Marginal Revenue

The Marginal Revenue indicates the increase in revenue when the quantity increases a little bit.

Example

Zoe's lemonade stand satisfies a demand characterized by:

$$ P = 10 - 2 Q $$

Her revenue is

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

The marginal revenue is the derivative of the revenue `R \left( Q \right)`:

$$ MR \left( Q \right) = \frac{d R \left( Q \right)}{d Q} = 10 - 2 \times 2 Q = 10 - 4 Q $$

Question

The inverse demand for lemonade is `P = 16 - 3Q`.

What is Zoe's marginal revenue?

First, her revenue is

$$ R \left( Q \right) = PQ = (16 - 3Q) Q = 16 Q - 3Q^2 $$

So her marginal revenue is

$$ MR \left( Q \right) = \frac{d R \left( Q \right)}{d Q} = 16 - 3 \times 2 Q = 16 - 6Q $$