Parallel session IARIW 2014

1. “Accounting for changes in the distribution of
household income by its sources”
Iryna Kyzyma
CEPS/INSTEAD Luxembourg and the University of Bremen
(in co-authorship with Alessio Fusco and Philippe Van Kerm,
CEPS/INSTEAD Luxembourg)
This research is funded through the AFR PhD grant scheme from the “Fonds national de la
Recherche (Luxembourg)” (2011-2014)

2. Motivation
Alarming trends in the distribution of disposable income in rich
2
OECD countries over recent decades:
• Increase in income inequality
• Increase in relative poverty, especially for certain age
categories (children, elderly)
• Rise of polarization indexes
Example - Luxembourg:
• one of the richest countries in the world with a remarkable
economic growth over the past 25 years
• relatively low but quickly increasing income inequality
• Profound increase in relative child poverty rates

3. Motivation (cont.)
The question is:
Why? Why we observe these trends? Which factors might be
potentially responsible for them?
and correspondingly…
How? How can we identify the contributions of these factors to
the overall change in distributional measures?
3

4. Literature
Stream 1 - ‘Global methods’: decomposition is performed for
pre-determined components whose contributions sum up to
the inequality to be explained (Shorrocks, 1982; Lerman &
Yitzhaki, 1985; Mussard & Pi Alperin; 2011; Araar, 2008)
Stream 2 – ‘Local methods’: decomposition is based on before /
after calculations (Cowell & Jenkins, 1995; Cancian & Reed,
1998; Fuest et al., 2010)
Stream 3 – ‘Distributional methods’: decomposition is
performed for the total income distribution (Dinardo et al.,
1996; Jenkins & Van Kerm, 2005; Rothe, 2012; Larrimore,
2013)
4

5. Ultimate goal of this paper
To develop a decomposition method which would allow to
decompose the overall change in the income distribution
between two points in time into three sets of components
capturing the contributions of:
(a) Changes in the population structure
(b) Shifts in the marginal distributions of different income
components
(c) Changes in their dependence structure
We propose to do it using the copula function
5

6. CDF of total income at one point in time
Consider that total income of each individual i in time period t,
is composed of a set of components, yk , so that:
(1)
The CDF of this total income can then be expressed as:
(2)
6



K
k
t
ik
t
yi y
1

y yk
( ) ... ( ,..., ) ...
 




1
1
0
1 1
0
K
k
k k
t
y
t F y g y y dy dy

7. CDF of total income at one point in time (cont.)
Sklar’s theorem says that the joint CDF of income components
can be expressed as a function of their marginal distributions,
F(y1), …, F(yk), and a dependence structure between them, C:
(3)
t G y y C F y F y
( ,..., ) ( ( ),..., ( )) 1 1 1 k
Substituting Equation 3 in Equation 2 will give us:
(4)
   
( ) ... ( ( ),..., ( ))
We can also condition everything on a set of covariates, X:
(5)
7
 

1 1
1
...
0
1
0
k
k
y y y
k
t
y
t
y
t
y
t F y dC F y F y
1 1
F y dC F y X F y X X dH X t
( ) ... ( ( | ),..., ( | ) | ) ( )
    
1
...
0
| 1 |
0
y y y
k
t
y X
t
y X
t
y
t
k
k
X
   

t
y
t
y
t
k
k


8. Overall decomposition
Consider the overall change in the CDF of total income, ΔF(y),
between a base period (t = 0) and a final period (t = 1):
(6)
From Equation (5) it follows that ΔF(y) can be decomposed into
three sets of contributions induced by:
(i) changes in the distribution of population sub-groups, H(X);
(ii) changes in the marginal CDFs of income components within
each population sub-group, Fy1|X, … , Fyk|X;
(iii) changes in their dependence structure, C|X:
(7)
8
( ) ( ) ( ) (1) (0) F y  F y  F y
( ) ( ( ), ( ), ( )) (0,1) (0,1) (0,1) F y F y F y F y X M D      

