Calculating the Elasticity of Demand: An Example

Go back to our earlier example:

quantity demanded = 100 − 5 × price.

Suppose a firm sets a price of $15 and sells 25 units. What is the elasticity of demand if we think of a change in price from $15 to $14.80? In this case, the change in the price is −0.2, and the change in the quantity is 1. Thus we calculate the elasticity of demand as follows:

1. The percentage change in the quantity is $\frac{1}{25}$ (4 percent).
2. The percentage change in the price is $-\frac{0.2}{15}=\frac{\text{1}}{\text{75}}$ (approximately −1.3 percent).
3. −(elasticity of demand) is $\frac{1/25}{1/75}=3.$

The interpretation of this elasticity is as follows: when price decreases by 1 percent, quantity demanded increases by 3 percent. This is illustrated in Figure 7.6 "The Elasticity of Demand".

One very useful feature of the elasticity of demand is that it does not change when the number of units changes. Suppose that instead of measuring prices in dollars, we measure them in cents. In that case our demand curve becomes

quantity demanded = 100 − 500 × price.

Make sure you understand that this is exactly the same demand curve as before. Here the slope of the demand curve is −500 instead of −5. Looking back at the formula for elasticity, you see that the change in the price is 100 times greater, but the price itself is 100 times greater as well. The percentage change is unaffected, as is elasticity.

Market Power

The elasticity of demand is very useful because it is a measure of the market powerThe extent to which a firm produces a product that consumers want very much and for which few substitutes are available. that a firm possesses. In some cases, some firms produce a good that consumers want very much—a good in which few substitutes are available. For example, De Beers controls much of the world’s market for diamonds, and other firms are not easily able to provide substitutes. Thus the demand for De Beers’ diamonds tends to be insensitive to price. We say that De Beers has a lot of market power. By contrast, a fast-food restaurant in a mall food court possesses very little market power: if the fast-food Chinese restaurant were to try to charge significantly higher prices, most of its potential customers would choose to go to the other Chinese restaurant down the aisle or even to eat sushi, pizza, or burritos instead.

Ellie’s company had significant market power. There were a relatively small number of drugs available in the country to treat high blood pressure, and not all drugs were identical in terms of their efficacy and side effects. Some doctors were loyal to her product and would almost always prescribe it. Some doctors were not very well informed about the price because doctors don’t pay for the medication. For all these reasons, Ellie had reason to suspect that the demand for her drug was not very sensitive to price.

The Elasticity of Demand for a Linear Demand Curve

The elasticity of demand is generally different at different points on the demand curve. In other words, the market power of a firm is not constant: it depends on the price that a firm has chosen to set. To illustrate, remember that we found −(elasticity of demand) = 3 for our demand curve when the price is $15. Suppose we calculate the elasticity for this same demand curve at $4. Thus imagine that that we are originally at the point where the price is $4 and sales are 80 units and then suppose we again decrease the price by 20 cents. Sales will increase by 1 unit:

1. The percentage change in the quantity is $\frac{1}{80}$ (1.25 percent).
2. The percentage change in the price is $-\frac{0.2}{4}=-\frac{\text{1}}{\text{20}}$ (5 percent).
3. −(elasticity of demand) is $\frac{1/80}{1/20}=\mathrm{0.25.}$

The elasticity of demand is different because we are at a different point on the demand curve.

When −(elasticity of demand) increases, we say that demand is becoming more elastic. When −(elasticity of demand) decreases, we say that demand is becoming less elastic. As we move down a linear demand curve, −(elasticity of demand) becomes smaller, as shown in Figure 7.7 "The Elasticity of Demand When the Demand Curve Is Linear".

Measuring the Elasticity of Demand

To evaluate the effects of her decisions on revenues, Ellie needs to know about the demand curve facing her firm. In particular, she needs to know whether the quantity demanded by buyers is very sensitive to the price that she sets. We now know that the elasticity of demand is a useful measure of this sensitivity. How can managers such as Ellie gather information on the elasticity of demand?

At an informal level, people working in marketing and sales are likely to have some idea of whether their customers are very price sensitive. Marketing and sales personnel—if they are any good at their jobs—spend time talking to actual and potential customers and should have some idea of how much these customers care about prices. Similarly, these employees should have a good sense of the overall market and the other factors that might affect customers’ choices. For example, they will usually know whether there are other firms in the market offering similar products, and, if so, what prices these firms are charging. Such knowledge is much better than nothing, but it does not provide very concrete evidence on the demand curve or the elasticity of demand.

