1

The Gini Decomposition: an Alternative Formulation with an Application to Tax Reform

Maria Monti

Alessandro Santoro

Università degli Studi of Milan, Italy

Università di Milano Bicocca

maria.monti@unimi.it

alessandro.santoro@unimib.it

Abstract

In this paper, we do two things. First, we offer an alternative expression of the Gini decomposition,

and particularly of its 'residual' term R, that it is based on the concept of transvariation as introduced

by the Italian statistical school. This expression yields a better understanding of the idea that R

measures a between-group phenomenon that is generated by inequality within groups. In fact, it is

shown here that in general a change in inequality within groups may alter R, but only when the

overall intensity of transvariation is also changed. Second, we apply this expression to a tax reform

aimed at decreasing between-group inequality.

Keywords: Gini coefficient, inequality decomposition, Gini residual

Jel: D63

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1. Introduction

The Gini index is surely one of the most popular tools of economic analysis. Its use is widespread

among the public opinion and not only among the economic profession. The United Nations

evaluate the trend in global inequality using the Gini. When a newspaper needs to explain to its

readers the impact of a tax reform on income inequality it often report the Gini values of the tax

schedules. This popularity is due to the fact that the Gini index is, at once, easy to intepret and to

calculate.

However, these advantages are strongly limited by the failure of the Gini index to decompose

additively into a within-group and a between-group component. In a complex world, income

inequality cannot be measured ignoring the non-income features of households (or individuals).

Two households with the same income may differ with respect their composition, sex, age, and

many other features. Only few of these could be accounted for by equivalence scales which are, in

any case, arbitrary and debatable. Dividing the population in groups and distinguishing the within-

group from the between-group component of overall inequality is thus considered as a key feature

of an inequality index.

For a long time, the best-known decomposition of the Gini index has been the one proposed by

Mookherjee and Shorrocks (1982) (building on Bahattacharya and Mahalanobis (1967), Rao

(1969), Pyatt (1976)) where the Gini was shown to be equal to the sum of three terms: a within-

group component, a between-group component and a residual, usually denoted by R. This term is

defined by Mookherjee and Shorrocks (1982) as an “interaction effect” among groups, depending

upon the frequency and magnitude of overlaps between the incomes in different groups. At this

stage of research, it was impossible to interpret R with any precision and it was believed that “that

the way in which it reacts to changes in the group characteristics is so obscure that it can cause the

overall Gini value to respond perversely to such changes” (Mookherjee and Shorrocks (1982), p.

891).

Since then, much work has been devoted to the decomposition of the Gini index. With no claim to

be exhaustive, let us just mention the works by Silber (1989), Yitzhaki and Lerner (1991), Lambert

and Aronson (1993), Yitzhaki (1994) and Dagum (1997) (a more complete list of relevant papers

can be found in Lambert and Decoster (2005)). However, we can say that the contribution by

Lambert and Aronson (1993) has been the most influential for the interpretation of the Gini

decomposition, and particularly of the residual term. Using a geometric approach, Lambert and

Aronson (1993) interpret all of the three components of the Gini index directly and explicitly in

terms of areas on the Lorenz diagram. In particular, R is shown to be equal to (twice the) area

between a particular concentration curve and the Lorenz curve of incomes. The particular

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concentration curve is obtained in two steps: first, by ordering groups from the poorest to the richest

and, second, ordering households within each of these groups in order of their incomes.

This peculiar ‘lexicographic’ ordering emphasizes the link between R and the between-groups

overlapping. To see this point, consider that, in the lexicographic ordering, the richest household of

a given group is always close to the poorest household of the next-richer group. This means that, if

this was the actual ordering, i.e. the ordering shown by the actual Lorenz curve, no overlap would

occur between groups: for any pair of groups, every member of a given group would be richer than

a member of a poorer (on average) group. On the contrary, a positive value of R is generated by the

difference between the actual ordering and this ‘lexicographic’ ordering and it depends on

‘between-groups overlapping’.

This explanation is very attractive since it is based on conventional tools such as Lorenz and

concentration curves. However, to compute R directly (i.e. not as a residual) it is necessary to

complete a number of steps. This is probably the reason why Lambert and Decoster (2005) recently

claim that they find “perhaps for the first time (…) a transparent analytical expression for R”.

In this paper, we do two things. First, we offer an alternative formulation of the Gini

decomposition, and particularly of its third term R, that it is based on the concept of transvariation.

This concept was suggested by the Italian statistical school. It was introduced by Gini (1916) and,

further used by Deutsch and Silber (1997) and Dagum (1997). We show that the derived expression

of the overlapping term R, in its continuous form, is equivalent to the one recently proposed by

Lambert and Decoster (2005). Second, to prove that this formulation can be useful in empirical

analysis, we apply it to a case of a (hypothetical) tax reform.

