Answer: The firm will have economies of scale if it's output level is less than
15 units.
We have:
Long-run Total Cost curve:
[tex]LTC = 100q - 10q^{2}+\frac{1}{3}q^{3}[/tex]
and Long run Marginal Cost Curve as:
[tex]LMC = 100 -20q + q^{2}[/tex]
If the firm produces 'q' units then, we derive the firm's long term Average Cost Curve (AC) by dividing the long term cost curve by q.
[tex]AC = \frac{100q - 10q^{2}+\frac{1}{3}q^{3}}{q}[/tex]
[tex]\mathbf{AC = 100 - 10q+\frac{1}{3}q^{2}}[/tex]
A firm is said to have economies of scale as long as its Marginal Costs (MC) is lesser than the Average Costs (AC) i.e [tex]MC < AC[/tex].
Hence,
[tex]100 -20q + q^{2} < 100 - 10q+\frac{1}{3}q^{2}[/tex]
Solving for q from the equation above, we get
-10q +\frac{2}{3}q^{2} < 0
Dividing by q we get,
[tex]-10 +\frac{2}{3}q < 0[/tex]
[tex]\frac{2}{3}q < 10[/tex]
[tex]q < 10*\frac{3}{2}[/tex]
[tex]\mathbf{q < 15 units}[/tex]
