Question

(DSR Chp 10, U6) Consider a small town that has a population of dedicated pizza eaters but is able to accommodate only two pizza shops, Donna’s Deep Dish and Pierce’s Pizza Pies. Each seller has to choose a price for its pizza 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:

QP =12−PP +0.5PD.

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

YP = (PP −3)QP = (PP −3)(12−PP +0.5PD),

YD = (PD −3)QD = (PD −3)(12−PD +0.5PP).

(a) Use these profit functions to determine each firm’s best-reply rule and use the best-reply 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: YD = YP = (P −3)(12−P +0.5P) = (P −3)(12−0.5P). What price do they choose to maximize joint profits?

(c) Suppose the two stores are in a repeated relationship,trying to sustain the jointprofit-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?

Answer 1