We introduce steady state policies for a class of infinite horizon, deterministic, discounted dynamic programming models with compact–convex state spaces, compact action spaces and continuous cost and transition functions. Steady state policies are stationary policies under which a system moves to a steady state in a given number of steps. Because of their simple structure, it is sometimes easy to solve problems where one considers only steady state policies. Although such a restriction is sometimes costly (e.g., in many inventory models), there are cases where it is quite reasonable. In this paper we find conditions under which there exists a good steady state policy, i.e., one whose cost differs from the optimal cost by an amount whose bound is independent of the discount factor. (Such policies are optimal with respect to the average cost criterion.) We also give examples of models which satisfy these conditions. Our results depend on Brouwer’s fixed-point theorem.