There are two cities, A and B, joined by two routes, I and II. All roads are one-way roads. There are 100 travelers who begin in city A and must travel to city B. Route I links city A to city B through city C. This route begins with a road linking city A to city C which has a cost-of-travel for each traveler equal to 0.5 + x/200, where x is the number of travelers on this road. Route I ends with a highway from city C to city B which has a cost-of-travel for each traveler of 1 regardless of the number of travelers who use it. Route II links city A to city B through city D. This route begins with a highway linking city A to city D which has a cost-of-travel for each traveler of 1 regardless of the number of travelers who use it. Route II ends with a road linking city D to city B which has a cost-of-travel for each traveler equal to 0.5 + y/200, where y is the number of travelers on this road. These costs of travel are the value that travelers put on the time lost due to travel plus the cost of gasoline for the trip. Currently there are no tolls on these roads. So the government collects no revenue from travel on them.

(a) Draw the network described above and label the edges with the cost-of-travel needed to move along the edge. The network should be a directed graph as all roads are one-way.

(b) Travelers simultaneously chose which route to use. Find Nash equilibrium values of x and y.

(c) Now the government builds a new (one-way) road from city C to city D. The new road is very short and has 0 cost-of-travel. Find a Nash equilibrium for the game played on the new network.

(d) What happens to total cost-of-travel as a result of the availability of the new road?

(e) The government is unhappy with the outcome in part (c) and decides to impose a toll on users of the road from city A to city C and to simultaneously subsidize users of the highway from city A to city D. They charge a toll of 0.125 to each user, and thus increase the cost-of-travel by this amount, for users of the road from city A to city C. They also subsidize travel, and thus reduce the cost-of-travel by this amount, for each user of the highway from city A to city D by 0.125. Find a new Nash equilibrium. [If you are curious about how a subsidy could work you can think of it as a negative toll. In this economy all tolls are collected electronically, much as New York State attempts to do with its E-ZPass system. So a subsidy just reduces the total amount that highway users owe.]

(f) As you will observe in solving part (e) the toll and subsidy in part (e) were designed so that there is a Nash equilibrium in which the amount the government collects from the toll just equals the amount it loses on the subsidy. So the government is breaking even on this policy. What happens to total cost-of-travel between parts (c) and (e)? Can you explain why this occurs? Can you think of any break-even tolls and subsidies that could be placed on the roads from city C to city B, and from city D to city B, that would lower the total cost-of-travel even more?