We formulate real-time pricing in electric power systems with renewable energy as a linear model predictive control (MPC) problem with inequality constraints and demonstrate that a small change in its performance index yields drastic changes in distributions of renewable energy over a power network. We consider electric power systems consisting of consumers, suppliers, generators, renewable energy, and an independent system operator (ISO) and assume that average of electricity generated by renewable energy can be predicted over a certain time interval in the future. Consumers and suppliers are modeled by demand curves and supply curves, respectively, derived from their utility functions and cost functions. The ISO can manipulate electricity prices in each area at each time point to stabilize load frequencies of generators, to maximize total benefits of consumers and suppliers, and to balance demands and supplies. To achieve these objectives, the current electricity prices are determined by real-time optimization to minimize a performance index over a finite future, which is formulated as a MPC problem. We examine two types of performance indices, and simulation results show that a small difference in the performance indices results in contrasting responses in distributions of renewable energy over the power network.