Problem
The beltway is the portion of Interstate 495 that forms a circumferential highway around Washington D.C. The portion of I-495 in Maryland currently has no tolls or limits on the number of drivers that use it, leading to traffic congestion during peak usage times.
Consider the route between two exits, A and B. The time it takes a driver to get from A to B depends on the number of drivers on the road. Specifically, suppose the average time it takes (per driver) is given by the following:
AT=4+0.002N
Where N is the number of drivers on the road. So if there are 0 other drivers on the road, a driver would take 4 minutes to travel from A to B. If there are 1000 drivers on the road, it would take 6 minutes, etc.
Suppose the next best alternative route for drivers is to take side roads, which take a time of 20 minutes. We will assume the side roads are dispersed so that the time it takes on these side roads is always 20 minutes, regardless of the number of drivers.
What toll would the state want to set on the beltway to have the efficient (time minimizing) number of drivers using the interstate?