Question

Suppose the demand for pizza in a small town is p = 10 – Q.  There are only two firms, A and B, and each has a cost function C = 2 + q.  

a. Find each firm’s profit if they behave like Cournot duopoly. [10 pts.]

b. Find each firm’s profit if they behave like Stackelberg du0poly with firm A being the leader. [10 pts.]

c. Find each firm’s profit if they both think they are the Stackelberg leader firm. [10 pts.]

Answer 1

Solution -

Firm A's profit is p = [10 - (q_A+ q_B)] q_A- 2 - q_A. Maximizing with respect to its own output yields q_A= 4.5 - q_B/2. Similarly, firm B's best response is q_B= 4.5 - q_A/2. The equilibrium occurs when both firms produce 3 units. Price is 10 - 3 - 3 = 4.

Firm B's profit is p = [10 - (q_A+ q_B)] q_B- 2 - q_B. Maximizing with respect to its own output yields firm B's best response q_B= 4.5 - q_A/2. Knowing this, firm A substitutes this into demand and maximizes its profit. p = [10 - q_A- (4.5 - q_A/2)] q_A - 2 - q_A. Maximizing with respect to q_A yields q_A = 4.5. Firm B responds by producing 2.25.