This sort of thing is usually fertile breeding ground for corruption and backhanders. The NHS has been dogged by scandals caused by this sort of pseudo-privatization.
It works for smaller and short term contracts, especially for things which the private sector also needs, but the larger the size of the contract the more likely it is that the price will be determined by backhanders and which ministers play golf with which chief execs.
For something the size of "supply a city with emergency services", you could practically guarantee that the city council leader's cushy post-political job would be determined by which company they picked.
tl;dr read up on the principal/agent problem.