# 24 The Price Elasticity of Demand

### Learning Objectives

1. Explain the concept of price elasticity of demand and its calculation.
2. Explain what it means for demand to be price inelastic, unit price elastic, price elastic, perfectly price inelastic, and perfectly price elastic.
3. Explain how and why the value of the price elasticity of demand changes along a linear demand curve.
4. Understand the relationship between total revenue and price elasticity of demand.
5. Discuss the determinants of price elasticity of demand.

We know from the law of demand how the quantity demanded will respond to a price change: it will change in the opposite direction. But how much will it change? It seems reasonable to expect, for example, that a 10% change in the price charged for a visit to the doctor would yield a different percentage change in quantity demanded than a 10% change in the price of a Ford Mustang. But how much is this difference?

To show how responsive quantity demanded is to a change in price, we apply the concept of elasticity. The price elasticity of demand for a good or service, eD, is the percentage change in quantity demanded of a particular good or service divided by the percentage change in the price of that good or service, all other things unchanged. Thus we can write

Equation 3.1

$e_D = \frac{\% \ change \ in \ quantity \ demanded}{\% \ change \ in \ price}$

Because the price elasticity of demand shows the responsiveness of quantity demanded to a price change, assuming that other factors that influence demand are unchanged, it reflects movements along a demand curve. With a downward-sloping demand curve, price and quantity demanded move in opposite directions, so the price elasticity of demand is always negative. A positive percentage change in price implies a negative percentage change in quantity demanded, and vice versa. Sometimes you will see the absolute value of the price elasticity measure reported. In essence, the minus sign is ignored because it is expected that there will be a negative (inverse) relationship between quantity demanded and price. In this text, however, we will retain the minus sign in reporting price elasticity of demand and will say “the absolute value of the price elasticity of demand” when that is what we are describing.

Be careful not to confuse elasticity with slope. The slope of a line is the change in the value of the variable on the vertical axis divided by the change in the value of the variable on the horizontal axis between two points. Elasticity is the ratio of the percentage changes. The slope of a demand curve, for example, is the ratio of the change in price to the change in quantity between two points on the curve. The price elasticity of demand is the ratio of the percentage change in quantity to the percentage change in price. As we will see, when computing elasticity at different points on a linear demand curve, the slope is constant—that is, it does not change—but the value for elasticity will change.

## Computing the Price Elasticity of Demand

Finding the price elasticity of demand requires that we first compute percentage changes in price and in quantity demanded. We calculate those changes between two points on a demand curve.

Figure 3.1 “Responsiveness and Demand” shows a particular demand curve, a linear demand curve for public transit rides. Suppose the initial price is $0.80, and the quantity demanded is 40,000 rides per day; we are at point A on the curve. Now suppose the price falls to$0.70, and we want to report the responsiveness of the quantity demanded. We see that at the new price, the quantity demanded rises to 60,000 rides per day (point B). To compute the elasticity, we need to compute the percentage changes in price and in quantity demanded between points A and B.

Figure 3.1 Responsiveness and Demand

The demand curve shows how changes in price lead to changes in the quantity demanded. A movement from point A to point B shows that a $0.10 reduction in price increases the number of rides per day by 20,000. A movement from B to A is a$0.10 increase in price, which reduces quantity demanded by 20,000 rides per day.

