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

What is Zoe's profit is she produces `Q=5`L 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`

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