Event-based prospective memory (PM) tasks require participants to substitute an atypical PM response for an ongoing task response when presented with PM targets. Responses to ongoing tasks are often slower with the addition of PM demands ("PM costs"). Prominent PM theories attribute costs to capacity-sharing between the ongoing and PM tasks, which reduces the rate of processing of the ongoing task. We modeled PM costs using the Linear Ballistic Accumulator and the Diffusion Decision Model in a lexical-decision task with nonfocal PM targets defined by semantic categories. Previous decision modeling, which attributed costs to changes in caution rather than rate of processing (Heathcote et al., 2015; Horn & Bayen, 2015), could be criticized on the grounds that the PM tasks included did not sufficiently promote capacity-sharing. Our semantic PM task was potentially more dependent on lexical decision resources than previous tasks (Marsh, Hicks, & Cook, 2005), yet costs were again driven by changes in threshold and not by changes in processing speed (drift rate). Costs resulting from a single target focal PM task were also driven by threshold changes. The increased thresholds underlying nonfocal and focal costs were larger for word trials than nonword trials. As PM targets were always words, this suggests that threshold increases are used to extend the time available for retrieval on PM trials. Under nonfocal conditions, but not focal conditions, the nonword threshold also increased. Thus, it seems that only nonfocal instructions cause a global threshold increase because of greater perceived task complexity.