The extent to which the rain rate from shallow, liquid-phase clouds is microphysically influenced by aerosol, and therefore drop concentration Nd perturbations, is addressed through analysis of the precipitation susceptibility, So. Previously published work, based on both models and observations, disagrees on the qualitative behavior of So with respect to variables such as liquid water path L or the ratio between accretion and autoconversion rates. Two primary responses have emerged: (i) So decreases monotonically with increasing L and (ii) So increases with L, reaches a maximum, and decreases thereafter. Here we use a variety of modeling frameworks ranging from box models of (size-resolved) collision-coalescence, to trajectory ensembles based on large eddy simulation to explore the role of time available for collision-coalescence tc in determining the So response. The analysis shows that an increase in tc shifts the balance of rain production from autoconversion (a Nd-dependent process) to accretion (roughly independent of Nd), all else (e.g., L) equal. Thus, with increasing cloud contact time, warm rain production becomes progressively less sensitive to aerosol, all else equal. When the time available for collision-coalescence is a limiting factor, So increases with increasing L whereas when there is ample time available, So decreases with increasing L. The analysis therefore explains the differences between extant studies in terms of an important precipitation-controlling parameter, namely the integrated liquid water history over the course of an air parcel's contact with a cloud.