What's The Point Of Government Sector Unions?

While some conservatives claims that government workers are better paid than private sector workers, many left-liberals like Menzie Chinn claims that if you control for education levels and other relevant factors, government workers are actually lower paid than their private sector counterparts.

But if the largely unionized government workers are really underpaid compared to the mostly non-unionized private sector workers, then what is the point of having these unions? If they don't receive a higher pay or other privileges, then what's the point for government workers to be unionized?

Well, Paul Krugman answer to this question is: to ensure that some of the pay that the taxpayers send to government workers goes with the unions as middle men to Democratic election campaigns, or in other words to make governments subsidize the Democrats. Krugman of course expresses this very differently as he favors it, but that is in effect what he is saying, when he like Chinn says that unions haven't given government workers better pay and is instead meant to help counter the the power of the "oligarchy" (which Krugman claims is represented by Republicans like Wisconsin governor Scott Walker).