What happens to the housing market in the case of rent control?
Rent control is the classic example, used in practically every textbook, of a price ceiling. Price ceilings always (according to economists) lead to a shortage of the good or service on which they are placed.
To see a graphic representation of why this is, please follow the amosweb.com link below.
If you look there, you will see that a price ceiling is placed (if it is to do any good) under the equilibrium price and quantity. When this is done, the new price (the one set by the rent control) will cross the demand curve at a quantity that is much higher than the quantity at which the line crosses the supply curve. This (demand > supply) is the definition of shortage.
If you think about this, it makes sense. If you artificially lower the price of something, more people will want it (because it's cheap) and fewer people will want to sell it (because they make less money when the sell it). This leads to a shortage.