Restrictions and identification in a multidimensional risk-sharing problem

We consider $$H$$ H expected utility maximizers that have to share a risky aggregate multivariate endowment $$X\in {\mathbb {R}}^{N}$$ X ∈ R N and address the following two questions: does efficient risk-sharing imply restrictions on the form of individual consumptions as a function of $$X$$ X ? Can one identify the individual utility functions from the observation of the risk-sharing? We show that when $$H\ge \frac{2N}{N-1}$$ H ≥ 2 N N - 1 efficient risk sharings have to satisfy a system of nonlinear PDEs. Under an additional rank condition, we prove an identification theorem. Copyright Springer-Verlag Berlin Heidelberg 2014