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In this paper we provide an alternative framework to tackle the first-bestPrincipal-Agent problem under CARA utilities. This framework leads to both aproof of existence and uniqueness of the solution to the Risk-Sharing problemunder very general assumptions on the underlying contract space. Our analysisrelies on an optimal decomposition of the expected utility of the Principal interms of the reservation utility of the Agent and works both in a discrete timeand continuous time setting. As a by-product this approach provides a novel wayof characterizing the optimal contract in the CARA setting, which is as analternative to the widely used Lagrangian method, and some analysis of theoptimum.

Analysis of the Risk-Sharing Principal-Agent problem through the Reverse-H{\"o}lder inequality