Question 4 Two firms compete as a Stackelberg duopoly. The demand they face is P =...

Question 4 Two firms compete as a Stackelberg duopoly. The demand they face is P = 40 − Q. The cost function for each firm is C(Q) = 4Q. What are the profits of the two firms? I believe the answer is πL = $162; πF = $81. however I need clear steps to understand how to understand the process.

Solutions

Expert Solution

P = 40 - Q = 40 - Q1 - Q2 [Since Q = Q1 + Q2 where Firm 1 is assumed Leader and Firm 2 is assumed Follower]

C1 = 4Q1, so MC1 = dC1/dQ1 = 4

C2 = 4Q2, so MC2 = dC2/dQ2 = 4

For Firm 2,

Total revenue (TR2) = P x Q2 = 40Q2 - Q1Q2 - Q2²

Marginal revenue (MR2) = TR2/Q2 = 40 - Q1 - 2Q2

Equating MR2 and MC2,

40 - Q1 - 2Q2 = 4

Q1 + 2Q2 = 36 ......(1) [Reaction function, firm 2]

Firm 1 considers this reaction function as known. For firm 1,

Q2 = (36 - Q1) / 2 = 18 - 0.5Q1

TR1 = P x Q1 = 40Q1 - Q1² - Q1Q2

TR1 = 40Q1 - Q1² - Q1 x (18 - 0.5Q1)

TR1 = 40Q1 - Q1² - 18Q1 + 0.5Q1²

TR1 = 22Q1 - 0.5Q1²

MR1 = dTR1/dQ1 = 22 - Q1

Equating MR1 and MC1,

22 - Q1 = 4

Q1 = 18

Q2 = 18 - (0.5 x 18) = 18 - 9 = 9

Q = 18 + 9 = 27

P = 40 - 27 = 13

Profit, Firm 1 = Q1 x (P - MC1) = 18 x (13 - 4) = 18 x 9 = 162

Profit, Firm 2 = Q2 x (P - MC2) = 9 x (13 - 4) = 9 x 9 = 81

Rahul Sunny answered 1 year ago

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