The government's workforce is more educated than the private workforce. For instance, the government's "college plus" level is 54%, while all private workforce is 35%. "Some college" is 14% of government workers, 19% of the private workforce. So this is important to control for.
This is part of an argument the government workers are not underpaid. As a central planner, you are convinced that you can measure a worker's value by looking at characteristics such as education level.
Two points I would make.
1. The government can never know the value of a government worker. We do not know the value of the output that they produce. The socialist calculation problem is very real.
2. We might be able to make a guess about the opportunity cost of a government worker. That is, what does the worker forego in order to work for the government? To make this guess, we would want to look at the willingness of people to take government jobs and the willingness of people to leave government jobs. If the government has a hard time filling positions, then those positions are likely to be underpaid relative to opportunity cost. If the government has a hard time retaining workers, then those workers are likely to be underpaid relative to opportunity cost.
It seems to me that no matter how many studies one does of pay relative to education or other characteristics, they will only be convincing to dedicated believers in central planning. If you held a gun to my head and made me a central planner, I suppose that I would try to allocate labor by measuring characteristics of workers and aligning those characteristics to jobs.
However, if I were hiring for the government in the context of a market, and I could pay workers whatever I want, then I would only keep salaries high for workers based on my guess about their opportunity cost and value added (the latter being almost impossible to measure, but assume we come up with something). My guess is that I could cut pay for many categories of government workers without causing them to seek jobs elsewhere. My fear is not that government workers would quit, but that they would stay and cause problems to express their resentment.