Question

Given a long-run total cost function: C(q) = q2 + 1, determine the range of output level where the firm exhibits economies of scale, no economies of scale and diseconomies of scale respectively.

Answer 1

Solution:

Economies of scale occurs where with increase in quantity, long run average cost decrease, no economies where it is minimum, and discontinued of scale where long run average cost increase with quantity.

Long run average cost, LRAC = total cost/quantity

LRAC = (q² + 1)/q = q + 1/q

Now, dLRAC/dq = 1 - 1/q²

So, if with increase in quantity, LRAC decreases it must be:

dLRAC/dq < 0

1- 1/q² < 0

1 < 1/q²

q² < 1 or q < 1 (as q is quantity so it can never be negative).

Thus, economies of scale occurs for quantity range of (0, 1), no economies at q = 1, and diseconomies of scale for quantity range of (1, infinity).