A monopolist faces inverse demand $P = 300 - 6 Q$ and has marginal cost $M C = 120 + 6 Q$. What price should this monopolist charge to maximize profit?

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Quizplus · Accepted Answer

The monopolist's profit is given by:$$\pi = P \cdot Q - MC \cdot Q$$Substituting the inverse demand and marginal cost functions, we get:$$\pi = (300 - 6Q) \cdot Q - (120 + 6Q) \cdot Q$$$$\pi = 300Q - 6Q^2 - 120Q - 6Q^2$$$$\pi = 180Q - 12Q^2$$To maximize profit, we take the derivative of the profit function with respect to quantity and set it equal to zero:$$\frac{d\pi}{dQ} = 180 - 24Q = 0$$$$Q = 7.5$$Substituting this quantity back into the inverse demand function, we get:$$P = 300 - 6(7.5) = 240$$Therefore, the monopolist should charge a price of $240 to maximize profit.

A Monopolist Faces Inverse Demand P=300−6QP = 300 - 6 QP=300−6Q

A Monopolist Faces Inverse Demand $P = 300 - 6 Q$