5.10 On a particular busy bridge, congestion occurs if 400 vehicles cross it per minute. We can divide the drivers of these cars into four groups: A, B, C, and D. Each group has 100 drivers. Each driver in Group i has the following value of crossing the bridge: vi if 100 or fewer autos cross, vi - 1 if between 101 and 200 cars cross, vi - 2 if between 201 and 300 cars cross, and vi - 3 if more than 300 cars cross. Suppose that vA = +5, vB = +4, vC = +3, and vD = +2. The marginal cost of crossing the bridge, not including the marginal cost of congestion, is zero.

a. If the price of crossing equals a driver’s marginal private cost—the price in a competitive market—how many cars will cross?

b. In the social optimum, which groups of drivers will cross? That is, which collection of groups crossing will maximize the sum of the drivers’

utilities?