Question

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).

Hint:

In a perfectly competitive market, firms produce at the point where price equals marginal cost (P = MC).

Answer 1

Correct Answer: 9904

To find the number of firms in the industry at equilibrium, we first calculate the equilibrium output per firm. The marginal cost (MC) of the firm is the derivative of the total cost function: \[ MC = \frac{dC}{dq} = 25 - 6q + 4.5q^2 \] In a perfectly competitive market, firms produce where \( P = MC \). Using the demand function, the price is: \[ P = 2500 - 0.25Q \] At equilibrium, \( P = MC \), so: \[ 2500 - 0.25Q = 25 - 6q + 4.5q^2 \] Since all firms are identical, the total output \( Q \) is the output of one firm \( q \) multiplied by the number of firms \( n \), i.e., \( Q = nq \). Solving this system yields: \[ n = 9904 \] Thus, the number of firms in the industry at equilibrium is \( 9904 \).