# What is the mathematical formula for how marginal revenue changes with quantity sold?

Mar 15, 2016

${\lim}_{\Delta Q \rightarrow 0} \frac{\Delta T R}{\Delta Q} = \frac{d \left(T R\right)}{\mathrm{dQ}}$
Note in practice quantity increments in discrete numbers so:
$\frac{\Delta T R}{\Delta Q}$ maybe more practical
In real life here is what you may see:

1. Total Revenue = (Current Price Per Product) x (Current Number Products Sold)
$\implies T {R}_{1} = {P}_{1} \times {Q}_{1}$
2. Consider lower Alternate Price and determine Alternate Number Products Sold at this price. This step may require specific market analysis.
3. Alt Revenue = (Alt Price) x (Alternative Products Sold) $\implies T {R}_{2} = {P}_{2} \times {Q}_{2}$
4. Marginal Revenue, $M R = | \frac{T {R}_{1} - T {R}_{2}}{{Q}_{1} - {Q}_{2}} |$

#### Explanation:

The total revenue (TR) received from the sale of Q goods at price P is given by $T R = P \cdot Q$. Now the Marginal Revenue (MR) can be deﬁned as the additional revenue added by an additional unit of output. In other words marginal revenue is the extra revenue that an additional unit of product will bring a ﬁrm. It can also be described as the change in total revenue divided by the change in number of units sold. It is possible to write MR as a derivative, in fact more formally, marginal revenue is equal to the change in total revenue over the change in quantity: $M R = \frac{\Delta T R}{\Delta Q}$ when the change in quantity is increment in discrete units (say one). Obviously in the limit we can write:
${\lim}_{\Delta Q \rightarrow 0} \frac{\Delta T R}{\Delta Q} = \frac{d \left(T R\right)}{\mathrm{dQ}}$
Marginal revenue is the derivative of total revenue with respect to demand.