Most Canadians don’t have to pay for a flu shot, although we do pay for some other types of vaccinations, such as the Hepatitis B vaccine. However, it is not uncommon for residents of other countries, such as the United States, to have to pay for their flu shots. So, what if individual Canadians did have to pay for flu vaccines? In order for Flunomics, a hypothetical flu vaccine distributor, to know whether it could raise its revenue by significantly raising the price of its flu vaccine when it was most needed, it would have to know the price elasticity of demand for flu vaccinations.

Figure 6-1 shows a hypothetical demand curve for flu vaccinations. At a price of $20 per vaccination, consumers would demand 10 million vaccinations per year (point A); at a price of $21, the quantity demanded would fall to 9.9 million vaccinations per year (point B).

Figure 6-1, then, tells us the change in the quantity demanded for a particular change in the price. But how can we turn this into a measure of price responsiveness? The answer is to calculate the price elasticity of demand, also known as the own price elasticity of demand because it relates the change in the quantity demanded of a product to a change in its own price (as opposed to the prices of other products).

The price elasticity of demand is the ratio of the percentage change in quantity demanded to the percentage change in price as we move along the demand curve. As we’ll see later in this chapter, the reason economists use percentage changes is to obtain a measure that doesn’t depend on the units in which a good is measured (say, a child-size dose versus an adult-size dose of vaccine). But before we get to that, let’s look at how elasticity is calculated.

To calculate the price elasticity of demand, we first calculate the percentage change in the quantity demanded and the corresponding percentage change in the price as we move along the demand curve. These are defined as follows:

and

In Figure 6-1, we see that when the price rises from $20 to $21, the quantity demanded falls from 10 million to 9.9 million vaccinations, yielding a change in the quantity demanded of 0.1 million vaccinations. So the percentage change in the quantity demanded is

The initial price is $20 and the change in the price is $1, so the percentage change in price is

To calculate the price elasticity of demand, we find the ratio of the percentage change in the quantity demanded to the percentage change in the price:

In Figure 6-1, the price elasticity of demand¹ is therefore

The law of demand says that demand curves are downward sloping, so price and quantity demanded always move in opposite directions. In other words, a positive percentage change in price (a rise in price) leads to a negative percentage change in the quantity demanded; a negative percentage change in price (a fall in price) leads to a positive percentage change in the quantity demanded. This means that the price elasticity of demand is, in strictly mathematical terms, a negative number. However, it is inconvenient to repeatedly write a minus sign. So when economists talk about the price elasticity of demand, they usually drop the minus sign and report the absolute value of the price elasticity of demand. In this case, for example, economists would usually say “the price elasticity of demand is 0.2,” taking it for granted that you understand they mean minus 0.2. We follow this convention here.

The larger the price elasticity of demand, the more responsive the quantity demanded is to the price. When the price elasticity of demand is large—when consumers change their quantity demanded by a large percentage compared with the percentage change in the price—economists say that demand is highly elastic.

As we’ll see shortly, a price elasticity of 0.2 indicates a small response of quantity demanded to price. That is, the quantity demanded will fall by a relatively small amount when price rises. This is what economists call inelastic demand. And inelastic demand is exactly what Flunomics needed for its strategy to increase revenue by raising the price of its flu vaccines.

Price elasticity of demand compares the percentage change in quantity demanded with the percentage change in price. When we look at some other elasticities, which we will do shortly, we’ll learn why it is important to focus on percentage changes. But at this point we need to discuss a technical issue that arises when you calculate percentage changes in variables.

The best way to understand the issue is with a real example. Suppose you were trying to estimate the price elasticity of demand for gasoline by comparing gasoline prices and consumption in different countries. Because of higher taxes, gasoline usually costs more in Canada than in the United States (approximately thirty cents more per litre in 2013). So what is the percentage difference between Canadian and American gas prices?

Well, it depends on which way you measure it. In 2013, the average price of a litre of gas in Canada was about $1.23 while the average price of a litre of gas in the United States was closer to $0.91. Because the price of a litre of gas in Canada was $0.32 higher than in the United States, it was 35% ($0.32/$0.91 × 100%) higher. Because the price of gasoline in the United States was $0.32 lower, it was 26% ($0.32/$1.23 × 100%) lower.

This is a nuisance: we’d like to have a percentage measure of the difference in prices that doesn’t depend on which way you measure it. To avoid computing different elasticities for rising and falling prices we use the midpoint method.

The midpoint method replaces the usual definition of the percentage change in a variable, X, with a slightly different definition:

where the average value of X is defined as

When calculating the price elasticity of demand using the midpoint method, both the percentage change in the price and the percentage change in the quantity demanded are found using this method. To see how this method works, suppose you have the following data for some good:

To calculate the percentage change in quantity going from situation A to situation B, we compare the change in the quantity demanded—a fall of 200 units—with the average of the quantity demanded in the two situations. So we calculate

In the same way, we calculate

So in this case we would calculate the price elasticity of demand to be

again dropping the minus sign.

The important point is that we would get the same result, a price elasticity of demand of 1, whether we go up the demand curve from situation A to situation B or down from situation B to situation A.

To arrive at a more general formula for price elasticity of demand, suppose that we have data for two points on a demand curve. At point 1 the quantity demanded and price are (Q₁, P₁); at point 2 they are (Q₂, P₂). Then the formula for calculating the price elasticity of demand is:

As before, when finding a price elasticity of demand calculated by the midpoint method, we drop the minus sign and use the absolute value.