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Rents should be limited by marginal utility of living in costly city. If you can have higher wage by living in costly city compared to small town, than it make sense to accept higher rent. But if you receive extra income regardless of where you live, then it is hard to say what effect it would have on prices. It may even cause people to prefer cheaper small towns as marginal utility from wage difference would be lower.

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That has to assume that most people would choose lower-paying jobs if the extra income presents. But I don't think it is necessarily true. It is also unlikely that people would right away leave their current job if rents are on the rise, at least not until the net gain gets surpassed by the alternatives. The extra income probably gives the tenants more 'tolerance' to the rent increase and that is simply how subsidies interact with market.

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