Many answers here seem to come from wishful thinking bordering on financial illiteracy.
Money has a cost. Whether you use it to build houses or indirectly invest in companies through stock markets or your own business or treasury bonds, all expect yield vs risk to match other investment opportunities. If you make it tough for new investors to enter housing markets, then new money will stop flowing into housing markets as they can find better yields else where. That means lower new supply and higher rents. If you assume your toothpaste company is ripping you off by increasing prices, then the answer is not to make it difficult for new toothpaste companies to enter the market (or add more taxes to the existing ones). The answer is to make it easy for other companies to enter the market and ensure its profitable for them to do so.
Money has a cost. Whether you use it to build houses or indirectly invest in companies through stock markets or your own business or treasury bonds, all expect yield vs risk to match other investment opportunities. If you make it tough for new investors to enter housing markets, then new money will stop flowing into housing markets as they can find better yields else where. That means lower new supply and higher rents. If you assume your toothpaste company is ripping you off by increasing prices, then the answer is not to make it difficult for new toothpaste companies to enter the market (or add more taxes to the existing ones). The answer is to make it easy for other companies to enter the market and ensure its profitable for them to do so.