# Consider the pizza stores introduced in Exercise U2, Donna’s Deep Dish and Pierce’s Pizza Pies….

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:

Then profits per week (Y, in thousands of dollars) for

each firm are:

(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:

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?