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                      American Economic & Social Review; Vol. 5, No. 2; 2019 
                                       ISSN 2576-1269     E-ISSN 2576-1277 

Published by Centre for Research on Islamic Banking & Finance and Business, USA 
 

     12 
 

 

The “Deciles Implied Inequality Indices” for Expressing Income Distribution 
and Measuring Public Financial Burden of Equalization Policies 

 

 
Bijan Bidabad 

Professor 

Economics and Chief Islamic Banking Advisor 
Bank Melli Iran, Tehran, Iran         
Email: bijan@bidabad.com 

 

Abstract 

Different ways have been proposed to measure income inequality; there is no best way to calculate the inequality 

index that expresses income distribution as it is. Popular inequality indices provide information about some points 

on the distribution function and analyze the inequality of income without reference to the amount of the budget 

needed to improve the income distribution. In this paper, we propose a set of “Deciles Implied Inequality Indices”. 

By using this index, we can show how much transfer payment is needed as a redistribution policy to achieve a desired 

income distribution consistent with the perceived economic goals of the society. That is, we try to find a fiscal-

compensation-based index for reducing inequality. By using “Deciles Implied Inequality Indices”, we may measure 

how much income (tax and subsidy) may be redistributed to reach the targeted income distribution policy. 

Keywords: Income distribution, Inequality Index, Lorenz curve, Public Finance, Equalization Policy.  
 

Keywords: JEL: C63 

1. Introduction 

For better distribution of income and welfare, the government might redistribute income through tax, subsidy, or 
transfer payments to achieve equity and promote the utility of every member of the society.  
Popular inequality indices provide comparative information about the income or wealth distribution without 
responding to the question of what the amount of the money needed to improve the distribution. In this paper, we 
define a series of income inequality indices; namely, the “Deciles Implied Inequality Indices” to show how much 
transfer payments or redistribution policies are needed to achieve a desired distribution of income in accordance of 
the economic equity goal of the policy-maker. 

There are different measures of income inequality (Atkinson (1970); Cowell (1977); Sen (1973)). Popular 
measures are such as the Gini index; the decile ratio; the proportions of total income earned by the bottom 50%, 
60%, and 70% of the households; the Robin Hood index; the Atkinson index; and Theil's entropy measure.  
The Lorenz curve may illustrate the degree of income inequality. The income inequality shown by a Lorenz curve 
can be measured and expressed by the Gini index. The Gini index is the ratio of the area between the Lorenz curve 
and the angle bisector 45° line, to the total area below the 45° line. This index expresses the distribution inequality 
just by one figure. Gini index is calculated by a ratio that its numerator is the area between the Lorenz curve and the 
diagonal line, and the denominator is the area under the uniform distribution line. This ratio is between 0 and 1.  

Kakwani (1980) by some recalculation of the Gini index measures the length of the Lorenz curve as an 
inequality index. The Robin Hood index is equivalent to the maximum vertical distance between the Lorenz curve 
and the line of equal incomes. The Atkinson (1970) index is one of the few inequality measures that explicitly 
incorporate normative judgments about social welfare. It is derived by calculating the so-called equity-sensitive 
average income, which is defined as that level of per capita income, which wether enjoyed by everybody would make 
total welfare precisely equal to the overall well-being generated by the actual income distribution. Theil (1967) 
entropy measure derives from the notion of entropy in information theory.  



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There is no single "best" measure of income inequality. Some measures such as the Atkinson index are more 
bottom-sensitive than others are; i.e., more strongly correlated with the extent of poverty. The measures perform 
differently under various types of income transfers. For instance, the Gini index is much less sensitive to income 
transfers between households, if they lie near the middle of the income distribution compared to the tails. The Robin 
Hood index is insensitive concerning income transfers between households on the same side of the mean income, and 
so on. While there are alternative methods, there is no best way to calculate the inequality index, mainly 
concentrating on fiscal view. That is, they generally analyze the distribution without inferring about the amount of 
fund needed to correct income inequality. 

