Question

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:

1) q₁ = 22.5 − q₂/4

2) q₁ = 30 − q₂/3

3) q₁ = 45 − q₂/2

4) q₁ = 30 − q₂/2

5) None of the other answers is correct.

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

1) q₁ = q₂ = 45

2) q₁ = q₂ = 22.5

3) q₁ = q₂ = 18

4) q₁ = q₂ = 30

5) None of the other answers is correct.

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

1) π₁ = π₂ = 759.4 (to 1 dp)

2) π₁ = π₂ = 450

3) π₁ = π₂ = 1012.5

4) π₁ = π₂ = 900

5) None of the other answers is correct.

Answer 1

In the duopoly market in which the inverse demand function is given by

P = 96 - Q = 96 - (q1+q2)

Where, q1 is the output of firm 1 and q2 is the output of firm 2.

Firm 1's cost function is

Hence, Marginal Cost of Firm 1 is

MC1 = dc(q1)/dq1

or, MC1 = 6+q1.........(1)

And, Firm 2's cost function is

Hence, Marginal Cost of Firm 2 is

MC2 = dc(q2)/dq2

or, MC2 = 6+q2.........(2)

Hence, from the inverse demand function,

P=96-q1-q2,

The Total Revenue of Firm 1 is

TR1 = P.q1 = (96-q1-q2).q1

or,

Hence, the Marginal Revenue of Firm 1 is

MR1 = dTR1/dq1

or, MR1 = 96 - 2q1 - q2........(3)

And, the Total Revenue of Firm 2 is,

TR2 = P.q2 = (96-q1-q2).q2

or, TR2 =

Hence, Marginal Revenue for Firm 2 is,

MR2 = dTR2/dq2

or, MR2 = 96 - q1 - 2q2.........(4)

Now let us remember equations (1), (2), (3) and (4) to answer the follwoing questions.

(Q1) If the firms are in a Cournot Competition, they will maximize profit taking each other's decision in account.

If firm 1 is maximizing profit, then the Marginal Revenue must be equal to Marginal Cost.

Hence, from equations (1) and (3) we get

MR1 = MC1

or, 96 - 2q1 - q2 = 6 + q1

or, 3q1 + q2 = 90..........(5)

or, q1 = 30 - q2/3.........BF1

This is the Cournot best response function for firm 1.

Hence, the Cournot best response function for firm 1 is: q1 = 30-q2/3. (Answer is Option 2)

(Q2) Now, we will find the best response function for firm 2.

In the Cournot competition, if the firm 2 is also maximizing profit, then similarly,

MR2 = MC2

or, 96 - q1 - 2q2 = 6 + q2

or, q1 + 3q2 = 90.........(6)

or, q2 = 30 - q1/3.........BF2

This is the Cournot best response function for firm 2.

Now, the firms will select their production decision where the best response functions intersect. Hence, we need to solve their best response functions.

Now, from equation (5) and (6), we get

2q1+q2 = 90.......(5) and

q1+2q2 = 90.......(6)

Solving (5) and (6), we get

q1* = 30 and q2* = 30

Hence, the outputs of the two firms in the Cournot Nash equilibrum are q1*=q2*=30.

(Answer is Option 4)

(Q3) The market price is

P* = 96-q1*-q2* = 96-30-30

or, P* = $36

Now, for firm 1, Total Revenue is

TR1 = P*.q1* = $36×30 = $1080

Total Cost of firm 1 is

  

or, TC1 = 6×30 + 0.5×(30×30) = $630

Hence, Toral Profit of firm 1 is

π1 = TR1 - TC1 = $1080 - $630

or, π1 = $450

Now, for firm 2, Total Revenue is

TR2 = P*.q2* = $36×30 = $1080

or, TC2 = 6×30 + 0.5.(30×30) = $630

Hence, total Profit of firm 2 is

π2 = $1080 - $630 = $450

The profit of the two firms in Cournot Nash equilibrum will be π1 = π2 = $450. (Answer is Option 2)

Hope the solution is clear to you my friend.