Proper Welfare Weights for Social Optimization Problems

Social optimization problems are often used in economics to study important issues. In a social optimization problem, the sum of individual weighted utilities is maximized over all feasible allocations that satisfy certain constraints. In this paper, we provide a mechanism that determines the set of proper individual weights to be applied to social optimization problems. To do this, 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. 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. 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. Finally, we provide several numerical examples that illustrate our methodology. When households are not ex ante identical, the examples suggest that using the proper weights can substantially affect the allocations.