Since the skewness of income distribution is persistently exhibited for different statistical populations, the 
Lorenz curve becomes a method to analyze the skew distributions. Therefore, the Pearsonian family distributions are 
rival functions to explain income distribution. There is also a relation between the area under the Lorenz curve and 
the corresponding probability distribution function (see, Kendall and Stuart (1977)). That is, when the probability 
distribution function is known, we may find the corresponding Lorenz curve and Gini index as well. 
Estimation of the Lorenz curve has some difficulties. For this estimation, we should define an appropriate functional 
form that can accept different curvatures. There is another problem, that is, to create a large-size dataset for 
estimating the corresponding parameters of the Lorenz curve, a large amount of computation on raw sample income 
data is inevitable. These problems, despite their computational difficulties, make the significance of the estimated 
parameters weak (see, Bidabad and Bidabad (1989)). To avoid this, Bidabad (1989a,b) proposed the estimation of 
the functional form of the Lorenz curve by defining and using continuous L1 norm smoothing. The continuous L1 
norm estimation problems of linear one and two parameter models were solved by Bidabad (1988a,b).  

The viewpoint of this paper is to introduce a set of “Deciles Implied Inequality Indices”, which satisfies the 
policy implications needs. These indices can be used for reducing the degrees of inequality. We show how to use 
Lorenz Curve to measure the distribution of income and to calculate the amount of money needed to be levied on 
rich and then transferred to the poor to promote income distribution of the society. However, some more 
mathematical sophistication to estimate the Lorenz Curve regarding “Implied Inequality Index” was given by 
Bidabad and Shahrestani (2008). In that paper, we introduced a shortcut to use the probability density function of 
population income to estimate the Lorenz function parameters. That is the continuous L1 norm smoothing method 
used to estimate the regression parameters of the Lorenz curves corresponding to two different probability density 
functions: (a) log-normal function that is more suitable for a broader range of income but is not integrable and (b) 
Pareto density distribution function that is integrable.  
 
2. Lorenz Curve  

The Lorenz curve is a simple case of the Concentration Surfaces in statistics that are used to show the shape and 
distribution of inequality in data. In other words, the Lorenz Curve is a representation of the cumulative income 
distribution function. It explains what portion of the total income is received by the bottom percentages of the 
households. The Lorenz curve for a random variable with probability density function f(v) may be defined as an 
ordered pair.  

    )
)(

)(
),((

VE

vVVE
vVVP


          Rv                                                                                                  (1) 

 For a continuous density function f(v), (30) can be written as, 

    )))((),(()
)(

)(
,)(( vxyvx

dwwwf

dwwwf
dwwf

v

v





 



                                                                                    (2) 

Taguchi (1972a,b,c,73,81,83,87,88) multiplies the second element of (1) by P(V|V≤v) which is not correct; his 
definition of (2) is equivalent to ours. We denote (2) by ordered pair (x(v),y(x(v))) where x(v) and y(x(v)) are its 
elements. "x" is a function which maps "v" to x(v) and "y" is a function which maps x(v) to y(x(v)). The function 
y(x(v)) is simply the Lorenz curve function. We may use the form introduced by Gupta (1984) and a modified 
version of Bidabad and Bidabad (1989), which benefits from certain properties as the explicit function of the Lorenz 
curve. 



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Gupta (1984) proposed the functional form, 
         y= xAx-1         A>1                                                                           (3) 
The modified version of Bidabad and Bidabad (1989) suggests the following functional form: 
         y= xBAx-1       B≥ 1, A≥ 1                                                                   (4) 
To estimate the above functions we need discrete data from the population, to construct relevant x and y vectors to 
estimate "A" of (3) or "A" and "B" of (4).  

 

 
3. Implied-Inequality-Index  

Most inequality indices are concentrated on the statistical aspect of the income distribution. That is, they generally 
analyze the distribution without inferring about the amount of fund needed to correct income inequality. In this 
section, we will introduce an inequality index, which shows how much money should be transferred from the upper-
income group to the lower group to achieve the desired distribution of income. Bidabad and Shahrestani (2010) 
introduced an implied inequality index using L1 norm estimation of the Lorenz curve. Here we extend this index to 
be more qualified for income redistribution policies and implications. 

Suppose there is a personal income υ at which, the half of the total income of the population belongs to 

those who have less than υ, and the other half of the income belongs to those who have a higher income than υ. That 
is: 

     





v

v

dwwwfdwwwf )()(                                                                                                                  (5)          

By definition, we have: 

    









v

v

wwfdwwwfdwwwf )()()(                                                                                      (6) 

That is: 

      2)(  

v

dwwwf                                                                                          (7) 

On the other hand: 

     21
)(

)(











dwwwf

dwwwf
v

                                                                                                                              (8) 

According to (2) this is a point on the Lorenz curve with the following ordered pair: 

     )21,)(( 

v

dwwwf                                                                                                                                (9) 

