Non-Orthogonal Multiple Access (NOMA) is an essential enabling technology that is expected to help future broadband wireless networks meet their higher system throughput requirements. However, in addition, NOMA should also aim to provide a desired trade-off between system throughput and user fairness, as fairness is an equally important aspect that should go hand in hand with system throughput. In order to achieve such a desired trade-off, in this paper, we derive optimal power allocation (PA) coefficients at the NOMA transmitter. We formulate and solve a joint sum rate and fairness optimization problem that, along with the usual transmitter power budget and Quality of Service (QoS) constraints, also includes the minimum transmit power gap between users, required for successful signal decoding of a user in a successive interference cancellation (SIC) receiver, a constraint ignored at large in the literature. We use the weighted sum method to convert the joint objective optimization problem into a single-objective optimization problem to make it analytically solvable and also provide the desired trade-off between the conflicting objectives. We present simulation results to validate our analytical results.