This paper shows that the optimal policy and value functions of a Markov Decision Process (MDP), either discounted or not, can be captured by a finite-horizon undiscounted Optimal Control Problem (OCP), even if based on an inexact model. This can be achieved by selecting a proper stage cost and terminal cost for the OCP. A very useful particular case of OCP is a Model Predictive Control (MPC) scheme where a deterministic (possibly nonlinear) model is used to reduce the computational complexity. This observation leads us to parameterize an MPC scheme fully, including the cost function. In practice, Reinforcement Learning algorithms can then be used to tune the parameterized MPC scheme. We verify the developed theorems analytically in an LQR case and we investigate some other nonlinear examples in simulations.