Thus, we define implied-inequality-index (iii) as  
v

dwwf )(  when υ satisfies (5).  That is, 

      
v

dwwfiii )(     when v satisfies 21
)(

)(











dwwwf

dwwwf
v

                                                                (10) 

To find iii, (7) should be solved for υ and its value be replaced in (10). As iii approaches ½, distribution 
becomes more symmetric. If iii tends to 1, distribution tends to be fully right-skewed, indicating high (right) 
inequality and as iii tends to 0, distribution tends to be left-skewed, and distribution tends to (left) high inequality. 
The values of iii less than ½, the iii has no economic implication for income distribution. Let us define the cost of 
equalization as:  

         C= [iii-½]×N×μ                                                                                                                                (11) 
This expression means that to equalize the distribution of income without changing the average income of 

the society, the amount of C in terms of money should be transferred from higher income earner to the lower income 

earner, where N and μ are the population size and average income of the society. 
We may normalize this index by dividing the equalization cost by total income of the society and find an inter-
societies comparable index. That is: 



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Relative cost of equalization = [(iii-½)×N×μ ] / ( N×μ) = (iii-½)                                                           (12) 
 
4. Numerical Example 

To illustrate, the following table 1 of income distribution for a hypothetical society is used. Consider a society of 
400 households with a total income of the society equal to $2000 where 280 poor income earners receive half of it 
($1000) and 120 richer earn another 50% ($1000) of the society’s income. These values can be simply understood 
from table 1. At the half of total income of the society ($1000), the bottom 70% of the population earns 50% of 
society’s income, and 30% of the top of the population earn other 50% of the total income of the society. 
According to table 1, we have: 
 
         N = 400                                           (Number of households) 

    υ =  μ = 2000/400=5                       (Average income) 

    μ lower   = 1000/280 = 3.57               (Average income of lower category) 

    μ upper   = 1000/120 = 8.33,              (Average income of upper category)  
    iii = 280/400 = 0.7                          (implied inequality index) 

    C = (0.7-0.5) × 400×5= $400         (Cost of equalization)  
  
That is, if we collect total tax of $400 from the top 30% of the population and transfer it to the lower 70% of 

the income earners, the average income of both groups will be the same: 
 
        (1000+400)/280 = (1000-400)/ 120 = 5 

   Relative cost of equalization = 0.7-0.5 = 0.2 = %20  
 
That is the cost of such equalization is 20% of the total income of the society.   

According to table 1, we may depict the iii and the relative cost of equalization on the Lorenz curve as follows. This 
index is illustrated by using columns (4) and (9) of table 1. Figure 1 shows the results, as well. 
In table 1, the column (1) depicts dollar values of income categories, and the column (2) shows the number or 
frequencies of households in each income category of column (1).  Columns (3), (4), and (5) are for cumulative 
frequencies, relative frequencies, and relative cumulative frequencies.  Column (6) shows the number of lower and 
higher income earners.  Column (7) shows the multiplication of the paired elements of the columns (1) and (2).  
Column (8) cumulates (7) and (9) shows the relative cumulative income.  The column (10) shows half of the total 
income of society. 
 

Table 1: Income distribution for a hypothetical society 
Income 

w 
Frequency 

f 
Cumulative 
Frequency 

F 

Relative 
Frequency 

Relative 
Cumulative 
Frequency  

Half 
Income 
Earner 

w . f 
 

(1)*(2) 

Cumulative 
Income 

Relative 
Cumulative 

Income 

Half 
Income 

($) (Numbers) (Numbers) (%) (%) (Numbers) ($) ($) ($) ($) 

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 

1 17 17 4.3% 4.3% 280 17 17 0.9%  
 
 
 

1000 

2 20 37 5.0% 9.3% 40 57 2.9% 

3 95 132 23.8% 33.0% 285 342 17.1% 

4 82 214 20.5% 53.5% 328 670 33.5% 

5 66 280 16.5% 70.0% 330 1000 50.0% 

6 30 310 7.5% 77.5% 120 180 1180 59.0%  
 
 
 
 
 
 

7 21 331 5.3% 82.8% 147 1327 66.4% 

8 18 349 4.5% 87.3% 144 1471 73.6% 

9 17 366 4.3% 91.5% 153 1624 81.2% 

10 14 380 3.5% 95.0% 140 1764 88.2% 

11 11 391 2.8% 97.8% 121 1885 94.3% 

12 4 395 1.0% 98.8% 48 1933 96.7% 



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Figure 1: Implied inequality index iii 
 

 
This figure depicts the information of table 1. The implied inequality index (iii) and the relative cost of equalization are shown 

as corresponding parts of the Lorenz curve 
 

5. Deciles-Implied-Inequality-Indices  

In this section, we will introduce deciles implied inequality indices, which can be simply generalized to quantiles or 
percentiles indices. They show how much money should be transferred from the upper-income deciles to the lower 
deciles or quantiles or percentiles groups to achieve the desired distribution of income.  

