Question #70201

A monopolist faces a demand curve Q = A – P and has a cost function C(Q) = cQ. Derive
the optimal monopoly price and its total profit.

Expert's answer

Maximizing profit:

MR=MC.

MC = C’(Q) = (CQ)’ = C.

Demand function:

Q=A-P

P=A-Q.

TR=AQ-Q^2

MR=A-2Q

A-2Q=C

-2Q=C- A

Q=1/2(A-C) – optimal quantity.

P=A-1/2(A- C) = A-1/2A+1/2C=1/2(A+C)

Total profit is:

TP=(P-AC)*Q

AC=cQ/Q=C.

TP=(1/2A+1/2C-C)*(1/2A- 1/2C)= (1/2A-1/2C)^2

MR=MC.

MC = C’(Q) = (CQ)’ = C.

Demand function:

Q=A-P

P=A-Q.

TR=AQ-Q^2

MR=A-2Q

A-2Q=C

-2Q=C- A

Q=1/2(A-C) – optimal quantity.

P=A-1/2(A- C) = A-1/2A+1/2C=1/2(A+C)

Total profit is:

TP=(P-AC)*Q

AC=cQ/Q=C.

TP=(1/2A+1/2C-C)*(1/2A- 1/2C)= (1/2A-1/2C)^2

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