The Dual Theory of Measuring Social Welfare and Inequality

Winter School (University of Verona), Canazei, 12-16 January 2009

Rolf AabergeResearch Department, Statistics Norway

Outline

1 MOTIVATION

2 Expected and rank-dependent utility theories of social welfare

3

Normative theories for ranking Lorenz curves 4

Statistical characterization of income distributions and Lorenz curves

5Ranking 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

L u F t dt uu

( ) ( ) , , 10 1

0

1

where 1F (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

1

1

1 2

i

jj

n

jj

Xi

L( ) , i , ,...,n,n

X

where 1 2 nX ,X ,...,X 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 ( ) 1F t to a person with income ( ) 1F 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.

ProblemConsider a set of income distributions

1 2, ,..., sF F F

How should we rank and summarize differences between these distributions?

Introduce an ordering relation which justifies the statement

i jF F

Expected utility based theory of social welfare

[0,1]

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 .Then implies characterizes the following family of social welfare functions

1 2F F

1 3 2 3(1 ) (1 )F F F F

where u(x) is a positive increasing concave function of x.

0

( ) ( ) ( ) ( )W F Eu X u x dF x

Expected utility based measures of inequality

By introducing the equally distributed equivalent income for F defined by Atkinson (1970) proposed the following measure of inequality

1( ) ( ( ))F u W F

( )( ) 1 ,

FI F

where is the mean of F.

Rank-dependent utility based theory of social welfare

[0,1]

1 2F F

1 1 1 11 3 2 3(1 ) (1 ) ,F F F F

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 . Then implies then it follows from Yaari (1987, 1988) that the ordering relation is characterized by the following family of rank-dependent social welfare functions

11

0 0

( ) ( ( )) ( ) ( )PW F xdP F x P t F t dt

where P(t) is an increasing concave function of t.

Rank-dependent measures of inequality

Since and obeys the Pigou-Dalton transfer principleYaari (1988) proposed the following family of rank-dependent measures of inequality

( )PW F ( )PW F

11

0

( ) 1( ) 1 1 ( ) ( )P

P

W FJ F P t F t dt

Mehran (1976) introduced the JP-family by relying on descriptive arguments.

Statistical characterization of income distributions and Lorenz curves

( , ) ( , )F L M

( , ) ( , ) ( , ( ) : 1,2,... )kF L M C F k

where

1 11

0 0

( ) ( ) (1 ( )) , 1, 2,...k kkC F u dM u k u M u du k

Gini’s Nuclear Family

1

2

0 0

1G C (F) 2 u 1 M(u) du F(x) 1 F(x) dx

1

1

0 0

1C (F) (1 M(u))du F(x)log F(x)dx

1

2 23

0 0

1C (F) 3 u (1 M(u))du F(x) 1 F (x) dx

2

Bonferroni:

Gini:

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)

0.224 (0.002)

0.177 (0.002)

1987

0.330 (0.003)

0.224 (0.003)

0.177 (0.002)

1988

0.327 (0.003)

0.223 (0.002)

0.176 (0.002)

1989

0.340 (0.004)

0.233 (0.004)

0.186 (0.004)

1990

0.343 (0.003)

0.232 (0.002)

0.183 (0.002)

1991

0.340 (0.003)

0.232 (0.003)

0.185 (0.003)

1992

0.348 (0.003)

0.23 (0.003)

0.18 (0.002)

1993

0.352 (0.005)

0.240 (0.005)

0.191 (0.005)

1994

0.366 (0.003)

0.249 (0.002)

0.199 (0.002)

1995

0.358 (0.003)

0.247 (0.003)

0.198 (0.003)

1996

0.364 (0.004)

0.255 (0.004)

0.207 (0.004)

1997

0.371 (0.004)

0.260 (0.004)

0.212 (0.004)

1998

0.355 (0.003)

0.249 (0.003)

0.202 (0.003)

Average of (1986-92)

0.337 (0.001)

0.229 (0.001)

0.181 (0.001)

Average of (1993-98)

0.361 (0.002)

0.250 (0.001)

0.201 (0.001)

Percentage change, (1986-92) - (1993-1998) 7.14 9.07 10.96

*Standard deviation in parentheses

Normative theories for ranking Lorenz curves

By defining the ordering relation on the set of Lorenz curves Lrather 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

0

( )( ) 1 ( ) ( ) 1 P

P

W FJ L P u dL u

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

where Q´(t) is a positive increasing function of t.

1

0

1*QJ ( L ) Q ( L( u ))du,

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

( ) ( ) ,1 2L u L 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) P 1 P 2J L J L for all increasing concave P

(v) 1 2* *Q QJ ( L ) J ( L ) for all increa sin g conv .ex Q

D E F IN IT IO N 2 .4 A . A L o ren z cu rv e L 1 is sa id to seco n d -d eg ree u p w a rd d o m in a te a L o ren z cu rv e L 2 if

0

u

1

0

u

2L (t) d t L (t) d t fo r a ll u 0 ,1

an d th e in eq u ality h o ld s stric tly fo r so m e u 0 ,1 . D E F IN IT IO N 2 .4 B . A L o ren z cu rv e L 1 is sa id to seco n d -d eg ree d o w n w a rd d o m in a te a L o ren z cu rv e L 2 if

1 1

2 1

u u

1 - L (t) d t 1 - L (t) d t fo r a ll u 0 ,1

an d th e in eq u ality h o ld s stric tly fo r so m e u 0 ,1 .

T H E O R E M 2 . 2 A . L e t L 1 a n d L 2 b e m e m b e r s o f L . T h e n t h e f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t , ( i ) L 1 s e c o n d - d e g r e e u p w a r d d o m i n a t e s L 2

( i i ) P 1 P 2J L J L f o r a l l i n c r e a s i n g a n d c o n c a v e P w i t h n e g a t i v e s e c o n d a n d p o s i t i v e t h i r d d e r i v a t i v e s .

T H E O R E M 2 . 2 B . L e t L 1 a n d L 2 b e m e m b e r s o f L . T h e n t h e f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t ,

( i ) L 1 s e c o n d - d e g r e e d o w n w a r d d o m i n a t e s L 2 ( i i ) P 1 P 2J L J L f o r a l l i n c r e a s i n g a n d c o n c a v e P w i t h n e g a t i v e s e c o n d a n d t h i r d d e r i v a t i v e s .

The principles of first-degree downside and upside positional transfer sensitivity

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 s tI h 0 when s t [ , , 1 s tI h 0 when s t ].

Let I be an inequality measure and let t(,h) denote the change in I resulting from a transfer from a

person with income 1F (t h) to a person with income 1F (t) that leaves their ranks in the income

distribution F unchanged. Thus, tI ,h is a negative number. Furthermore, let 1 s,tI ,h be

defined by

1 s,t t sI ,h I ,h I ,h .

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) P 1 P 2J L J L for all increa sin g concave P with positve third derivative

(iii) P 1 P 2J L J L 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) P 1 P 2J L J L for all increa sin g concave P with negative third derivative

(iii) P 1 P 2J L J L 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 i

th-degree upward dominates L2 (ii) P 1 P 2J L J L for all P with derivatives between second and ith order that alternate in sign

1,...,3,2,0)(1 )(1 ijtP jj

(iii) P 1 P 2J L J L 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 i

th-degree downward dominates L2 (ii) P 1 P 2J L J L for all increasing P with negative derivatives of order two and up to i+1

P t j ij( ) ( ) , , , .. . , 0 2 3

(iii) P 1 P 2J L J L for all P being such that JP obeys the principle of (i-1)th-degree UPTS