This will vary a lot from time and place; but as a former civil servant (State and Federal) I can tell you that what usually happens is, massive lay offs in some sectors but more employees (often transfers or "bumbs" from the layoffs) in others.
That's because someone has to process all those extra unemployment claims, welfare checks, applications for disability, foster kids, applications for public housing etc., etc.. If the forms are not processed right, if the investigations are not done; then the next head lines are "thousands allowed benefits they don't qualify for."
When people don't get their claims processed in a "reasonable time" (what that is varies) States get faced with a combination of class action law suits and extremely negative press coverage with headlines like:
"Sally X worked every day of her life until the lay off, now she and her five kids and disabled husband face eviction because the program that was supposed to.......fill in the blank, is six months behind in applications..."
Finally, humans do have a breaking point, even civil servants. When I worked in Worker's Compensation for the State of Colorado during the oil crash there, the State "saved" money by not replacing people and of course laying them off (eventually I was laid off too). But before then, claims that by law had to be processed within x number of weeks had back up to folders stacked on the floor, to the tune of about 6 months worth. When the other clerk moved away, of course she was not replaced and the piles became so large we had to make pathways between them (this was when computers were limited to data bases).
After several weeks of this, and dealing with lawyers coming serving summons on how our department would be sued because it had failed to meet its statutory requirements for processing cases in a timely manner (including being told by one loud mouth, that "of course when YOU are put on the stand to testify,") etc; the other clerk said no job was worth her life. The stress was destroying her and her home life, she was going to quit and move back to another State with her baby and live with a sister.
After that, I got "bumped" by someone with 10 years more experience, but who was totally incapable of doing my job (I know I tried to train her). When I visited a few years later, before moving to California; my former boss admitted that the women was still just sitting there collecting a pay check and they had to work around her.
That's the sort of real problems that exist in the civil service a combination of policies that look good on paper (but really cost more money by shorting staff to the point that nothing can get done, people give up and just start pushing paper around) and work rules that make it almost impossible to get rid of someone who can't to their job.
The best you can hope for with a long-term employee who can't do the work is to get them to transfer elsewhere or just work around them (effectively causing a larger staff shortage).
But really, what usually happens is that some civil servants wind up on unemployment (and eventually homeless) like everyone else, but many other simply transfer over (or are even hired) to man the new services required by the economic downturn.
expatriate Californian living in rural Ireland with husband, dogs, horses. garden and many, many cats