This paper studies a multi-period demand response management problem in the smart grid where multiple utility companies compete among themselves. The user-utility interactions are modeled by a noncooperative game of a Stackelberg type where the interactions among the utility companies are captured through a Nash equilibrium. It is shown that this game has a unique Stackelberg equilibrium at which the utility companies set prices to maximize their revenues (within a Nash game) while the users respond accordingly to maximize their utilities subject to their budget constraints. Closed-form expressions are provided for the corresponding strategies of the users and the utility companies. It is shown that the multi-period scheme, compared with the single-period case, provides more incentives for the users to participate in the game. A necessary and sufficient condition on the minimum budget needed for a user to participate is provided.