here is a link to an empirical study conducted by a professor of mine. It may help answer your question.
A small exerpt from section IV
Based on the extant evidence to date, a reasonable conclusion is that
the average effect of unions upon productivity is small and, if
anything, more likely to be negative than positive
In short, union firms clearly display substantially slower
productivity growth than do nonunion firms, but most (if not all) of
this difference is associated with effects attributable to industry
differences, since union firms are located in industries or sectors
with slow growth. As with the evidence on productivity, it is
concluded that there exists no strong evidence that unions have a
causal effect on productivity growth.