Each morning during rush hour, 10,000 people want to travel from New Jersey to New York City. If a person takes the commuter train, the trip lasts 40 minutes. If x thousand people per morning drive to...

Each morning during rush hour, 10,000 people want to travel from New Jersey to New York City. If a person takes the commuter train, the trip lasts 40 minutes. If x thousand people per morning drive to New York, it takes 20 + 5x minutes to make the trip. This problem illustrates a basic fact of life: If people make their decisions individually, they will cause more congestion than is actually necessary.

a. Show that if people make their decisions individually, an average of 4000 people will travel by road from New Jersey to New York. Here you should assume that people will divide up between the trains and roads in a way that makes the average travel time by road equal to the travel time by train. When this “equilibrium” occurs, nobody has an incentive to switch from the road to the train or vice versa.

b. Show that the average travel time per person is minimized if 2000 people travel by road.