Question: Firm X has a marginal cost of cx = 2, and the market demand is p = 22qx. The firm has to pay an entry fee of $48 before it can produce and sell its product.



Part A: Monopoly with entry choice (20 pts in total for Part A and B)

Firm X has a marginal cost of cx = 2, and the market demand is p = 22qx. The firm has to pay an entry fee of $48 before it can produce and sell its product.

(a) If X enters, how much is its optimal operating profit? (You do not need to include the entry fee when calculating the operating profit.) (2 pts)

(b) Will X enter? (2 pts)



Part B: Cournot Competition with entry choice.

Now suppose that before X's entry, it learns that there is a potential competitor, firm Y, whose marginal cost is Cy = 7. We will analyze whether X still wants to enter. A knows that Y will also need to pay an entry fee of $20. If a firm does not enter, it does not pay the entry fee and does not earn any operating profit. If a firm enters, it pays the entry fee, and its operating profit depends on the competition. If only one firm enters, it will choose the price and quantity as a monopolist. If both enter, the two firms will play a Cournot game, with the market demand being p = 22 (qx + qx).
—
First consider what will happen in a Cournot competition if both firms enter.
(c) For any given qy, write down firm X's best response, i.e., X's optimal choice of qx (as a function of qy). (2 pts)
(d) For any given qx, write down firm Y's optimal choice of qy (as a function of qx). (2 pts)
(e) Solve the two best response functions simultaneously to obtain the equilibrium qx and qy. (2 pts)
(f) How much is the equilibrium price? (2 pts)
(g) How much is each company's operating profit in the Cournot equilibrium? (2 pts)
Now consider the two firms' entry choices.
(h) Will firm Y enter? Will firm X enter? (3 pts)
(i) In the entry equilibrium, how much is the equilibrium price? (3 pts)



Part C: Entry Game with Stackelberg Capacity Competition

Now consider a sequential entry game with capacity competition. First, firm Y chooses whether to enter and chooses its capacity qy. Then, firm X chooses whether to enter and chooses qx. As before, the market demand is p = 22 - (qx + qy). Entry costs and marginal costs are also the same as before.
(j) Suppose Y enters and chooses qy. For what values of qy is it optimal for firm X to enter? (5pts)
(Hint: Suppose firm X enters. What quantity qx does firm X choose? What is the price? What is the operating profit of firm X?)
(k) Given firm X's best response, what does firm Y choose? (5pts)
(1) How do consumer surplus, producer surplus, and total surplus compare in Parts A, B, and C? Provide intuition. (5pts)

https://apaxresearchers.com/storage/files/2023/11/21/9667-2pS_19_00_40_sample-exam-mgm-7004.pdf