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Smart grid is a visionary user-centric system that will elevate the conventional power grid system to one which functions more cooperatively, responsively, and economically. Dynamic demand side management is one of the key issues that enable the implementation of smart grid. In this paper, we use the framework of dynamic games to model the distribution demand side management. The market price is characterized as the dynamic state using a sticky price model. A two-layer optimization framework is established. At the lower level, for each player (such as one household), different appliances are scheduled for energy consumption. At the upper level, the dynamic game is used to capture the interaction among different players in their demand responses through the market price. We analyze the N-person nonzero-sum stochastic differential game and characterize its feedback Nash equilibrium. A special case of homogeneous users is investigated in detail and we provide a closed-form solution for the optimal demand response. From the simulation results, we demonstrate the use of demand response strategy from the game-theoretic framework and study the behavior of market price and demand responses to different parameters.