9. Accounting for changes in the population structure
To account for changes in the population structure between two
points in time, DiNardo, Fortin, Lemieux (1996) re-weighting
procedure can be used:
(8)
t
Pr( 1)
t X
Pr( 0 | )
dF X t
( | 0)
dH X
( )
1



to construct the counterfactual CDF of total income in period t=1
and derive :
(9)
9
Pr( 0)
Pr( 1| )
( | 1)
( )
0





 
t
t X
dF X t
dH X

   
( ) [ ... ( ( | ),..., ( | ) | ) ( ) (1)
F y dC F y X F y X X dH X
      

...
0
(1)
1 |
(1)
|
(1)
|
0
(0,1)
1 1
1
k
k
X
y y y
X y X y X k
y
X
  ...
 
... ( ( | ),..., ( | ) | ) ( )] (0)
0
(1)
dC F y X F y X X dH X
1 |
(1)
|
(1)
|
0
1 1
1
k
k
X
y y y
X y X y X k
y
    

( ) (0,1) F y X 

10. Accounting for changes in the marginal
distributions of income components
Recall that the CDF of total income is:
(10)
1 1
F y dC F y X F y X X dH X t
    
Then, we can derive the contribution of the change in the marginal
CDFs of all income components, , as follows:
(11)
10
( ) ... ( ( | ),..., ( | ) | ) ( )
1
...
0
| | 1 |
0
y y y
k
t
y X
t
y X
t
X
y
t
k
k
X
   

( ) (0,1) F y M 
   
( ) [ ... ( ( | ),..., ( | ) | ) ( ) (1)
F y dC F y X F y X X dH X
      

...
0
(1)
1 |
(1)
|
(1)
|
0
(0,1)
1 1
1
k
k
X
y y y
X y X y X k
y
M
  ...
 
... ( ( | ),..., ( | ) | ) ( )] (1)
0
(0)
dC F y X F y X X dH X
1 |
(0)
|
(1)
|
0
1 1
1
k
k
X
y y y
X y X y X k
y
    


11. Accounting for changes in the marginal
distributions of income components (cont.)
The overall marginal effect, , can be partitioned into a set
of components which:
(i) capture contributions of changes in the marginal CDFs of
income sources separately from each other (first order effects)
and
(ii) contributions resulting from all possible interactions between
them (higher order effects):
(12)
11
( ) (0,1) F y M 
(0,1) (0,1) (0,1)
(0,1) (0,1)
F ( y ) F  F  ...
  F  F
M M 1
k j M
2 1
all
k
k
j
k
j C
M
j C
M
K
k
   

12. Accounting for changes in the marginal
distributions of income components (cont.)
The first-order effects can be identified by constructing k
counterfactual situations replacing in each of them the marginal
CDF given X of only one income component to its analog in the
base period :
(13)
  ...
 k

F y dC F y X F y X X dH X
And then taking the difference between the actual CDF of total
income in the final period and counterfactual CDF separately for
each component
In a similar way we can derive the contributions attributed to
interactions between marginal CDFs of income components
12
( ) ... ( ( | ),..., ( | ) | ) ( )] (1)
0
(1)
1 |
(0)
|
(1)
|
0
1 1
1 1
k
X
y y y
X y X y X k
y
C
 M    


13. Accounting for changes in the dependence
structure (copula)
The total copula contribution:
(14)
( ) [ ... ( ( | ),..., ( | ) | ) ( ) (1)
F y dC F y X F y X X dH X
      
  ...
 
dC F y X F y X X dH X
If needed, can be partitioned further in a set of contributions
induced by pairs or higher-order combinations of income
sources
13
... ( ( | ),..., ( | ) | ) ( )] (1)
0
(1)
1 |
(1)
|
(0)
|
0
1 1
1
k
k
X
y y y
X y X y X k
y
    

   

...
0
(1)
1 |
(1)
|
(1)
|
0
(0,1)
1 1
1
k
k
X
y y y
X y X y X k
y
D

14. Combining all parts together
Recall that the total change in income distribution is:
(15)
Hence, we have the contributions of:
-> changes in the population structure, ;
-> changes in the marginal CDFs of income sources and their
interactions, ;
-> changes in the dependence structure, ;
-> all possible interactions between these three factors
14
( ) ( ( ), ( ), ( )) (0,1) (0,1) (0,1) F y F y F y F y X M D      
( ) (0,1) F y X 
( ) (0,1) F y M 
( ) (0,1) F y D 