As before, suppose there is a personal income υ at which a decile or quantile of the total income of the 

population belongs to those who have less than υ, and the other income belongs to those who have a higher income 

than υ. That is for the Di (ith decile, i=1, 2, … , 10): 

     
1 2

1 9

( ) ( ) ... ( )
v v

wf w dw wf w dw wf w dw
v v




                                                                               (13)          

By definition, we have: 

    
1 2

1 9

( ) ( ) ( ) ... ( )
v v

wf w dw wf w dw wf w dw wf w dw
v v


 

 
                                                  (14) 

That is: 

13 3 398 0.8% 99.5% 39 1972 98.6%  
1000 14 2 400 0.5% 100.0% 28 2000 100.0% 

 400  100%  400 2000   2000 



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1

2

1

9

( ) 10

( ) 2 10

( ) 10 10

v
wf w dw

v
wf w dw

v

wf w dw
v























                                                                                       (15) 

On the other hand: 

     

1

2

1

9

( )
1 10

( )

( )

1 10
( )

( )

1 10
( )

v
wf w dw

wf w dw

v
wf w dw

v

wf w dw

wf w dw
v

wf w dw



































                                                                                                                         (16) 

According to (2) these points are on the Lorenz curve with the following ordered pairs: 

     ( ( ) , 10), 1,2,...,10
iv
wf w dw i i


                                                                                    (17) 

Thus, we define decile-implied-inequality-index (diiii) as ( )
iv
f w dw

  when υi satisfies (13).  That is, 

     ( )
i

i

v
iii f w dwd 
      when v satisfies 

( )
10

( )

iv
wf w dw

i
wf w dw











                                                          (18) 

To find diiii, (17) should be solved for υi, and its value be replaced in (18). As diiii approaches i/10, 
distribution becomes more symmetric. If diiii tends to 1, distribution tends to be fully right-skewed, indicating high 
(right) inequality and as diiii tends to 0, distribution tends to be left-skewed, and distribution tends to left high 
inequality. The values of diiii less than i/10, however, have no economic implication for income distribution. Let us 
define the cost of equalization policy as:  

         Ci= [diiii-i/10]×N×μ                                                                                                                         (19) 
The above expression means that to equalize the distribution of income without changing the average 

income of the society, the amount of Ci should be transferred from higher income earner to lower income earner 

(lower than the ith decile), where N and μ are the population size and average income of the society. 
We may normalize this index by dividing the equalization cost by total income of the society and find an inter-
societies comparable index. That is: 

Relative decile cost of equalization = [(diiii-i/10)×N×μ ] / ( N×μ) = (diiii-i/10)                                       (20) 
 



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Figure 2. The deciles 3 and 8 Implied Inequality Indices (d3iii and d8iii) 

 
This figure depicts the two deciles (3 and 8) of Decile Implied Inequality Indices and the relative cost of equalization as 

corresponding parts of the Lorenz curve 

 
6. Conclusion 

To have a better understanding and policy arrangements about the equality of income distribution, it is not enough 
to know the traditional inequality indices. The redistribution policies or income distribution monitoring need to deal 
with specific budget guidelines to promote the society to a better distributional position. While there are alternative 
methods for presenting income distribution, there is no best way to present the inequality index, mainly 
concentrating on fiscal viewpoints. That is, they generally analyze the statistical distribution without inferring about 
the amount of fund needed to promote income equality. The view of this paper is to introduce a set of Deciles 
Implied Inequality Indices, which satisfies these policy implications needs. The designed Deciles Implied Inequality 
Indices are good fiscal guideposts for monitoring or performing equalization policies of society's income.  

 
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0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0% 10%

20%

30%

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50%

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equalization  

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 www.cribfb.com/journal/index.php/aesr                              American Economic & Social Review                                Vol. 5, No. 2; 2019 
 

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http://economics.dal.ca/RePEc/dal/wprarch/howgini.pdf

