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Questions › GATE › GATE 2023 › Q58
2 marksMicroeconomics › Monopolistic competition and oligopoly › Cournot competitionHardabout 3 min

GATE 2023 Question 58: Monopolistic competition and oligopoly

A duopoly faces the inverse market demand function p=120−Qp=120-Qp=120−Q, where ppp is the unit price (in Rs.) of the good being sold by firms A and B, and QQQ is the total output. Firm A has a constant marginal cost of Rs. 20, which is exactly half of firm B's constant marginal cost. There is no fixed cost for both the firms. If there exists a Cournot-Nash equilibrium, QQQ is ______ (in integer).

Answer and worked solution

Answer: 60

Firm A's marginal cost is 20; Firm B's is twice that, so MCB=40MC_B = 40MCB​=40.

Firm A's problem: πA=(120−qA−qB)qA−20qA\pi_A = (120-q_A-q_B)q_A - 20q_AπA​=(120−qA​−qB​)qA​−20qA​. FOC: 120−2qA−qB−20=0⇒qA=100−qB2120-2q_A-q_B-20=0 \Rightarrow q_A = \dfrac{100-q_B}{2}120−2qA​−qB​−20=0⇒qA​=2100−qB​​.

Firm B's problem: πB=(120−qA−qB)qB−40qB\pi_B = (120-q_A-q_B)q_B - 40q_BπB​=(120−qA​−qB​)qB​−40qB​. FOC: 120−qA−2qB−40=0⇒qB=80−qA2120-q_A-2q_B-40=0 \Rightarrow q_B = \dfrac{80-q_A}{2}120−qA​−2qB​−40=0⇒qB​=280−qA​​.

Substituting Firm B's best response into Firm A's:

qA=100−80−qA22=200−80+qA22=120+qA4q_A = \frac{100 - \frac{80-q_A}{2}}{2} = \frac{\frac{200-80+q_A}{2}}{2} = \frac{120+q_A}{4}qA​=2100−280−qA​​​=22200−80+qA​​​=4120+qA​​

4qA=120+qA⇒3qA=120⇒qA=404q_A = 120+q_A \quad\Rightarrow\quad 3q_A = 120 \quad\Rightarrow\quad q_A = 404qA​=120+qA​⇒3qA​=120⇒qA​=40

qB=80−402=20q_B = \frac{80-40}{2} = 20qB​=280−40​=20

Q=qA+qB=40+20=60Q = q_A+q_B = 40+20 = 60Q=qA​+qB​=40+20=60

Therefore the Cournot-Nash equilibrium total output is 60.

The trap: 50

Students sometimes apply the symmetric-firm Cournot shortcut formula directly despite the firms having different marginal costs here, or make an algebra slip substituting one best-response function into the other, producing a plausible but incorrect total output.

Plain-text version

A duopoly faces the inverse market demand function p=120-Q, where p is the unit price (in Rs.) of the good being sold by firms A and B, and Q is the total output. Firm A has a constant marginal cost of Rs. 20, which is exactly half of firm B's constant marginal cost. There is no fixed cost for both the firms. If there exists a Cournot-Nash equilibrium, Q is (in integer).

Monopolistic competition and oligopoly has come up 104 times in 7 exams since 2004.

What this question tests

This question is on Cournot competition, part of Monopolistic competition and oligopoly in Microeconomics.

Firms choosing quantities simultaneously, reaction curves, and equilibrium as the number of firms or costs change.

TestStop has 31 past-paper questions on this subtopic; the nearest in year are below.

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