Debtors have to make even on their mortgage, at least, so an increase in prices also drags rental rates up - either because the landlord bought later and has higher costs; or because the increase in prices drives up property tax and thus increases costs on existing properties.

If the market worked, you'd see people building multi-family units everywhere until supply caught up with demand, and then you'd see a huge crater in prices as people who took a bath on real estate speculation were overrun with the resulting supply glut. This doesn't happen, for a host of various reasons. Governments want housing to be simultaneously affordable and an investment, which is impossible. Hence most cities wind up building a sort of shadow immigration system, through rent control, selective property tax moratoriums, and so on. People who have lived in a city all their life enjoy lower rents, subsidized by people who just moved in and have to buy at market rate.