Question

Assume that the marginal cost of a competitive firm is given by;

MC = 6Q2 – 4Q – 12 and the marginal Revenue of the firm is given by,

MR = – 2Q. Then find,

A) Functions of TVC, AVC, AC and AR

B) The firms profit maximization level of output.

C) How much profit firm can generate?

D) Does the firm generate profit or incur loss at 5 units of output?

Answer 1

(a)

$MC=6Q^{2}4Q-12\\MR=-2Q\\TC=\smallint 6Q^{2}-4Q-12\space \space dq\\=\frac{6Q^{3}}{3}-\frac{4Q^{2}}{2}12Q$

$TC=2Q^{3}-2Q^{2}-12Q+C\\TVC=2Q^{3}-2Q^{2}-12Q$

$AVC=\frac{TVC}{Q}=\frac{2Q^{3}-2Q^{2}-12Q}{Q}\\=2Q^{2}-2Q-12$

$AC=\frac{TC}{Q}\\=2Q^{2}-2Q-12+\frac{C}{Q}$

$TR=\smallint -2Q\space \space dq\\=\frac{-2Q^{2}}{2}=-Q^{2}+C$

$AR=\frac{TR}{Q}=\frac{Q^{2}+C}{Q}=-Q+\frac{C}{Q}$

(b)

$MR=MC\\6Q^{2}-4Q-12=-2Q\\6Q^{2}-4Q+2Q=12\\6Q^{2}+2Q=12\\6Q^{2}+2Q-12=0$

$Q=1.25733\\Q\approx1$

(c)

$profit=TR-TC\\=[-Q^{2}]-[2Q^3-2Q^2-12Q]\\=[-1]-[2-2-12]\\=[-1]-[-12]\\=-1+12\\=11$

(d)

at Q=5

$profit=TR-TC\\=[-Q^{2}]-[2Q^3-2Q^2-12Q]\\=[-1]-[2-2-12]\\=[-1]-[-12]\\=-1+12\\=11$

$[-25]-[250-50-60]\\=[-25]-[140]\\=-165\\LOSS$