Differentiability of the Value Function: A New Characterization

This paper characterizes the differentiability of the value function. We provide a characterization of the necessary and sufficient conditions for the differentiability of the value function. This generalizes the well-known differentiability result of Benveniste and Scheinkman (1979) which shows that the concavity restriction on the return function and the convex graph restriction on the constraint correspondence are sufficient to prove the differentiability. In addition to generalization, our proof is quite simple and different from that of Benveniste and Scheinkman in not using the concavity assumptions. We also show the differentiability of the indirect function in the envelope theorem under quite weak assumptions. This generalizes the established results regarding the differentiability of the support function and that of the cost function.