Policy functions of strictly concave optimal growth models

We consider discrete time optimal growth models in the reduced form. The maintained assumptions are that the production technology set is convex and that preferences are additively separable with a continuous and strictly concave (reduced form) utility function. Using the dynamic programming approach we derive two new optimality conditions for models of this class. The first one is an inequality which has to be satisfied by the optimal value function and the optimal policy function of any strictly concave optimal growth model. As an application of this condition we derive Hölder continuity and Lipschitz continuity, respectively, of the optimal policy functions in strongly concave optimal growth models. (These properties were originally proven by Luigi Montrucchio using a completely different approach.) The second optimality condition is a condition that has to be satisfied by a continuous mapping h from a convex set X into itself if this mapping is the optimal policy function of any strictly concave optimal growth model with a given discount factor p and the state space X. This condition is stated in terms of a dominance relation between two probability measures on X. We also derive a reformulation of this condition for the special case of probability measures with finite support and illustrate its application by showing that the tent map (one of the most famous examples from chaos theory) cannot be an optimal policy function unless the discount factor is smaller than $\text{1}\text{√6}$.