-yes, it's a problem, but I think it's a mistake to call them -"unproductive." They are directly unproductive in that they don't -contribute directly to individual capitalist profits, but they are -indirectly productive, in that they help the capitalist class as a -whole.
This is a bizarre definition of "productive" to begin with. A health care worker in a public hospital does the same work as one in a private hospital-- and they both contribute value to society. I ride the subway every day and the work done to keep that subway going is as productive as when I ride a private bus to go to a different city.
No reasonable definition of productivity should depend on whether the work is paid for with tax dollars or private corporate funds. There may be useful distinctions between work that produces goods versus those based on servicing the workers who do that production, but such categories would treat private and public health care in the same category.
No dobut some important analysis hinges on public versus private ownership, but "productivity" doesn't seem like a useful term in that context.
nathan