Restrictions and identification in a multidimensional risk-sharing problem

We consider H expected utility maximizers that have to share a risky aggregate multivariate endowment X∈RN and address the following two questions: does efficient risk-sharing imply restrictions on the form of individual consumptions as a function of X ? Can one identify the individual utility functions from the observation of the risk-sharing? We show that when H≥2NN−1 efficient risk sharings have to satisfy a system of nonlinear PDEs. Under an additional rank condition, we prove an identification theorem.