A one-price monopolist faces a demand of P = 107 – 0.015Q and has a total cost f

Question

A one-price monopolist faces a demand of P = 107 – 0.015Q and has a total cost function C(Q) = 5000ln(Q) + 30Q.

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a) Calculate the profit of the monopolist

b) Prove that regulating the monopoly to produce with no deadweight loss will drive them out of business. Use elements from the calculations here to find the deadweight loss with a monopoly

c) Draw a picture as part of an explanations why the regulator may choose a price 39.33. Find the new DWL at this level of production

Explanation / Answer

P = 107 – 0.015Q

Calculate MR as follows:

TR=P.Q=(107 – 0.015Q )Q=107q-0.015Q2
MR=107-0.03Q

C(Q) = 5000ln(Q) + 30Q.
Thus, MC=5000/Q+30

A monopolist's profit maximizing quantity is where MC=MR

5000/Q+30=107-0.03Q

5000/Q+0.03Q=107-30
5000+0.03Q2=77Q
0.03Q2-77Q+5000=0

Q = (2500,66.66)
P=107-0.015Q
P=107-0.015(2500)
P=69.5

But regulating a monopoly with no deadweight loss would mean setting Price=MC

107 – 0.015Q=5000/Q+30
0.015Q-77Q+5000=0
Q=(5067.5,65.77)

P=107-0.015(5067.5)
P=30.99

MC=5000/Q+30
MC=5000/5067.5 + 30
MC=30.98667

MR=107-0.03Q
MR=107-0.03(5067.5)
MR=-45.05

This would make the firm shut down its operations

Deadweight loss=2500×5067.5=12668750

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