# 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*}

### Question

Zoe's satisfies an inverse demand P = 331 - 3Q.

Her production costs are 13 Q+ 91.

What quantity of lemonade should Zoe produce to maximize profit?

Zoe maximizes profit when producing Q=53L of lemonade.

With a cost function 13 Q+ 91, Zoe's marginal cost is

$$MC \left( Q \right) = \frac{d C ( Q )}{d Q} = 13$$

With an inverse demand P = 331 - 3Q, Zoe's revenue is

$$R \left( Q \right) = (331 - 3Q) Q = 331 Q - 3Q^2$$

Her marginal revenue is

$$MR \left( Q \right) = \frac{d R ( Q )}{d Q} = 331 - 3 \times 2 Q = 331 - 6Q$$

The profit-maximizing quantity satisfies

\begin{align*} MC \left( Q \right) &= MR \left( Q \right) \\ 13 &= 331 - 6Q \\ 6 Q &= 331 - 13 \\ 6 Q &= 318 \\ Q &= 53 \\ \end{align*}