I think that building more housing helps, but a lack of supply is not the only cause of high housing prices. Wealth inequality is also a huge driver, possibly the primary driver. If wealth is allocated in a society in a sufficiently lopsided way, then a small number of people can buy up literally all the assets in the society. Regular people are bidding against a small cadre of ultra-wealthy people who can't possibly spend all their money on consuption, so they use it to buy assets like other people's homes. In the aggregate, increasing the housing supply won't bring down housing prices if wealth inequality is also growing above a certain rate.