This letter proposes a model for tracking the equilibrium point of the real-time locational marginal price (LMP) based residential demand response program, where elastic demand is modeled as a monotonously decreasing linear function of the LMP. The resulting bi-level model contains both primary and dual variables, making it difficult to solve. Using duality, the dual model is formulated as a convex quadratic problem which is tractable to solve and find the global optimum. Furthermore, the condition for the existence of the equilibrium point is given. Numerical results on the IEEE 30-bus system verifies the effectiveness of the demand response model.