Two carts selling coconut milk (from the coconut) are located at O and l, 1 mile apart on the beach in Rio de Janeiro. (They are the only two coconut milk carts on the beach.) The carts-Cart O and Cart 1-charge prices p0 and p 1 , respectively, for each coconut. Their customers are the beach 0oers uni formly distributed along the beach between O and 1. Each beach goer will purchase one coconut milk in the course of her day at the beach and. in ad dition to the price, each will incur a transport cost of 0.5 times d2, where di the distance (in miles) from her beach blanket to the coconut cart. In chis system, Cart O sells to all of the beach goers located between O and x, and Cart 1 sells to all of the beach goers located between x and 1, where x is the location of the beach goer who pays the same total price if she goes to O or l. Location x is then defined by the expression
Po + 0.5x2 = p1 + 0.5(1 - x)2.
The two carts will set their prices to maximize their bottom-line profit fig ures, B; profits are determined by revenue (the cart's price times its number of customers) and cost (the carts each incur a cost of $0.25 per coconut times the number of coconuts sold).
(a) Determine the expression for the number of customers served at each cart. (Recall that Cart O gets the customers between O and x, or just x, while Cart 1gets the customers betv.reen x and l, or 1 - x.)
(b) Write out profit functions for the two carts and find the two best response rules for their prices.
(c) Graph the best-response rules, and then calculate (and show on your graph) the Nash equilibrium price level for coconuts on the beach.