1. Presentation Comments
K. Decancq & C. Zoli:
Long term social welfare: mobility, social
status and inequality
Presentation and comments for the 33rd IARIW General
Conference
Florent BRESSON
CERDI, CNRS – Université d’Auvergne
August 29, 2014
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
2. Presentation Comments
Introduction
Context
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
3. Presentation Comments
Introduction
Context
Crisis has strengthen concerns about both rising inequalities and
decreasing mobility (Stiglitz, 2012; Piketty, 2013; Krueger’s “Great
Gatsby curve”).
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
4. Presentation Comments
Introduction
Context
Crisis has strengthen concerns about both rising inequalities and
decreasing mobility (Stiglitz, 2012; Piketty, 2013; Krueger’s “Great
Gatsby curve”).
Since Dalton (1920), many authors have investigated the normative
foundations of social aversion to inequality (Atkinson, 1970; Kolm,
1976), and more recently the same efforts have been performed
concerning intergenerational or intragenerational mobility (Atkinson,
1981; Dardanoni, 1993), so as to provide social indices based on
well-behaved social welfare functions.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
5. Presentation Comments
Introduction
Context
Crisis has strengthen concerns about both rising inequalities and
decreasing mobility (Stiglitz, 2012; Piketty, 2013; Krueger’s “Great
Gatsby curve”).
Since Dalton (1920), many authors have investigated the normative
foundations of social aversion to inequality (Atkinson, 1970; Kolm,
1976), and more recently the same efforts have been performed
concerning intergenerational or intragenerational mobility (Atkinson,
1981; Dardanoni, 1993), so as to provide social indices based on
well-behaved social welfare functions.
However, only few studies (Shorrocks, 1978; Kanbur & Stiglitz, 1986)
have considered the desired properties for multiperiod social welfare
functions that make it possible to take simultaneously inequality and
mobility into account.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
6. Presentation Comments
Introduction
Contribution
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
7. Presentation Comments
Introduction
Contribution
Ï Axiomatically characterize the set of intertemporal social
evaluation functions (SEF) that show concern with respect
to both inequality and mobility.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
8. Presentation Comments
Introduction
Contribution
Ï Axiomatically characterize the set of intertemporal social
evaluation functions (SEF) that show concern with respect
to both inequality and mobility.
Ï (In the spirit of Sen, 1973) Propose a family of rank
dependent SEF that are based on generalized Gini indices
weighing scheme and are sensitive to cross-sectional
inequalities and exchange mobility.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
9. Presentation Comments
Introduction
Contribution
Ï Axiomatically characterize the set of intertemporal social
evaluation functions (SEF) that show concern with respect
to both inequality and mobility.
Ï (In the spirit of Sen, 1973) Propose a family of rank
dependent SEF that are based on generalized Gini indices
weighing scheme and are sensitive to cross-sectional
inequalities and exchange mobility.
Ï Propose a Gini-based relative mobility index.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
10. Presentation Comments
Introduction
Contribution
Ï Axiomatically characterize the set of intertemporal social
evaluation functions (SEF) that show concern with respect
to both inequality and mobility.
Ï (In the spirit of Sen, 1973) Propose a family of rank
dependent SEF that are based on generalized Gini indices
weighing scheme and are sensitive to cross-sectional
inequalities and exchange mobility.
Ï Propose a Gini-based relative mobility index.
Ï Provide three decompositions
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
11. Presentation Comments
Introduction
Contribution
Ï Axiomatically characterize the set of intertemporal social
evaluation functions (SEF) that show concern with respect
to both inequality and mobility.
Ï (In the spirit of Sen, 1973) Propose a family of rank
dependent SEF that are based on generalized Gini indices
weighing scheme and are sensitive to cross-sectional
inequalities and exchange mobility.
Ï Propose a Gini-based relative mobility index.
Ï Provide three decompositions
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
12. Presentation Comments
Introduction
Contribution
Ï Axiomatically characterize the set of intertemporal social
evaluation functions (SEF) that show concern with respect
to both inequality and mobility.
Ï (In the spirit of Sen, 1973) Propose a family of rank
dependent SEF that are based on generalized Gini indices
weighing scheme and are sensitive to cross-sectional
inequalities and exchange mobility.
