Home Tools PROBB.pdf x G) ' . [EJG'JBQ :/2lk@@1%''?E'/azf>ll 628% 3. Congestion and 'Ibll Booths. Sometimes very long queues build up at toll booths. You nd yourself, say, waiting 5 minutes to pay 55050, An often heard 'kvetch' is that such tolls are the cause of long commutes and should be eliminated. This question asks you to apply some game theory to this problem, In eect, you are going to design tolls that minimize traic problems! The whole question is quite hard, but each part is not too bad so do not panic: just take it a step at a time. Suppose that a very large (say, innite) number of commuters must travel, one per car, from A to B There are two possible routes, 1 and 2. Let x be the proportion of cars that use route 1. There is a public debate over whether to put a toll on route 1, The time it takes each car to get from A to B, depends on the proportion of other cars on that route. The time, in hours, each car takes on Route 1 is given by T (x) 7 38 if there is no toll 1 ' 4x if there is a toll The time each car takes on route 2 is given by T2('I) = 1 + 2(1 :5) The utility of a commuter who chooses route 2 is T2 (1') Her utility if she chooses route 1 is T1(.L') if there is no toll, and T1(.'L') 17 if there is a toll of $11 In principle, 17 could be negative. (21) Consider the game among the commuters simultaneously choosing which route to take. Find the Nash-equilibrium proportion :12 that choose route 1 when there is no toll. How long does the commute take each car on route 1, and on route 2'? [Hint: in equilibrium, what must be true of the utilities of a commuter who chooses route 1 and one who chooses route 2? You might want to use a calculator.] (b) Find the Nashequilibrium u: when there is a toll, as a function of the toll charge p (0) Consider a social planner who can set a: and who aims to minimize average commuting time T(m) := $T1(i) + (1 ;::)T2(w). (We assume that the social planner does not care about the money that changes hands at the toll booth) Find the social planner's optimum in when there is a toll What are the corresponding commutes on each route, and the average commuting time T? (d) Suppose that the social planner can not direct the traic it, but she can set the toll charge p. She could also choose no tolli Consider the following game. First, the social planner sets the toll (or chooses no toll). Then, after observing the toll charge (or no toll), the commuters simultaneously choose their routes (much as in parts (a) and (b)) Consider the subgame perfect equilibrium outcome of this game. Does the social planner set a toll and, if so, what toll does she choose? [Hint: from part (b), you already know the relation between p and :c if there is a toll. From part (c), if there is to be a toll, you already know the at the social planner would like to choose, and the T that this induces And, from part (a), you can gure out the T without a toll] (e) Assuming that the time spent at the toll booth is 0.2x, in the subgameperfect equilibrium outcome from part (d), how long will each car on route 1 spend at the booth