Question

You are a monopolist with the following demand and cost conditions:

Q = 50 - .5P and

C(Q) = 50 + Q².

a. Determine the profit-maximizing output and price.
b. Show your total revenue, total cost and profits.

Answer 1

a)

the firm maximize profit at MR=MC

the inverse demand curve is

Q=50-0.5P

0.5P=50-Q

P=100-2Q

MR=100-4Q ...... An MR curve is double sloped than the inverse demand curve

MC is the change in total cost and found by differentiation

MC=dTC/dQ=2Q

equating MR=MC

100-4Q=2Q

6Q=100

Q=16.6666667=16.67

P=100-2*16.6666667

P=66.6666666=66.67

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b)

Total revenue =P*Q

=16.6666667*66.6666666

=1111.11111

=1111.11

total cost =50+Q^2

=50+16.6666667^2

=327.777779

=327.78

Profit =TR-TC

=1111.11111-327.777779

=783.333331

=783.33

the profit is $783.33