Consider the following problem with commuter train that has more passengers than
seats. The train starts its evening journey at 6:05 pm, but the train is ready for
travelers to board already 30 minutes earlier.
Eectively, the rail company delivers two types of services, transportation and a
seat. While there is a price charged for transportation, there is no extra-charge for
traveling with a seat rather than standing. Unfortunately, there are fewer seats than
passengers every day. Seats are distributed on a rst-come rst serve basis (i.e, the
rst 600 people who come get a seat and the remaining 400 have to stand). In the
following, we consider only the seat allocation problem.
The marginal willingness to pay for a seat is given by MWP = 1000􀀀x (measured
in cent). That is, the person who is most willing to pay for a seat would be willing
to pay $10, the next one $ 9.99 and so on.
(a) Assume that the 600 seats are given to the 600 travelers who have the highest
willingness to pay for a seat. What is the total welfare generated in this market?
(b) Assume now that the 600 seats are just randomly distributed among the 1000
passengers. What is the total welfare generated in this market? (Hint: What
is the average willingness to pay for a seat among all customers?)
(c) Assume that, if seat availability was not an issue, then all individuals would
like to come to the train 5 minutes before it departs (i.e., at 6:00 pm). Each
minute that an individual comes earlier leads to a utility cost equivalent to 25c.
The reason why people might want to come earlier is, of course, to get seats.
In equilibrium, which people will choose to come so early that they get the
seats? When do they come?1 When will the other people come? Which welfare
is generated in this arrangement?2
(d) Suppose that the welfare with a random assignment of seats (i.e., the scenario
in question (b)) is bigger than the welfare in scenario (c) where some people
come early. Is there a practical way to implement a random assignment of seats
rather than the waiting solution?
(e) How could one implement the ecient allocation from question (a) in a practical
way?
1Hint: Suppose (as will be true in equilibrium) that there is a time t minutes before 6 such that all
people who come until that time get a seat, and everyone who comes after has to stand. Consequently,
there must be exactly 600 people willing to \pay" the waiting cost associated with t and 400 who feel that
getting a seat is not worth coming so early.
2Hint: To answer this last question, you have to deduct the total social waiting cost from the surplus
generated by the seat allocation