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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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