We believe that the formulation of R that we propose here is useful for both its interpretation and

calculation.

The paper is organized as follows. In section 2, we define the concept of transvariation and we use

it to obtain a new expression of R in discrete terms. In section 3, we derive the continuous

formulation of R, showing that it is equivalent to the one recently obtained by Lambert and

Decoster (2005). In section 4, we apply the new expression of R in the context of a revenue-neutral

tax reform aimed at helping families who rent the house where they live by taxing house-owners.

Section 5 concludes.

2 The concept of transvariation and a new expression of R (discrete terms)

Let us consider a population of n individuals. Let yi be the income of household i and

1

nii

y n=

= ∑µ the population mean income. Let us also consider k groups in which the population is

partitioned and let us denote by (j,h) a pair of different groups whose size are denoted respectively

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by nj and by nh and whose means are denoted respectively by µj and by µh. A typical income of a

household belonging to the jth group is denoted by yji while a typical income of a household

belonging to the hth group is denoted by yhl..

The concept of transvariation is expressed by Gini (1916, p.3, our translation) in the following

terms “there is transvariation between the jth and the hth group with respect to the mean µ when,

among the nj·nh differences yji-yhl that may be calculated among the two groups, some have an

opposite sign with respect to the sign of the difference µj - µh. The difference yji-yhl with opposite

sign is defined transvariation, and ji hly y− is the intensity of the transvariation”.

Deutsch and Silber (1997) show the link which exists between the transvariation as defined by

Gini and the degree of overlapping between two distributions. Thus, the transvariation concept may

be usefully employed here, as it emerges in the decomposition proposed by Dagum (1997) that we

rearrange as follows:

gbw GGG += (1)

where Gw is the conventional Gini-within component that is the weighted sum of the Gini’s within

groups1 and Ggb is defined as the Gini gross-between inequality component. This second component

is specified by Dagum as the sum

Ggb=Gnb+Gt (2)

where

( )1

2 1

k jnb jh j h h j jhj h

G G p s p s D−

= == +∑ ∑

( )( )1

2 11

k jt jh j h h j jhj h

G G p s p s D−

= == + −∑ ∑

with

1 1 ; ; 1;

k kj j j j j j jj j

p n n s n n p sµ µ= =

= = = =∑ ∑ 1 11

k kj hj h

p s= =

=∑ ∑

1 1

1( )

j hn n jhjh ji hli l

j h j h j hG y y

n n µ µ µ µ= =

∆= − =

+ +∑ ∑

and Djh is the relative economic affluence (R.E.A.) i.e the directional economic distance ratio

between the jth and the hth groups. The term Gjh is the extended Gini ratio between the jth and the hth

group as defined by Dagum (1980) where ∆jh is the absolute mean difference between the two

groups. Dagum (1997) emphasizes the difference between Gnb, that he defines as the net-between

Gini component, and Gt, that it is interpreted as the component more directly connected to the

concept of transvariation. However, for our purposes it is more useful to rewrite (1) as follows

G=Gw+ ( )1

2 1

k j jhj h h jj h

j hp s p s

µ µ−

= =

∆+

+∑ ∑ (3)

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where

hj

n

i

n

lhlji

jh nn

yyj h

∑∑= =

−=∆ 1 1 (4)

Using the transvariation concept, we are now able to decompose the absolute mean difference

between the two groups (i.e. ∆jh) into two parts.

Let us consider the jth and the hth groups. We suppose that incomes have the same frequency within

each group2, so that income frequencies are respectively 1/nj and 1/nh. Without loss of generality,

we assume that the jth group average income µj is greater than the average income of the hth group µh

( )j hµ µ> . Let us write

'hly , "

jiy if yhl> y ji

and

"hly , '

jiy if yhl< y ji.

Now it can be shown (see Gini (1916) and Appendix 1) that

jhhjhj

n

i

n

lhlji Tnnyy

j h

2)(1 1

+−=−∑∑= =

µµ (5)

where

( )∑

>

−≡jh

jihljh yyT )( '''

is the sum of all transvariations between the two groups and thus it is directly related to the intensity

of transvariations as defined by Gini (see before). It can be said that Tjh represents the ‘amount of

overlapping’ between the two groups. Clearly, Tjh is non-zero if (and only if) there is some

‘between-groups overlapping’.

Using (5), (4) rewrites immediately as

jhhjjh p2+−=∆ µµ (6)

where

hj

jhjh nn

Tp ≡

can be interpreted as the ‘average amount of overlapping’ obtained using as the weights the joint

income frequency 1 j hn n .

We remark that to construct pjh one has to consider two elements: (i) the differences between

incomes of the individuals, belonging to the more affluent group, poorer than individuals belonging

to the less affluent group, (ii) the relative frequency of the incomes involved in the differences3.

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We now use (6) to obtain a new expression for R in discrete terms.

