Session 8 a presentation decancq-zoli

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

Introduction
Context

Introduction
Context
Crisis has strengthen concerns about both rising inequalities and
decreasing mobility (Stiglitz, 2012; Piketty, 2013; Krueger’s “Great
Gatsby curve”).

Introduction
Context
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.

Introduction
Context
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.

Introduction
Contribution

Introduction
Contribution
Ï Axiomatically characterize the set of intertemporal social
evaluation functions (SEF) that show concern with respect
to both inequality and mobility.

Introduction
Contribution
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.

Introduction
Contribution
Ï Propose a Gini-based relative mobility index.

Introduction
Contribution
Ï Provide three decompositions

Introduction
Contribution
Ï Provide three decompositions (in fact two since the last is
a special case).

Notations
Concepts and notations

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.

Notations
The multiperiod income profile

Notations
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
.

Notations
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
.

Notations
Additional concepts and notations

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.

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.

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.

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 ).

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.

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.

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.

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.

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
#
.

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 )

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 )

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 ).

Results
Decompositions
Considering the case ®(·) =¯(·) = 1
2 v(·), decompositions of W
can be performed:

Results
Decompositions
can be performed:
1. Decomposition into per period contributions that depend
on mean income, periodic inequality and a reranking effect.

Results
Decompositions
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.

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.

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.

Conclusion

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,

Conclusion
Ï Propose a family of generalized Gini intertemporal social
welfare functions,

Conclusion
Ï Propose a family of generalized Gini intertemporal social
welfare functions,
Ï Introduce a (new?) mobility index.

(Minor) Comments

(Minor) Comments
Ï Preliminary version ⇒ polishing is needed (references,
unexplained notations,. . . ).

(Minor) Comments
Ï Little is said about M.

(Minor) Comments
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 ).

(Minor) Comments
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.

(Minor) Comments
Ï Assume that there is no costs of intertemporal income
variability or that costs are outweighted by gains related to
reranking.

(Minor) Comments
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 .

(Minor) Comments
reranking.
Ï Extension to more than 2 periods. Perfect mobility with
T > 2 periods?

(Minor) Comments
reranking.
Ï Extension to more than 2 periods. Perfect mobility with
T > 2 periods?
Ï Ex-ante or ex-post evaluation ?

That’s all folks!

Session 8 a presentation decancq-zoli

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