As noted above, price elasticity of demand measures the responsiveness or sensitivity of consumers to changes in the price of a good or service. We will begin this section by presenting a formal (mathematical) definition of price elasticity and then show how price elasticity can be used to predict the change in sales when price rises or falls or to predict the percentage reduction in price needed to stimulate sales by a given percentage amount.

Consumer responsiveness to a price change is measured by the **price elasticity of demand (E)**, defined as

E = %ΔQ / %ΔP = Percentage change in quantity demanded / Percentage change in price

Since price and quantity demanded are inversely related by the law of demand, the numerator and denominator always have opposite algebraic signs, and the price elasticity is always negative. The price elasticity is calculated for movements along a given demand curve (or function) as price changes and all other factors affecting quantity demanded are held constant. Suppose a 10 percent price decrease (%ΔP = -10%) causes consumers to increase their purchases by 30 percent (%ΔQ = +30%). The price elasticity is equal to -3 (= +30%/-10%) in this case. In contrast, if the 10 percent decrease in price causes only a 5 percent increase in sales, the price elasticity would equal -0.5 (= +5%/-10%). Clearly, the smaller (absolute) value of E indicates less sensitivity on the part of consumers to a change in price.

When a change in price causes consumers to respond so strongly that the percentage by which they adjust their consumption (in absolute value) *exceeds* the percentage change in price (in absolute value), demand is said to be **elastic** over that price interval. In mathematical terms, demand is elastic when |%ΔQ| exceeds |%ΔP|, and thus |E| is greater than 1. When a change in price causes consumers to respond so weakly that the percentage by which they adjust their consumption (in absolute value) is *less than* the percentage change in price (in absolute value), demand is said to be** inelastic** over that price interval. In other words, demand is inelastic when the numerator (in absolute value) is smaller than the denominator (in absolute value), and thus |E| is less than 1. In the special instance in which the percentage change in quantity (in absolute value) *just equals* the percentage change in price (in absolute value), demand is said to be **unitary elastic**, and |E| is equal to 1. Table 6.1 summarizes this discussion.

**Predicting the Percentage Change in Quantity Demanded **

Suppose a manager knows the price elasticity of demand for a company’s product is equal to -2.5 over the range of prices currently being considered by the firm’s marketing department. The manager is considering decreasing price by 8 percent and wishes to predict the percentage by which quantity demanded will increase. From the definition of price elasticity, it follows that

-2.5 = %ΔQ / -8%

so, with a bit of algebraic manipulation, %ΔQ = +20% (= -2.5 * -8%). Thus the manager can increase sales by 20 percent by lowering price 8 percent. As we mentioned in the introduction, price elasticity information about industry demand can also help price-taking managers make predictions about industry- or marketlevel changes. For example, suppose an increase in industry supply is expected to cause market price to fall by 8 percent, and the price elasticity of industry demand is equal to -2.5 for the segment of demand over which supply shifts. Using the same algebraic steps just shown, total industry output is predicted to increase by 20 percent in this case.

**Predicting the Percentage Change in Price**

Suppose a manager of a different firm faces a price elasticity equal to -0.5 over the range of prices the firm would consider charging for its product. This manager wishes to stimulate sales by 15 percent. The manager is willing to lower price to accomplish the increase in sales but needs to know the percentage amount by which price must be lowered to obtain the 15 percent increase in sales. Again using the definition of price elasticity of demand, it follows that

-0.5 = +15% / %ΔP

so, after some algebraic manipulation, %ΔP = -30% (= 15%/-0.5). Thus this manager must lower price by 30 percent in order to increase sales by 15 percent. As we explained in the case of predicting percentage changes in quantity demanded, elasticity of industry demand can also be used to make predictions about changes in market-determined prices. For example, suppose an increase in industry supply is expected to cause market output to rise by 15 percent, and the price elasticity of *industry* demand is equal to -0.5 for the portion of demand over which supply shifts. Following the algebraic steps shown above, market price is predicted to fall by 30 percent. As you can see, the techniques for predicting percentage changes in quantity demanded and price can be applied to both individual firm demand curves or industry demand curves.

As you can see, the concept of elasticity is rather simple. Price elasticity is nothing more than a mathematical measure of how sensitive quantity demanded is to changes in price. We will now apply the concept of price elasticity to a crucial question facing managers. How does a change in the price of the firm’s product affect the total revenue received?