Question

Given TC = 100 + 60Q – 12Q2 + Q3. Find
a. The equations of the TVC, AVC, and MC functions.
b. The level of output at which AVC and MC are minimum, and prove that the AVC and MC curves are U-shaped.
c. Find the AVC and MC for the level of output at which the AVC curve is minimum.

Answer 1

From the Total cost TC ;If Q=0,TC=100 and TFC=100

Therefore TVC=60Q-12Q²+Q³

AVC=$\frac{TVC} {Q}$ =$\frac{(60Q−12Q2+Q3)} {Q}$

AVC=60-12Q+Q²

MC=$\frac{d(TC)} {dq}$ =$\frac{d(60Q-12Q^2+Q^3)} {dq}$

MC=60-24Q+3Q²

b) For Minimum Output AVC ;$\frac{d(AVC)} {dq} =0$

$\frac{d(60-12Q+Q^2) }{dq} =0$

$\frac{d(AVC)} {dQ}$ =-12 +2Q=0

Q=6

Hence AVC is minimum when output, Q=6

For Minumum output MC; 60-24Q+3Q²

$\frac{d(MC)} {dQ}$ =-24+6Q=0

Q=4

Hence MC is minimum when output, Q=4

To prove that AVC and MC curves are U shaped ;

$\frac{d(MC)} {dQ}$ =-24+6Q

$\frac{d2(MC)} {d2(Q)}$ =6>0, hence the curves are U shaped.

c) AVC=60 - 12Q+Q²

^ 60-12(6)+6²

^60-72+36

=24

The output level Q=6 at which AVC function is minimum, AVC and MC is 24.