Question

Assume that the cost function for the i^th firm in an industry is given by
C_i = 0.25\(q^2_i\) + 2q_i + 5, i = 1, 2, ..., 150,
where C_i and q_i are cost and output for the i^th firm, respectively.
Let the aggregate inverse demand function be P = 10-0.01Q, where P is the unit price and Q is the aggregate output.
Assuming perfect competition, the equilibrium quantity is _____ (in integer).

Answer 1

Correct Answer: 4

To find the equilibrium quantity under perfect competition, we first equate marginal cost (MC) to the market price (P) derived from the inverse demand function. The cost function given for each firm is \(C_i = 0.25q_i^2 + 2q_i + 5\). The marginal cost is the derivative of the cost function with respect to output \(q_i\):
MC = \(\frac{dC_i}{dq_i}\) = \(0.5q_i + 2\).
In perfect competition, MC = P. Thus, \(0.5q_i + 2 = 10 - 0.01Q\).
Since each firm is small and acts independently, assume \(Q = nq\), where \(n = 150\) (number of firms) and \(q_i = q\), the firm's output.
Substituting \(Q = 150q\), we get:
\(0.5q + 2 = 10 - 0.01(150q)\).
Simplifying the equation gives:
\(0.5q + 2 = 10 - 1.5q\).
Rearranging terms:
\(0.5q + 1.5q = 8\).
\(2q = 8\).
\(q = 4\).
Hence, for each firm, the equilibrium output is 4. Consequently, \(Q = 150 \times 4 = 600\).