You have purchase property to construct something on. So high property values works against your own argument.
> You need a real example, not a contrived one
No I don't, as I did not make the original argument you inquired on. That poster needs a real example, I however, only need to entertain how that possibly could be.
The basis of construction is that you're converting fewer units into more, on the same piece of property. If a single-family home is a million dollars then you can buy one for a million dollars, build a 50-unit condo tower on the lot at a cost of $180,000/unit, apportion the original million dollars across 50 new units so the cost per unit is $20,000, and you're creating new units at a cost of $200,000/unit while selling into a market where a unit goes for a million dollars.
> I however, only need to entertain how that possibly could be.
But that's only interesting if it's something that might plausibly happen in practice. You can make anything possibly happen given an arbitrary set of unrealistic constraints.
>Suppose you built twice as many housing units as you had residents. Rents wouldn't decrease? Why not?
I gave you an example of how that would not happen: in an uncompetitive market such as a monopoly, additional supply doesn't impact price if it is simply hoarded. Now we could argue whether that's realistically to happen or not, but I see not why that is relevant to your question.
Let's consider whether this can even possibly happen. In order for a monopolist to own all the housing, there would have to be somebody with that much money.
The largest wealth fund in the world is China's, with a total value of around $2.75 trillion US dollars. This is the entire fund across all asset classes. Jeff Bezos with his paltry $200 billion can't even hold a candle to it. By contrast, the total value of US housing is north of $47 trillion dollars. So if you were to double the amount of housing in the US, no one in the entire world exists who could afford to buy all of it. Even the entire government of China isn't within an order of magnitude of it.
The construction costs are much lower than property values, so construction would be highly profitable. Every wealthy person in the world would be lining up to invest in a hundred construction companies because they would be yielding a 400% year-over-year return until the monopoly is broken.
> All I need is one possible example to demonstrate how that can be.
You need a real example, not a contrived one. "What if the emperor of the world declared that rents have to stay high?" Then you don't have a market anymore and we're no longer talking about what happens in markets at all.