted to present an analysis of three formulaic methods to calculate the Gini Coefficient; two based on area ratios under a Lorenz curve and the other based on covariance formulas. The numerical integration method of the Trapezium rule in comparison to Method 2 is extremely unreliable as it inevitably results in a positive bias for the Lorenz curve, and a negative bias for the gini coefficient. This is because, the method creates the curve with straight line segments that would lie above parabolic lines connecting the data points (as seen in Method 2). This results in a greater area underneath the Lorenz curve for Method 1, and thus a smaller Gini coefficient. When comparing methods 2 and 3, despite the fact that the values of the Gini coefficient using Methods 2 and 3 were lower than the governmentally defined value of G=0.510, Method 2 seems to be more ineffective in accurately measuring the value since the value it predicted had a greater discrepancy from the actual value, than the one predicted by the covariance formula. A reason for this could be that formulating the Lorenz curve L ( x ) from a data set of size n
=
6
results in a curve that estimates income proportions( y) for all the
unspecified segments / population proportions of the Indian society( x ) . In the case of my investigation, where the data points were restricted to quintile income data this gives great room for uncertainties and inaccurate estimations of the income disparity within these quintiles. On the other hand, since the Gini coefficient based on the covariance formula
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was solely derived from the relationship between the 5 x andy coordinates, its value of G=0.495 was closer to the actual value. With empirical evidence form my investigation, the Gini coefficient seems to be most accurately calculated using the covariance-based method. However, with better access to a wide range of income proportions and data points, most governments opt to employ the Lorenz curve to determine the coefficient instead (Method 2). The most distinct difference between Method 2 and Method 3 is that the Lorenz curve is an extremely contextualised and direct manner to calculate the Gini. This is because, it was created primarily to act as a graph of the cumulative frequencies of income proportions and population proportions, which along with the well defined rules for the curve suggest that it was solely meant for this purpose. On the other hand, the covariance formula in Method 3, is used as an inference to the Gini Coefficient, generally indicating the type of relationships between two random variables. This allows Method 3, to provide measurements for various other areas of interest such as the magnitude of positive or negative correlation between any two variables. This trait of Method 3, can be used to better understand the degree of inequality in a country, by filling in loopholes that may exist as a result of the coefficient being a simplistic consideration of income distribution. Overall, the Gini coefficient does have limitations as a measure of inequality. One of the major ones is that the coefficient is not additive across various segments of a population and fails to ignore the nuances of income disparity that may exist within each segment. For a better judgement of a nation’s inequality degree, the coefficient is used in conjunction with other indices of income inequality such as the Theil Index, which is additive over various population segments and measures. It identifies the share of inequality attributable to the between region components, and is measurement based on General Entropy formulae, mitigating some of the limitations of the Gini coefficient. /cite/ Assumptions and Limitations In the investigation, the use of the Gini coefficient as a tool to compare income inequalities of multiple countries was not explored. This could have been a possible extension of the investigation, which could also lend for a deeper understanding of its relevance in modern economic inequality and its reliability as such. Additionally, the scope of the research, as a result of limited access to census data regarding India’s income proportions, was limited. Yet, for the sake of comparison and exploration, the results were assumed to be conclusive and were compared to the actual value of the coefficient published by the Indian government to determine the reliability of each method.
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Conclusion The investigation enabled us to determine the various implications and calculations of the Gini coefficients that may vary numerically depending on the nuances of each method. Working with the Gini coefficient and with so many areas of mathematics that were novel to me has allowed me appreciate the idea of inequality, the sharing of monetary resources and applied mathematics in the modern day. I was astonished at how drastic the difference was between the lowest and highest quintiles of India’s earning population, an insight which would have not been as revelatory without mathematically deriving them through Lorenz curves. quantitative and empirical analysis of social issues such as income inequality allowed me to broaden my perspective on the implications and severity of this prevalent issue.
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Bibliography Bourne, Murray. "The Gini Coefficient of Wealth Distribution." Intmathcom RSS. N.p., 24 Feb. 2010. Web. 07 Mar. 2017. Nair, Remya. "IMF Warns of Growing Inequality in India and China." Http:// www.livemint.com/. Livemint, 03 May 2016. Web. 07 Mar. 2017. "Finding Residuals." Interactivate: Finding Residuals. CSERD, n.d. Web. 23 Mar. 2017. Weisstein, Eric W. "Vandermonde Matrix." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/VandermondeMatrix.html. 23 Mar. 2017. Lubrano, Michael. "The Econometrics of Inequality and Poverty." (n.d.): n. pag. Http:// www.vcharite.univ-mrs.fr/PP/lubrano/cours/Lecture-4.pdf. Sept. 2016. Web. 24 Mar. 2017.