Ï Propose a Gini-based relative mobility index.
Ï Provide three decompositions (in fact two since the last is
a special case).
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
13. Presentation Comments
Notations
Concepts and notations
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
14. Presentation Comments
Notations
Concepts and notations
Ï N set of n individuals.
Ï T = {1,2} set of periods.
Ï xt
i income of individual i at time t .
Ï pt
X
(i ) : indicates position of i at time t in X.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
15. Presentation Comments
Notations
The multiperiod income profile
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
16. Presentation Comments
Notations
The multiperiod income profile
The whole joint distribution described by the multiperiod income
profile:
X =
¡
X1;X2;PX
¢
where:
Ï X t is the instantaneous income distribution at time t . X t is
a (strictly) ordered vector
³
xt
[1]
, xt
[2]
, . . . , xt
[n]
´
with
xt
1
< xt
2
< . . . < xt
[n],
Ï PX , the (exchange) mobility matrix, is an n ×n permutation
matrix such that:
PX (i1, i2) =
(
1 if ∃i ∈N s. t. p1
X
(i ) = i1 and p2
X
(i )= i2
0 otherwise
.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
17. Presentation Comments
Notations
The multiperiod income profile
Let suppose a society with three individuals with income pairs
(1,5), (3,4), (2,6).
1 5
3 4
2 6
| {z }
standard notation
→X =
1
2
3
;
4
5
6
;
0 1 0
0 0 1
1 0 0
| {z }
the authors’ notation
.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
18. Presentation Comments
Notations
Additional concepts and notations
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
19. Presentation Comments
Notations
Additional concepts and notations
Ï X the set of all income profiles for populations of size n
and two time periods.
Ï X(PX ) the set of all income profiles with mobility matrix PX .
Ï X
t
is equal distribution at time t .
Ï μ(X t ) is mean income at time t .
Ï W :X →R is the social evaluation function.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
20. Presentation Comments
The axiomatic framework
M-IND
Axiom 1. (Mobility preserving Independence (M-IND))
For all X,Y ,Z in X with PX = PY =PZ ,
W(X) ÊW(Y )⇔W(X +Z)ÊW(Y +Z).
⇒ sequences of rank-preserving income increments preserve
welfare orderings.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
21. Presentation Comments
The axiomatic framework
NORM
Axiom 2. (Normalization (NORM))
For all X in X s.t. PX = I if X1→X
1
and X2→X
2
with
X
1
= X
2
= μ·1n, then W(X)→¸·μ where ¸Ê 0.
⇒ money metric evaluation.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
22. Presentation Comments
The axiomatic framework
MON
Axiom 3. (Monotonicity (MON))
For all X,Y , s.t. PX = PY and xt
i
= yt
i for all i , t except for
xt
h
= yt
h
+" where " > 0, t ∈ {1,2} it holds W(X) >W(Y ).
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
23. Presentation Comments
The axiomatic framework
IME
Axiom 4. (Irrelevance of Mobility for Equal distributions
(IME))
t
t
For all X,Y in X, if X t →X
and Y t →X
for all t , then
W(X)−W(Y )→0.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
24. Presentation Comments
The axiomatic framework
IMEG
Axiom 5. (Irrelevance of Mobility for Equal Groups
(IMEG))
For all X,Y ,Z in X, for all subgroups A and B with NA ∪NB = N,
At At
At
if X →X
and Y At →X
for all t , then
W(X A,ZB)−W(Y A,ZB)→0.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
25. Presentation Comments
The axiomatic framework
MIA
Axiom 6. (Multiperiod Inequality Aversion (MIA))
For all X,Y in X with PX = PY , W(X) ÊW(Y ) if X can be
obtained from Y by a sequence of multiperiod Pigou-Dalton
transfers.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
26. Presentation Comments
The axiomatic framework
MPREF
Axiom 7. (Mobility preference (MPREF))
For all X,Y in X with X t = Y t for all t ∈T , if Y can be obtained
from X by a finite sequence of correlation increasing switches,
then W(X) ÊW(Y ).
