Mr Always-grumbles-about-correlation-studies reporting in here. Maybe it's a nice causal study, but the jump from "moderates compensation of top executives" to "reduces equality across the whole income spectrum" is a big stretch.
Update: also, I went and scanned the linked article (because I'm that type). They're credible authors, but they themselves explicitly say: "In reporting correlations between unionization and managerial pay, we are not necessarily able to establish whether unions cause differences.... We try a variety of strategies to understand these issues, however each is imperfect."
"In reporting correlations between unionization and managerial pay, we are not necessarily able to establish whether unions cause differences"
Yes, but that's a different correlation vs causation issue. The point is, the phenomenon of unions equalizing pay, on the company (micro) level (which has been reproduced in other countries and time periods), can be considered to be a causal explanation for the phenomenon of rising inequality when unions decline, in the society at large.
Update: also, I went and scanned the linked article (because I'm that type). They're credible authors, but they themselves explicitly say: "In reporting correlations between unionization and managerial pay, we are not necessarily able to establish whether unions cause differences.... We try a variety of strategies to understand these issues, however each is imperfect."