# Profit Maximization

The monopolist maximizes profit when its marginal cost equals its marginal revenue.

## Example

Zoe, in her lemonade stand, faces the inverse demand P = 10 - 2Q. So her revenue is $$R ( Q ) = PQ = (10 - 2Q)Q = 10 Q - 2Q^2$$

Her marginal revenue is $$MR ( Q ) = \frac{d R ( Q )}{d Q} = 10 - 4 Q$$

Zoe faces production costs equal to C ( Q ) = 2Q. So her marginal costs are $$MC ( Q ) = \frac{d C ( Q )}{d Q} = 2$$

The quantity Q that maximizes Zoe's profit solves \begin{align*} MC ( Q ) &= MR ( Q ) \\ 2 &= 10 - 4 Q \\ 4 Q &= 10 - 2 \\ 4 Q &= 8 \\ Q &= 2 \end{align*}

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