Answer to Question #263028 in Microeconomics for mujtaba syed

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

given

$SRTC = 8 + \frac{1}{2}Q^2 \\ MC = \frac{\delta TC}{\delta q}=q.$

1a.

TC =8 as its independent of output produced

VC=$\frac{1}{2}Q^2$ as it varies with level of production

$AVC=\frac{VC}{q}=\frac{1}{2}q\\ ATC=\frac{TC}{q}=\frac{8}{q}+\frac{1}{2}q$

2b.

(i) AVC is minimum where

$AVC=MC\\\frac{1}{2}q=q\\q=0$

(ii)ATC is minimum where

$ATC=MC\\\frac{8+0.5q^2}{q}=q\\0.5^2=8\\q=4\space units$

ATC(at q=2) $=\frac{8}{2}+\frac{1}{2}(2)=5$

ATC(at q=8)$=\frac{8}{8}+\frac{1}{2}(8)=5$