The multi-carrier proportional fair scheduling (MC-PFS) problem in a multi-user system has been shown to be NP-hard. Carrier by carrier proportional fair scheduling (CC-PFS) is commonly used instead to allocate resources in real-time. Considering the sub-optimal nature of CC-PFS and its popularity, this paper formulates the optimization beyond CC-PFS as a constrained maximum sum rate problem (with users' data rates scheduled by CC-PFS as the constraint) and tackles the problem in two ways. Firstly, the problem is shown to be equivalent to the well-studied generalized assignment problem (GAP) under the assumption of infinitely backlogged data. By considering traffic arrivals, the problem becomes a nonlinear integer programming problem which can be solved by outer approximation (OA) algorithms. Secondly, the problem is shown to be equivalent to a classical trading problem, and a low complexity heuristic algorithm that can be run in real-time is developed based on trading resource blocks (RBs). Using a system scheduling many video call users, we show that in about half of the time slots, the sum rate can be improved over CC-PFS while each user transmits at least as many bits as scheduled by CC-PFS. The heuristic algorithm captures about 30% of the throughput improvement found by OA. Finally, the reason for the improvements over CC-PFS is found to be traffic arrival.