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, by formulating and solving a joint sum rate and fairness optimization problem. To the best of our knowledge, such a work is missing in the literature. Additionally, along with the usual transmitter power budget and Quality of Service (QoS) constraints, we also consider the minimum transmit power gap between users, that is required for successful signal decoding of a user in a SIC receiver, a constraint ignored at large in the literature. The weighted sum method is used 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. As part of the validation process, we present simulation results and also compare the performance of the derived system with a system that considers only the sum rate objective.