Every morning, 6,000 commuters must travel from East Potato to West Potato. Commuters all try to minimize the time it takes to get to work. There are two ways to make the trip. One way is to drive straight across town, through the heart of Middle Potato. The other way is to take the Beltline Freeway that circles the Potatoes. The Beltline Freeway is entirely uncongested, but the drive is roundabout and it takes 45 minutes to get from East Potato to West Potato by this means. The road through Middle Potato is much shorter, and if it were un-congested, it would take only 20 minutes to travel from East Potato to West Potato by this means. But this road can get congested. In fact, if the number of commuters who use this road is N, then the number of minutes that it takes to drive from East Potato to West Potato through Middle Potato is 20 + N/100.
(a) Assuming that no tolls are charged for using either road; in equilibrium how many commuters will use the road through Middle Potato? __________ What will be the total number of person-minutes per day spent by commuters travelling from East Potato to West Potato? ____________________
(b) Suppose that a social planner controlled access to the road through Middle Potato and set the number of persons permitted to travel this way so as to minimize the total number of person-minutes per day spent by commuters travelling from East Potato to West Potato. Write an expression for the total number of person-minutes per day spent by commuters travelling from East Potato to West Potato as a function of the number N of commuters permitted to travel on the Middle Potato road. __________ How many commuters per day would the social planner allow to use the road through Middle Potato? _________________
In this case, how long would it take commuters who drove through Middle Potato to get to work? ___________. What would be the total number of person-minutes per day spent by commuters travelling from East Potato to West Potato? __________________
(c) Suppose that commuters value time saved from commuting at $w per minute and that the Greater Potato metropolitan government charges a toll for using the Middle Potato road and divides the revenue from this toll equally among all 6,000 commuters. If the government chooses the toll in such a way as to minimize the total amount of time that people spend commuting from East Potato to West Potato, how high should it set the toll? ___________ How much revenue will it collect per day from this toll? ________ Show that with this policy every commuter is better off than he or she was without the tolls and evaluate the gain per consumer in dollars.