Question

Suppose that there is only one firm, the monopolist and many consumers in the market. Assume that the market demand function is given by P = 210−2Q^d and the monopolist’s cost function is given by C(Q) = 10Q.

(a) Setup the monopolist’s profit maximization problem.
(b) Solve the monopolist’s profit maximization problem and find the market equilibrium price P∗ and quantity Q∗.

Now, assume that the government eliminates all the entry cost of entering the market
as the government wants to regulate the monopoly market. As a result, there are now
Firm 1 and Firm 2. Assume that both Firm 1 and Firm 2 have identical technology.
Thus, the cost of functions of Firm 1 and Firm 2 are C1(q1) = 10q1 and C2(q2) = 10q2.
Assume that Firm 1 and Firm 2 are competing over quantity.

(c) Setup the profit maximization problems of Firm 1 and Firm 2.
(d) Find the best responses of Firm 1 and Firm 2, BR1(q2) and BR2(q1).
(e) Draw the reaction curves of Firm 1 and Firm 2.
(f) Find the NE. What is the market equilibrium price?
Now, assume that Firm 1 and Firm 2 compete over prices.
(g) Find the NE in this case. Please show all relevant works.

Answer 1

(a) Total revenue (TR) = P x Q = 210Q - 2Q²

Profit (Z) = TR - TC = 210Q - 2Q² - 10Q = 200Q - 2Q²

Profit maximization problem is:

Maximize Z = 200Q - 2Q²

Subject to Q >= 0

(b) Profit is maximized when dZ/dQ = 0.

200 - 4Q = 0

4Q = 200

Q = 50

P = 200 - (2 x 50) = 200 - 100 = 100

(c) P = 210 - 2q1 - 2q2 (Since Q = q1 + q2)

For firm 1,

TR1 = P x q1 = 210q1 - 2q1² - 2q1q2

Profit (Z1) = TR1 - C1 = 210q1 - 2q1² - 2q1q2 - 10q1 = 200q1 - 2q1² - 2q1q2

Profit maximization problem is:

Maximize Z1 = 200q1 - 2q1² - 2q1q2

Subject to q1 >= 0, q2 >= 0

For firm 2,

TR2 = P x q2 = 210q2 - 2q1q2 - 2q2²

Profit (Z2) = TR2 - C2 = 210q2 - 2q1q2 - 2q2² - 10q2 = 200q2 - 2q1q2 - 2q2²

Profit maximization problem is:

Maximize Z2 = 200q2 - 2q1q2 - 2q2²

Subject to q1 >= 0, q2 >= 0

(d)

For firm 1, Profit is maximized when Z1/q1 = 0

200 - 4q1 - 2q2 = 0

4q1 + 2q2 = 200 (Best response, firm 1)

For firm 2, Profit is maximized when Z2/q2 = 0

200 - 2q1 - 4q2 = 0

2q1 + 4q2 = 200 (Best response, firm 1)

NOTE: As per Answering Policy, 1st 4 parts are answered.