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.
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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