⇒ Maximum mobility is achieved with complete rank reversal.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
27. Presentation Comments
Results
Main results
Theorem 1
For all X =
¡
X1;X2;PX
¢
in X, W satisfies M-IND if and only if
there exists functions !1
PX
and !2
PX
and an increasing and
continuous function VPX such that:
W(X) =VPX
"
X
i
!1
PX
¡
p1
X
(i ),p2
X
¢
· x1
(i )
i
+
X
i
!2
PX
¡
p1
X
(i ),p2
X
¢
· x2
(i )
i
#
.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
28. Presentation Comments
Results
Main results
Theorem 2
For all X =
¡
X1;X2;PX
¢
in X, W satisfies M-IND, MON, NORM
and IME if and only if there exists °1,°2 >0 such that:
W(X) = °1 ·
X
i
w1
PX
¡
p1
X
(i ),p2
X
¢
· x1
(i )
i
+°2 ·
X
i
w2
PX
¡
p1
X
(i ),p2
X
¢
· x2
(i )
i
,
where: °1+°2 = ¸,
X
i
w1
PX
¡
p1
X
(i ),p2
X
¢
=
(i )
X
i
w2
PX
¡
p1
X
(i ),p2
X
¢
= 1, and
(i )
wt
PX
¡
p1
¢
>0 ∀i , t and PX .
X (i ),p2
X (i )
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
29. Presentation Comments
Results
Main results
Theorem 3 (core result)
For all X =
¡
X1;X2;PX
¢
in X, W satisfies M-IND, MON, NORM
and IMEG if and only if there exists a °1,°2 > 0 such that:
W(X) = °1 ·
X
i
£
®1 ¡
¢
+¯1 ¡
p1
X (i )
¢¤
p2
X (i )
· x1
i
+°2 ·
X
i
£
®2 ¡
p1
X
¢
+¯2 ¡
(i )
p2
X
¢¤
(i )
· x2
i
,
where: °1 +°2 = ¸,
X
i
®t ¡
p1
X
¢
+
(i )
X
i
¯t ¡
p2
X
¢
=1 ∀t , and
(i )
®t ¡
¢
+¯t ¡
p1
X (i )
¢
> 0 ∀i and t .
p2
X (i )
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
30. Presentation Comments
Results
Main results
Theorem 4
For all X =
¡
X1;X2;PX
¢
in X, W in Theorem 3 satisfies MIA and
MPREF if and only if:
Ï ®t
¡
p1
X
(l )
¢
+¯t
¡
p2
X
(l )
¢
Ê ®t
¡
p1
X
(k)
¢
+¯t
¡
p2
X
(k)
¢
∀t ∈T if p1
X
(l ) <
p1
X
(k) and p2
X
(l ) < p2
X
(k), and
Ï ®2(·) is non-increasing in p1
X
(i ) and ¯1(·) is non-increasing in
p2
X
(i ).
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
31. Presentation Comments
Results
Decompositions
Considering the case ®(·) =¯(·) = 1
2 v(·), decompositions of W
can be performed:
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
32. Presentation Comments
Results
Decompositions
Considering the case ®(·) =¯(·) = 1
2 v(·), decompositions of W
can be performed:
1. Decomposition into per period contributions that depend
on mean income, periodic inequality and a reranking effect.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
33. Presentation Comments
Results
Decompositions
Considering the case ®(·) =¯(·) = 1
2 v(·), decompositions of W
can be performed:
1. Decomposition into per period contributions that depend
on mean income, periodic inequality and a reranking effect.
2. Decomposition into periodic components (mean income
and inequality) and the mobility effect.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
34. Presentation Comments
Results
The specific case of the Gini weighing scheme
Using the Gini inequality index weighing scheme one then
obtains the relative Gini mobility index:
M(X) =
1
2
·
P
i
£
p1
X
(i )−p2
X
¤
·
(i )
¡
°2
£
μ(X2)−x2
i
¤
−°1
£
μ(X1)−x1
i
¤¢
°1 ·μ(X1) ·G(X1)+°2 ·μ(X2) ·G(X2)
,
with 0 ÉM É1.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
35. Presentation Comments
Results
The specific case of the Gini weighing scheme
Using the Gini inequality index weighing scheme one then
obtains the relative Gini mobility index:
M(X) =
1
2
·
P
i
£
p1
X
(i )−p2
X
¤
·
(i )
¡
°2
£
μ(X2)−x2
i
¤
−°1
£
μ(X1)−x1
i
¤¢
°1 ·μ(X1) ·G(X1)+°2 ·μ(X2) ·G(X2)
,
with 0 ÉM É1. Noting G(X t ) the Gini index for distribution X t .
