There are 240 automobile drivers per minute who are considering using the E-ZPass lanes of the Interstate 78 toll bridge over the Delaware River that connects Easton, Pennsylvania, and Phillipsburg, New Jersey. With that many autos, and a 5 mph speed restriction through the E-ZPass sensors, there is congestion. We can divide the drivers of these cars into groups A, B, C, and D. Each group has 60 drivers. Each driver in Group i has the following value of crossing the bridge: vi if 60 or fewer autos cross, vi - 1 if between 61 and 120 autos cross, vi - 2 if between 121 and 180 autos cross, and vi - 3 if more than 180 autos cross. Suppose vA = $4, vB = $3, vC = $2 and vD = $1. 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 per minute will cross? Which groups 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?