In this paper, a novel dynamic power allocation scheme is proposed to downlink and uplink non-orthogonal multiple access (NOMA) scenarios with two users for more flexibly meeting various quality of service requirements. The exact expressions for the outage probability and the average rate achieved by the proposed scheme, as well as their high signal-to-noise ratio approximations, are established. Compared with the existing works, such as NOMA with fixed power allocation and cognitive radio inspired NOMA, the proposed scheme can: 1) strictly guarantee a performance gain over conventional orthogonal multiple access; and 2) offer more flexibility to realize different tradeoffs between the user fairness and system throughput. Monte Carlo simulation results are provided to demonstrate the accuracy of the developed analytical results and the performance gain of the proposed power allocation scheme.