This paper investigates the energy efficiency optimization for movable antenna (MA) systems by considering the time delay and energy consumption introduced by MA movement. We first derive the upper bound on energy efficiency for a single-user downlink communication system, where the user is equipped with a single MA. Then, the energy efficiency maximization problem is formulated to optimize the MA position, and an efficient algorithm based on successive convex approximation is proposed to solve this non-convex optimization problem. Simulation results show that, despite the overhead caused by MA movement, the MA system can still improve the energy efficiency compared to the conventional fixed-position antenna (FPA) system.