Two carts selling coconut milk (from the coconut) are located at 0 and 1, 1 mile apart on the beach.

Two carts selling coconut milk (from the coconut) are
located at 0 and 1, 1 mile apart on the beach in Rio de Janeiro. (They are the
only two coconut-milk carts on the beach.) The carts—Cart 0 and Cart 1—charge
prices p0 and p1, respectively, for each coconut. One thousand beachgoers buy
coconut milk, and these customers are uniformly distributed along the beach
between carts 0 and 1. Each beachgoer will purchase one coconut milk in the
course of her day at the beach, and in addition to the price, each will incur a
transport cost of 0.5 3 d 2 , where d is the distance (in miles) from her beach
blanket to the coconut cart. In this system, Cart 0 sells to all of the
beachgoers located between 0 and x, and Cart 1 sells to all of the beachgoers
located between x and 1, where x is the location of the beachgoer who pays the
same total price if she goes to 0 or 1. Location x is then defined by the
expression:

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The two carts will set their prices to maximize their
bottom-line profit figures, B; profits are determined by revenue (the cart’s
price times its number of customers) and cost (each cart incurs a cost of $0.25
per coconut times the number of coconuts sold).

(a) For each cart, determine the expression for the
number of customers served as a function of p0 and p1. (Recall that Cart 0 gets
the customers between 0 and x, or just x, while Cart 1 gets the customers
between x and 1, or 1 2 x. That is, cart 0 sells to x customers, where x is
measured in thousands, and cart 1 sells to (1 2 x) thousand.)

(b) Write the profit functions for the two carts. Find
the two best-response rules for each cart as a function of their rival’s price.

(c) Graph the best-response rules, and then calculate
(and show on your graph) the Nash equilibrium price level for coconut milk on
the beach.

 

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