Many economic models mix continuous and discrete choice problems, such as a consumption/savings trade-off alongside a labor participation decision. These problems have non-differentiable and non-concave value functions. We show that the value function is differentiable at optimal choices if the underlying utility function is the upper envelope of differentiable functions. Hence, we obtain first-order conditions that are necessary for optimality. We do not make any concavity assumptions.

Our theorem is in the tradition of the envelope theorems by Benveniste and Scheinkman (1979) or Milgrom and Segal (2002). However, whereas the former states a theorem for concave objective functions, the latter uses strong assumptions such as equidifferentiability of the underlying functions. We do neither need left- and right-derivatives, nor equicontinuity, and allow for non-concave utility functions.