The reason housing prices are high and there is a housing shortage in desirable areas are simple: government keeps pushing up demand for housing in such areas through various housing subsidies (low income rent programs, Section 8, government support of mortgages), while at the same time discouraging the creation of new supply through price controls (rent control, affordable housing unit requirements, special taxes on developers) and regulations (zoning, usage restrictions, etc.).
I know, dear Elizabeth, you're just a greedy lawyer and a rabble rousing politician, but please, learn some basic economics: you and people like you are responsible for the housing shortage. And restricting the ability of people to rent out their places for short periods, as on AirBnB, will make the housing shortage worse. In fact, the reason AirBnB is likely so popular in the first place is because AirBnB hosts don't have to deal with all the other rental regulation bullshit people like you have created; in a free housing market, AirBnB would be much less attractive, since landlords could get similar income without all the risk associated with an unpredictable succession of short term renters. So, if you restrict AirBnB rentals, people will probably either leave their apartments empty, or they will convert them into expensive luxury condos. See, Elizabeth, you can certainly stop people from engaging in some economic transactions by wielding your big senatorial stick, but you cannot force them to engage in economic transactions against their will.