1 Two stores, Allfoods (A) and Barks&Sensor (B), are located on the two opposite ends of a
straight road. The length of the road is 1 mi. The road is populated by a continuum of
consumers (a unit mass) that are distributed uniformly along this road (alternatively, you may
assume that there is a single consumer living on the road, but the location of the consumer is
uniform random and is not observed by A and B). In order to buy from a store, each consumer
has to travel to that store and back. The cost of a return trip is equal to the distance from
consumer's home to the store in miles.
The stores sell identical nondivisible products and each consumer wants to buy at most one
unit of this product. The value of the product for the consumer is $v$. The cost of producing
one unit of the product is $c_i$ for store $i \in \{A, B\}$. The stores maximize profits and the
consumers maximize the value of the product net of expenses.
(a) Assume that $c_i = 0$ for store $i \in \{A, B\}$, and that $v = \infty$ (since the value is ill-
defined, you can assume that the consumers minimize expenses, rather than maximize
value). Suppose the two stores set their prices simultaneously (the consumers can see
the prices before they decide which store to attend). Formalize this as a game. Suggest
an equilibrium notion.
(b) Find an equilibrium.
(c) Now assume that $v$ is finite. Find an equilibrium.
