The Dual Theory of Measuring Social Welfare and Inequality

1
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

2
Outline
1
MOTIVATION
2
Expected and rank-dependent utility theories of
social welfare
Statistical characterization of income
distributions and Lorenz curves
3
Normative theories for ranking Lorenz curves
4
Ranking Lorenz curves and measuring inequality
when Lorenz curves intersect
5
3
The Lorenz curve
4
Principle of transfers
5
Problem
Consider a set of income distributions
How should we rank and summarize differences
between these distributions?
Introduce an ordering relation
which justifies the statement
6
Expected utility based theory of social welfare
7
Expected utility based measures of inequality
8
Rank-dependent utility based theory of social
welfare
where P(t) is an increasing concave function of t.
9
Rank-dependent measures of inequality
Since and
obeys the Pigou-Dalton transfer principle Yaari
(1988) proposed the following family of
rank-dependent measures of inequality
10
Statistical characterization of income
distributions and Lorenz curves
11
(No Transcript)
12
where
13
Ginis Nuclear Family
Bonferroni
Gini
Aaberge, R. (2007) Ginis Nuclear Family,
Journal of Economic Inequality, 5, 305-322.
14
(No Transcript)
15
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
16
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.
17
Complete axiomatic characterization of the Gini
coefficient
18
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?

19
(No Transcript)
20
(No Transcript)
21
(No Transcript)
22
The principles of first-degree downside and
upside positional transfer sensitivity
23
Illustration of DPTS and UPTS
24
(No Transcript)
25
Lorenz dominance of i-th degree