15. Estimation
Recall Sklar’s theorem:
(16)
t t t F y y y  C F y F y
( , ,..., ) ( ( ),..., ( )) 1 2 1
The copula function, C, in turn can be estimated as:
(17)
1t kt kt 1t t kt C u u F F u F u   
-1 is a quantile function of income component k, so that
where Fk
, 0 < rk < 1 (18)
15
( ,..., ) ( ( ),..., ( )) 1
1 1
1
t
k
t t t t
k
 
( ) 1
k y k y F r
k

16. Application
• Country: G.D. Luxembourg
• Data source: Socio-economic Panel “Liewen zu Lëtzebuerg”
• Extracted for the years: 1987 and 2010
• Income information: simulated gross values of income
components
• Definition of total net household income:
Total net income = Eh + Es + Eo + CI + ST – ITC
• Adjustments: All income components are expressed in Euros,
adjusted for prices of 2005 and the number of individuals
living in the household
16

17. Changes in the distribution of total disposable income in
Luxembourg between 1987 and 2010
Source: PSELL 1 and PSELL 3 cross-sectionally weighted data, authors’ calculations.
17
.00001 .00002 .00003 .00004 .00005
Density
0 50000 100000 150000 200000
Income (in Euros)
2010
1987

18. Changes in income inequality and poverty measures in
Luxembourg between 1987 and 2010
Source: PSELL 1 and 3 cross-sectionally weighted data, own calculations.
18
Indexes 1987 2010 Change (2010 to 1987)
Absolute Relative, %
Mean income 22728.04 37284.52 +14556.48 + 64.04
Median income 20894.39 32885.66 +11991.27 + 57.39
Standard deviation 10973.05 23213.67 +12240.62 +111.55
P90/P10 2.904 3.400 +0.496 + 17.08
P90/P50 1.669 1.834 +0.165 + 9.89
P50/P10 1.740 1.854 +0.114 + 6.55
Gini 0.241 0.273 +0.032 + 13.28
Theil index 0.098 0.135 +0.037 + 37.75
Poverty rate (%) 11.62 14.40 +2.78 + 23.92

19. Changes in the marginal distributions of different
income sources in Luxembourg
100000 150000 200000
100000 150000 200000
Earnings of other household members
100000 150000 200000
Source: PSELL 1 and 3 cross-sectionally weighted data, author’s calculations.
19
0
50000
0 .2 .4 .6 .8 1
Income in Euros
1987
2010
Earnings of household head
0
50000
Income in Euros
0 .2 .4 .6 .8 1
Quantiles
1987
2010
Earnings of spouse
0
20000 40000 60000 80000
Income in Euros
0 .2 .4 .6 .8 1
Quantiles
1987
2010
0
50000
Income in Euros
0 .2 .4 .6 .8 1
Quantiles
1987
2010
Capital income

20. Changes in the marginal distributions of different
income sources in Luxembourg (cont.)
100000 150000
100000 150000 200000
Source: PSELL 1 and 3 cross-sectionally weighted data, author’s calculations.
20
0
50000
Income in Euros
0 .2 .4 .6 .8 1
Quantiles
1987
2010
Transfer income
0
50000
Income in Euros
0 .2 .4 .6 .8 1
Quantiles
1987
2010
Income taxes

21. Decomposition results: changes in the population
structure
Source: PSELL 1 and 3 cross-sectionally weighted data, own calculations.
21
0
100000 150000 200000 250000
50000
0 .2 .4 .6 .8 1
Income quantiles
1987
2010
Counterfactual

22. Decomposition results: changes in marginal distributions
of income components (aggregate decomposition)
Source: PSELL 1 and 3 cross-sectionally weighted data, own calculations.
22
0
100000 150000 200000 250000
50000
0 .2 .4 .6 .8 1
Income quantiles
1987
2010
Counterfactual

