Gardens in the Bronx

Twenty-five years ago, sections of the Bronx were in terrible condition: there were many abandoned buildings and vacant lots, and there was a lot of crime. To keep things from becoming even worse, the city government encouraged people to start using the vacant lots (which belonged to the city) for gardens. A lot of people did, and there are now hundreds of gardens throughout the Bronx, and they are very popular. However, the city is now short of affordable housing and the government is considering destroying the gardens to build new housing units.

Suppose that there are 100 gardens, and that each one could be converted to a housing unit that could be rented to a low income household for $10,000 per year. In addition, suppose that 100,000 people live in or near the Bronx, and that each person’s annual benefit from Q gardens is given by: MBi =1 - Q/80.

1. How many gardens, if any, should be converted to housing? How many should be preserved? Be sure to show your work.