An available bit rate (ABR) service allows applications to access a time-varying network capacity. In a basic ABR service the available network capacity is divided "fairly" amongst active connections, without regard to the utility that each application derives from the bandwidth allocation. The goal of this paper is to improve both the global and individual utility obtained by applications using ABR, as compared to standard max-min bandwidth allocation. The key to our novel ABR algorithm is to relax the requirement for short-term fair bandwidth allocation, allowing an allocation that trades bandwidth between applications to increase the overall utility. To increase the utility of each individual application, we also re-allocate the bandwidth on the usual ABR time interval. This re-allocation increases the long-term average utility obtained by each individual application; it also ensures that the longer term average bandwidth allocated to each application is equal to the max-min allocation.