A negative-ion source extraction model has been formulated and implemented that explicitly considers the motion of positive ions and the volume generation of negative ions. It is found that (1) for high-beam currents, the beam current is limited by a transverse space-charge limit, meaning that an increase in negative-ion density at the extraction sheath will result in a lower-beam current (this result is universally observed at high-beam current); (2) there is a saddle point with a potential barrier preventing most volume-produced negative ions from being extracted [the combination of (1) and (2) indicates that most of the negative ions being created do not find their way into the beam]; (3) the introduction of cesium may cause an increase in the transverse space-charge limit; (4) cesium also results in an increase in the fraction of volume-produced negative ions that are extracted; (5) cesium may also result in reduction of extracted electrons by producing a less negative bias on the plasma electrode with respect to the plasma, thus allowing the transverse space-charge limit budget to be taken up virtually totally by the ions. [The combination of (3)–(5) represents the way an actual increase in the beam current can be achieved]; (6) a strong ion time scale sheath instability due to violation of the Bohm criteria produces an anomalous ion temperature, which increases with the beam current, as routinely seen in measurements; and (7) the introduction of cesium may result in a reduction in this instability. These insights may lead to improvements in volume negative-ion sources.