but when you say the solution "isn't to clamp down on the supply" i lose the thread of your argument.
clamping down on the supply is exactly what a public sector union does. union work rules and other union-favorable city regulations exclude or limit non-union workers who might otherwise be hired to carry out various city functions.
working for the city or the department of water and power can be a very good deal for the worker, but city residents pay more in taxes and see less service as a result. merely unionizing the labor force helps some people but hurts others.
indeed, a case can be made that, because police officers are so highly paid and benefitted, they are scarce. and because they are scarce, there's more property crime, and murder, than there would be otherwise. it seems quite plausible that some city residents pay a very high price because of this public sector unionization.
in politics, this leads to a strategy wherein city residents who live in "electorally unimportant" areas (i.e. poor areas with lower voter turnout) receive lower levels of government service than city residents in areas that vote a lot.