Answer to Question #224773 in Macroeconomics for Tuji Yusuf

Question

given a short run cost function as tc =1/3q^3-2q^2+60q+100 then find
A. what is the minimum value of avc & mc
B. Function or expression of TVC, AVC, AFC, ATC and MC

Answer 1

$a)\\ TVC=\frac{1}{3}q^3−2q^2+60q\\AVC=\frac{1}{3}q^2−2q+60\\ TC=\frac{q^3}{3}−2q^2+60q+100$

$MC=\frac{d}{dq}[\frac{q^3}{3}−2q^2+60q+100]$

$MC=\frac{1}{3}.\frac{d}{dq}[q^3]−2.\frac{d}{dq}[q^2]+60.\frac{d}{dq}[q]+\frac{d}{dq}[100]$

$\\MC=\frac{3q^2}{3}−(2×2q)+(60×1)+0\\MC=q^2−4q+60$

For minimum value, 

$AVC=MC\\ \frac{1}{3}q^2−2q+60=q^2−4q+60\\\frac{1}{3}q^2−q^2−2q+4q+60−60=0\\\frac{q^2−3q^2}{3}+2q=0\\−2q^2+6q=0\\2q^2−6q=0\\2q(q−3)=0\\q=3$

Minimum value is q=3

$b)\\ TVC=\frac{1}{3}q^3−2q^2+60q\\AVC=\frac{TVC}{q}= \frac{1}{3}q^2−2q+60\\FC=100\\AFC=\frac{100}{q}\\ATC=\frac{TC}{q}=\frac{1}{3}q^2−2q+60+\frac{100}{q}\\ATC=\frac{1}{3}q^2−2q+\frac{100}{q}+60\\MC=q^2−4q+60$