In a certain city where all parking is controlled by the city, it is possible to provide parking facilities in the downtown area at a constant marginal capital investment of $10,000 per space. Costs of operation can by neglected. There are three equal periods during the day of eight hours each, and spaces are rented only for complete eight hour periods. During the peak period of each of 250 days per year, the demand for parking is given by P= a-bQ, where P is the price per period for a parking space. During the other two off peak periods of those 250 days, the spaces demanded are half that in the peak period, for each possible price. On other days demand is zero. Assume that the interest rate is 10 percent and the facilities do not depreciate.
If a=$16, b= 0.08, and existing spaces are 120, what would be the socially optimal prices during the three periods? What would be the most optimal number of spaces, and what are these corresponding prices?