Applications of the (α,β)-decomposition to study
the wage inequalities between men and women
I. Notations and definitions:
The global α-Gini index :
𝛼
│𝑥 −𝑥 │
𝐺 𝛼 (𝐱, n) = ∑𝑛𝑖=1 ∑𝑛𝑟=1 2𝑛𝑖2 𝜇𝛼𝑟(𝐱) , ∀ 𝒙𝜖 𝑅+ , and 𝜖 𝑅+ ,
1
𝛼
with 𝜇𝛼 (𝐱) = (𝑛 ∑𝑛𝑖=1 𝑥𝑖 ) .
The components of the (α,β)-decomposition :
𝑛𝑘
𝛼
𝑘
∑𝑛𝑟=1
The within-group index: 𝐺𝑘𝑘
(𝐱, 𝑛) = ∑𝑖=1
𝑛𝑘
𝛼
ℎ
∑𝑛𝑟=1
The between-group index: 𝐺𝑘ℎ
(𝐱, 𝑛) = ∑𝑖=1
The β–directional distance: Dkh (  ) 
pkh

 1

 nk nh





d kh  pkh
d kh  pkh
𝛼
│𝑥𝑖𝑘 −𝑥𝑟𝑘 │
𝛼 𝜇 𝛼 (𝐱)
2𝑛𝑘
𝑘
;
𝛼
│𝑥𝑖𝑘 −𝑥𝑟ℎ │
𝛼)
𝑛𝑘 𝑛ℎ (𝜇𝑘𝛼 +𝜇ℎ
with d kh

;
 1

 nk nh



( xik  xrh )  and



i 1 yij  yrh

nk


 .
(
x

x
)


rh
ik

i 1 yrh  yij

nk
The weighted within and between-group components :
𝛼
𝐺𝑤𝛼 (𝐱, n) = ∑𝑛𝑖=1 𝐺𝑘𝑘
𝑝𝑘 𝑠𝑘𝛼
;
𝑗−1 𝛼
𝛼 (𝐱,
𝐺𝑔𝑏
n) = ∑𝑘𝑗=2 ∑ℎ=1 𝐺𝑘ℎ
(𝑝ℎ 𝑠𝑘𝛼 + 𝑝𝑘 𝑠ℎ𝛼 )
𝛼 (𝐱,
𝛼 (𝐱,
Note that : 𝐺𝑔𝑏
n) = 𝐺𝑛𝑏
n) + 𝐺𝑡𝛼 (𝐱, n) such as ,
𝑗−1 𝛼
𝛼 (𝐱,
𝐺𝑛𝑏
n) = ∑𝑘𝑗=2 ∑ℎ=1 𝐺𝑘ℎ
𝐷𝑘ℎ (𝛽)(𝑝ℎ 𝑠𝑘𝛼 + 𝑝𝑘 𝑠ℎ𝛼 )
is the net between-group component, and,
𝑗−1 𝛼
𝐺𝑡𝛼 (𝐱, n) = ∑𝑘𝑗=2 ∑ℎ=1 𝐺𝑘ℎ
[1 − 𝐷𝑘ℎ (𝛽)](𝑝ℎ 𝑠𝑘𝛼 + 𝑝𝑘 𝑠ℎ𝛼 )
transvariation.
is the between-group component of
II. Data :
Individual’s
code
1
2
3
4
5
6
7
8
Gender
Wages
1
1
1
1
2
2
2
2
950
1100
1200
2500
1000
1050
2600
3500
α =1, β=1 yields Dagum’s result :
Segmentation variable
Name of the analysis
Number of groups
Value of alpha
Name of the group
Description of the group
Modality code
Size of the group
Total income of the group
Mean income of the group
Share of the group /total
Income of the group/Total income
Variance
Filter
gender
(alpha,beta)-Decomposition
2
1
nk
Rk
Mk
Pk=nk/n
Sk=Rk/R
Value of
beta
GTT
Total
8
13900
1737,5
1
1
966250
gender
L5:Beta-directional economic distance D
1
G1
G2
Modality 1
Modality 2
1
2
4
4
5750
8150
1437,5
2037,5
0,5
0,5
0,413669065 0,586330935
384218,75 1126718,75
gender
gender
1
2
G1
G2
G1
0,000000
0,581818
G1
G2
G1
593,75
1031,25
1131,25
593,75
1131,25
Delta Matrix
Delta Vector(DELTA kk)
G2
0,000000
G2
L6:Within-group GINI ratio (Gkk vector)
0,272482
0,206522
0,277607
L7:Weighted Within-group GINI ratio (PSGkk vector)
0,124101
0,042716
0,081385
G1
0,206522
0,296763
G2
G1
G2
L8:Between-group GINI ratio (Gkh Matrix)
0,277607
L8a : Gross between-group GINI ratio contribution (Gbbkh
Matrix)
L9:Net between-group GINI ratio contribution (Gbkh Matrix)
L10 : Between-group transvariation (Gtkh Matrix)
0,148381
G1
G2
G1
0,042716
0,148381
0,086331
G1
G2
G1
0,000000
0,086331
0,062050
G1
G2
G1
0,085432
0,062050
G2
0,081385
G2
0,000000
G2
0,162770
α = 2, β=1 yields to Chameni’s result :
Segmentation variable
Name of the analysis
Number of groups
Value of alpha
Name of the group
Description of the group
Modality code
Size of the group
Total income of the group
Mean income of the group
Share of the group /total
Income of the group/Total income
Variance
Filter
Gender
(alpha,beta)-Decomposition
2
2
nk
Rk
Mk
Pk=nk/n
Sk=Rk/R
Value of
beta
GTT
Total
8
13900
1737,5
1
1
966250
Gender
L5:Beta-directional economic distance D
1
G1
G2
Modality 1
Modality 2
1
2
4
4
5750
8150
1437,5
2037,5
0,5
0,5
0,413669065 0,586330935
384218,75 1126718,75
Gender
Gender
1
2
G1
G2
G1
0,000000
0,581818
G1
G2
G1
593,75
1031,25
1131,25
593,75
1131,25
Delta Matrix
Delta Vector(DELTA kk)
G2
0,000000
G2
L6:Within-group GINI ratio (Gkk vector)
0,280058
0,185936
0,271407
L7:Weighted Within-group GINI ratio (PSGkk vector)
0,125123
0,031818
0,093305
G1
0,185936
0,300900
G2
G1
G2
L8:Between-group GINI ratio (Gkh Matrix)
0,271407
L8a : Gross between-group GINI ratio contribution (Gbbkh
Matrix)
L9:Net between-group GINI ratio contribution (Gbkh Matrix)
L10 : Between-group transvariation (Gtkh Matrix)
0,154935
G1
G2
G1
0,031818
0,154935
0,090144
G1
G2
G1
0,000000
0,090144
0,064791
G1
G2
G1
0,063635
0,064791
G2
0,093305
G2
0,000000
G2
0,186610
α = 3, β=2 yields to the following result :
Segmentation variable
Name of the analysis
Number of groups
Value of alpha
Name of the group
Description of the group
Modality code
Size of the group
Total income of the group
Mean income of the group
Share of the group /total
Income of the group/Total income
Variance
Filter
Gender
(alpha,beta)-Decomposition
2
3
nk
Rk
Mk
Pk=nk/n
Sk=Rk/R
Value of
beta
GTT
Total
8
13900
1737,5
1
1
966250
Gender
L5:Beta-directional economic distance D
2
G1
G2
Modality 1
Modality 2
1
2
4
4
5750
8150
1437,5
2037,5
0,5
0,5
0,413669065 0,586330935
384218,75 1126718,75
Gender
Gender
1
2
G1
G2
G1
0,000000
0,869350
G1
G2
G1
176269,5313
711738,25
639863,25
176269,5313
639863,25
Delta Matrix
Delta Vector(DELTA kk)
G2
0,000000
G2
L6:Within-group GINI ratio (Gkk vector)
0,322270
0,182734
0,287286
L7:Weighted Within-group GINI ratio (PSGkk vector)
0,141688
0,025871
0,115817
G1
0,182734
0,331514
G2
G1
G2
L8:Between-group GINI ratio (Gkh Matrix)
0,287286
L8a : Gross between-group GINI ratio contribution (Gbbkh
Matrix)
L9:Net between-group GINI ratio contribution (Gbkh Matrix)
L10 : Between-group transvariation (Gtkh Matrix)
0,180582
G1
G2
G1
0,025871
0,180582
0,156989
G1
G2
G1
0,000000
0,156989
0,023593
G1
G2
G1
0,051742
0,023593
G2
0,115817
G2
0,000000
G2
0,231634
III. Remarks and interpretations :
In the examples proposed above the wage distribution has been shared out according to 2
groups, the women’s group on one hand and the men’s one on the other hand.

