Question

If the demand function faced by a firm is:

Q = 90 – 2P

TC = 2 + 57Q – 8Q² + Q³

Determine the level of output at which the firm maximizes the profit.

Determine the best level of output for the above question by the MR and MC approach.

Answer 1

a) The firm will maximize profits at a point where MR = MC

TR = PQ = (45 - Q/2)$\times$ Q

=> MR = 45 - Q

TC = 2 + 57Q - 8Q² + Q³

=> MC = 57 - 16Q + 3Q²

MR = MC

=> 45 - Q = 57 - 16Q + 3Q²

=> 3Q² - 15Q + 12 = 0

=> Q² - 5Q + 4 =0

=> Q² - Q - 4Q + 4 =0

=> Q(Q-1) -4(Q-1) = 0

=> Q = 4 or Q = 1

Thus, the answer is Q = 4

b) Q = 90 – 2P

$P=45-1/2Q$

$MR=dP/dQ=-1/2$

$MC=dTC/dQ=57-16Q+3Q^2$

Now MC=MR

$57-16Q+3Q^2=-1/2$

$6Q^2-32Q+115=0$

On solving quadratically,

$Q_1=2.67 and \space Q_2$ $=2.67$

so 2.67 is the best level of output