Social welfare and the unrepresentative representative consumer

If, for all prices, income distribution is optimal for a planner with a social welfare function, then aggregate demand is the same as that of a single "representative consumer" whose preferences over aggregate consumption are the same as the planner's. This paper shows that the converse is false. Aggregate demand may be the demand function of a representative consumer although the income distribution is not optimal for any social welfare function. The representative consumer may be Pareto inconsistent, preferring situation A to B when all the actual consumers prefer B to A. We give conditions under which existence of a representative consumer implies that the income distribution satisfies first order conditions for optimality. Satisfying the first order optimality conditions for an additively separable social welfare function is essentially equivalent to aggregate demand for every pair of consumers having a symmetric Slutsky matrix.