A firm may be able to make use of existing sales data to develop a more concrete measure of the elasticity of demand. For example, a firm might have past sales data that show how much they managed to sell at different prices, or a firm might have sales data from different cities where different prices were charged. Suppose a pricing manager discovers data for prices and quantities like those in part (a) of Figure 7.8 "Finding the Demand Curve". Here, each dot marks an observation—for example, we can see that in one case, when the price was $100, the quantity demanded was 28.

The straight-line demand curves that appear in this and other books are a convenient fiction of economists and textbook writers, but no one has actually seen one in captivity. In the real world of business, demand curves—if they are available at all—are only a best guess from a collection of data. Economists and statisticians have developed statistical techniques for these guesses. The underlying idea of these techniques is that they fit a line to the data. (The exact details do not concern us here; you can learn about them in more advanced courses in economics and statistics.) Part (b) of Figure 7.8 "Finding the Demand Curve" shows an example. It represents our best prediction, based on available data, of how much people will buy at different prices.

If a firm does not have access to reliable existing data, a third option is for it to generate its own data. For example, suppose a retailer wanted to know how sensitive customer demand for milk is to changes in the price of milk. It could try setting a different price every week and observe its sales. It could then plot them in a diagram like Figure 7.8 "Finding the Demand Curve" and use techniques like those we just discussed to fit a line. In effect, the store could conduct its own experiment to find out what its demand curve looks like. For a firm that sells over the Internet, this kind of experiment is particularly attractive because it can randomly offer different prices to people coming to its website.

Finally, firms can conduct market research either on their own or by hiring a professional market research firm. Market researchers use questionnaires and surveys to try to discover the likely purchasing behavior of consumers. The simplest questionnaire might ask, “How much would you be willing to pay for product x?” Market researchers have found such questions are not very useful because consumers do not answer them very honestly. As a result, research firms use more subtle questions and other more complicated techniques to uncover consumers’ willingness to pay for goods and services.

Ellie decided that she should conduct market research to help with the pricing decision. She hired a market research firm to ask doctors about how they currently prescribed different high blood pressure medications. Specifically, the doctors were asked what percentage of their prescriptions went to each of the drugs on the market. Then they were asked the effect of different prices on those percentages. Based on this research, the market research firm found that a good description of the demand curve was as follows:

quantity demanded = 252 − 300 × price.

Remember that the drug was currently being sold for $0.50 a pill, so

quantity demanded = 252 − 300 × 0.5 = 102.

The demand curve also told Ellie that if she increased the price by 10 percent to $0.55, the quantity demanded would decrease to 87 (252 − 300 × 0.55 = 87). Therefore, the percentage change in quantity is $\frac{87-102}{102}=-\mathrm{14.7.}$ From this, the market research firm discovered that the elasticity of demand at the current price was

$$-\text{(elasticity of demand)}=\frac{\text{percentage change in quantity}}{\text{percentage change in price}}=\frac{14.7}{10}=\mathrm{1.47.}$$

Marginal Revenue

Earlier we suggested that a good strategy for pricing is to experiment with small changes in price. So how do small changes in price affect the revenue of a firm? Suppose, for example, that a firm has set the price at $15 and sells 25 units, but the manager contemplates decreasing the price to $14.80. We can see the effect that this has on the firm’s revenues in Figure 7.10 "Revenues Gained and Lost".

The firm will lose 20 cents on each unit it sells, so it will lose $5 in revenue. This is shown in the figure as the rectangle labeled “revenues lost.” But the firm will sell more units: from the demand curve, we know that when the firm decreases its price by $0.20, it sells another unit. That means that the firm gains $14.80, as shown in the shaded area labeled “revenues gained.” The overall change in the firm’s revenues is equal to $14.80 − $5.00 = $9.80. Decreasing the price from $15.00 to $14.80 will increase its revenues by $9.80.

Look carefully at Figure 7.10 "Revenues Gained and Lost" and make sure you understand the experiment. We presume throughout this chapter that a firm must sell every unit at the same price. When we talk about moving from $15.00 to $14.80, we are not supposing that a firm sells 25 units for $15 and then drops its price to $14.80 to sell the additional unit. We are saying that the manager is choosing between selling 25 units for $15.00 or 26 units for $14.80.

Figure 7.11 "Calculating the Change in Revenues" explains this idea more generally. Suppose a firm is originally at point A on the demand curve. Now imagine that a manager decreases the price. At the new, lower price, the firm sells a new, higher quantity (point B). The change in the quantity is the new quantity minus the initial quantity. The change in the price is the new price minus the initial price (remember that this is a negative number). The change in the firm’s revenues is given by

change in revenues = (change in quantity × new price) + (change in price × initial quantity).

