That doesn't mean prices won't change because of it.
As a simplified example, let's say poor person A is renting an apartment from rich person B for $500/month. The new system gives everyone $500/month basic income, but to pay for this rich person B has to pay $1000/month more in tax (of which he gets back half as basic income), while poor person A doesn't pay more tax. So essentially poor person A is getting $500/month from rich person B. Obviously it wouldn't be that simple in reality, but roughly speaking that's what a universal basic income covered by taxes would mean - money flows from rich to poor, in varying degrees.
Now sticking to the simplified two-person version, rich person A thinks "well poor person B could afford my $500/month rent before, now their income has gone up $500/month, so if I charge them $1000/month rent I know they can still afford it". The end result would be wasting time on behalf of the government and indeed the individuals to update everything while money split would remain identical, the only difference being that $500 now flows from B to A (via tax and basic income) then back to B (via rent increase).
Obviously in the non-simplified world it wouldn't be as easy as saying "that person is now getting some money from my taxes so I can charge them that much more", and numbers wouldn't be exact... but as a rough theory, pricing definitely could alter at least to some extent.
All that said: I'm not an economist, I don't know if that would happen, or to what extent. And my personal non-educated opinion is that basic income would be a good thing anyway.
I would doubt that that would happen. If I was person A, and person B raised my rent by $500/mo, I would find a person C to rent from for much less than $1000/mo, rather than continuing to rent from person B.
Prices might go up a bit, but not by that much. The poverty rate in the United States is about 15%. That means that, assuming that the cost of that $500/mo will be spread (unevenly) across the other 85% of the population, or a bit under $100 per taxpayer. That also assumes that the mincome program would exist alongside existing welfare programs, where in reality it would most likely replace them.
Your example doesn't work because it's a monopoly situation. That is, in your world there's only one landlord, so given that housing is a fundamental need and he controls 100% of the supply of housing, he can always raise his prices.
Now, that's not to say that a guaranteed income wouldn't necessarily cause inflation in some goods and services. It probably would! But the standard answer to "why doesn't the provider of a good or service just increase the cost of their service to the maximum disposable income of the buyer" is competition.
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.
