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Why did you write that "State and local workers are doing important things"? Was that just a flinch away from the Matt Yglesias crowd, always willing to beat up anyone who thinks government is overgrown?
I think we can objectively conclude that half or more of government workers are not doing anything important.
Worse, large numbers of workers in the private sector produce only "regulatory compliance," so most of them are effectively government workers, and of those a large fraction are not doing anything important.
The manufacturing sector and the service sectors are in many ways mirror images of one another, which is why it is so difficult to claim which one holds the unsustainable jobs. And then there is the issue that manufacturing and production create the wealth to provide services in the first place, which government so aptly obscures.
A sustainable equilibrium may not look like either of the present sectors. For example, people came out of the Great Depression through incremental steps, in which they gradually adapted new forms of both capital and goods. But that adaptation took place in highly expansive ways. Adaptations in the years ahead are more likely to take place in internal ways which actually shrink the (private) living spaces we use. Can present day financing evolve towards such a reality?
I think if you're looking for a definition for sustainable public-sector jobs that is reasonably neutral with regard to differing opinions about how large government should be, I would use the following:
"A sustainable public sector job is one for which the governing authority would pay the full compensation (including fully funding deferred compensation of any type) directly from current tax revenue without borrowing directly or indirectly."
Essentially, if you need the job done and the public is willing (in as much as we can ascribe a will to the public) to pay the full cost of that job being done out of present taxes, then the job is sustainable. This fairly reliably excludes jobs created by stimulus programs, whose purpose is to borrow money to employ people, as well as excluding government employees who are funded by persistent deficits.
It is neutral to the size of government, though not neutral to how it is paid for. High tax/large government is acceptable, as is low tax/small government. Big deficits don't fit in though. I suppose big surpluses would, though those aren't common. It is my intuition that this fits within the PSST structure as I understand it, but you may disagree. I think you might disagree that a high tax/large government system fits well within your framework. If so, I suppose you could take this as a necessary but insufficient condition.