An optoelectronic device composed of a heterojunction phototransistor (HPT) in series with a light-emitting diode (LED) has previously been shown to switch from a low-current state to a high-current state through a region of negative differential resistance (NDR) when a voltage greater than the breakover voltage is applied. We call this device a light-amplifying optical switch (LAOS). The NDR has been qualitatively discussed by several authors who attribute the NDR to either optical feedback, electrical feedback, or a combination of both. This paper presents a quantitative circuit model for the LAOS that includes optical and electrical feedback, nonlinear HPT current gain, the current leakage shunting the HPT, and the Early effect. It is shown that the HPT must have nonlinear gain and either the Early effect or leakage in order for the current-voltage characteristics of the LAOS to have an NDR region. The breakover voltage for the NDR is shown to be a strong function of the optical and electrical feedback and the Early effect. The results of the model are compared to measured values obtained from experimental LAOS devices fabricated from layers of lattice-matched InGaAs/InP grown by gas-source molecular beam epitaxy. By electrically contacting each of the layers, the HPT and LED parameters were measured and used in the model. The model indicates that the breakover voltage of our LAOS devices (with modest input light) is primarily determined by electrical feedback since the calculations show that the optical coupling between the LED and the HPT is weak. For low input-light levels, avalanche breakdown of the HPT appears to initiate the breakover into the NDR region. Avalanche breakdown has not previously been considered as a contributing factor in LAOS-type devices.