In: Economics

# Two firms operate in a Cournot duopoly and face an inverse demand curve given by P...

Two firms operate in a Cournot duopoly and face an inverse demand curve given by P = 200 - 2Q, where Q=Q1+Q2 If each firm has a cost function given by C(Q) = 20Q, how much output will each firm produce at the Cournot equilibrium?

a. Firm 1 produces 45, Firm 2 produces 45.

b. Firm 1 produces 30, Firm 2 produces 30

c. Firm 1 produces 45, Firm 2 produces 22.5

d. None of the above.

## Solutions

##### Expert Solution

The inverse demand function is given as:

P = 200 - 2Q1 - 2Q2

The cost function is given as:

C(Q) - 20Q

The marginal cost (MC) is:

MC(Q) = 20

The total revenue for firm 1 is:

PQ1 = (200 - 2Q1 - 2Q2)(Q1)

TR1 = 200Q1 - 2Q12 - 2Q1Q2

The marginal revenue for firm 1 is:

MR1 = 200 - 4Q1 - 2Q2

Similarly, the total revenue for firm 2 is:

PQ2 = (200 - 2Q1 - 2Q2)(Q2)

TR2 = 200Q2 - 2Q22 - 2Q1Q2

The marginal revenue for firm 2 is:

MR2 = 200 - 4Q2 - 2Q1

Equating MR1 and MC1:

200 - 4Q1 - 2Q2 = 20

4Q1 + 2Q2 = 180

2Q1 + Q2 = 90

Q2 = 90 - 2Q1 (Equation 1)

Equating MR2 and MC2:

200 - 4Q2 - 2Q1 = 20

4Q2 + 2Q1 = 180

2Q2 + Q1 = 90 (Equation 2)

Solving equation 1 and equation 2:

2(90 - 2Q1) + Q1 = 90

180 - 4Q1 + Q1 = 90

-3Q1 = -90

Q1 = 30

Substituting Q1 = 30 in equation 1:

Q2 = 90 - 2Q1

Q2 = 90 - 2(30)

Q2 = 90 - 60

Q2 = 30

Answer: Q1 = Q2 = 30.

So, the firm 1 will produce 30 units and firm 2 will also produce 30 units. Therefore, the correct answer is 'Option B'.

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