Total revenue test

The mathematical link between them comes from the formula of the price elasticity of demand:

${\displaystyle E_{d}=-\left(\left(Q_{2}-Q_{1}\right)/\left(P_{2}-P_{1}\right)\right)\cdot \left(P_{1}/Q_{1}\right)}$

where ${\displaystyle P}$ stands for Price, ${\displaystyle Q}$ for quantity demanded, ${\displaystyle \left(Q_{2}-Q_{1}\right)}$ for change in quantity demanded, and ${\displaystyle \left(P_{2}-P_{1}\right)}$ for change in price.^[1] Here the minus sign converts the result to a non-negative number, as is conventional but not universal.

Using the idea of limits for infinitesimal changes in price and therefore in quantity, the formula becomes

${\displaystyle E_{d}=-{\frac {dQ}{dP}}\cdot {\frac {P}{Q}}}$

Total revenue is given by ${\displaystyle TR=P\cdot Q}$.

Since quantity demanded Q is a function of price P, ${\displaystyle Q=f(P),}$ total revenue can be rewritten as

${\displaystyle TR=P\cdot f(P).}$

The derivative of total revenue with respect to P is thus:

${\displaystyle {\frac {dTR}{dP}}=1\cdot f(P)+P\cdot f'(P)}$

But ${\displaystyle Q=f(P)}$, so ${\displaystyle {\frac {dTR}{dP}}=f'(P)\cdot P+Q}$.

After both multiplying and dividing by ${\displaystyle Q}$, the equation can be rewritten as:

${\displaystyle {\frac {dTR}{dP}}=Q\left(f'(P)\cdot {\frac {P}{Q}}+1\right).}$

The last step is to substitute the elasticity of demand for ${\displaystyle -f'(P)\cdot {\frac {P}{Q}}}$ to obtain:

${\displaystyle {\frac {dTR}{dP}}=Q(-E_{d}+1)=Q(1-E_{d})}$.

To find the elasticity of demand using the mathematical explanation of the total revenue test, it's necessary to use the following rule:

If demand is elastic, ${\displaystyle E_{d}>1\!\ }$, then ${\displaystyle {\dfrac {dR}{dP}}<0\!\ }$: price and total revenue move in opposite directions. If demand is inelastic, ${\displaystyle E_{d}<1\!\ }$, then ${\displaystyle {\dfrac {dR}{dP}}>0\!\ }$: price and total revenue change in the same direction. If demand is unit elastic, ${\displaystyle E_{d}=1}$, then ${\displaystyle {\frac {dR}{dP}}=0}$: an increase in price has no influence on the total revenue.

Total revenue, the product price times the quantity of the product demanded, can be represented at an initial point by a rectangle with corners at the following four points on the demand graph: price (P₁), quantity demanded (Q₁), point A on the demand curve, and the origin (the intersection of the price axis and the quantity axis).

The area of the rectangle anchored by point A is the measure of total revenue.

When the price changes the rectangle changes. The change in revenue caused by the price change is called the price effect, and the change In revenue in the opposite direction caused by the resulting quantity change is called the quantity effect.

When the price changes from ${\displaystyle P_{1}}$ to ${\displaystyle P_{2}}$ the magnitude of the price effect is represented by the rectangle ${\displaystyle P_{1}P_{2}CA}$ and the magnitude of the quantity effect is given by rectangle ${\displaystyle Q_{1}Q_{2}BC}$. The price effect is the loss of revenue from selling the original quantity at the lower price; the quantity effect is the added revenue earned at the new price on the newly induced units sold.

So, if the area of the rectangle giving the price effect is greater than the area of the rectangle giving the quantity effect, demand is inelastic: ${\displaystyle E_{d}<1\!\ }$. If the reverse is true, demand is elastic: ${\displaystyle E_{d}>1\!\ }$. If the sizes are equal, demand is unit elastic (or unitary elastic).^[2]