We measure the percentage change between two points as the change in the variable divided by the average value of the variable between the two points. Thus, the percentage change in quantity between points A and B in Figure 3.1 “Responsiveness and Demand” is computed relative to the average of the quantity values at points A and B: (60,000 + 40,000)/2 = 50,000. The percentage change in quantity, then, is 20,000/50,000, or 40%. Likewise, the percentage change in price between points A and B is based on the average of the two prices: ($0.80 +$0.70)/2 = $0.75, and so we have a percentage change of −0.10/0.75, or −13.33%. The price elasticity of demand between points A and B is thus 40%/(−13.33%) = −3.00. This measure of elasticity, which is based on percentage changes relative to the average value of each variable between two points, is called arc elasticity. The arc elasticity method has the advantage that it yields the same elasticity whether we go from point A to point B or from point B to point A. It is the method we shall use to compute elasticity. For the arc elasticity method, we calculate the price elasticity of demand using the average value of price, $$\bar{P}$$ , and the average value of quantity demanded, $$\bar{Q}$$. We shall use the Greek letter Δ to mean “change in,” so the change in quantity between two points is ΔQ and the change in price is ΔP. Now we can write the formula for the price elasticity of demand as Equation 3.2 $\displaystyle e_D = \frac{\Delta Q / \bar{Q}}{\Delta P / \bar{P}}$ The price elasticity of demand between points A and B is thus: $e_D = \frac{\frac{20,000}{(40,000+60,000)/2}}{\frac{- \ 0.10}{( \ 0.80 + \ 0.70)/2}} = \frac{40 \% }{-13.33 \% } = -3.00$ With the arc elasticity formula, the elasticity is the same whether we move from point A to point B or from point B to point A. If we start at point B and move to point A, we have: $e_D = \frac{\frac{-20,000}{(60,000+40,000)/2}}{\frac{ \ 0.10}{( \ 0.80 + \ 0.70)/2}} = \frac{-40 \% }{13.33 \% } = -3.00$ The arc elasticity method gives us an estimate of elasticity. It gives the value of elasticity at the midpoint over a range of change, such as the movement between points A and B. For a precise computation of elasticity, we would need to consider the response of a dependent variable to an extremely small change in an independent variable. The fact that arc elasticities are approximate suggests an important practical rule in calculating arc elasticities: we should consider only small changes in independent variables. We cannot apply the concept of arc elasticity to large changes. Another argument for considering only small changes in computing price elasticities of demand will become evident in the next section. We will investigate what happens to price elasticities as we move from one point to another along a linear demand curve. ### Heads Up! Notice that in the arc elasticity formula, the method for computing a percentage change differs from the standard method with which you may be familiar. That method measures the percentage change in a variable relative to its original value. For example, using the standard method, when we go from point A to point B, we would compute the percentage change in quantity as 20,000/40,000 = 50%. The percentage change in price would be −$0.10/$0.80 = −12.5%. The price elasticity of demand would then be 50%/(−12.5%) = −4.00. Going from point B to point A, however, would yield a different elasticity. The percentage change in quantity would be −20,000/60,000, or −33.33%. The percentage change in price would be$0.10/$0.70 = 14.29%. The price elasticity of demand would thus be −33.33%/14.29% = −2.33. By using the average quantity and average price to calculate percentage changes, the arc elasticity approach avoids the necessity to specify the direction of the change and, thereby, gives us the same answer whether we go from A to B or from B to A. ## Price Elasticities Along a Linear Demand Curve What happens to the price elasticity of demand when we travel along the demand curve? The answer depends on the nature of the demand curve itself. On a linear demand curve, such as the one in Figure 3.3 “Price Elasticities of Demand for a Linear Demand Curve”, elasticity becomes smaller (in absolute value) as we travel downward and to the right. Figure 3.3 Price Elasticities of Demand for a Linear Demand Curve The price elasticity of demand varies between different pairs of points along a linear demand curve. The lower the price and the greater the quantity demanded, the lower the absolute value of the price elasticity of demand. Figure 3.3 “Price Elasticities of Demand for a Linear Demand Curve” shows the same demand curve we saw in Figure 3.2 “Responsiveness and Demand”. We have already calculated the price elasticity of demand between points A and B; it equals −3.00. Notice, however, that when we use the same method to compute the price elasticity of demand between other sets of points, our answer varies. For each of the pairs of points shown, the changes in price and quantity demanded are the same (a$0.10 decrease in price and 20,000 additional rides per day, respectively). But at the high prices and low quantities on the upper part of the demand curve, the percentage change in quantity is relatively large, whereas the percentage change in price is relatively small. The absolute value of the price elasticity of demand is thus relatively large. As we move down the demand curve, equal changes in quantity represent smaller and smaller percentage changes, whereas equal changes in price represent larger and larger percentage changes, and the absolute value of the elasticity measure declines. Between points C and D, for example, the price elasticity of demand is −1.00, and between points E and F the price elasticity of demand is −0.33.