By first, we substitute (6) in (3), and thus we obtain the Dagum decomposition of the Gini index in

the following alternative form4

( )1

22 1

k j j h j hw j h

n nG G

n

µ µ

µ−

= =

−= + ∑ ∑ +

122 1

2k j j h

jhj h

n np

n µ−

= =∑ ∑ (7)

Then, we compare (7) with the Gini index decomposition (8) proposed by Mookherjee and

Shorrocks (1982)

w BG G G R= + + (8)

where the Gini-between component (GB) is written as

12B j h j h

j h

G p p λ λ≡ −∑∑

and ; . j j j jp n n λ µ µ= =

In Appendix 2 we show that

( )1

22 1

k j j h j hB j h

n nG

n

µ µ

µ−

= =

−= ∑ ∑ (9)

so that5

1

22 12

k j j hjhj h

n nR p

n µ−

= == ∑ ∑ (10)

As it was recalled above, Lambert and Aronson (1993) show that R is equal to (twice the) area

between a particular concentration curve and the Lorenz curve of incomes.

Here, we have obtained an analytical expression of this area in discrete terms, as the weighted sum

of the “average amount” of the overlapping between each pair of groups, pjh, with weights being

equal to 2j hn n

n µ. This expression is suitable for empirical application as it will be shown in section 4.

Moreover, equation (10) sheds some light also on the interpretation of R. When frequencies and

overall means are unaltered, R will not be changed unless the overall intensity of transvariation is

also changed.

This is an important finding, as it can be appreciated by considering two groups. By looking at (10)

it is evident that any transfer within a given group will not alter R unless it changes the sum of all

transvariations between the two groups. For example, a progressive transfer does not change R if

the rank of the income-units involved in the transfer is preserved. Note that this is more general than

the result, obtained by Lambert and Decoster (2005, p.7), that R will be unaltered when the range of

incomes across which the transfer takes place is absent for the other group.

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3. A comparison with Lambert and Decoster (2005)

In this section we assume that the income distribution functions of the two groups are continuous

on (-∞, +∞) and that they are denoted by Fj(y) and Fh(y), with densities ( )jf y and ( )hf y . Assuming

incomes are all positive, we introduce here a new notation for group means

( ) ( )

( ) ( )0

0

j j j

h h h

M y yf y dy

M y yf y dy

µ

µ

∞

∞

= =

= =

∫∫

and , w.l.g., we postulate that:

( ) ( )j hM y M y> .

It can be shown6 that

,j h jh jhd p∆ = + (11)

where

djh ( ) ( ) ( )j h h j hM F y Y M F y Y M Y⎡ ⎤⎡ ⎤= + −⎣ ⎦ ⎣ ⎦ (12)

pjh ( ) ( ) ( )j h h j jM F y Y M F y Y M Y⎡ ⎤⎡ ⎤= + −⎣ ⎦ ⎣ ⎦ (13)

Expression (13) is just the continuous counterpart of the 'average amount of overlapping'

discussed above. The derivation of this expression hinges on the use of joint density functions as the

weighting factors (see Appendix 3).

It is immediate to rewrite (11) as

( ) ( ) ( )2 2 ( ) ( )jh j h h j h jM F y Y M F y Y M Y M Y⎡ ⎤⎡ ⎤∆ = + − +⎣ ⎦ ⎣ ⎦ (14)

and, adding and subtracting 2Mj(y), as

( ) ( ) 2jh j h jhM Y M Y p⎡ ⎤∆ = − +⎣ ⎦ (15)

which is equivalent to (6).

By repeating the same reasoning that we followed in previous section to derive expression (10)

from expression (6) we obtain the continuous counterpart of R

R=1

22 12

k j j hj h

n nn µ

−

= =∑ ∑ ( ) ( )( )( )j h h j jM F y Y M F y Y M Y⎡ ⎤⎡ ⎤ + −⎣ ⎦ ⎣ ⎦ (16)

However, equation (16) is not so simple to use. Therefore, we need to look for an alternative

expression for R.

Proposition: When the population is divided in two groups, h and j, and the income distribution

functions of the two groups, denoted by Fj(y) and Fh(y), with densities ( )jf y and ( )hf y , are

continuous on (-∞, +∞), the Gini index decomposes as follows

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G=Gw+GB+R

where

( )122 1

2k j j h

jhj h

n nR p

n µ−

= == ∑ ∑ = 1

22 12

k j j hj h

n nn µ

−

= =∑ ∑ ( ) ( )( )1j hF y F y dy∞

−∞−∫

as claimed by Lambert and Decoster (2005).

