Distribution of benefits for transferring drivers to transit during a congested morning commute

A residential suburb has N = 30,000 commuters who drive alone to jobs in a central business district (CBD) with uniformly distributed desired arrival times to the CBD from 8:00am to 9:00am. All car traffic must cross a bridge with limited capacity μ = 20,000 veh/hr to enter the

CBD, but road capacity upstream and downstream can be assumed sufficient to serve demand without delay. Assume that all commuters value their time of delay at β = 30 $/hr, and that the out-of-pocket cost of driving a car is cd = $4 per trip.

A. Initially, the only way to get to work is by car. Using Vickrey's model of the morning commute with earliness time valued at ½ the rate of queuing delay (e = 0.5) and lateness time unacceptable (L = ∞), draw to scale a queuing diagram for a typical day. Show on it the maximum queuing delay, and the maximum earliness. (Note: free flow travel time can be ignored; explain why.) What is the average generalized cost per commuter? The city government is considering starting a high-speed ferry service to relieve congestion and reduce the cost of the morning commute. How many commuters (n), uniformly distributed between 8:00am to 9:00am, must be diverted to the ferry to eliminate the delay caused by the bridge bottleneck?

B. Write an expression of the generalized cost per passenger in terms of the ferry ship's scheduled headway h << 1 (hrs/trip), the prorated cost per commute period for the ferry terminal infrastructure cf ($/period) and the variable cost per revenue ferry trip cv ($/trip) (Assume: fares reflect the ferry's costs; the ferry carries the n commuters identified in part A; a ferry trip excluding the waiting delay takes the same amount of time as a car trip exclusive of the bottleneck delay; headway are so short that people do not bother checking the schedule; and the number of seats on the ferry always exceeds the demand.)

C. What h* minimizes the average ferry rider's generalized cost if cf = 20,000 $/period, cv =

2,000 $/ trip, and all the costs of ferry operation are passed on to ferry riders as fare? What is the average generalized cost to each ferry rider, and how does it compare to the average cost to each driver on the now uncongested bridge? Who is better off? What is the average generalized cost per commuter (averaging across ferry riders and drivers)?

D. Could the solution of part C be an equilibrium solution if commuters were free to choose their transportation mode? If your answer is "no", suggest a revenue-neutral toll and ferry fare policy that would achieve the solution we have assumed, and discuss the role of crosssubsidies.

E. Extra credit: Determine whether the average generalized cost across all commuters could be reduced by changing n.