Question

Assuming the about run cost function of a firm is given by:TC=4Q³+3Q²+10Q+30.

a)find the expression of TFC and TVC.

b) derive the expression of AFC,AVC,AC and MC.

c) find the levels of output that minimize MC and AVC.

d) find the minimum values of MC and AVC

Answer 1

a).i) Expression of T FC will be ; =30

ii)Expression of TVC will be; 4Q³+3Q²+10Q

b)(i) MC=$\frac{change in TC}{change in Q}$ =MC=$\frac{dTC}{dQ}$ =12Q²+6Q=10

(ii)AVC=Q=Fl

TC=FC+VC=FC=WL

AVC IS $\therefore$ $\frac{VC}{Q}$ =$\frac{4Q^3+3Q^2+10Q}{Q}$ =4Q²+3Q+10

(iii)AFC=TC-AVC=$\frac{FC}{Q}$ =$\frac{30}{Q}$

(IV)AC=AFC+AVC

$\frac{30}{Q}$ +4Q²+3Q+10=30+4Q³+3Q²+10Q

iv)Expression of AVC will be;$\frac{VC}{Q}$ =$\frac {4Q^3}{Q}$ +$\frac {3Q^2}{Q}$ +$\frac{10Q}{Q}$ =4Q²+3Q+10

(c). level of minimizing MC=$\frac{dMC}{dQ}$ =24Q+6

finding the value of Q= -$\frac{6}{24}$ =-$\frac{1}{4}$

level of minimizing AVC =$\frac{dAVC}{dQ}$ =8Q+3

Q=$-\frac{3}{8}$

(d).finding the minimum values of MC and AVC

minimum value of MC=$\frac{dMC}{dQ}$ =0

$\implies$ 24Q+6=0

$\therefore$ Q=$-\frac{6}{24}$ =$-\frac{1}{8}$ thus output of MC is minimum at this point.

minimum value of AVC=$\frac{dAVC}{dQ}$ =0

$\implies$8Q+3=0

$\therefore$ Q=$-\frac{3}{8}$ thus output of AVC is minimum at this point.