Implicatures serve as an important testing ground for examining the process of integrating semantic and pragmatic information. Starting with Bott & Noveck (2004), several studies have found that implicature computation is costly. More recently, attention has shifted toward identifying contextual cues that modulate this processing cost. Specifically, it has been hypothesized that calculation rate and processing cost are a function of whether the Question Under Discussion (QUD) supports generating the implicature (Degen 2013; Degen & Tanenhaus 2015). In this paper, we present a novel elicitation task establishing what the relevant QUDs are for a given context (Experiment 1). In Experiment 2, a sentence-picture verification study, we extend earlier findings about the effect of QUDs on scalar inference to a different kind of pragmatic inference: it-cleft exhaustivity. For both inferences, we find that under QUDs that bias toward calculation, there is no increase in reaction times, but under QUDs that bias against calculating the inference we observe longer reaction times. These results are most compatible with a constraint-based account of implicature, where QUD is one of many cues. Additionally, we explore whether our findings can be informative in narrowing down precisely what aspect of the inferential process incurs a cost.