If house prices cross a threshold where first time buyers can't afford to get onto the property ladder, they're forced to rent - driving up competition for stock & thus rental prices. (This appears to be affecting more & more people in the UK).
These people pulling the ladder up behind them might be in for a rude surprise.
A sketch of my argument: as we are sometimes reminded to bear in mind, prices are driven by market conditions and affordability, not costs. If prices shoot up, why will the rent increase? Is it not already set to the highest value the market can bear, or something very close to that? We'll assume it must be, because it would be silly to do anything else (and the tenancy market is liquid enough). But then, if prices are at the right level already, how will people pay for increased rent?
If salaries increase as well, then this is just ordinary inflation. House prices go up, rents have go up, salaries go up - well, people have been warning about this for years! On the other hand, if prices increase, and rents go up, and salaries don't, how are people going to be able to afford to pay for any of this? The obvious answer, of course, is that they won't.
Prices are what the market will bear. If everybody's costs go up, or supply is limited, then rent can sure go up. Renting is an inflexible demand - unless you want to go homeless.