# PRICE ELASTICITY OF DEMAND - Managerial Economics

## Elasticity of Demand in Managerial Economics

The most widely used elasticity measure is the price elasticity of demand, which measures the responsiveness of the quantity demanded to changes in the price of the product, holding constant the values of all other variables in the demand function.

Price Elasticity Formula

Using the formula for point elasticity, price elasticity of demand is found as

where ΔQP is the marginal change in quantity following a one-unit change in price, and P and Q are price and quantity, respectively, at a given point on the demand curve.

The concept of point price elasticity can be illustrated by referring to Equation 5.3:

Q = 8,500 – 5,000P + 3,500PV + 150I + 1,000A

The coefficient for the price variable indicates the effect on quantity demanded of a one-unit change in price:

At the typical values of PV = $3, I =$40,000, and A = $20,000, the demand curve is calculated as Q = 8,500 – 5,000P + 3,500(3) + 150(40) + 1,000(20) = 45,000 – 5,000P This demand curve relation can be used to calculate _P at two points: (1) where P1 =$7 and Q1 = 10,000 and (2) where P2 = $8 and Q2 = 5,000. This implies _P1 = –3.5 and _P2 = –8 because Therefore, a 1 percent increase in price from the$7 movie ticket price level results in a 3.5 percent reduction in the quantity demanded. At the $8 price level, a 1 percent increase results in an 8 percent reduction in the quantity demanded. This indicates that movie ticket buyers, like most consumers, become increasingly price sensitive as average price increases. This example illustrates how price elasticity tends to vary along a linear demand curve, with _P increasing in absolute value at higher prices and lower quantities. Although price elasticity always varies along a linear demand curve, under certain conditions it can be constant along a curvilinear demand curve. This point will be illustrated in a later section. When evaluating price elasticity estimates, recognize that price elasticities are uniformly negative. This is because the quantity demanded for all goods and services is inversely related to price. In the previous example, at a$7 price, a 1 percent increase in price leads to a 3.5 percent decrease in the quantity of movie tickets demanded. Conversely, a 1 percent decrease in price leads to a 3.5 percent increase in the quantity demanded. For expository convenience, the equation for price elasticity is sometimes multiplied by –1 to change price elasticities to positive numbers. Therefore, when price elasticities are reported as positive numbers, or in absolute value terms, it is important to remember the underlying inverse relation between price and quantity.

Using the arc elasticity concept, the equation for price elasticity is

This form is especially useful for analyzing the average sensitivity of demand to price changes over an extended range of prices. For example, the average price elasticity over the price range from $7 to$8 is

This means that, on average, a 1 percent change in price leads to a 5 percent change in quantity demanded when price is between $7 and$8 per ticket.

Price Elasticity and Total Revenue

One of the most important features of price elasticity is that it provides a useful summary measure of the effect of a price change on revenues. Depending on the degree of price elasticity, a reduction in price can increase, decrease, or leave total revenue unchanged. A good estimate of price elasticity makes it possible to accurately estimate the effect of price changes on total revenue.

For decision-making purposes, three specific ranges of price elasticity have been identified. Using |_P| to denote the absolute value of the price elasticity, three ranges for price elasticity are

With elastic demand, |_P| > 1 and the relative change in quantity is larger than the relative change in price. A given percentage increase in price causes quantity to decrease by a larger percentage. If demand is elastic, a price increase lowers total revenue and a decrease in price raises total revenue. Unitary elasticity is a situation in which the percentage change in quantity divided by the percentage change in price equals –1. Because price and quantity are inversely related, a price elasticity of –1 means that the effect of a price change is exactly offset by the effect of a change in quantity demanded. The result is that total revenue, the product of price times quantity, remains constant. With inelastic demand, a price increase produces less than a proportionate decline in the quantity demanded, so total revenues rise. Conversely, when demand is inelastic, a price decrease generates a less than proportionate increase in quantity demanded, so total revenues falls. These relations are summarized in Table.