The first term is positive: it is the extra revenue from selling the extra output. The second term is negative: it is the revenue lost because the price has been decreased. Together these give the effect of a change in price on revenues, which we call a firm’s marginal revenueThe extra revenue from selling an additional unit of output, which is equal to the change in revenue divided by the change in sales..

Marginal Revenue and the Elasticity of Demand

Given the definitions of marginal revenue and the elasticity of demand, we can write

$$\text{marginal revenue}=\text{price}\times \left(1-\frac{\text{1}}{-\text{(elasticity of demand)}}\right)\text{.}$$

It may look odd to write this expression with two minus signs. We do this because it is easier to deal with the positive number: −(elasticity of demand). We see three things:

1. Marginal revenue is always less than the price. Mathematically, $\left(1-\frac{\text{1}}{-\text{(elasticity of demand)}}\right)<\text{1}\text{.}$ Suppose a firm sells an extra unit. If the price stays the same, then the extra revenue would just equal the price. But the price does not stay the same: it decreases, meaning the firm gets less for every unit that it sells.
2. Marginal revenue can be negative. If –(elasticity of demand) < 1, then $\frac{\text{1}}{-\text{(elasticity of demand)}}>\text{1,}$ and $\left(1-\frac{\text{1}}{-\text{(elasticity of demand)}}\right)<\text{0}\text{.}$ When marginal revenue is negative, increased production results in lower revenues for a firm. The firm sells more output but loses more from the lower price than it gains from the higher sales.
3. The gap between marginal revenue and price depends on the elasticity of demand. When demand is more elastic, meaning −(elasticity of demand) is a bigger number, the gap between marginal revenue and price becomes smaller.

These three ideas are illustrated in Figure 7.12 "Marginal Revenue and Demand". The demand curve shows us the price at any given quantity. The marginal revenue curve lies below the demand curve because of our first observation: at any quantity, marginal revenue is less than price.When a demand curve is a straight line, the marginal revenue curve is also a straight line with the same intercept, but it is twice as steep. The marginal revenue curve intersects the horizontal axis at 50 units: when output is less than 50 units, marginal revenue is positive; when output exceeds 50, marginal revenue is negative. We explained earlier that a linear demand curve becomes more inelastic as you move down it. When the demand curve goes from being relatively elastic to relatively inelastic, marginal revenue goes from being positive to being negative.

Earlier, we showed that when a firm sets the price at $15, −(elasticity of demand) = 3. Thus we can calculate marginal revenue at this price:

$$\text{marginal revenue}=\text{price}\times \left(1-\frac{\text{1}}{-\text{(elasticity of demand)}}\right)=15\left(1-\frac{1}{3}\right)=15\times \frac{2}{3}=10.$$

What does this mean? Starting at $15, it means that if a firm decreases its price—and hence increase its output—by a small amount, there would be an increase in the firm’s revenues.

When revenues are at their maximum, marginal revenue is zero. We can confirm this by calculating the elasticity of demand at $10. Consider a 10 percent increase in price, so the price increases to $11. At $10, sales equal 50 units. At $11, sales equal 45 units. In other words, sales decrease by 5 units, so the decrease in sales is 10 percent. It follows that

$$-\text{(elasticity of demand)}=-\frac{\text{percentage change in quantity}}{\text{percentage change in price}}=1.$$

Plugging this into our expression for marginal revenue, we confirm that

$$\text{marginal revenue}=\text{price}\times \left(1-\frac{\text{1}}{-\text{(elasticity of demand)}}\right)=10\left(1-\frac{1}{1}\right)=10\times 0=0.$$

At $10, a small change in price leads to no change in revenue. The benefit from selling extra output is exactly offset by the loss from charging a lower price.

We can thus divide the demand curve into two parts, as in Figure 7.13 "Marginal Revenue and the Elasticity of Demand". At low quantities and high prices, a firm can increase its revenues by moving down the demand curve—to lower prices and higher output. Marginal revenue is positive. In this region, −(elasticity of demand) is a relatively large number (specifically, it is between 1 and infinity) and we say that the demand curve is relatively elastic. Conversely, at high quantities and low prices, a decrease in price will decrease a firm’s revenues. Marginal revenue is negative. In this region, −(elasticity of demand) is between 0 and 1, and we say that the demand curve is inelastic. Table 7.3 represents this schematically.