It is interesting to note that the (1, 1)-decomposition strictly corresponds to Dagum’s Gini
decomposition in subgroup.

Imposing α=2 and β=1 permits to retrieve Chameni’s decomposition in subgroup of the
coefficient of variation squared.

The various components must be carefully manipulated. Only the standard Gini index is
included in [0,1], since α> 1 implies that 𝐺 𝛼 (𝐱, n)ϵ [0; ∞[. However the β-directional
distances may be compared for all real positive value of β.

The β parameter represents the decision’s maker sensibility towards between-group inequality.
Its value impacts directly the net between-group term and de facto the β-directional economic
distance. The more important β is, the more the redistributive action will be.

Although the values of the various coefficients are not comparable, their respective
contributions to the global α –Gini may be appraised. Note that when β=2 the distance
between the women’s wage distribution and the men’s one increases. The decision maker is
more sensitive to wage discrepancies that men and women do not have in common (i.e. to the
net between-group inequalities). Whatever the value of α we notice that the inequalities are
generated by wage gaps between genders. And the more important β is, the higher the
between-group contribution is.

To sum up :
𝐺 𝛼 (𝐱, n)
𝐺𝑤𝛼 (𝐱, n)
𝐺𝑤𝛼 /𝐺 𝛼
𝛼 (𝐱,
𝐺𝑔𝑏
n)
𝛼
𝐺𝑔𝑏 /𝐺 𝛼
𝛼 (𝐱,
𝐺𝑛𝑏
n)
𝛼
𝐺𝑛𝑏 /𝐺 𝛼
𝐺𝑡𝛼 (𝐱, n)
𝐺𝑡𝛼 /𝐺 𝛼
𝐷𝑘ℎ (𝛽)
α=1,β=1
0,27248201
0,12410071
45,5%
0,14838129
54,5%
0,08633094
31,7%
0,06205036
22,8%
0,58181816
α=2 , β=1
0,28005797
0,12512293
44,7%
0,15493505
55,3%
0,09014402
32,2%
0,06479102
23,1%
0,58181816
α=3, β=2
0,32226984
0,141688
44,0%
0,18058185
56,0%
0,1569889
48,7%
0,02359294
7,3%
0,86935043