On determining "close equals groups" in decomposing redistributive and reranking effects

Recently van De Van, Creedy and Lambert (2001) and Urban and Lambert (2008) have reconsidered the original Aronson, Johnson and Lambert (1994) decomposition of the redistributive effect in order to identify the optimal bandwidth that should be used in decomposing the redistributive effect when groups with close pre-tax incomes are considered. The methodology proposed by van De Van, Creedy and Lambert (2001) suggests choosing as the optimal bandwidth the one which maximizes the ratio between the potential effect V (which depends on the bandwidth) and the actual redistributive effect RE (which is invariant). Urban and Lambert (2008) discuss a set of further possible decompositions of the redistributive effect together with a decomposition of the Atkinson-Plotnick-Kakwani index into three terms. In this paper we want to throw some more light on the behavior of three of the main decompositions analyzed by Urban and Lambert (2008) in order to look for criteria to choose a bandwidth which allows the three different definitions of potential redistributive effect to be assumed as coherent as possible values and, in the meanwhile, to catch as much as possible of the potential vertical effect. We suggest looking for the bandwidth where the ratio between the maximum distance among the different potential vertical effect definitions and the minimum among the different potential vertical effects is minimum