Income Inequality and Social Welfare

Report
The Dual Theory of Measuring
Social Welfare and Inequality
Rolf Aaberge
Research Department, Statistics Norway
Winter School (University of Verona), Canazei, 12-16 January 2009
Outline
1
MOTIVATION
2
Expected and rank-dependent utility theories of social welfare
3
Statistical characterization of income distributions and Lorenz curves
4
Normative theories for ranking Lorenz curves
5
Ranking Lorenz curves and measuring inequality
when Lorenz curves intersect
The Lorenz curve
The Lorenz curve L for a cumulative income distribution F with mean  is defined by
1
L( u ) 

u

F 1 ( t ) dt , 0  u  1 ,
0
where F1 (t)  inf x : F(x)  t is the left inverse of F. Thus, the Lorenz curve L(u) shows the share of
total income received by the poorest 100 u per cent of the population.
The empirical counterpart of L is defined by
i
i
L( ) 
n
X
j
X
j
j 1
n
j 1
, i  1, 2,...,n,
where X 1 , X 2 ,..., X n is ordered incomes of n units (individuals or households).
Principle of transfers
DEFINITION 2.1. (The Pigou-Dalton principle of transfers.) Consider a discrete income distribution
F. A transfer   0 from a person with income F 1 (t ) to a person with income F 1 ( s ) , where the
transfer is assumed to be rank-preserving, is said to reduce inequality in F when s  t and to raise
inequality in F when s>t.
Problem
Consider a set of income distributions
F1 , F2 ,..., Fs
How should we rank and summarize differences between
these distributions?
Introduce an ordering relation
which justifies the statement Fi
Fj
Expected utility based theory of
social welfare
Given suitable continuity and dominance assumptions for the social planner’s preference
ordering
defined on the family of income distributions F, it can be demonstrated (see e.g.
Fishburn, 1982) that the following axiom,
Axiom (Independence). Let F1, F2 and F3 be members of F and let   [0,1] .Then F1
implies  F1  (1   ) F3  F2  (1   ) F3
characterizes the following family of social welfare functions

W ( F )  Eu ( X )   u ( x) dF ( x)
0
where u(x) is a positive increasing concave function of x.
F2
Expected utility based measures of
inequality
By introducing the equally distributed equivalent income

for F defined by
 ( F )  u 1 (W ( F ))
Atkinson (1970) proposed the following measure of inequality
 (F )
I (F )  1 
,

where  is the mean of F.
Rank-dependent utility based theory
of social welfare
Replacing the independence axiom with the following alternative dual independence axiom
Axiom (Dual independence). Let F1,F2 and F3 be members of F and let   [0,1] . Then
F1
F2 implies  F11  (1   ) F31  F21  (1   ) F31 ,
then it follows from Yaari (1987, 1988) that the ordering relation
following family of rank-dependent social welfare functions

1
0
0
is characterized by the
WP ( F )   xdP ( F ( x))   P(t ) F 1 (t ) dt
where P(t) is an increasing concave function of t.
Rank-dependent measures of
inequality
Since WP ( F )   and WP ( F ) obeys the Pigou-Dalton transfer principle
Yaari (1988) proposed the following family of rank-dependent measures of inequality
J P (F )  1 
WP ( F )

 1
1

1
1

P
(
t
)
F
(t ) dt

0
Mehran (1976) introduced the JP-family by relying on descriptive arguments.
Statistical characterization of income
distributions and Lorenz curves
F  (  , L)  (  , M )
F  (  , L)  (  , M )  (  , Ck ( F ) : k  1, 2,...)
where
1
1
0
0
Ck ( F )   u k dM (u )  k  u k 1 (1  M (u ))du , k  1, 2,...
Gini’s Nuclear Family

1
Bonferroni:
Gini:
1
C1 (F)  (1  M(u))du  
F(x)log F(x)dx
0
0



1
1
G  C2 (F)  2 u 1  M(u) du 
F(x) 1  F(x)  dx
0
0



1


1
C3 (F)  3 u (1  M(u))du 
F(x) 1  F2 (x) dx
2 0
0

2

Aaberge, R. (2007): Gini’s Nuclear Family, Journal of Economic Inequality, 5, 305-322.
Table 2. Trend in income inequality in Norway, 1986-1998*
Year
C1
C2=G
C3
1986
0.331
(0.002)
0.330
(0.003)
0.327
(0.003)
0.340
(0.004)
0.343
(0.003)
0.340
(0.003)
0.348
(0.003)
0.352
(0.005)
0.366
(0.003)
0.358
(0.003)
0.364
(0.004)
0.371
(0.004)
0.355
(0.003)
0.337
(0.001)
0.361
(0.002)
0.224
(0.002)
0.224
(0.003)
0.223
(0.002)
0.233
(0.004)
0.232
(0.002)
0.232
(0.003)
0.23
(0.003)
0.240
(0.005)
0.249
(0.002)
0.247
(0.003)
0.255
(0.004)
0.260
(0.004)
0.249
(0.003)
0.229
(0.001)
0.250
(0.001)
0.177
(0.002)
0.177
(0.002)
0.176
(0.002)
0.186
(0.004)
0.183
(0.002)
0.185
(0.003)
0.18
(0.002)
0.191
(0.005)
0.199
(0.002)
0.198
(0.003)
0.207
(0.004)
0.212
(0.004)
0.202
(0.003)
0.181
(0.001)
0.201
(0.001)
9.07
10.96
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
Average of (1986-92)
Average of (1993-98)
Percentage change,
(1986-92) - (19937.14
1998)
*Standard deviation in parentheses
Normative theories for ranking
Lorenz curves
By defining the ordering relation
on the set of Lorenz curves L
rather than on the set of income distributions F, Aaberge (2001)
demonstrated that a social planner who supports the
Von Neumann – Morgenstern axioms will rank Lorenz curves
according to the criterion
1
J P ( L)  1   P(u )dL (u )  1 
0
WP ( F )

