Consider the pizza stores introduced in Exercise U2, Donna's Deep Dish and Pierce's Pizza Pies. Suppose that they are not constrained to choose from only two possible prices, but that they can choose a specific value for price to maximize profits. Suppose further that it costs $3 to make each pizza {for each store) and that experience or market surveys have shown that the relation between sales {Q) and price {P) for each firm is as follows:

Q_plerce = 12 - P_Plerce + 0.5P_donna

Then profits per week (Y, in thousands of dollars) for each firm are:

Y_Pterce = (P_pierce - 3) Q_pierce = (P_pierce - 3) (12 - P_Plerce + 0.5P_donna)

Y_donna = (P_donna - 3) Q_donna = (P_donna - 3) (12 - P_donna + 0.5 P_pierce)

(a) Use these profit functions to determine each firm's best-response rule, as in Chapter 5, and use the best-response rules to find the Nash equilibrium of this pricing game. What prices do the firms choose in equilibrium? How much profit per week does each firm earn?
(b) If the firms work together and choose a joint best price, P, then the profit of each will be:

Y_donna = Y_pierce = (P - 3) (12 - P + 0.5 P) = (P - 3) (12 - 0.5 P).

What price do they choose to maximize joint profits?

(c) Suppose the two stores are in a repeated relationship, trying to sustain the joint profit-maximizing prices calculated in part {b). They print new menus each month and thereby commit themselves to prices for the whole month. In any one month, one of them can defect from the agreement. If one of them holds the price at the agreed level, what is the best defecting price for the other? What are its resulting profits? For what interest rates will their collusion be sustainable by using grim-trigger strategies?