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We present a non-cooperative game-theoretical study of the power and rate control problem in IEEE 802.11 WLANs where network participants choose appropriate transmission power and data rate to achieve maximum throughput with minimum energy consumption. In such game-theoretical study, the central question is whether a Nash equilibrium (NE) exists, if so, whether the network operates efficiently at the NE. In this paper, we show the existence and uniqueness of the NE and the convergence to the NE under best response strategy. However, the unique NE is inefficient, i.e., neither social optimal nor Pareto optimal. Motivated by this fact, we propose both linear and non-linear pricing scheme to improve efficiency. We demonstrate that by wisely choosing the parameters, the game converges to an efficient NE. Finally, we examine the convergence to the NE under a practical rate update scheme: the subgradient rate update. Both analytical and numerical results show that the proposed rate control scheme can lead the network to the social optimal equilibrium.