We consider resource allocation over a wireless downlink where Base Station (BS) power consumption is minimized while upholding a set of required link rates. A Power and Resource Allocation Including Sleep (PRAIS) method is proposed that combines resource sharing, Power Control (PC), and Discontinuous Transmission (DTX), such that downlink power consumption is minimized, which can be formed into a convex optimization problem. Unlike conventional approaches that aim at minimizing transmit power, in this work the BS mains supply power is chosen as the relevant metric. Based on a linear power model, which maps a certain transmit power to the necessary mains supply power, we quantify the fundamental limits of PRAIS in terms of achievable BS power savings. The fundamental limits are numerically evaluated on link level for four sets of BS power model parameters representative of envisaged future hardware developments. We establish an expected lower limit for PRAIS of 27W to 68W depending on load per link for BSs installed in 2014, which provides a 61% to 34% gain over conventional resource allocation schemes.