Properly weighting social welfare functions

When it comes to evaluating optimal policies, one has to define a social welfare criterion. That is problematic as soon as there is heterogeneity. A popular criterion is Pareto Optimality, but this is a weak criterion in the sense that it is not very restricting. Or one can look at a political equilibrium, but this ignores how much people care about various policy outcomes. Another way that makes microeconomic theoreticians cringe is to add up the utility of everyone. They cringe because utility functions are only defined up to a Paretian transformation, and thus not comparable across individuals. But sometimes you have to find a way, and it is commonly assumed that all individuals have the same utility function, but potentially different utilities. Yet, adding utilities up has the drawback to the optimal policy will always be about equalizing income and consumption across individuals, because poorer ones have a higher marginal utility of consumption. But not all policies should be primarily about redistribution. The typical solution to this problem is to apply so-called Negishi weights, which essentially freezes the initial distribution of income, and thus allows to concentrate on the purpose on the policy.

Alexis Anagnostopoulos, Eva Carceles-Poveda and Yair Tauman offer a different solution to this problem. While Negishi amounts to weigh each individual by the inverse of her marginal utility at the maximal outcome, this results relies on the existence of complete markets. Under incomplete markets, the set of weights may be different. To give credit to the precise formulation of the problem, I quote the authors here:

We first define for every set of individual weights and for every social welfare function the contribution of every individual to the total welfare through the individual’s initial endowments. We then provide an axiomatic approach to the notion of the per unit contribution of every good and every individual, where the contribution of an individual to the total welfare is the total contribution of his initial endowments. We then define a set of individual weights to be proper iff the weighted utilities of every individual from this allocation are proportional to the contribution of the individual to the total welfare as defined by this set of weights.

The axiomatic approach consists of four axioms that characterize an elegant family of contribution mechanisms. The first axiom asserts that the per unit contribution should be independent of the units of measurement of the goods. The second asserts that if two (or more) goods play the same role in the welfare function, they should have the same per unit contribution. The third axiom asserts that if the welfare function can be broken into different components, then the per unit contribution of a given good is the sum of the per unit contributions arising from the different components. The last axiom guarantees that the per unit contribution is a continuous mapping with respect to an appropriate norm.

It is shown that every contribution mechanism that satisfies these four axioms is uniquely determined by a non negative measure on the unit interval. The selection of a specific contribution mechanism (or equivalently the selection of a specific nonnegative measure on the unit interval) determines for a given economy and a given set of weights a proper constrained efficient allocation and a proper set of weights.

This is a very exciting paper that should lay the foundation for a better assessment of policies than the silly adding up of utilities that is typically done.