1. The Dual Theory of Measuring Social Welfare and Inequality Winter School (University of Verona), Canazei, 12-16 January 2009 Rolf Aaberge Research Department, Statistics Norway

2. Outline 1MOTIVATION 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 5 Ranking Lorenz curves and measuring inequality when Lorenz curves intersect

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

12. where

13. Gini’s Nuclear Family Bonferroni: Gini: Aaberge, R. (2007): Gini’s Nuclear Family, Journal of Economic Inequality, 5, 305-322.

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?

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

23. Illustration of DPTS and UPTS

25. Lorenz dominance of i-th degree