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Question 3: Purely Continuous Simultaneous-move Games

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Question 3: Purely Continuous Simultaneous-move Games

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 cartsβ€”Cart 0 and Cart 1β€”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 Γ— 𝑑2, 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 π‘₯, and Cart 1 sells to all of the beachgoers located between π‘₯ and 1, where π‘₯ is the location of the beachgoer who pays the same total price if she goes to 0 or 1. Location π‘₯ is then defined by the expression:

𝑝0 + 0.5π‘₯2 = 𝑝1 + 0.5(1 βˆ’ π‘₯)2

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).

  1. 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 π‘₯, or just π‘₯, while Cart 1 gets the customers between π‘₯ and 1, or 1 βˆ’ π‘₯. That is, cart 0 sells to π‘₯ customers, where π‘₯ is measured in thousands, and cart 1 sells to (1 βˆ’ π‘₯) thousand.)

  2. 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.

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