On a linear demand curve, the price elasticity of demand varies depending on the interval over which we are measuring it. For any linear demand curve, the absolute value of the price elasticity of demand will fall as we move down and to the right along the curve.

## The Price Elasticity of Demand and Changes in Total Revenue

Suppose the public transit authority is considering raising fares. Will its total revenues go up or down? Total revenue is the price per unit times the number of units sold1. In this case, it is the fare times the number of riders. The transit authority will certainly want to know whether a price increase will cause its total revenue to rise or fall. In fact, determining the impact of a price change on total revenue is crucial to the analysis of many problems in economics.

We will do two quick calculations before generalizing the principle involved. Given the demand curve shown in Figure 3.3 “Price Elasticities of Demand for a Linear Demand Curve”, we see that at a price of $0.80, the transit authority will sell 40,000 rides per day. Total revenue would be$32,000 per day ($0.80 times 40,000). If the price were lowered by$0.10 to $0.70, quantity demanded would increase to 60,000 rides and total revenue would increase to$42,000 ($0.70 times 60,000). The reduction in fare increases total revenue. However, if the initial price had been$0.30 and the transit authority reduced it by $0.10 to$0.20, total revenue would decrease from $42,000 ($0.30 times 140,000) to $32,000 ($0.20 times 160,000). So it appears that the impact of a price change on total revenue depends on the initial price and, by implication, the original elasticity. We generalize this point in the remainder of this section.

The problem in assessing the impact of a price change on total revenue of a good or service is that a change in price always changes the quantity demanded in the opposite direction. An increase in price reduces the quantity demanded, and a reduction in price increases the quantity demanded. The question is how much. Because total revenue is found by multiplying the price per unit times the quantity demanded, it is not clear whether a change in price will cause total revenue to rise or fall.

We have already made this point in the context of the transit authority. Consider the following three examples of price increases for gasoline, pizza, and diet cola.

Suppose that 1,000 gallons of gasoline per day are demanded at a price of $4.00 per gallon. Total revenue for gasoline thus equals$4,000 per day (=1,000 gallons per day times $4.00 per gallon). If an increase in the price of gasoline to$4.25 reduces the quantity demanded to 950 gallons per day, total revenue rises to $4,037.50 per day (=950 gallons per day times$4.25 per gallon). Even though people consume less gasoline at $4.25 than at$4.00, total revenue rises because the higher price more than makes up for the drop in consumption.

Next consider pizza. Suppose 1,000 pizzas per week are demanded at a price of $9 per pizza. Total revenue for pizza equals$9,000 per week (=1,000 pizzas per week times $9 per pizza). If an increase in the price of pizza to$10 per pizza reduces quantity demanded to 900 pizzas per week, total revenue will still be $9,000 per week (=900 pizzas per week times$10 per pizza). Again, when price goes up, consumers buy less, but this time there is no change in total revenue.

Now consider diet cola. Suppose 1,000 cans of diet cola per day are demanded at a price of $0.50 per can. Total revenue for diet cola equals$500 per day (=1,000 cans per day times $0.50 per can). If an increase in the price of diet cola to$0.55 per can reduces quantity demanded to 880 cans per month, total revenue for diet cola falls to $484 per day (=880 cans per day times$0.55 per can). As in the case of gasoline, people will buy less diet cola when the price rises from $0.50 to$0.55, but in this example total revenue drops.

In our first example, an increase in price increased total revenue. In the second, a price increase left total revenue unchanged. In the third example, the price rise reduced total revenue. Is there a way to predict how a price change will affect total revenue? There is; the effect depends on the price elasticity of demand.

