Restrictions and identification in a multidimensional risk-sharing problem

We consider <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$H$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>H</mi> </math> </EquationSource> </InlineEquation> expected utility maximizers that have to share a risky aggregate multivariate endowment <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$X\in {\mathbb {R}}^{N}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>X</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </mrow> </math> </EquationSource> </InlineEquation> and address the following two questions: does efficient risk-sharing imply restrictions on the form of individual consumptions as a function of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$X$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>X</mi> </math> </EquationSource> </InlineEquation>? Can one identify the individual utility functions from the observation of the risk-sharing? We show that when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$H\ge \frac{2N}{N-1}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>H</mi> <mo>≥</mo> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </math> </EquationSource> </InlineEquation> 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