23. Direct contributions of different income components (1)
(a) Contribution of earnings of household head
100000 150000 200000 250000
(b) Contribution of earnings of spouse
(c) Contribution of other household members
100000 150000 200000 250000
(d) Contribution of capital income
Source: PSELL 1 and 3 cross-sectionally weighted data, own calculations.
23
0
100000 150000 200000 250000
50000
0 .2 .4 .6 .8 1
Income quantiles
1987
2010
Counterfactual
0
50000
Income (in Euros)
0 .2 .4 .6 .8 1
Income quantiles
1987
2010
Counterfactual
0
100000 150000 200000 250000
50000
0 .2 .4 .6 .8 1
Income quantiles
1987
2010
Counterfactual
0
50000
Income (in Euros)
0 .2 .4 .6 .8 1
Income quantiles
1987
2010
Counterfactual

24. Direct contributions of different income components (2)
(e) Contribution of transfer income
100000 150000 200000 250000
Source: PSELL 1 and 3 cross-sectionally weighted data, own calculations.
24
0
100000 150000 200000 250000
50000
0 .2 .4 .6 .8 1
Income quantiles
1987
2010
Counterfactual
0
50000
Cumulative frequency
0 .2 .4 .6 .8 1
Income (in Euros)
1987
2010
Counterfactual
(f) Contribution of taxes

25. Decomposition results: Accounting for changes in copula
Source: PSELL 1 and 3 cross-sectionally weighted data, own calculations.
25
0
100000 150000 200000 250000
50000
0 .2 .4 .6 .8 1
Income quantiles
1987
2010
Counterfactual

26. Summarizing the contributions of different factors
Decomposition components P90/P10 P90/P50 P10/P50 Gini
index
Poverty
rate
1. Population structure +0.168 +0.102 +0.004 +0.0105 -0.081
2. Marginal CDFs of income components including:
(i) Direct contributions:
Earnings of household head +0.198 +0.041 -0.020 +0.0050 +1.095
Earnings of spouse +0.089 +0.020 -0.008 -0.0024 -0.169
Earnings of other household members +0.148 +0.087 +0.003 +0.0063 -0.082
Capital income +0.024 +0.027 +0.004 +0.0023 -0.089
Transfer income -1.682 -0.271 +0.123 -0.0780 -8.816
Income taxes -0.204 -0.070 +0.011 -0.0060 -0.660
Sum of all direct contributions -1.427 -0.166 +0.113 -0.0728 -8.721
(ii) Interactions:
Sum of all second-order interactions between components +1.411 -0.224 -0.197 +0.0537 +11.645
Sum of all higher-order interactions between components +0.145 +0.434 +0.077 +0.0320 -2.706
Sum of all contributions induced by marginal CDFs of
income components and their interactions +0.129 +0.044 -0.007 +0.0129 +0.218
3. Dependence structure (copula) +0.046 +0.050 +0.008 +0.0074 +0.805
4. Interaction between population structure and
-0.181 -0.057 +0.011 -0.0097 -0.716
marginal CDFs of income components
5. Interaction between population structure and copula -0.170 -0.062 +0.008 -0.0048 -0.724
6. Interaction between marginal CDFs and copula +0.364 +0.052 -0.050 +0.0237 +3.471
7. Interaction between population structure, marginal
+0.207 +0.054 -0.015 +0.0048 +0.284
CDFs of income components and copula
Total change due to all factors (1 through 7) +0.563 +0.183 -0.041 +0.0391 +3.257
Source: PSELL 1 and 3 cross-sectionally weighted data, own calculations. 26

27. Conclusions
• The distribution of total income has become more dispersed and
skewed to the right in Luxembourg between 1987 and 2010
• Changes in the population structure, marginal distributions of
income components and copula, if considered separately, had a
disequalizing effect on total income distribution
• There are also large interactive effects between different groups of
factors
• If disaggregated, the marginal distributions of the earnings of
household head, earnings of other members of household and
capital income are associated with the increase in inequality
• Contrarily, changes in the marginal distributions of transfers and
taxes as well as earnings of spouses impose equalizing contributions
on the distribution of total disposable income over time
27

28. Thank you for your attention!
28