Proof Our starting point is the following

( ) ( ) ( )2 min ,jh j h j hM y M y M y y⎡ ⎤∆ = + − ⎣ ⎦ (17)

As it is well known, when two independent and equally distributed variables X1 and X2 are

considered, the probability function of Y=min(X1,X2) is defined as

( ) ( ) ( ) ( )1 2 1 21 , 1YF y P X y X y P X y P X y= − ≥ ≥ = − ≥ ≥

( ) ( ) 21 1YF y F y⎡ ⎤= − −⎣ ⎦ (18)

Since frequency distribution functions are independent but generally not equal, we rewrite (18) as

( ) ( ) ( )1 1 1j hG y F y F y⎡ ⎤ ⎡ ⎤= − − −⎣ ⎦⎣ ⎦ (19)

and then we obtain (see Appendix 4)

( ){ }min ,j hM Y Y = ( ) ( )( )1h j hM y F F y dy∞

−∞− −∫ (20)

Substituting (20) in the expression of the absolute mean difference ∆jh (17) one has

( ) ( ) ( ) ( )( )2 1jh j h j hM y M y F y F y dy∞

−∞⎡ ⎤∆ = − + −⎣ ⎦ ∫ (21)

Then, comparing the expressions (21) and (15) one has

( ) ( )( )1jh j hp F y F y dy∞

−∞= −∫ (22)

and the proposition is proved by substitution of (22) in (10).■

As Lambert and Decoster (2005, p. 6) stress, the expression of R shown above is transparent and

useful for empirical application, as it can be said also for its discrete counterpart (10) obtained in

previous section.

4. Application to a tax reform

This section is devoted to an empirical application of the discrete version of the Gini decomposition

illustrated in previous sections. In particular, we focus here on the calculation of R for a simple case

where the population is divided in two groups and a tax reform is implemented. The reform we

depict here is of the following kind: we imagine that the State wants to help families who do not

own the house where they live and who therefore pay a rent for it. We interpret total monthly

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equivalent consumption as a proxy for total monthly equivalent income and note that house-owners

are on average richer than house-renters.

The additional features of the reform are the following:

1) the State offers a tax refund of a given percentage of equivalent income of the house-renters;

2) this tax refund generates a loss of revenue which is financed by an increase in taxation on the

house-owners (i.e. the reform is pure or revenue-neutral);

3) the increase in taxation is such that the equivalent income of house-owners is reduced by the

same percentage.

Features 1) and 3) are such that the reform, by definition, does not alter the Gini coefficient within

each group (but does alter the Gini-within component). This makes the reform highly abstract since:

i) given the objective to help families who rent their house, it would make more sense to grant a tax

credit proportional to the amount of rents, rather than to total income;

ii) it is not easy to figure out the technicalities which would make it possible to obtain, via the tax

system, a proportional increase (decrease) of equivalent income among families who rent (own) the

house where they live.

The reason why we choose such an artificial reform is that we want to give some insights on the

properties of R as calculated in section 3. In particular, we show here that R can be further

decomposed in two components: a 'between-alike' component and two 'within-alike' components.

The starting point of this decomposition is the overlapping set, which includes incomes from the

richest house-renter and the poorest house-owners.

4.1. The dataset

The dataset is Indagine sui consumi delle famiglie italiane nell'anno 2000, i.e. is the survey on

consumption by Italian families in year 2000, issued by Istat, the Italian Institute of Statistics. It

comprises data about 23.728 households which are representative of approximately 21millions of

Italian families. For each household, information is available on:

-the level of monthly consumption of a number of food and non-food commodities;

-the level of investment in durables;

-the composition of the household, age, profession and other socio-economic variables.

The distribution of the sample with respect to the number of household components is described in

Table 1 below.

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Table 1: Description of the sample

Type of household Number in sample %

single 4.736 19,96

2 comp. 5.979 25,20

3 comp. 5.513 23,23

4 comp. 5.448 22,96

5 comp. 1.610 6,79

6 comp. 334 1,41

7+ compon. 108 0,46

All families 23.728 100,00

For each household we calculate the total monthly consumption as the sum of food and non-food

commodities thus excluding investment on durables as well as savings. We interpret this as monthly

income. Then we apply the equivalence scale which is used by Istat in the computation of the

relative poverty line. For simplicity, we express this scale adopting the single as the reference type,

so that the coefficient increases by 2/3 for the second component, by 0,55 for the third, by 0,5 for

the fourth and so on. The coefficients of this scale as well as the resulting average values of

equivalent monthly income are reported in Table 2.

Table 2: Average equivalent income

Type of household Coefficient of equivalence Average (in €) equiv. income

single 1 1.322

2 comp. 1,666667 1.141

3 comp. 2,216667 1.092

4 comp. 2,716667 982

5 comp. 3,166667 851

6 comp. 3,6 790

7+ comp. 4 811

All families - 1.103

The average equivalent income decreases as the number of the components of the family increases

except for the last aggregation (families with at least 7 components being apparently richer than

families with 6 components) whose statistical reliability, however, is severely limited by the low

number of sampled households.

