Question

Suppose a firm operating in a perfectly competitive industry has costs in the short run given by:

SRTC = 8 + 1/2Q^2 and therefore MC = q.

1. (a) Derive expressions for fixed costs (FC), those that do not vary with output, variable costs (VC), those that do vary with output, average variable cost (AVC), and average total cost (ATC).
2. (b) At what quantity is AVC at its minimum (at what AVC level)? At what quantity is ATC at its minimum (at what ATC level)? Calculate ATC at q = 2 and q = 8 and sketch MC, AVC and ATC betweenq=0andq=8.

Answer 1

1(a) Fixed cost is found when Q=0

given

$TC=8+\frac{1}{2}Q^2\\=8+\frac{1}{2}0^2\\FC=8$

(ii) variable cost=TC - FC

$VC=8+\frac{1}{2}Q^2-8$

$VC=\frac{1}{2}Q^2$

(iii) $AVC=\frac{VC}{Q}$

$=\frac{\frac{1}{2}Q^2}{Q}$

$=\frac{1}{2}Q$

(iv) $ATC=\frac{TC}{Q}$

$ATC=\frac{8+\frac{1}{2}Q^2}{Q}$

$=\frac{8}{Q}+\frac{1}{2}Q$

2(b) At what quantity

it is where Marginal revenue (MR) = Marginal Cost (MC)

ATC at q = 2 and q = 8

$At \ q=2, \ ATC=\frac{8+\frac{1}{2}(2)^2}{2}$

$=5$

$At \ q=8, \ ATC=\frac{8+\frac{1}{2}(8)^2}{8}$

$=5$

graph