Concavity and Differentiability of Value Function with CRS Return Functions

Abstract

This paper investigates concavity and differentiability of the value function of a dynamic optimization problem when involved functions and correspondences exhibit CRS property. For the purpose, the relationship between the value function and the solution of the associated Bellman equation is investigated beforehand. As a byproduct of these investigations, the followings are obtained: a strictly quasi-concave CRS function is strictly concave when at least one of the independent variable is fixed in a 2 or higher dimensional case, and quasi-concave CRS function is concave.