Next, we construct the overall distribution of equivalent income and we calculate the Gini

coefficient of this distribution as illustrated in Table 3.

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Table 3: Gini coefficient of equivalent income before the reform

Mean 1.103

Covariance 181

Gini coefficient of equivalent income 32,94%

4.2 The decomposition of the Gini before the reform

The reform we consider here aims at subsidizing families who rent the house where they live by

taxing families who own the house where they live. Such a policy can decrease between-group

inequality since the house-renters group are on average poorer than house-owners (see Table 4).

Table 4: Average equivalent income of owner and house-renters, before the tax reform

Group Number of families Average equivalent income

House-owners 19.390 1.143

House-renters 4.338 926

All families 23.728 1.103

We now present a further decomposition of R which is particularly useful when the population is

divided in two groups as it happens here. Since we observe that the constant income frequency

assumption is verified, we rewrite (10) as

22

jhR Tn µ

= (23)

where j is the group of house-owners and h is the group of house-renters.

Let us denote with x the income of the house-renters and with y the income of house-owners. Also, l

indicates a house-renter (hr), l=1,...,nR, and i indicates a house-owner (ho), i=1,...,nO. With a little

change of notation, here we write

{ }∑

>

−=il

iljh yxT )( (24)

where the set {l>i}is obtained form the overlapping set defined as follows

L={ maxmin ,... li xy } (25)

where maxjx is the maximum income of the hr's and min

iy is the minimum income of the ho's.

Clearly, R=0 if the overlapping set is void, i.e. if (and only if) miniy > max

jx . If the overlapping set is

not void, then its subset {l>i} can be obtained by considering all pairs of ho's and hr's such that the

income of the former is lower.

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Now, let us focus on the summation term in [24]. It is immediate that this summation can be

rewritten as follows

{ }{ }

))()()()(()( iLL

lLL

il ilil yyMxMxyMxMyx −+−+−=−∑ ∑

> >

(26)

where )( LxM is the mean equivalent income of hr's belonging to the overlapping set and )( LyM is

the mean equivalent income of ho's belonging to the overlapping set. It can be shown that the

summation can be rewritten in terms of the number of transvariations in the following way:

{ }

{ } ∑∑∑ −+−+−=− >> i

iL

ij

Lll

LLil

ilil yyMyntxMxxntyMxMNyx ))()(())()(())()(()( (27)

where N{l>i} is the total number of transvariations (i.e the number of cases where a house-renter as

an income higher than a house-owner), nt(xl) is the number of transvariations where xl is involved

(i.e. the number of cases where xl is higher than a house-owner) and, similarly, nt(yi) is the number

of transvariations where yi is involved (i.e. the number of cases where yi is lower than a house-

renter)7. This expression is particularly useful for empirical applications since all of its terms can be

calculated using a statistical package8.

Substituting in (24) we obtain a new decomposition of R in three components:

WoWrB RRRR ++= (28)

where

22n

RB µ= { } ))()(( LL

ij yMxMN −>

22n

RWr µ= ∑ −

j

Ljj xMxxnt ))()((

22n

RWo µ= ∑ −

ii

Li yyMynt ))()((

RB can be defined as a 'between-alike' component of R, since its sign depends on the comparison of

subgroup means, i.e group means in the overlapping set. RWr and RWr can be interpreted respectively

as the 'within-alike' component of the subgroup of house-renters and for that of house-owners.

Intuitively, the link between these 'within-alike' components and within-group variability may be

found by disaggregating further these components so to insulate the absolute difference from the

means.

We report the value of this three components, jointly with the value of GB and GW in Table 5

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Table 5: Decomposition of the Gini before the tax reform

Component Value

GB 2,94%

GW 23,51%

R 6,5%

-of which RB -2,2%

-of which RWr 4,34%

-of which RWo 4,32%

Gini coefficient 32,94%

Note that the negative sign of the RB component derives from the fact that the overlapping set is

very large (more than 98% of the cases are included) and therefore we have )()( LL yMxM < . The

magnitude of RB, however, depends not only on the difference between subgroup means, but also on

the number of transvariations (see Table 6).

Table 6: Decomposition of RB, before the reform

Number of transvariations 33.331.936M(xL) 926 M(yL) 1.127

RB 2,2%

4.3 The decomposition of the Gini after the reform

The reform is specified along the lines described above: the equivalent income of all house-renters

is increased by 10% while the equivalent-income of all house-owners is decreased by

approximately 1,7%. This reform is (approximately) revenue-neutral taking into account monetary

(rather than equivalent) flows (see Table 7).

Table 7: Parameters of the tax reform

Average increase in equivalent income for house-renters 92,6

Total increase in equivalent income for house-renters 401.661

Revenue loss (total monetary cost of the subsidy) 738.505

Average decrease in equivalent income for house-owners 19,3

Total decrease in equivalent income for house-owners 373.900

Revenue increase 738.504

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The reform is also rank-preserving between groups, as shown in Table 8.

