Aggregation and pairwise aggregation of demand when the distribution of income is fixed

This paper derives necessary and sufficient conditions for “pairwise aggregation” of the demand functions of a group of consumers (conditions under which the mean demand for each pair of consumers satisfies the Slutsky restrictions) when the distribution of income is fixed. The sufficient conditions imply existence of a “representative” competitive consumer whose demand is the mean demand of the group. The necessary conditions imply that such a representative consumer exists for every fixed income distribution only if the consumers have homothetic preferences or, alternatively, if for each price vector, all the consumers' income expansion paths lie in the same plane.