Question

The firm's long-run total cost is given by ltc = 100q – 10q2 + (1/3)q3, and long-run marginal cost is given by lmc = 100 – 20q + q2. at what output level does the firm have economies of scale

Answer 1

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:

`LTC = 100q - 10q^(2)+(1)/(3)q^(3)`

and Long run Marginal Cost Curve as:

`LMC = 100 -20q + q^(2)`

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.

`AC = (100q - 10q^(2)+(1)/(3)q^(3))/(q)`

`\mathbf{AC = 100 - 10q+(1)/(3)q^(2)}`

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
`MC < AC`.

Hence,

`100 -20q + q^(2) < 100 - 10q+(1)/(3)q^(2)`

Solving for q from the equation above, we get

-10q +\frac{2}{3}q^{2} < 0

Dividing by q we get,

`-10 +(2)/(3)q < 0`

`(2)/(3)q < 10`

`q < 10*(3)/(2)`

`\mathbf{q < 15 units}`