6.5 The French autoroute network is composed of toll motorways with entrances marked as such.

A motorist entering the network obtains a ticket from a toll booth and pays a toll when leaving the motorway. Given the number of vehicles traveling these routes, congestion at both entrances and exits is common. Suppose that four groups, each of 50 motorists, use an autoroute. The price each group is willing to pay, v, differs among groups and falls as the amount of congestion,

c, increases in the following way: vij = vi max-cj

; i, j = 0, …, 3, where vi max = 0.4 - 0.1 * i euros per kilometer is the maximum group i is willing to pay to use the motorway and cj = 0.06 * j euros per kilometer is the loss in value as more groups use it. The marginal cost of using the autoroute, not including the marginal cost of congestion, is zero.

a. If the toll charged for using the autoroute equals a motorist’s marginal private cost—the price in a competitive market—how many vehicles per minute will use it? Which groups will use it?

b. In the social optimum, which groups of drivers will use the autoroute? That is, which collection of groups using it will maximize the sum of the motorists’ utilities? M