Question

A monopoly firm is faced with the following demand function

P = 26 – 0.5Q. The Marginal Cost function for the firm is given

by 6 + 6Q and the total fixed cost is 4.

Determine;

a) The profit maximizing output.

b) The level of supernormal profit if any.

c) The output level at the break-even point

Answer 1

$Soln,$

$p=26-0.5Q$

$MC=6+6Q$

$TFC=4$

a. $MC=MR$

P=AR

$P=26-0.5Q$

$TR=P.Q(26-0.5Q)Q$

$TR=26Q-0.5Q^2$

$MR=\frac {\delta TR}{\delta Q}=26-Q$

$MR=26-Q$

Max at MC=MR

$=6+6Q=26-Q= \frac{ 7Q}{7}=\frac {20}{7}$

$Q=2.86 units.$

b.$TR$ greater than TC

$TR=26-0.5Q^2$ BUT $Q=2.86$

$26(2.86)-0.5(2.86)^2$

74.36−4.0898=78.4498

$TR=78.4498$

$TC=\int(MC) dQ$ BUT $MC=6+6Q$

$TC=4+6Q+6Q^2=4+17.16+49.07=70.23$

$TC=70.23$

$TR-TC$

$=78.4498-70.23 =8.2122$

c. At break even point TC=TR

$TC=4+6Q+6Q^2$

$TR=26Q-0.5Q^2$

THUS, $26Q-0.5Q^2=4+6Q+6Q^2$

$=6.5Q^2-20Q+4=0$

=42.25-40Q+8=0

$40Q=50.25$

$=1.26$