# 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.

$$P = 10 - 2Q$$

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 - 10Q and the costs of production are 30 Q.

What is Zoe's profit is she produces Q=5L of lemonade.

The revenue is

$$R \left( Q \right) = PQ = (40 - 10Q) Q = 40 Q - 10Q^2$$

The cost is

$$C ( Q ) = 30 Q$$

So the profit function is

\begin{align*} \Pi ( Q ) &= R (Q ) - C ( Q) \\ &= 40 Q - 10Q^2 - 30 Q \\ &= 10 Q - 10 Q^2 \end{align*}

When Q=5

$$\Pi ( 5 ) = 10 \times 5 - 10 \times 5^2 = -200$$