Table 8: Average equivalent income of owner and house-renters, after the tax reform

Group Number of families Average equivalent income

House-owners 19.390 1.123

House-renters 4.338 1.019

All families 23.728 1.104

The decomposition of the Gini after the tax reform is reported in Table 9.

Table 9: Decomposition of the Gini after the tax reform

Component Value

GB 1,42%

GW 23,2%

R 8,13%

-of which RB -1,2%

-of which RWr 4,6%

-of which RWo 4,7%

Gini coefficient 32,74%

Table 10: Decomposition of RB, after the reform

Number of transvariations 37.925.252

M(xL) 1.019 M(yL) 1.113

RB -1,2%

In sum, the tax reform has reduced the inequality, but this reduction has emerged as the net effect of

two opposite changes: a decrease in GB and an increase (in absolute terms) in R, while GW has kept

(almost) constant. Clearly, this is not surprising since the reform was designed in order to reduce

between-group inequality while keeping within-group inequality constant without dramatically

changing average values.

However, the advantage of the decomposition proposed here is that we can clearly see what

originates the increase in R. In this case, the 'within-alike' components of R has not been altered

dramatically. On the contrary, the absolute value of the 'between-alike' component of R (i.e. RB) has

been reduced. This result could not be taken for granted since it stems, in turn, from two opposite

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changes: an increase in the number of transvariations (from approximately 33 millions to

approximately 38 millions) and a decrease in the difference between subgroup means.

5. Concluding remarks

In this paper we have obtained a new expression for the 'residual' R of the Gini decomposition

which is based on the concept of transvariation, introduced by Gini himself and developed further

by the Italian statistical school, namely by Dagum, throughout the 20th century. This expression is

useful for both the interpretation and the calculation of R.

When the population of income units is divided in groups, a transvariation between two groups is

said to exist when an income-unit belonging to the poorer group has an income higher than that of

an income-unit belonging to the richer group. The intensity of a transvariation is thus measured by

the difference between these two incomes.

We show here that the area between the particular 'concentration curve' obtained by using the

'lexicographic ordering' proposed by Lambert and Aronson (1993) and the Lorenz curve of incomes

is directly connected with the intensity of transvariations. More precisely, this area is equal to a

weighted sum of the 'average amount' of the overlapping between each pair of groups, where the

'amount of overlapping' is the sum of all transvariations between each pair.

This expression allows to reach a better understanding of the idea that R measures "a between-group

phenomenon (...) that is generated by inequality within groups" (Lambert and Aronson, 1993,

p.1224). In fact, it is shown here that in general a change in inequality within groups may alter R,

but only when the overall intensity of transvariation is also changed. This implies, for example, that

a transfer within a group does not change R if the rank of the income-units involved in the transfer

is preserved. Moreover, both the discrete and the continuous expression of R obtained here are

transparent and useful for empirical applications.

Future work may further emphasize the importance of the concept of transvariation for the

interpretation of the nature and implications of the Gini decomposition.

NOTES 1Dagum writes the Gini between component as

1 1 1 ; ; 1

k k kw jj j j j j j j j j jj j j

G G p s p n n s n n p sµ µ= = =

= = = = =∑ ∑ ∑

Generally this component is written as in Mookherjee and Shorrocks (1982) 2

W j j jjG p Gλ= ∑ and . j j j jp n n λ µ µ= =

16

Being Gj=Gjj= 2j

jµ∆

Gw and GW has the same formal expression

21 1

2

k kj j j jw j j j Wj j

j

n nG G p G

n nµ

λµ µ= =

∆= = =∑ ∑

2This hypothesis is not substantial: it is assumed only for convenience for the discrete case and will be

removed in the continuous case. 3The importance of this second component will be better appreciated in next section. 4Recall that

G=Gw+ ( )1

2 1

k j jhj h h jj h

j hp s p s

µ µ−

= =

∆+

+∑ ∑

2jh j h jhpµ µ∆ = − +

( )( ) ( )1 1

2 1 2 1

1 12k j k j

w j h j h h j jh j h h jj h j hj h j h

G G p s p s p p s p sµ µµ µ µ µ

− −

= = = == + − + + +

+ +∑ ∑ ∑ ∑

Observing that ( ) ( ) 2j h h j j h j hp s p s n n nµ µ µ⎡ ⎤+ = +⎣ ⎦ , one has

( )1

22 1

k j j h j hw j h

n nG G

n

µ µ

µ−

= =

−= + ∑ ∑ +

122 1

2k j j h

jhj h

n np

n µ−

= =∑ ∑ .

5Clearly (10) can be further simplified under the constant income frequency hypothesis: see equation (23). 6See Dagum (1997) and Appendix 3. 7Clearly, it holds that { }

{ } { }∑∑

>∈>∈> ==

ilii

illlil yntxntNT )()( .

