The progressivity of equalization payments in federations

We investigate the conditions under which an inequality averse and additively separable welfarist central government would choose to set up a progressive equalization payments scheme in a federation with local public goods. A progressive equalization payments scheme is a list of per capita net (possibly negative) subsidies – one such net subsidy for every jurisdiction – that are decreasing with respect to jurisdictions per capita wealth. We examine this question in a setting where the case for progressivity is a priori the strongest, namely: all citizens have the same utility function, inhabitants of a given jurisdiction have the same wealth and are not able to move across jurisdictions and there is no cross-jurisdiction competition in the setting of tax rates. We show that the central government favors a progressive equalization payments scheme for all distributions of wealth and population sizes if and only if its objective function is additively separable between each jurisdiction's per capita wealth and number of inhabitants. When interpreted for a mean of order r social welfare function, and assuming the absence of congestion in the local public good, this condition is shown to be equivalent to the requirement that the individual indirect utility function be additively separable between wealth public good price and be raised at the power 1/r before its agregation by means of the mean-of-order r social welfare function. Some implications of this restriction to the case where the individual's direct utility function is additively separable are also derived.