Faculty at prestigious institutions dominate scientific discourse, producing a disproportionate share of all research publications. Environmental prestige can drive such epistemic disparity, but the mechanisms by which it causes increased faculty productivity remain unknown. Here, we combine employment, publication, and federal survey data for 78,802 tenure-track faculty at 262 PhD-granting institutions in the American university system to show through multiple lines of evidence that the greater availability of funded graduate and postdoctoral labor at more prestigious institutions drives the environmental effect of prestige on productivity. In particular, greater environmental prestige leads to larger faculty-led research groups, which drive higher faculty productivity, primarily in disciplines with group collaboration norms. In contrast, productivity does not increase substantially with prestige for faculty publications without group members or for group members themselves. The disproportionate scientific productivity of elite researchers can be largely explained by their substantial labor advantage rather than inherent differences in talent.