Revenue

The monopolist's revenue is market price multiplied by the quantity that the monopolist produces. $$R = PQ$$

Example

The price for lemonade $$P = 10 - 2 Q$$ Zoe is the only lemonade seller on her street. She controls the quantity of lemonade available on the market.

Her revenue is $$R \left( Q \right) = PQ = \left( 10 - 2Q \right) Q = 10 Q - 2 Q^2$$ If Zoe produces Q=2 liters of lemonade, her revenue is equal to \$12: $$R \left( 2 \right) = 10 \times 2 - 2 \times 2^2 = 20 - 8 = 12$$

Question

The price for lemonade follows $$P = 34 - 9 Q$$ What is Zoe's revenue function? What is her revenue if she produces 4 liters of lemonade?

Zoe's revenue function is \begin{align*} R \left( Q \right) &= P Q \\ &= \left( 34 - 9 Q \right) Q \\ &= 34 Q -9 Q^2 \end{align*}

If Q=1, then her revenue is \begin{align*} R \left( 1 \right) &= 34 \times 1 - 9 \times 1^2 \\ &= 34 - 9 \\ &= 25 \end{align*}