Problem:
The equilibrium or "user optimum" scenario
Each morning during rush hour, 10,000 people want to travel from New Jersey to New York City. If a person takes the subway, the trip lasts 40 minutes. If x thousand people per morning drive to NYC, it takes 20+5x minutes to make the trip. This problem illustrates a basic fact of life: if people are left to their own devices, they will cause more congestion than need actually occur!
a) Show that if people are left to their own devices, an average of 4,000 people will travel by road from NJ to NYC. Here you should assume that people will divide up between subway and road in a way that makes the average travel time by road equal to the average travel time by subway. When this "equilibrium" occurs, nobody has an incentive to switch from road to subway or vice versa.
This is known as the "user optimal" scenario.
b) Show that the average travel time per person is minimized if 2,000 people travel by road.
This is known as the "system optimal" scenario.
c) What percent worse than system optimal is the user-optimal scenario?