Two firms with cost Cᵢ(qᵢ) = 10qᵢ compete in prices for a homogeneous good. Firm i faces demand 100 - pᵢ if pᵢ pⱼ, and (100 - pᵢ)/2 if pᵢ = pⱼ. Prices take only integer values. Nash equilibrium/equilibria is/are (A) p₁ = 10, p₂ = 10 (B) p₁ = 12, p₂ = 12 (C) p₁ = 40, p₂ = 40 (D) p₁ = 11, p₂ = 11

2 marksMicroeconomicsMonopolistic competition and oligopolyBertrand competitionHardabout 3 min

IIT JAM 2021 Question 32: Monopolistic competition and oligopoly

Two firms with cost $C_i(q_i) = 10q_i$ compete in prices for a homogeneous good. Firm $i$ faces demand $100 - p_i$ if $p_i < p_j$ , $0$ if $p_i > p_j$ , and $\tfrac{100 - p_i}{2}$ if $p_i = p_j$ . Prices take only integer values. Nash equilibrium/equilibria is/are

Select all that apply.

Answer and worked solution

Answer: (A) and (D)

Marginal cost is $10$ . At $p_1 = p_2 = 10$ each firm earns zero; undercutting is impossible (integer prices, and $p < 10$ loses money), so $(10,10)$ is a Nash equilibrium, statement (A).

At $p_1 = p_2 = 11$ each earns $(11-10)\cdot\tfrac{100-11}{2} = 44.5$ . Undercutting to $10$ captures the whole market but at zero margin, giving zero profit, which is worse, so no firm deviates and $(11,11)$ is also a Nash equilibrium, statement (D). At $12$ or $40$ a one-rupee undercut is strictly profitable, so those are not equilibria. The correct options are (A) and (D).

The trap: (C)

p_1 = 40,\ p_2 = 40

Prices well above cost (12, 40) invite a profitable one-unit undercut; only cost pricing and the just-above-cost integer 11 survive as equilibria.

Plain-text version

Two firms with cost Cᵢ(qᵢ) = 10qᵢ compete in prices for a homogeneous good. Firm i faces demand 100 - pᵢ if pᵢ < pⱼ, 0 if pᵢ > pⱼ, and (100 - pᵢ)/2 if pᵢ = pⱼ. Prices take only integer values. Nash equilibrium/equilibria is/are (A) p₁ = 10, p₂ = 10 (B) p₁ = 12, p₂ = 12 (C) p₁ = 40, p₂ = 40 (D) p₁ = 11, p₂ = 11

Also written as: Two firms with cost C_i(q_i) = 10q_i compete in prices for a homogeneous good. Firm i faces demand 100 - p_i if p_i < p_j, 0 if p_i > p_j, and (100 - p_i)/2 if p_i = p_j. Prices take only integer values. Nash equilibrium/equilibria is/are

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

What this question tests

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

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