Price elasticity can range from completely inelastic, where _P = 0, to perfectly elastic, where _P = –∞. To illustrate, consider first an extreme case in which the quantity demanded is independent of price so that some fixed amount, Q*, is demanded regardless of price. When the quantity demanded of a product is completely insensitive to price, ΔQP = 0, and price elasticity will equal zero, irrespective of the value of P/Q. The demand curve for such a good or service is perfectly vertical, as shown in Figure.

The other limiting case, that of infinite price elasticity, describes a product that is completely sensitive to price. The demand curve for such a good or service is perfectly horizontal, as shown in Figure. Here the ratio ΔQP = –∞ and _P = –∞, regardless of the value of P/Q.

The economic as well as mathematical properties of these limiting cases should be understood. Afirm faced with a vertical or perfectly inelastic demand curve could charge any price and still sell Q* units. Theoretically, such a firm could appropriate all of its customers’ income or wealth. Conversely, a firm facing a horizontal or perfectly elastic demand curve could sell an unlimited quantity of output at the price P*, but it would lose all sales if it raised prices by even a small amount. Such extreme cases are rare in the real world, but monopolies that sell necessities such as pharmaceuticals enjoy relatively inelastic demand, whereas firms in highly competitive industries such as grocery retailing face highly elastic demand curves.

Relationship Between Price Elasticity and Total Revenue

Completely Inelastic Demand Curve: εP = 0

Uses of Price Elasticity Information

Price elasticity information is useful for a number of purposes. Obviously, firms are required to be aware of the price elasticity of demand when they price their products. For example, a profitmaximizing firm would never choose to lower its prices in the inelastic range of the demand curve. Such a price decrease would decrease total revenue and at the same time increase costs, because the quantity demanded would rise. A dramatic decrease in profits would result. Even over the range in which demand is elastic, a firm will not necessarily find it profitable to cut price. The profitability of a price cut in the elastic range of the demand curve depends on whether the marginal revenues generated exceed the marginal cost of added production. Price elasticity information can be used to answer questions such as

• What is the expected impact on sales of a 5 percent price increase?
• How great a price reduction is necessary to increase sales by 10 percent?
• Given marginal cost and price elasticity data, what is the profit-maximizing price?

The importance of price elasticity information was illustrated during 2000–2001 in California when electric utilities were forced to raise prices dramatically because of a rapid increase in fuel costs. The question immediately arose: How much of a cutback in quantity demanded and, hence, how much of a reduction in future capacity needs would these price increases cause? In other words, what was the price elasticity of electricity? In view of the long lead times required to build electricity-generating capacity and the major economic dislocations that arise from power outages, this was a critical question for both consumers and producers of electricity.

Price elasticity information has long played a major role in the debate over national energy policy. Some industry and government economists argue that the price elasticity of demand for energy is sufficiently large that an equilibrium of demand and supply will occur following only modest price changes. Others argue that energy price elasticities are so low that unconscionable price increases are necessary to reduce the quantity demanded to meet pending supply shortfalls. Meanwhile, bouts of falling oil prices raise fears among some that low oil prices may increase Western reliance on imported oil. These same issues have also become a focal point in controversies surrounding nuclear energy, natural gas price deregulation, and alternative renewable energy sources. In this debate on energy policy, the relation between price and quantity supplied—the price elasticity of supply—is also an important component. As with most economic issues, both demand and supply sides of the marketplace must be analyzed to arrive at a rational decision.

Completely Elastic Demand Curve: εP = –∞

Another example of the importance of price elasticity information relates to the widespread discounts or reduced rates offered different customer groups. The Wall Street Journal offers students bargain rates; airlines, restaurants, and most hotel chains offer discounts to vacation travelers and senior citizens; large corporate customers get discounts or rebates on desktop computers, auto leases, and many other items. Many such discounts are substantial, sometimes in the range of 30 percent to 40 percent off standard list prices. The question of whether reduced prices attract sufficient additional customers to offset lower revenues per unit is directly related to the price elasticity of demand.

Additional uses of price elasticity information are examined in later chapters. At this point, it becomes useful to consider some other important demand elasticities.