Answer to Question #268108 in Microeconomics for Ahona

Question #268108

Suppose that two identical firms produce widgets and that they are the only firms in the

market. They have identical constant marginal costs equal to 30 and no fixed costs. Price

is determined by the following demand curve : P=150-Q where Q=Q1+Q2

a. Find the Cournot–Nash equilibrium outputs. Calculate the price and profit of

each firm in this equilibrium.

b. What is the market equilibrium price and quantity when each firm behaves as a

Bertrand duopolist choosing prices? What are firms’ profits?

c. If firm 1 is a leader and firm 2 is a follower, how will the outcome in (a)

changes?

d. Draw a table to compare price ,quantity and profit among the three oligopoly

models , perfect competition and monopoly model.


1
Expert's answer
2021-11-18T10:21:09-0500

Solution:

a.). Cournat-Nash equilibrium:

P=150-Q where Q = Q1 + Q2

P = 150 – (Q1 + Q2)

TR1 = [150 – (Q1 + Q2)] Q1 = 150Q1 – Q12 – Q2

TR1 = 150Q1 – Q12 – Q2


MR1 ="\\frac{\\partial TR_{1} } {\\partial Q_{1} }" = 150 – 2Q1 – Q2

MR1 = 150 – 2Q1 – Q2

Set MR1 = MC

MC = 30

150 – 2Q1 – Q2 = 30

150 – 30 - 2Q1 – Q2 = 0

120 - 2Q1 – Q2 = 0

Q1 = 60 – 0.5Q2

TR2 = [150 – (Q1 + Q2)] Q2 = 150Q2 – Q1Q2 – Q22

TR2 = 150Q2 – Q1Q2 – Q22

MR2 = "\\frac{\\partial TR_{2} } {\\partial Q_{2} }" = 150 – Q1 – 2Q2

MR2 = 150 – Q1 – 2Q2

Set MR2 = MC

MC = 30

150 – Q1 – 2Q2 = 30

150 – 30 - Q1 – 2Q2 = 0

120 - Q1 – 2Q2 = 0

Q2 = 60 – 0.5Q1

Substitute Q2 in Q1:

Q1 = 60 – 0.5Q2 = 60 – 0.5(60 – 0.5Q1) = 60 – 30 – 0.25Q1

Q1 – 0.25Q1 = 30

0.75Q1 = 30

Q1 = 40

Q2 = 60 – 0.5(40) = 60 – 20 = 40

P = 150 – (Q1 + Q2) = 150 – (40 + 40) = 150 – 80 = 70

Price (P) = 70

Profit (Firm 1) = (P "\\times" Q1) – (30Q1) = (70 "\\times" 40) – (30 "\\times" 40) = 2,800 – 1,200 = 1,600

 

Profit (Firm 2) = (P "\\times" Q2) – (30Q2) = (70 "\\times" 40) – (30 "\\times" 40) = 2,800 – 1,200 = 1,600

 

b.). Betrand:

Profit (Firm 1) = (150 – Q1) Q1 – 30Q1 = 150Q1 – Q12 – 30Q1

"\\frac{\\partial \\pi{1} } {\\partial Q_{1} }" = 150 – 2Q1 – 30


150 – 2Q1 – 30 = 0

120 = 2Q1

Q1 = 60

Price = 150 – Q = 150 – 60 = 90

Profit = (90 "\\times" 60) – (30 "\\times" 60) = 5,400 – 1,800 = 3,600

 

Profit (Firm 2) = (150 – Q2) Q2 – 30Q2 = 150Q2 – Q22 – 30Q2

"\\frac{\\partial \\pi {2}} {\\partial Q_{2} }" = 150 – 2Q2 – 30


150 – 2Q2 – 30 = 0

120 = 2Q2

Q2 = 60

Price = 150 – Q = 150 – 60 = 90

Profit = (90 "\\times" 60) – (30 "\\times" 60) = 5,400 – 1,800 = 3,600


c.). The leader will move first and then the follower moves sequentially, thus the outcome will change.



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