Question

In a perfectly competitive market, suppose the market demand curve is given by P = 10 + W - Q, where P is the market price, W is the average wealth of the consumers in the market, and Q is the industry output. The total cost function for a representative firm is given by C(q) = q³ - 2q² + 5q, where q is the output of a firm. If W = 80, then the total number of firms in this industry in the long-run will be ______ (in integer)

Answer 1

Correct Answer: 86

Step 1: Market demand function
The demand equation is: \[ P = 90 - Q \] where \(P\) is the price and \(Q\) is the total industry output.

Step 2: Firm’s cost and marginal cost
The cost function is: \[ C(q) = q^3 - 2q^2 + 5q \] The marginal cost is: \[ MC = \frac{dC(q)}{dq} = 3q^2 - 4q + 5 \]

Step 3: Profit maximization condition
In equilibrium: \[ P = MC \] Substituting: \[ 90 - Q = 3q^2 - 4q + 5 \] Since \(Q = nq\), \[ 90 - nq = 3q^2 - 4q + 5 \]

Step 4: Long-run zero profit condition
In the long run, price also equals average cost (AC): \[ AC = \frac{C(q)}{q} = q^2 - 2q + 5 \] Therefore: \[ P = AC \quad \Rightarrow \quad 90 - nq = q^2 - 2q + 5 \]

Step 5: Equating MC and AC
For equilibrium: \[ 3q^2 - 4q + 5 = q^2 - 2q + 5 \] Simplifying: \[ 2q^2 - 2q = 0 \quad \Rightarrow \quad q(q-1)=0 \] Thus, \(q = 0\) (not feasible) or \(q = 1\).

Step 6: Finding the number of firms
Substituting \(q=1\): \[ 90 - nq = 3(1)^2 - 4(1) + 5 \] \[ 90 - nq = 4 \] \[ nq = 86 \quad \Rightarrow \quad n = 86 \]

Final Answer:
The total number of firms in the industry in the long run is: \[ \boxed{86} \]