A note on separability and intra-household resource allocation in a collective household model

We consider a collective model of a household in which earn member has a utility function satisfying the weak separability condition. We show that the separability at the individual level carries over to the household level and that the allocation of private goods in any Pareto-efficient allocation can be supported as a Pareto-efficient allocation of the private sub-problem. We also provide the necessary and sufficient condition for the Pareto weight for the private sub-problem to move in the same direction as the household Pareto weight.