The effects of lowering taxes on rental income is complicated.
If you lower the taxes on rental income, more people will buy houses (and build new ones!) to take advantage of the reduction in taxes. However, they will still have to rent the apartments/houses out at market prices. The rental price will go down because now there is a greater supply of houses, and renting out an apartment at a lower rent is equally profitable as before (since you don't have to pay as much in taxes).
The cost of purchasing a house, however will increase, because there are more landlords competing to purchase the house with you.
Overall, the real problem is zoning single family homes everywhere, and rent control that disincentivizes building new housing.
If you lower the taxes on rental income, more people will buy houses (and build new ones!) to take advantage of the reduction in taxes. However, they will still have to rent the apartments/houses out at market prices. The rental price will go down because now there is a greater supply of houses, and renting out an apartment at a lower rent is equally profitable as before (since you don't have to pay as much in taxes).
The cost of purchasing a house, however will increase, because there are more landlords competing to purchase the house with you.
Overall, the real problem is zoning single family homes everywhere, and rent control that disincentivizes building new housing.