Question

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.

Answer 1

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.