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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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Solution:

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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