2 marks, −0.667MicroeconomicsMonopolistic competition and oligopolyCournot competitionModerateabout 3 min

GATE 2021 Question 25: Monopolistic competition and oligopoly

Consider a Cournot type $n$ -firm natural spring oligopoly where the market demand for natural spring water is given by $P(Q)=a-Q$ , $a>0$ . The $n$ firms are symmetric. Each firm incurs a bottling cost of $C_i=cq_i$ , $c>0$ and $a>c$ . The equilibrium market price will be

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Answer and worked solution

Answer: (B)

Firm $i$ 's profit: $\pi_i = (a-Q)q_i - cq_i$ , where $Q=\sum q_j$ . FOC with respect to $q_i$ (treating rivals' outputs as given): $a-Q-q_i-c=0$ .

By symmetry, all $n$ firms produce the same quantity $q^*$ , so $Q=nq^*$ . Substituting:

$a - nq^* - q^* - c = 0 \Rightarrow (n+1)q^* = a-c \Rightarrow q^* = \frac{a-c}{n+1}$

Total output: $Q^* = nq^* = \dfrac{n(a-c)}{n+1}$ .

Equilibrium price:

$P^* = a - Q^* = a - \frac{n(a-c)}{n+1} = \frac{a(n+1)-n(a-c)}{n+1} = \frac{an+a-na+nc}{n+1} = \frac{a+nc}{n+1}$

$= \frac{a}{n+1}+\frac{nc}{n+1}$

This matches option (B) exactly.

The trap: (C)

\dfrac{na}{(n+1)}+\dfrac{nc}{(n+1)}

Option (C) looks structurally similar but has an extra factor of $n$ in the numerator of the first term ( $na$ instead of $a$ ) — a student who makes an algebra slip while simplifying $a - \frac{n(a-c)}{n+1}$ (e.g., mishandling the distribution of terms) could easily land on this nearby but incorrect expression.

Plain-text version

Consider a Cournot type n -firm natural spring oligopoly where the market demand for natural spring water is given by P(Q)=a-Q, a>0. The n firms are symmetric. Each firm incurs a bottling cost of Cᵢ=cqᵢ, c>0 and a>c. The equilibrium market price will be (A) ((n+1))/a+(n(n+1))/c (B) a/((n+1))+nc/((n+1)) (C) na/((n+1))+nc/((n+1)) (D) ((a-c))/((n+1))+c/(n(n+1))

Also written as: Consider a Cournot type n -firm natural spring oligopoly where the market demand for natural spring water is given by P(Q)=a-Q, a>0. The n firms are symmetric. Each firm incurs a bottling cost of C_i=cq_i, c>0 and a>c. The equilibrium market price will be

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.

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