Question

50 factories behave in competitive manner and have identical cost functions given by C(x)=x² /2.  There is one monopolist that has 0 marginal cost. The demand for product that factories produce is D(p)=1,000-50p

(a) What is the supply curve of one of the competitive factories? Find total supply from this competitive sector at price p.

(b) Find the monopolist’s profit maximizing output. What is the monopolist's profit-maximizing price? How much output will this competitive sector provide at this price? What will be the total amount of output sold in this industry?

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Answer 1

a)

Given

C(x)=x^2/2

Marginal Cost=MC=dC(x)/dx=x

A competitive firm sets its output such that MC=p to maximize profit. So,

MC=p

x=p (supply curve of a single competitive firm)

There are 50 such firms, total supply from competitive sector is given as

Qc=50*x=50*p (supply curve of competitive sector)

b)

Market demand is given as

D(p)=1000-50p

Residual demand for monopolist is given as

Q=D(p)-Qc=1000-50p-50p=1000-100p

or

1000-Q=100p

p=10-0.01Q

Total Revenue of monopolist=TR=p*Q=10Q-0.01Q^2

Marginal Revenue of monopolist =MR=dTR/dQ=10-0.02Q

A monopolist will maximize its profit by producing such that such that

MR=MC

10-0.02Q=0

Q=10/.02=500 (Monopolist's optimal output)

p=10-0.01*Q=10-0.01*500=$5 (profit maximizing price)

Output of competitive sector=Qc=50p=50*5=250

Total industry output=Qc+Q=250+500=750