1 answer

3. Suppose the two firms cannot collude and instead compete in the Cournot Model in the...

Question:

3. Suppose the two firms cannot collude and instead compete in the Cournot Model in the market described in question 1 (marke

3. Suppose the two firms cannot collude and instead compete in the Cournot Model in the market described in question 1 (market demand is still Q = 18 – P) with the same cost (C(Q)=Q2). a. Set up firm 1's profit maximization. b. Solve for firm 1's best response function. C. Solve for firm 1's quantity, firm 2's quantity, the equilibrium market quantity, and price. Show your work. d. Is this a Nash equilibrium? e. Do consumers prefer the Cournot competition equilibrium over the collusion of the two firms in question 3? f. Do the two firms prefer Cournot competition over colluding (assuming the collusion agreement is to split joint profits equally)?

Answers

P = 18 - Q = 18 - Q1 - Q2 [since Q = Q1 + Q2]

C1 = 0.5Q12

C2 = 0.5Q22

(a)

For firm 1,

TR = P x Q1 = 18Q1 - Q12 - Q1Q2

Profit (Z1) = TR1 - TC1 = 18Q1 - Q12 - Q1Q2 - 0.5Q12 = 18Q1 - 1.5Q12 - Q1Q2

Firm 1's optimization problem is:

Maximize Z1 = 18Q1 - 1.5Q12 - Q1Q2

Subject to Q1 >= 0, Q2 >= 0.

(b)

Firm 1 maximizes profit when Z1/Q1 = 0

18 - 3Q1 - Q2 = 0

3Q1 + Q2 = 18...........(1) [Best response, firm 1]

(c)

For firm 2,

TR2 = P x Q2 = 18Q2 - Q1Q2 - Q22

Profit (Z2) = TR2 - TC2 = 18Q2 - Q1Q2 - Q22 - 0.5Q22 = 18Q2 - Q1Q2 - 1.5Q22

Firm 2's optimization problem is:

Maximize Z2 = 18Q2 - Q1Q2 - 1.5Q22

Subject to Q1 >= 0, Q2 >= 0.

Firm 2 maximizes profit when Z2/Q2 = 0

18 - Q1 - 3Q2 = 0

Q1 + 3Q2 = 18...........(2) [Best response, firm 2]

(2) x 3 yields:

3Q1 + 9Q2 = 54.........(3)

3Q1 + Q2 = 18.........(1)

(3) - (1) yields:

8Q2 = 36

Q2 = 4.5

Q1 = 18 - 3Q2 [from (2)] = 18 - (3 x 4.5) = 18 - 13.5 = 4.5

Q = 4.5 + 4.5 = 9

P = 18 - 9 = 9

(d)

The firms do not have an incentive to deviate from their respetive output at the equilibrium price, since both are maximizing profit at this price-output combination. So this is a Nash equilibrium.

NOTE: As HOMEWORKLIB's Policy, only 1st 4 parts can be answered. You need to post (e) and (f) in separate question.

.

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