## Elastic, Unit Elastic, and Inelastic Demand

To determine how a price change will affect total revenue, economists place price elasticities of demand in three categories, based on their absolute value. If the absolute value of the price elasticity of demand is greater than 1, demand is termed price elastic. If it is equal to 1, demand is unit price elastic. And if it is less than 1, demand is price inelastic.

## Relating Elasticity to Changes in Total Revenue

When the price of a good or service changes, the quantity demanded changes in the opposite direction. Total revenue will move in the direction of the variable that changes by the larger percentage. If the variables move by the same percentage, total revenue stays the same. If quantity demanded changes by a larger percentage than price (i.e., if demand is price elastic), total revenue will change in the direction of the quantity change. If price changes by a larger percentage than quantity demanded (i.e., if demand is price inelastic), total revenue will move in the direction of the price change. If price and quantity demanded change by the same percentage (i.e., if demand is unit price elastic), then total revenue does not change.

When demand is price inelastic, a given percentage change in price results in a smaller percentage change in quantity demanded. That implies that total revenue will move in the direction of the price change: a reduction in price will reduce total revenue, and an increase in price will increase it.

Consider the price elasticity of demand for gasoline. In the example above, 1,000 gallons of gasoline were purchased each day at a price of $4.00 per gallon; an increase in price to$4.25 per gallon reduced the quantity demanded to 950 gallons per day. We thus had an average quantity of 975 gallons per day and an average price of $4.125. We can thus calculate the arc price elasticity of demand for gasoline:  Percentage change in quantity demanded = -50/975 = -5.1% Percentage change in price=0.25/4.125=6.06% Price elasticity of demand = -5.1%/6.06% = -.084 The demand for gasoline is price inelastic, and total revenue moves in the direction of the price change. When price rises, total revenue rises. Recall that in our example above, total spending on gasoline (which equals total revenues to sellers) rose from$4,000 per day (=1,000 gallons per day times $4.00) to$4037.50 per day (=950 gallons per day times $4.25 per gallon). When demand is price inelastic, a given percentage change in price results in a smaller percentage change in quantity demanded. That implies that total revenue will move in the direction of the price change: an increase in price will increase total revenue, and a reduction in price will reduce it. Consider again the example of pizza that we examined above. At a price of$9 per pizza, 1,000 pizzas per week were demanded. Total revenue was $9,000 per week (=1,000 pizzas per week times$9 per pizza). When the price rose to $10, the quantity demanded fell to 900 pizzas per week. Total revenue remained$9,000 per week (=900 pizzas per week times $10 per pizza). Again, we have an average quantity of 950 pizzas per week and an average price of$9.50. Using the arc elasticity method, we can compute:

 Percentage change in quantity demanded = -100/950 = -10.5% Percentage change in price = $1.00/$9.50 = 10.5% Price elasticity of demand = -10.5%/10.5% = -1.0

Demand is unit price elastic, and total revenue remains unchanged. Quantity demanded falls by the same percentage by which price increases.

### Answers to Try It! Problems

1. The absolute value of price elasticity of demand tends to be greater when more time is allowed for consumers to respond. Over time, riders of the commuter rail system can organize car pools, move, or otherwise adjust to the fare increase.
2. Using the formula for price elasticity of demand and plugging in values for the estimate of price elasticity (−0.5) and the percentage change in price (5%) and then rearranging terms, we can solve for the percentage change in quantity demanded as: eD = %Δ in Q/%Δ in P; −0.5 = %Δ in Q/5%; (−0.5)(5%) = %Δ in Q = −2.5%. Ridership falls by 2.5% in the first few months.
3. Using the formula for price elasticity of demand and plugging in values for the estimate of price elasticity over a few years (−1.5) and the percentage change in price (5%), we can solve for the percentage change in quantity demanded as eD = %Δ in Q/%Δ in P; −1.5 = %Δ in Q/5%; (−1.5)(5%) = %Δ in Q = −7.5%. Ridership falls by 7.5% over a few years.
4. