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Cournot Equilibrium Assignment
Question: How to do Cournot Equilibrium and duopoly problems in Microeconomics.
What is Optimal Price P1* And Optimal Profit Π1* for duopolist 1
What is (optimal) quantity Q2 for duopolist 2
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Given the inverse demand function for duopoly firm 1
P1 = 200 - 4q1 - 2q2
SRTC = 5q12 + 500
The duopoly firm 2 wants to maintain one -third of the share of the market, i.e.
q2 = (q1+q2)/3
Provided, that firm 2 is more focused on acquiring the market share it could be written as
(q1)/2 = q2
The firm 1 will try to maximize its profit provided that Firm 2 wants one-third part of the quantity
Then, the profit function becomes
Π = TR1 - TC1
Π = P1.q1 - TC1
It could be written as
Π = [200 - 4q1 - 2q2] q1 - 5(q1)2 - 500
Π =[200 - 4q1 - 2(q1/2)]q1 - 5(q1)2 - 500
Differentiating profit function w.r.t q
dΠ/ dq1 = 200 - 8q1 - 2q1 - 10q1
Putting foc equal to zero
dΠ/ dq1 = 0
200 - 20q1 = 0
q1 = 10
q2 = q1/2
q2 = 10/2
q2 = 5
The optimal price becomes,
P1* = 200 - 4 (10) - 2(5)
P1* = 150
The profit for firm 1 becomes
Π*= P1.q1 - 5q12 - 500
Π* = 150 (10) - 500 - 500
Π*= 1500 - 1000
Π* = 500
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