Could a tax on rental income that depends on price per square meter induce more housing with lower rents? The background of this question is: In Germany rental income is generally taxed the same as regular business income. In some cities, there seem to be housing shortages across different incomes.
Are there countries where rental income is taxed by square meter/footage? Could it provide housing relief?

In the following tables there are three models of taxation shown. The first tries to model the current situation where the taxation is related to the gained income and not to the size and price of a flat. The following tables try to show the situation when taxation is based on price per m².
Why do you think that additional taxes on more expensive apartments (progressive taxes) would lead to lowering the prices and offering them cheaper than market dictates? 

Adding taxes on top would just make these apartments even more expensive for the consumer, since the owner will push down the additional cost to the consumer rather than take a haircut. Especially if these apartments are laying around empty anyway. 

If I am willing to sell product A for say 1.000$ but there are no buyers - will an increase in the tax help me sell it? 
If you on the other hand gave me a tax discount, I might want to sell it cheaper, since I may be left with the same after-tax profit.