The action that attempts to control the high cost of rent for lower-income families is called price ceilings. As a result of this action, few people with the greatest need benefit from the program.
The price ceiling is a price control imposed by the government in which the price charged is greater or less than the equilibrium price determined by the market forces of demand and supply. Governments use price ceilings to protect consumers from conditions that could make products prohibitively expensive.
They often result in a shortage of localized supply if the ceiling is too low. They increase demand because, by definition, they make a good or service cheaper than the market would have set. A Price ceiling cannot alter the supply curve in a positive way, it always creates shortages, although sometimes the offer is sufficiently elastic to absorb it.