8A syntax file for SPSS®is available upon request.

17

REFERENCES

Amerio, L. Analisi matematica, Vol.2, UTET, 1977, Torino. (In Italian)

Bhattacharya, N. and Mahalanobis, B. (1967). “Regional disparities in household consumption in

India”, Journal of the American Statistical Association, vol. 62, pp. 143-161..

Dagum, C. (1980). “Inequality Measures between Income Distributions with Applications”,

Econometrica, vol. 48, pp. 1791-180.

Dagum, C. (1997). “A New Approach to the Decomposition of the Gini Income Inequality Ratio”,

Empirical Economics, vol. 22, pp. 551-531.

Deutsch, J. and Silber, J. (1997). “Gini’s “Transvariazione” and the Measurement of Distance

Between Distributions”, Empirical Economics, vol. 22, pp.547-554.

Gini, C. (1959). “Il concetto di transvariazione e le sue prime applicazioni”. (1916).Reproduced by

Corrado Gini Memorie di metodologia statistica. Volume secondo: Transvariazione, pp 1-55,

Università degli Studi di Roma. (in Italian)

Lambert, P. and Aronson, J.R. (1993). “Inequality decomposition Analysis and the Gini Coefficient

Revisited”, The Economic Journal, vol.103, pp. 1221-1227.

Lambert, P. and Decoster,A. (2005). “The Gini Coefficient reveals more” Working paper series.

February 2005

Mookherjee, D. and Shorrock, A. (1982). “A Decomposition Analysis of the Trend in UK Income

Inequality”, The Economic Journal, vol.92, pp. 886-902.

Pyatt, G. (1976). “On the Interpretation and desegregations of Gini Coefficient”, Economic Journal,

vol.86, pp.243-255.

Rao, V. (1969). “The Decomposition of the Concentration ratio”, Journal of the Royal Statistical

Society, vol. 132, pp. 418-25.

Silber, J.(1989). “Factor Components,Population Subgroups and the Computation of the Gini Index

of Inequality”, The Review of Economics and Statistics, Vol. 71, pp 107-115.

Yitzahaki, S. and Lerman,R. (1991).”Income stratification and income inequality”, Review of

income and wealth, vol 37, pp. 313-329.

Yitzahaki, S. (1994). “Economic distance and overlapping of distributions”, Journal of

Econometrics, vol. 61, pp. 147-159.

18

APPENDICES

Appendix 1.

1 1j hn n

ji hli ljh

j h

y y

n n= =

−∆ = ∑ ∑

Let be ( ) ( )j hM y M y> and w the number of difference where yhl>yji

We write ' " and if hl ji hl jiy y y y>

" ' and if hl ji hl jiy y y y<

Then we have

( )( )

( ) ( ) ( )( )

( ) ( ) ( )

' " ' "1 1

' " ' " ' "

' "1 1

' "

2

2

2

j h

j h j h

h j h j

j l

n nji hl hl ji ji hli l w w n n w n n w

ji ji hl hl hl jin n n n w

n nh ji j hl hl jii l w

h j j j h h hl jiw

y y y y y y

y y y y y y

n y n y y y

n n M y n n M y y y

= = − −

= =

− = − + − =

= + − + − − =

= − + − =

= − + −

∑ ∑ ∑ ∑ ∑ ∑∑ ∑ ∑

∑ ∑ ∑∑

And jh∆ rewrites as

( ) ( )( )' "2 hl jiw

jh j hh j

y yM y M y

n n

−∆ = − +

∑

Appendix 2

Mookherjee and Shorrocks (1982) show that the Gini index can be written as

2 12j j j j h j hj

j h

G p G p p Rλ λ λ= + − +∑ ∑∑ (A.1)

with

2 1 1

122

n nj hj h

G y yn µµ = =

∆= − =∑ ∑

2 1 1

122

j jn n jj i hi h

jj j

G y yn = =

∆= − =∑ ∑ µµ

and . j j j jp n n λ µ µ= =

Defining

19

2W j j jj

G p Gλ= ∑ and 12B j h j h

j h

G p p λ λ= −∑∑

expression (A.1) rewrites as

W BG G G R= + +

Comparing expression (7) in the paper

( )1

22 1

k j j h j hw j h

n nG G

n

µ µ

µ−

= =

−= + ∑ ∑ +

122 1

2k j j h

jhj h

n np

n µ−

= =∑ ∑

with expression (A.1), we observe GW =Gw.(see note 1) Moreover, in (A.1) we consider groups

ranked in the non increasing order of the average incomes, then taking in account only positive

differences between average incomes, the term GB in (A.1) and the second term of expression (7)

are equivalent.

