A Calculus Proof of Proposition 1 in "On the Moral Hazard Problem Without the First-Order Approach"

In Kadan and Swinkels (2012a) we establish a general expression for the effect simultaneously tightening the minimum payment and individual rationality constraints in a general moral hazard problem. The result there does not rely on the validity of the first order approach (FOA), allows for multidimensional signals and effort, and requires no order structure on signals or differentiability of the distribution function. The proof of this result is established using a variational argument, which relies on differentiability and integrability results. In standard moral hazard problems, when the FOA is valid, this result becomes quite simple. Below we provide such a simple version of this result. We also provide a calculus proof of this result that does not hinge on a variational argument. The notation we use is as in Kadan and Swinkels (2012b).