Question

2. Suppose that a perfectly competitive firm’s total cost of producing output q is T C(q) = 5 + 5q 6q2 + 4q3. (a) Find the short-run supply curve of a firm in this industry. Be sure to identify the point at which the firm shuts down. (b) How much will the firm produce at p⇤ = 5?

Answer 1

We have the following information

Total Cost (TC) = 5 + 5q + 6q² + 4q³

Price = 5

In a perfectly competitive market structure the short-run supply curve of a firm is equal to the marginal cost (MC) curve.

MC = ΔTC/Δq = 5 + 12q + 12q²

In the short the shut-down point for a firm is that point where the price is equal to the minimum point of average variable cost (AVC)

From the TC function we can get the Variable Cost (VC) function as 5q + 6q² + 4q³

AVC = VC/q = (5q + 6q² + 4q³)/q

AVC = 5 + 6q + 4q²

Taking first derivative

ΔAVC/Δq = 6 + 8q

Equating it equal to zero we get

6 + 8q = 0

q = 8/6

q = 4/3

Taking the second derivative we get

Δ²AVC/Δq² = 8

Since, the second derivative is positive so, at q = 4/3 the AVC is minimum.

For price equal to 5 the equilibrium quantity can be determined by equating it to the MC

Price = MC

5 = 5 + 12q + 12q²

12q + 12q² = 0

12q(1 + q) = 0

So, either the q = 0 or the q = – 1. Since, quantity cannot be negative so at price 5 the equilibrium quantity is zero which means producers need higher price to start the production.