A note on the characterization of optimal allocations in OLG economies with multiple goods

We consider a pure exchange overlapping generations economy with finitely many commodities and consumers per period having possibly non-complete non transitive preferences. We provide a geometric and direct proof of the Balasko-Shell characterization of Pareto optimal allocation. To avoid some smoothness assumption, we replace the assumption on the curvature of the indifference surface by geometric properties of preferred sets. In particular, we use the notion of prox-regularity, introduced in variational analysis by Rockafellar and Poliquin. We provide preliminary results and explain how the multi-consumer case can be simplified by considering aggregate feasible Pareto improving transfer. We provide the proof and we show which assumptions are used for the "if" part and for the "only if" part of the criterion