Alternatively, ranking Lorenz curves by relying on
the dual independence axiom for Lorenz curves
rather than on the conventional independence axiom
is equivalent to employ the following
measures of inequality
1
J ( L )  1   Q( L( u ))du,
*
Q
0
where Q´(t) is a positive increasing function of t.
Complete axiomatic characterization
of the Gini coefficient
THEOREM 5. A preference relation
on L satisfies completeness,
transitivity, dominance as well as independence and dual independence
if and only if
can be represented by the Gini coefficient
Ranking Lorenz curves and measuring
inequality when Lorenz curves intersect
How robust is an inequality ranking
based on the Gini coefficient or a few
meausures of inequality?
DEFINITION 2.2. A Lorenz curve L1 is said to first-degree dominate a Lorenz curve L2 if
L1 (u )  L2 (u ) for all u  0, 1
and the inequality holds strictly for some u  0, 1 .
THEOREM 2.1. (Fields and Fei (1978), Yaari (1988), Aaberge (2001)). Let L1 and L2 be members of
L. Then the following statements are equivalent,
(i)
L1 first-degree dominates L2
(ii)
L1 can be obtained from L2 by a sequence of Pigou-Dalton progressive transfers
(iii)
L2 can be obtained from L1 by a sequence of Pigou-Dalton regressive transfers
(iv)
J P  L1   J P  L2  for all increasing concave P
(v)
J Q* ( L1 )  J Q* ( L2 )
for all increa sin g conv ex Q.
DEFINITION 2.4A. A Lorenz curve L1 is said to second-degree upward
dominate a Lorenz curve L2 if
u

u
L1 (t) dt 
0

L 2 (t) dt for all u  0,1
0
and the inequality holds strictly for some u 
0,1
.
DEFINITION 2.4B. A Lorenz curve L1 is said to second-degree downward
dominate a Lorenz curve L2 if
1
1
 1 - L (t) dt   1 - L (t) dt
2
u
1
for all u  0,1
u
u  0,1
and the inequality holds strictly for some
.
THEOREM 2.2A. Let L1 and L2 be members of L. Then the following statements
are equivalent,
(i) L1 second-degree upward dominates L2
(ii) J P  L1   J P  L2  for all increasing and concave P with negative second and
positive third derivatives.
THEOREM 2.2B. Let L1 and L2 be members of L. Then the following
statements are equivalent,
(i) LJ 1 second-degree
downward dominates L2
L

J
L



P
1
P
2
(ii)
for all increasing and concave P with negative second and
third derivatives.
The principles of first-degree downside and
upside positional transfer sensitivity
Let I be an inequality measure and let t(,h) denote the change in I resulting from a transfer  from a
person with income F1 (t  h) to a person with income F1 (t) that leaves their ranks in the income
distribution F unchanged. Thus, I t  , h  is a negative number. Furthermore, let 1Is,t  , h  be
defined by
1Is,t  , h    I t  , h    Is  , h  .
DEFINITION 2.5. Consider a discrete income distribution F, an inequality measure I that obeys the
Pigou-Dalton principle of transfers and rank-preserving transfers   0 from individuals with ranks
s  h and t  h to individuals with ranks s and t in F. Then the inequality measure I is said to satisfy
the principle of first-degree downside [upside] positional transfer sensitivity (first-degree DPTS
[UPTS]) if 1 I s ,t  , h   0 when s  t [ 1 I s ,t  , h   0 when s  t ].
Illustration of DPTS and UPTS
THEOREM 2.2A. Let L1 and L2 be members of L. Then the following statements are equivalent,
(i)
L1 second-degree upward dominates L2
(ii)
J P  L1   J P  L2  for all increa sin g concave P with positve third derivative
(iii)
J P  L1   J P  L2  for all P being such that JP obeys the principle of first-degree DPTS.
THEOREM 2.2B. Let L1 and L2 be members of L. Then the following statements are equivalent,
(i)
L1 second-degree downward dominates L2
(ii)
J P  L1   J P  L2  for all increa sin g concave P with negative third derivative
(iii)
J P  L1   J P  L2  for all P being such that JP obeys the principle of first-degree UPTS.
Lorenz dominance of i-th degree
THEOREM 3.2A. Let L1 and L2 be members of L. Then the following statements are equivalent,
(i)
L1 ith-degree upward dominates L2
(ii)
J P  L1   J P  L2  for all P with derivatives between second and ith order that alternate in sign
 1
(iii)
j 1

P ( j ) (t )0, j  2,3,..., i  1
J P  L1   J P  L2  for all increasing P being such that JP obeys the principle of (i-1)th-degree
DPTS.
THEOREM 3.2B. Let L1 and L2 be members of L. Then the following statements are equivalent,
(i)
L1 ith-degree downward dominates L2
(ii)
J P  L1   J P  L2  for all increasing P with negative derivatives of order two and up to i+1
P
(iii)
( j)
( t )  0, j  2,3,..., i

J P  L1   J P  L2  for all P being such that JP obeys the principle of (i-1)th-degree UPTS

similar documents