# Price Elasticity of Demand and Total Revenue

Information about the price elasticity of demand enables firms or suppliers to determine the impact of price changes on a firm's total revenue.
One beneficial use of the price elasticity of demand is to determine what impact changes in a good's or service's price will have on a firm's total revenue. Total revenue (TR) is the total amount earned by selling a good or service, and is equal to the price of a good (P) multiplied by the quantity of units sold (Q).
$\text{TR}=\text{P}\times \text{Q}$
The law of demand states that when price rises, the quantity sold will fall. The opposite case holds as well. Because these movements in price and quantity occur in opposite directions, the effect on total revenue can be ambiguous. However, when it is known whether the demand is elastic or inelastic, this ambiguity disappears. Knowledge of elasticity enables economists to determine whether, and by how much, the quantity demanded will change in response to a change in price, and this enables them to establish the impact on total revenue. For goods with elastic demand, firms should lower prices to increase total revenue by increasing the quantity demanded. For goods with inelastic demand, firms should increase price to raise total revenue. Calculating price elasticity helps a firm make these choices.

### Demand Is Elastic: Effect on Total Revenue

When demand is elastic, small changes in price correlate to large changes in quantity demanded.
Suppose that the absolute value of the price elasticity of demand (ED) is greater than 1, or $\lvert\text{E}_{\text{D}}\rvert\gt1$. Mathematically, this occurs because the percentage change in quantity demanded ($\% \Delta \text{Q}_{\text{D}}$ the numerator) is greater than the percentage change in price ($\% \Delta \text{P}$, the denominator).
$\lvert\%\Delta \text{Q}_{\text{D}}\rvert\gt\vert\%\Delta \text{P}\vert$
When demand is elastic, quantity demanded changes by a larger percentage than the percentage change in price; thus, by following the total revenue formula (quantity of goods sold multiplied by the price of the good), the quantity effect will be greater than the price effect.

For example, for a firm manufacturing keyboards, suppose that the price elasticity of demand is equal to 1.5. This means that a 10% increase in price will reduce the quantity demanded by 15%. In this case, total revenue will fall when there is an increase in price. If the price of a keyboard is raised from $100 to$110, the demand will drop by 15%.

In contrast, if the price of the keyboard were to fall by 10% (from $100 to$90), the quantity demanded would rise by 15%. In this case, even though the price is falling, total revenue will rise. If a company faces elastic demand, it will want to lower prices in order to increase total revenue.

### Demand Is Inelastic: Effect on Total Revenue

When demand is inelastic, changes in price have little effect on quantity demanded.
Inelastic demand has a price elasticity of less than 1 in absolute value, or $\lvert \text{E}_{\text{D}}\rvert \lt \ 1$. Mathematically, the percentage change in price ($\%\Delta\text{P}$, denominator) is greater than the percentage change in quantity demanded ($\%\Delta \text{Q}_{\text{D}}$ numerator).
$\lvert\%\Delta \text{Q}_{\text{D}}\rvert \lt \lvert\% \Delta \text{P}\rvert$
A company sells T-shirts, and the absolute value of the elasticity of demand for T-shirts was 0.625. Here, a 10% increase in price will result in only a 6.25% decline in quantity demanded, so total revenue will rise. For example, suppose the company sells 100 T-shirts for $10 each. If it increases its price for these T-shirts by 10%, the price will increase to$11. Suppose sales then decrease by 6.25%, or 93.75 T-shirts. The previous revenue was $1,000. The new revenue is higher, at$1,031.25. In contrast, a 10% decrease in price would lead to a 6.25% increase in quantity demanded but a fall in total revenue. If a firm faces inelastic demand, it will want to raise prices in order to increase total revenue.

### Demand Is Unit Elastic: Effect on Total Revenue

When demand is unit elastic, the percentage change in quantity demand is the same as the percentage change in price.

Unit elastic demand is a situation in which percentage change in quantity is equal to percentage change in price, and it occurs when the absolute value of the price elasticity of demand is equal to 1, or $\lvert \text{E}_{\text{D}}\rvert=1$.

In this case, the numerator and denominator are equal.
$\lvert\%\Delta \text{Q}_{\text{D}}\rvert=\lvert\%\Delta \text{P}\rvert$
With unit elasticity, a 10% rise in the price of a good or service will be exactly offset by a 10% decline in quantity demanded, leaving the total revenue unchanged. The same occurs when price falls by 10%; the quantity demanded rises by 10% and total revenue remains the same.

### Relationship between the Price Elasticity of Demand and Total Revenue

Demand Price Quantity Demanded Total Revenue
Elastic Rises Falls Falls
Falls Rises Rises
Inelastic Rises Falls Rises
Falls Rises Falls
Unit Elastic Rises Falls Unchanged
Falls Rises Unchanged

Knowing the type of demand that applies to their product helps firms understand how to maximize total revenue.