Appendix 3

We have to show that

,j h jh jhd p∆ = +

with

djh ( ) ( ) ( )j h h j hM F y Y M F y Y M Y⎡ ⎤⎡ ⎤= + −⎣ ⎦ ⎣ ⎦

pjh ( ) ( ) ( )j h h j jM F y Y M F y Y M Y⎡ ⎤⎡ ⎤= + −⎣ ⎦ ⎣ ⎦

quoted as (19) and (21) in Dagum (1997) and (12) and (13) in the paper

To avoid misunderstanding, we call

x the h-th group incomes with density function ( )hf x

z the j-th group incomes with density function ( )jf z

and we remember that ( )jf z ( )hf x = ( ),f z x

We write

( ) ( ), 0 0j h j hz x f z f x dzdx∞ ∞

∆ = −∫ ∫ = ( ) ( ) ( )0 0

z

j hf z dy z x f x dx∞

−∫ ∫ + ( ) ( ) ( )0 j hz

f z dy x z f x dx∞ ∞

−∫ ∫

By first we show that the first part in the sum ∆jh may be written as (12)

20

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

0 0 0 0 0

0 0 0 0 0

Remembering that for the integration order inversion one has (see Amerio pag.386)

( , )

z z z

j h j h h

z z

h j j h j h j h

f z dz z x f x dx f z dz z f x dx x f x dx

zF x f z dz f z dz x f x dx M F x Z f z dz x f x dx

f z x

∞ ∞

∞ ∞ ∞

⎡ ⎤− = − =⎢ ⎥⎣ ⎦

⎡ ⎤− = −⎣ ⎦

∫ ∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫

( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

0 0 0T

0 0 0 0

0

0 0

, ,

we have

1 ( )

Then

( )

Calling all incomes w

h z h h

x

z

j h h j h jx x

j h h h j

z

j h j h h h j

dzdx dz f z x dx dx f z x dz

f z dz x f x dx f x dx xf z dz xf x dx f z dz

x F z f x dx M X M F z X

f z dz z x f x dx M F x Z M X M F z X

∞ ∞ ∞ ∞ ∞

∞

∞

= =

= = =

⎡ ⎤− = + ⎣ ⎦

⎡ ⎤⎡ ⎤− = − +⎣ ⎦ ⎣ ⎦

∫ ∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫ ∫∫

∫ ∫

( ) ( )ith we obtain (12)

( )jh j h h h j

y

d M F y Y M Y M F Y Y⎡ ⎤⎡ ⎤= − +⎣ ⎦ ⎣ ⎦

Now we have to show that the second part of the sum ∆jh may be written as (13)

Remembering that

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 0 0 0

z

j h j h hzz x f z f x dzdx f z dz z x f x dx z x f x dx

∞ ∞ ∞ ∞⎡ ⎤− = − + −⎢ ⎥⎣ ⎦∫ ∫ ∫ ∫ ∫ =

( ) ( ) ( ) ( ) ( ) ( ) ( )0 0 0

( )z

j h j h j hzf z dz z x f x dx f z dz z x f x dx M Z M X

∞ ∞ ∞= − + − = −∫ ∫ ∫ ∫

Then

( ) ( ) ( ) ( )0

( )jh j h j hzd f z dz z x f x dx M Z M X

∞ ∞+ − = −∫ ∫

Substituting the obtained expression of djh and considering positive differences, one has

( ) ( ) ( )0 j hz

f z dy x z f x dx∞ ∞

−∫ ∫ = ( ) ( )j h h jM F x Z M F z X⎡ ⎤⎡ ⎤+ +⎣ ⎦ ⎣ ⎦ ( )jM Z−

and then we obtain (13)

( ) ( )jh j h h jp M F y Y M F y Y⎡ ⎤⎡ ⎤= + +⎣ ⎦ ⎣ ⎦ ( )jM Y−

Appendix 4

The probability function of Y=min(Yj,Yh,) is defined as

( ) ( ) ( )1 1 1j hG y F y F y⎡ ⎤ ⎡ ⎤= − − −⎣ ⎦⎣ ⎦

Then, remembering that M(X) may be written as

( ) ( ) ( )0

01M X F x dx F x dx

∞

−∞⎡ ⎤= − −⎣ ⎦∫ ∫

we write

21

( ){ } ( ) ( ) ( ) ( ){ }0

0min , 1 1 1 1 1j h j h j hM Y Y F y F y dy F y F y dy

∞

−∞⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= − − − − − −⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦∫ ∫

After some simple calculations one obtains

( ){ }min ,j hM Y Y = ( )( ) ( )( ) ( ) ( )( )0 0

0 01 1 1h j h h j hF y dy F F y dy F y dy F F y dy

∞ ∞

−∞ −∞− − − − − −∫ ∫ ∫ ∫

From which it derives

( ){ }min ,j hM Y Y = ( ) ( )( )1h j hM y F F y dy∞

−∞− −∫