2 marksMicroeconomicsPerfect competitionLong-run equilibrium, entry and exitHardabout 4 min

GATE 2021 Question 38: Perfect competition

The long-run cost function of all identical firms in a perfectly competitive industry is given by: $C=25q-3q^2+1.5q^3$

The market demand function is: $P=2500-0.25Q$

The number of firms in the industry at equilibrium is ______ (in integer).

Answer and worked solution

Answer: 9906

Step 1 - Find each firm's long-run output (minimum AC).

$AC(q) = \frac{C(q)}{q} = 25-3q+1.5q^2$

Minimizing: $\dfrac{d(AC)}{dq} = -3+3q=0 \Rightarrow q^*=1$ .

Step 2 - Find the long-run price (= minimum AC).

$AC(1) = 25-3+1.5 = 23.5$

In long-run competitive equilibrium, $P=AC_{min}=23.5$ .

Step 3 - Find market quantity from the demand curve. $P=2500-0.25Q$ . Setting $P=23.5$ :

$23.5 = 2500-0.25Q \Rightarrow 0.25Q = 2476.5 \Rightarrow Q = 9{,}906$

Step 4 - Number of firms. Since each firm produces $q^*=1$ :

$n = \frac{Q}{q^*} = \frac{9{,}906}{1} = 9{,}906$

Therefore the number of firms in the industry at equilibrium is 9,906.

The trap: 2476

Students sometimes stop midway and report an intermediate value (like $2500-23.5=2476.5$ , forgetting the final division by $0.25$ ) rather than completing the full solve for $Q$ , or mistakenly use the wrong price (e.g., the demand curve's vertical intercept) instead of the AC-minimizing price.

Plain-text version

The long-run cost function of all identical firms in a perfectly competitive industry is given by: C=25q-3q²+1.5q³ The market demand function is: P=2500-0.25Q The number of firms in the industry at equilibrium is (in integer).

Also written as: The long-run cost function of all identical firms in a perfectly competitive industry is given by: C=25q-3q^2+1.5q^3 The market demand function is: P=2500-0.25Q The number of firms in the industry at equilibrium is (in integer).

Perfect competition has come up 115 times in 7 exams since 2006.

What this question tests

This question is on Long-run equilibrium, entry and exit, part of Perfect competition in Microeconomics.

Why free entry drives profit to zero in the long run, and what that implies for the number of firms and industry supply.

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