Consider two firms selling differentiated varieties of a product, e.g., Coke and Pepsi. Each firmjchooses a pricepjfor its own variety. Since these varieties are close substitutes, the demand that each firm faces depends not only on its own price, but also the price of its competitor. Specifically, the demand forj's variety is given by

Dj(pj, pj) = max{????0,60 +pj2pj}????

Suppose that both firms can produce any amount of their variety atno cost.

1. (a)Find firmj's best response function.
2. (b)Assume that firms choose prices simultaneously and independently. Show that choosingpj= 18 is not rationalizable. [Hint: perform two rounds of iterated dominance]
3. (c)Assume that firms choose prices simultaneously and independently. Find theuniqueNash equilibirum of the game. [Hint: symmetric games typically have a symmetric equilibrium]
4. (d)Assume that firm 1 chooses its price first, and firm 2 chooses its price second after seeing firms 1's price. Find the Stackelberg equilibrium of the game.
5. (e)Compare the equilibrium profits of each firm under part (c), withe their profits on part (d).