Question

Suppose there are two firms in a market who each simultaneously choose a quantity.Firm 1's quantity is q1, and firm 2's quantity is q2. Therefore the market quantity is Q= q1+q2. The market demand curve is given by P=130-Q. Also, each firm has constant marginal cost equal to 25. There are no fixed costs. Marginal revenue of the firms are given by MR1=130-2q1-q2 & MR2=130-q1-2q2.

A) How much output will each firm produce in the Cournot equilibrium ?

B) What will be the market price of the good?

C) What is DWL that results from this duopoly?

D) How much profit does each firm make?

E) Suppose Firm 2 produced 20 units of output. How much output should Firm 1 produce in order to maximize profit?

Answer 1

MR1=130-2q1-q2

MR2 = 130-q1-2q2

MC1=MC2=25

A) Firm 1 would equate MR1 and MC1 in order to maximize its profits.

130-2q1-q2 = 25

(105-q2)/2 = q1

This is the reaction function of firm 1 (RF1)

Firm 2 would equate MR2 and MC2 in order to maximize its profits.

130-q1-2q2 = 25

(105-q1)/2 = q2

This is the reaction function of firm 2 (RF2)

Since the reaction functions are symmetrical,it implies that at the Cournot-Nash equilibrium both firms will produce the same level of output:

q1=q2=q*

(105-q*)/2 = q*

Therefore, each firm will produce 35 units in the Cournot equilibrium.

B) market price can be found out by putting the total quantity produced in the demand equation

P=130-Q

P=130-q1-q2

P=130-35-35

P = $60

Therefore, the market price = $60

C) If there was a perfect competition, then the profit-maximizing condition would have been P = MC

130-Q = 25

Qc = 105

Therefore, in case of perfect competition, 105 units would have been produced.

Now, deadweight loss for counot duopoply = 0.5 * (P - MC) * (Qc - Q) = 0.5 * (60 - 25) * (105 - 70) = 0.5*35*35 = 612.5.

D) Profits made by firm 1 = TR1-TC1 = pq* - 25q* (total cost = MC *quantity)

= 60*35- 25*35

=2,100-875

=1,225

Profits made by firm 2 = TR2-TC2 = pq* - 25q*

= 60*35- 25*35

=2,100-875

=1,225

Therefore, each firms makes a profit of $1,225

E) Suppose Firm 2 produced 20 units of output, that is, q2=20.

From what we found earlier,

Q = q1+q2 = 70

q1 = 70-q2 = 70-20 = 50

Firm 1 should produce 50 units in order to maximize profit.

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