Introduction and summary

The concept of the social welfare function was introduced by Bergson in his classic paper (1938) to express preferences over resource allocations to all individuals in society. Just as the commodity bundles of an individual are supposed to be compared by his or her individual preference ordering, so the alternative allocations of commodity bundles to all individuals can be ordered by a social welfare ordering over this entire space. As with any ordering satisfying certain regularity properties, the social welfare ordering can be represented by a real-valued function, the social welfare function. Its significance is essentially ordinal, but under additional conditions of separability the function representing an ordering can be chosen to be additive in an appropriate choice of variables.

The optimization of a social welfare function should determine the entire allocation of resources among individuals and therefore includes the normative problems of income distribution. To concentrate on this topic, I will confine attention to the special case where there is only one commodity in the economic system. A resource allocation is then simply a vector with a (nonnegative) component for every individual in the economy.

To combine attention to the simplest normative aspects, I will further abstract from incentive questions. It will simply be assumed that the total available of the one commodity is given and is not diminished by any transfers.