Sure competition is one of the many reasons it's way more complicated than my simple version, but it doesn't mean prices won't change to counter basic income. It's possible competition will force prices to remain where they are, but it's also possible that everyone raises prices, step by step in line with competition. Housing is one area I would assume would definitely go up in price, though not to 100% of the basic income. My biggest reason for thinking this is that housing costs in many (most?) areas are driven not by material costs but by demand. Give everyone basic income and the demand alone (i.e. competition between buyers/renters) would raise prices. This would happen if you, for example, gave everyone in NYC $1000/month and didn't tax anyone for it. Then add in motivation from owners who rent to make more money to balance out their tax losses..
But yes, my comment was a very, very simplified example, to counter "prices won't change, we're not printing more cash" arguments - not a direct tale of what could happen in the real world.
You haven't given any reason to believe that demand would go up. Demand doesn't increase because wealth increases.
Now, you MIGHT see increased demand in some areas! For example, some people who previously lived at home might look for an apartment if they had a higher income. Or people who are currently homeless might move into the housing market.
At the same time, some people would presumably graduate out the top of the lowest-income housing market, and put increased demand on the middle-income market. Or whatever. And of course we might say that it's a huge victory if a bunch of homeless people got out of the vicious cycle of homelessness, even if it does increase housing demand for the working poor.
The problem was not that your example was simplified, it was that it introduced a brand new element (monopoly) to the matter under discussion. If you want to make a simple example that actually shows some kind of demand-increase, rather than illustrating monopoly power, you should do so.
Cities like NYC and SF are a bit of an exception in the US - they're very desirable cities, and their housing stock is more or less filled. Getting a place to live, even with sufficient income, is a bit of a chore.
In other markets, there are houses that sit vacant. Bringing the minimum income up helps narrow the gap between the price floor on these properties, and the amount people are able to pay. That is, a basic income would help more efficiently distribute housing in less desirable areas.
If someone on basic income wants to move to NYC, they'll still need to work hard, and get enough money to compete with all the other people who want to live in NYC.
But yes, my comment was a very, very simplified example, to counter "prices won't change, we're not printing more cash" arguments - not a direct tale of what could happen in the real world.