Duality in Contracting

In a linear contracting environment the Fenchel transform provides a complete duality between the contract and the information rent. Through an appropriate generalised convexity this can be extended to provide a complete duality in the supermodular quasilinear contracting environment that covers the majority of applications. Using this framework, we provide a complete characterization of the allocation correspondences that can be implemented by a principal in this environment. We also address the question of when an allocation can be implemented by a menu of simple contracts. Along the way, a supermodular envelope theorem is proved, somewhat different in nature to the Milgrom Segal result.