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International audience; This paper deals with risk-sharing problems between many agents, each of whom having a strictly concave law invariant utility. In the special case where every agent's utility is given by a concave integral functional of the quantile of her individual endowment, we fully characterize the optimal risk-sharing rules. When there are many agents, these rules cannot be computed analytically. We therefore give a simple convergent algorithm and illustrate it on several examples.

A first version of this paper has been presented at the 11th Conference on Real Analysis and Measure Theory in Ischia (Italy, 2004). This version was presented at the Debreu Memorial Conference in Berkeley (USA, 2005); International audience; This paper studies production economies in a commodity space that is an ordered locally convex space. We establish a general theorem on the existence of equilibrium without requiring that the commodity space or its dual be a vector lattice. Such commodity spaces arise in models of portfolio trading where the absence of some option usually means the absence of a vector lattice structure. The conditions on preferences and production sets are at least as general as those imposed in the literature dealing with vector lattice commodity spaces. The main assumption on the order structure is that the Riesz-Kantorovich functionals satisfy a uniform properness condition that can be formulated in terms of a duality property that is readily checked. This condition is satisfied in a vector lattice commodity space but there are many examples of other commodity spaces that satisfy the condition, which are not vector lattices, have no order unit, and do not have either the decomposition property or its approximate versions.

The second welfare theorem and the core-equivalence theorem have been proved to be fundamental tools for obtaining equilibrium existence theorems, especially in an infinite dimensional setting. For well-behaved exchange economies that we call proper economies, this paper gives (minimal) conditions for supporting with prices Pareto optimal allocations and decentralizing Edgeworth equilibrium allocations as non-trivial equilibria. As we assume neither transitivity nor monotonicity on the preferences of consumers, most of the existing equilibrium existence results are a consequence of our results. A natural application is in Finance, where our conditions lead to new equilibrium existence results, and also explain why some financial economies fail to have equilibrium., Résumé. Le second théorème de l'économie du bien-être et le théorème d'équivalence coeur-équilibre sont des outils fondamentaux pour obtenir des théorèmes d'existence de l'équilibre, spécialement quand l'espace des biens n'est pas de dimension finie. Pour des économies d'échange que nous appelons propres, cet articles donne des conditions (minimales) de décentralisation par des prix des allocations Pareto-optimales et des équilibres d'Edgeworth. Comme nous ne supposons ni transitivité, ni monotonicité des préférences des consommateurs, la plupart des théorèmes existants d'existance de l'équilibre sont une conséquence de nos résultats. Une appication naturelle de ces résultats est en Finance où nos conditions conduisent à des résultats nouveaux, en même temps qu'elles expliquent pourquoi certains modèles financiers n'ont pas d'équilibre.