W can then be rewritten as:
W(X) = °1 ·μ(X1) ·
h
1−G(X1)
¢i
¡
1−M(X)
+°2 ·μ(X2) ·
h
1−G(X2)
¢i
¡
1−M(X)
.
so that @W
@M
Ê 0.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
36. Presentation Comments
Conclusion
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
37. Presentation Comments
Conclusion
Ï Show how basic properties, in particular M-IND and
IME(G), characterize the set of intertemporal
rank-dependant social evaluation functions that take both
inequality and exchange mobility into account,
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
38. Presentation Comments
Conclusion
Ï Show how basic properties, in particular M-IND and
IME(G), characterize the set of intertemporal
rank-dependant social evaluation functions that take both
inequality and exchange mobility into account,
Ï Propose a family of generalized Gini intertemporal social
welfare functions,
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
39. Presentation Comments
Conclusion
Ï Show how basic properties, in particular M-IND and
IME(G), characterize the set of intertemporal
rank-dependant social evaluation functions that take both
inequality and exchange mobility into account,
Ï Propose a family of generalized Gini intertemporal social
welfare functions,
Ï Introduce a (new?) mobility index.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
40. Presentation Comments
(Minor) Comments
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
41. Presentation Comments
(Minor) Comments
Ï Preliminary version ⇒ polishing is needed (references,
unexplained notations,. . . ).
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
42. Presentation Comments
(Minor) Comments
Ï Preliminary version ⇒ polishing is needed (references,
unexplained notations,. . . ).
Ï Little is said about M.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
43. Presentation Comments
(Minor) Comments
Ï Preliminary version ⇒ polishing is needed (references,
unexplained notations,. . . ).
Ï Little is said about M.
Ï Definition of Multiperiod Pigou-Dalton transfer does not fit
the interpretation that is given later (xt
k
+xt
l
= yt
k
+ yt
l
∀t ∈T ).
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
44. Presentation Comments
(Minor) Comments
Ï Preliminary version ⇒ polishing is needed (references,
unexplained notations,. . . ).
Ï Little is said about M.
Ï Definition of Multiperiod Pigou-Dalton transfer does not fit
the interpretation that is given later (xt
k
+xt
l
= yt
k
+ yt
l
∀t ∈T ).
Ï Distinction between direct and indirect effects of the
mobility matrix on the weighing scheme in W (Theorem 1)
is not clear.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
45. Presentation Comments
(Minor) Comments
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
46. Presentation Comments
(Minor) Comments
Ï Assume that there is no costs of intertemporal income
variability or that costs are outweighted by gains related to
reranking.
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
47. Presentation Comments
(Minor) Comments
Ï Assume that there is no costs of intertemporal income
variability or that costs are outweighted by gains related to
reranking.
Ï W not continuous. Marginal income increments in period t
that change PX may result in non-marginal variations of W
as they change the weight for the income at period t ′6= t .
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
48. Presentation Comments
(Minor) Comments
Ï Assume that there is no costs of intertemporal income
variability or that costs are outweighted by gains related to
reranking.
Ï W not continuous. Marginal income increments in period t
that change PX may result in non-marginal variations of W
as they change the weight for the income at period t ′6= t .
Ï Extension to more than 2 periods. Perfect mobility with
T > 2 periods?
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
49. Presentation Comments
(Minor) Comments
Ï Assume that there is no costs of intertemporal income
variability or that costs are outweighted by gains related to
reranking.
Ï W not continuous. Marginal income increments in period t
that change PX may result in non-marginal variations of W
as they change the weight for the income at period t ′6= t .
Ï Extension to more than 2 periods. Perfect mobility with
T > 2 periods?
Ï Ex-ante or ex-post evaluation ?
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality
50. Presentation Comments
That’s all folks!
Florent BRESSON CERDI, CNRS – Université d’Auvergne
K. Decancq & C. Zoli: Long term social welfare: mobility, social status and inequality