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?
The equilibrium or "user optimum" scenario is addressed.