We investigate the class of stochastic orders induced by Generalized Gini Functionals (GGF) or Yaari (1987) [The dual theory of choice under risk. Econometrica, 55, 95-115] dual functionals and identify the maximal classes of functionals associated with these orders. Our results are inspired by Marshall (1991) [Multivariate stochastic orders and generating cones of functions. In Stochastic Orders and Decisions under Risk, (Mosler, K. and Scarsini, M. eds.) IMS Lecture Notes Monograph Series vol. 19, 231-247] and are dual to those obtained for additive representations in Müller (1997) [Stochastic orders generated by integrals: a unified study. Advances in Applied Probability, 29, 414-428] and in Castagnoli and Maccheroni (1998) [Generalized stochastic dominance and unanimous preferences. In Generalized Convexity and Optimization for Economic and Financial Decisions, Giorgi, G. and Rossi, F. (eds.), 111-120. Bologna: Pitagora]. The closure of the convex hull generated by a given set of probability distortion functions (F) [or by a set of rank-dependent weighting functions (V)] identifies the maximal class of functionals associated with the stochastic orders that are consistent with F [or V]. Rank-dependent weighting functions obtained as convex combinations of indicator functions identify GGFs that can be considered the "basis" of relevant stochastic orders in decision theory and inequality measurement. As hinted by Wang and Young (1998) [Ordering risks: Expected utility theory versus Yaari's dual theory of risk. Insurance: Mathematics and Economics, 22, 145-161] and Zoli (1999, 2002) [Intersecting generalized Lorenz curves and the Gini index. Social Choice and Welfare, 16, 183-196. Inverse stochastic dominance, inequality measurement and Gini indices.Journal of Economics, Supplement # 9, P. Moyes, C. Seidl and A.F. Shorrocks (Eds.), Inequalities: Theory, Measurement and Applications, 119-161] the stochastic orders obtained are related to the class of inverse stochastic dominance (ISD) conditions introduced in Muliere and Scarsini (1989) [A note on stochastic dominance and inequality measures. Journal of Economic Theory, 49, 314-323]. Making use of our results we review some stochastic dominance conditions that can be applied in decision theory, inequality, welfare and poverty measurement. These conditions are associated with orders implied by first order ISD and implying second order ISD, as well as with orders implied by the latter
Inverse stochastic orders and generalized Gini functionals
ZOLI, Claudio
2005-01-01
Abstract
We investigate the class of stochastic orders induced by Generalized Gini Functionals (GGF) or Yaari (1987) [The dual theory of choice under risk. Econometrica, 55, 95-115] dual functionals and identify the maximal classes of functionals associated with these orders. Our results are inspired by Marshall (1991) [Multivariate stochastic orders and generating cones of functions. In Stochastic Orders and Decisions under Risk, (Mosler, K. and Scarsini, M. eds.) IMS Lecture Notes Monograph Series vol. 19, 231-247] and are dual to those obtained for additive representations in Müller (1997) [Stochastic orders generated by integrals: a unified study. Advances in Applied Probability, 29, 414-428] and in Castagnoli and Maccheroni (1998) [Generalized stochastic dominance and unanimous preferences. In Generalized Convexity and Optimization for Economic and Financial Decisions, Giorgi, G. and Rossi, F. (eds.), 111-120. Bologna: Pitagora]. The closure of the convex hull generated by a given set of probability distortion functions (F) [or by a set of rank-dependent weighting functions (V)] identifies the maximal class of functionals associated with the stochastic orders that are consistent with F [or V]. Rank-dependent weighting functions obtained as convex combinations of indicator functions identify GGFs that can be considered the "basis" of relevant stochastic orders in decision theory and inequality measurement. As hinted by Wang and Young (1998) [Ordering risks: Expected utility theory versus Yaari's dual theory of risk. Insurance: Mathematics and Economics, 22, 145-161] and Zoli (1999, 2002) [Intersecting generalized Lorenz curves and the Gini index. Social Choice and Welfare, 16, 183-196. Inverse stochastic dominance, inequality measurement and Gini indices.Journal of Economics, Supplement # 9, P. Moyes, C. Seidl and A.F. Shorrocks (Eds.), Inequalities: Theory, Measurement and Applications, 119-161] the stochastic orders obtained are related to the class of inverse stochastic dominance (ISD) conditions introduced in Muliere and Scarsini (1989) [A note on stochastic dominance and inequality measures. Journal of Economic Theory, 49, 314-323]. Making use of our results we review some stochastic dominance conditions that can be applied in decision theory, inequality, welfare and poverty measurement. These conditions are associated with orders implied by first order ISD and implying second order ISD, as well as with orders implied by the latter
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