It's the opposite: the cost of mortgage is fake-subsidized via tax exemption and since everyone thinks ownership is the "goal" the supply of rental properties asymptotes to the equivalent price. Plus because of the bias to home ownership there is less rental stock.
The reason I say it is "fake-subsidized" is straightforward: say you make $100K per year. A reasonable expense on housing is 25% of your income -- 25K per year or about 2K/month. Luckily for you, you can use the full 25K because you won't be paying income tax on the money when you buy the house. So you can buy more house...except all other buyers making 100K can do the same. If you had to pay after-tax dollars you could only pay, say, $1500/month -- but so would everyone else you're competing with. In essence the tax subsidy only helps real estate agents and those who want to live off the appreciation of their house...which is a risk (yes the long term trend is upwards, but not necessarily where you live, and not necessarily when you plan to retire).
And let's not forget that almost all the mortgages are held by the US government. It's a highly distorted market, and while I believe it developed with good intentions, it's not at all clear it's good for the majority of citizens.
