Why does the government have to build houses for every body? Why would it be cheaper for the government to build houses when they subcontract them out to private construction companies to build them anyways? Why would FHB's suddenly rush in to buy these homes? The prices would sure drop cause the place would resemble a ghetto.
I assume the original post was a reference to public housing, that is housing owned by government and made available to (mostly) low income people for them to live in.
Back in ye olde days, government did indeed build such houses and they did it using direct government employees. Then along came the idea that everything should be contracted out and the cost of doing literally any physical work went through the roof (or the quality dropped, one or the other depending on situation) when that happened.
Do it "in house" and you need to employ workers, buy materials (which for a government purchase in bulk will be relatively cheap) and build the houses, roads, dams or whatever.
Contract it out and you need an army of contract administrators and a legal team on both sides. Then you have a huge pile of paperwork. Then you need people to actually do the job and materials. Then you add a profit on top of all that - the work, materials, administration and so on.
What you end up with is a situation where labour productivity "on the tools" tends to go up, but where a lot of people spend a lot of time not actually using any tools. The end result is that some jobs cost a bit less, the occasional one costs an order of magnitude more, and on average you get either a modestly higher cost for the same quality, or the same cost for lower quality. Either way it's a worse deal so far as the taxpayer is concerned. Been there, seen this one first hand.
The real problem with governments contracting things is that to be blunt, they're generally a pushover. As long as all the I's are dotted and the T's crossed, nobody really cares about too much else. For political reasons government will want the job done, and the incredible amount of work required to award a tender which complies with all the rules means that no sane public servant would want to go back through the tender process again should there be a contractor. Unless they actually stop doing the work, which generally doesn't happen, then it's far more practical to simply hand over the money and keep the project moving so that's what they do. There are good contractors and bad, trouble being that the bad ones generally still get paid due to the difficulty changing part way through a project.
Working in the public service, I remember quite well tying up myself and two others for an entire week spending what would be around $10K in today's money on a single item. In due course the matter was resolved, meanwhile the supplier decided to raise the price since they knew it would take too long to repeat the purchasing process. So it ended up as $12K for the purchase plus another 5K in wages so $17K overall in order to buy something that retailed for $10K "off the shelf". In short, that's why things go wrong when government deals with private enterprise - too many rules and regulations means that government is an outright pushover.
So I can certainly see that government can't build cheap public housing for low income earners like they used to given that they've changed the way they do business. But they could always go back to the old model if they really wanted to as it's only ideology that stops them. Taxes could be reduced pretty much immediately if they did.
Go forward a couple of decades and everyone will have worked this out. It's a big cycle if you look at history with things like utilities, transport etc going from private to public and now mostly back to private. Once the pain of rising utility bills and the cost of taxpayer subsidies to privately owned operators of various things becomes an issue, and at some point it will, then things will start going back the other way over the next few decades. Then it will go back to private again. That's the lesson of history and I expect we'll repeat it. Tas bought back the railways not that long ago and WA has stopped outsourcing some things in the power industry so there's a bit of it happening already.
As for house prices, well if you have government buying existing houses so as to make them available as public housing then that's a very different situation to if government simply cleared the land, built a few houses here and there and sold the rest of the land to private owners. It's simple supply and demand there, with government having moved from being a source of supply to being a source of demand.