3. The San Francisco metro are lies on a large bay of water. There is one bridge crossing the bay from west to east (The Bay Bridge), and the only alternative way to get to the other side is to drive around the bay - a considerably longer distance. Not surprisingly, there is a good deal of traffic on the bridge, so much that traffic is slowed. To study this phenomenon, let q stand for the number of cars that cross the bridge, suppose that it takes to to cross the bridge and time to to drive around the bridge. Let the wage in San Francisco be w, and since w is high, assume that all costs of driving (e.g., gasoline) other than the time cost are negligible. The cost of driving around the bay is weto, and the cost of driving across the bridge if (q-1) other drivers are also driving across the bridge is wtq. a) Find the number of cars qe that cross the bridge in competitive market. b) Now suppose you are a benevolent social planner who values all drivers equally. Find the socially optimal number of cars qs crossing the bridge. (That is, maximize the total value of time saved by cars crossing the bridge.) a) Briefly explain the nature of the market failure here. Why do more cars cross the bridge than would be socially optimal? d) Now suppose that the city was to charge a "Pigouvian toll" of s dollars per car crossing the bridge. Find the optimal level of s