# Profit

The profit of a monopolist is its revenue minus its costs of production.

## Example

Zoe is the only lemonade seller on her street. People who drive home stop by to purchase liters of her lemonade.

The demand for lemonade is:

$$P = 10 - 2Q$$

When Zoe produces a quantity Q of lemonade, her revenue is

$$R \left( Q \right) = PQ = \left( 10 - 2Q \right) Q$$

Her cost follows the equation

$$C \left( Q \right) = 2 Q$$

Overall, her profit is

$$\Pi \left( Q \right) = R \left( Q \right) - C \left( Q \right) = \left( 10 - 2Q \right) Q - 2 Q = 8Q - 2 Q^2$$

If Zoe makes 2 liters of lemonade (Q = 2), her profit is

$$\Pi \left( 2 \right) = 8 \times 2 - 2 \times 2^2 = 8$$

She would make a profit of \$8.

### Question

The inverse demand is P = 40 - 3Q and the costs of production are 22 Q.

What is Zoe's profit is she produces Q=6L of lemonade.

The revenue is

$$R \left( Q \right) = PQ = (40 - 3Q) Q = 40 Q - 3Q^2$$

The cost is

$$C ( Q ) = 22 Q$$

So the profit function is

\begin{align*} \Pi ( Q ) &= R (Q ) - C ( Q) \\ &= 40 Q - 3Q^2 - 22 Q \\ &= 18 Q - 3 Q^2 \end{align*}

When Q=6

$$\Pi ( 6 ) = 18 \times 6 - 3 \times 6^2 = 0$$