Question

1.     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 Minimum 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.