Sufficient conditions for a "simple" Second Welfare Theorem with other-regarding preferences

We consider a pure exchange economy with consumption externalities in preferences. Using the notion of competitive equilibrium à la Nash, we point out that a simple condition for restoring the Second Welfare Theorem is that the set of Pareto optimal allocations is included in that of internal Pareto optimal allocations. We provide the Social Redistribution assumption to ensure such inclusion. This assumption is weaker than other relevant assumptions that have been studied in the literature. We then introduce the differential counterpart of Social Redistribution, called Directional Social Redistribution. This assumption entails an interesting consequence that relates social marginal utilities and supporting prices at a Pareto optimal allocation. Finally, we show that, for Bergson-Samuelson utility functions, Directional Social Redistribution is ensured by a specific property of the Jacobian matrix, which has a natural interpretation in terms of externalities