Pareto optimal uncertainty sharing under smooth ambiguity

We extend the classical syndicate model to a setting with ambiguous beliefs, where agents face both risk and ambiguity and have heterogeneous smooth ambiguity preferences. Our main theorem generalizes Borch's seminal result by providing necessary and sufficient conditions for Pareto optimality in this environment: an uncertainty-sharing contract is Pareto-optimal if and only if agents’ marginal utilities, weighted by ambiguity-adjusted belief densities, are proportional across states. The ambiguity-adjusted densities reflect how ambiguity aversion tilts evaluation toward adverse probabilistic models, with weights proportional to the marginal ambiguity attitude. The welfare interpretation is standard: Pareto-optimal contracts coincide with maximizers of a weighted social-welfare functional, and the proportionality constants equal the associated Pareto weights. The characterization nests Borch's rule, reducing to proportional marginal utilities when beliefs are unambiguous or agents are ambiguity neutral.