Question

Consumers prefer to buy higher quality products with reasonable prices. A monopolist supplies these products. This monopolist maximizes profits by selling output with better quality. The demand function facing the monopolist is given by

? = ?(50 − ?),

Where Q = output, P = price, and z = quality of the product the monopolist sells. Marginal cost of production is independent of quality and is constant at zero. Assume that product design costs rise with the quality level chosen such that the design cost ?(?) is: ?(?) = 5?2 . Find the monopolist’s profit maximizing level of output and quality. What is the monopolist’s profit maximizing price and profit

Answer 1

Total Revenue, TR, is given by

TR=P*Q=z*(50-Q)*Q=50zQ-zQ^2

TC=F(z)=5z^2

Profit==TR-TC=50zQ-zQ^2-5z^2

Differentiate with respect to Q, we get

d/dQ=50z-2zQ

Set d/dQ=0

50z-2zQ=0

Q=25 (Profit maximizing output)

Now differentiate with respect to z, we get

d/dz=50Q-Q^2-10z

Ser Q=25

d/dz=50Q-Q^2-10z=50*25-25^2-10z=625-10z

Set d/dz=0

625-10z=0

z=62.50 (Profit maximizing quality)

Profit==50zQ-zQ^2-5z^2=50*62.50*25-62.5*25^2-5*62.5^2=$19531.25