Question

Questions 1 through 3 refer to a duopoly market in which the inverse demand function is given by P = 96 − Q. Firm 1's cost function is c(q₁) = 6q₁ + 0.5q₁², and firm 2's cost function is c(q₂) = 6q₂ + 0.5q₂² (such that each firm has MC = 6 + q).

Q1.The Cournot best-response function for firm 1 will be:

a.q₁ = 30 − q₂/2

b.q₁ = 22.5 − q₂/4

c.None of the other answers is correct.

d.q₁ = 30 − q₂/3

e.q₁ = 45 − q₂/2

Q2.The outputs of the two firms in Cournot-Nash equilibrium will be:

a.q₁ = q₂ = 18

b.q₁ = q₂ = 45

c.q₁ = q₂ = 30

d.q₁ = q₂ = 22.5

e.None of the other answers is correct.

Q3.The profits of the two firms in Cournot-Nash equilibrium will be:

a.π₁ = π₂ = 450

b.None of the other answers is correct.

c.π₁ = π₂ = 1012.5

d.π₁ = π₂ = 759.4 (to 1 dp)

e.π₁ = π₂ = 900

Answer 1

1)

Given P=96-Q

where Q=q1+q2

Now take the case of firm 1

Total Revenue=TR1=P*q1=(96-q1-q2)*q1=96q1-q1^2-q1*q2

Marginal Revenue=MR1=dTR1/dq1=96-2q1-q2

Set MR1=MC1 for profit maximization

96-2q1-q2=6+q1

90-3q1-q2=0

3q1=90-q2

q1=30-(1/3)q2 (Response function of firm 1)

Correct option is

d. q1 = 30 − q2/3

2)

Now take the case of firm 2

Total Revenue=TR2=P*q2=(96-q1-q2)*q2=96q2-q1*q2-q2^2

Marginal Revenue=MR2=dTR2/dq2=96-q1-2q2

Set MR2=MC2 for profit maximization

96-q1-2q2=6+q2

90-3q2-q1=0

3q2=90-q1

Set q1=30-(1/3)q2 (Refer response function of firm 1)

3q2=90-30+(1/3)*q2

8/3q2=60

q2=(60*3/8)=22.50

q1=30-(1/3)q2=30-(1/3)*22.5=22.50

Correct option is

d. q1 = q2 = 22.5

3)

P=96-(q1+q2)=96-(22.5+22.5)=$51

Total Revenue of firm 1=TR1=P*q1=51*22.5=$1147.5

Total Cost=TC1=6q1+0.5q1^2=6*22.5+0.5*22.5^2=$388.125

Profit of firm 1=TR1-TC1=1147.5-388.125=$759.375 or say $759.4

Total Revenue of firm 2=TR2=P*q2=51*22.5=$1147.5

Total Cost=TC2=6q2+0.5q2^2=6*22.5+0.5*22.5^2=$388.125

Profit of firm 2=TR2-TC2=1147.5-388.125=$759.375 or say $759.4

Correct option is

d.π1 = π2 = 759.4 (to 1 dp)