The problem is some combination of corruption and regulatory requirements. If the bureaucrats write the contract in such a way that only one company can satisfy it, that company has no effective competition so they don't have to compete on price. Same thing if the contract requirements, or the paperwork required to enter into the bidding itself, are such that smaller entities need not apply and you again have insufficient competition to drive down prices.
But none of that goes away by bringing the project in house either, you just trade corruption and high costs by contractors for corruption and high costs by public sector unions, and in that case there is explicitly not even the possibility for competition.
So you still either have to solve the corruption and bureaucracy somehow (effective method unknown), or move the project entirely to the private sector, e.g. by using the money instead to make tax cuts for regular people or transfer payments so that instead of government housing projects you leave people with the money to buy market rate housing directly, or whatever else they need.
The fact that it's a natural monopoly makes it susceptible to a lot of the same problems as a government, but making it an independent entity actually mitigates them somewhat by reducing the size of the bureaucracy. One of the big problems with elected governments is that the Republican wants to install a corrupt transportation board and the Democrat wants to install a corrupt school board and you have to pick your poison. If you get to elect the directors responsible for local transportation separately from the directors responsible for local schools, that doesn't happen as much.
It also helps quite a lot to move as much of the spending as possible to as local a level as possible, because then you don't get wasteful pork motivated by a desire for inter-region wealth transfers.
The problem is that nothing is the “same thing as a free market”. Free market don't exist in the real world, and the market forces are way less powerful than what we would like. They are almost always dwarfed by other forces (transaction cost, network effect, economies of scale, and information asymmetry for instance).
Remember, the free market model comes from an era when chemistry was modeled with the 4 elements !
I fail to see how the age of the model has anything whatsoever to say to its validity. Not everything old is stupid; not everything new is right. If you want to disprove the free market model, you have to do a lot better than pointing out that it's old.
It is true that perfectly free markets don't exist in the world. That doesn't mean that the model isn't useful as an approximate ideal.
[1]: crisis for starter…
My prediction has never turned out to be false. It's a useful prediction.
Those are all free markets, too. Consider information asymmetry, for example. The less I know about something, the riskier it will be for me to buy it, hence the less I am willing to pay for it.
I.e. information asymmetry is called risk, and is accounted for in the pricing. Companies are incentivized to reduce risk for their customers, because then they can charge a higher price.
Privatization of government services is not the same thing as a free market, and hence free market forces are not applying to it.
In a free market, when a business contracts for services from another business, it is motivated by value received and price paid. When the government contracts for services, price and value are of much lesser concern. The bureaucrats negotiating the contracts have little care for what it costs, whereas political and social concerns are paramount.
It's not in the least surprising that privatizing who provides government services does not result in cost savings.