Envelope Theorem without Differentiability

We establish a new envelope theorem in which the choice variables are discrete and the objective function and the constraints are Lipschitz continuous with respect to the parameters. The parameters can be ?nite or in?nite dimensional vectors in a Banach space. In an application, we revisit the principal-agent problem and derive a weaker ?rst-order condition than the traditional one in the literature. In an insurance example, we use the condition to show an insurance contract that is discontinuous at some level of the loss.