Question

Consider a perfectly competitive industry where each firm has a long-run cost function:

C(q) = q³ – 20q² +120q

Long-run average costs are minimised at an output level of 10 units. The industry demand curve is given by:

Q = 1000 – 40p

where Q represents industry output and p represents the equilibrium price of industry output.

Required:

In the long-run equilibrium, this industry will sustain how many firms?

Answer 1

$C(q)=q^3-20q^2+120q$

$MC=3q^2-40q+120$

$Q=1000-40P$

To find P,

$10 units= 100-40P$

$\implies P=24.75$

Equate $P=MC$

$24.75=3q^2-40q+120$

$3q^2-40q+95.25=0$

Solving the above quadratic equation, we get:

$q=10.23$

If each perfectly competitive firm is producing 3 units of output and the market output is 10 units, then there will be $\frac{10}{3}=3$ firms in the industry.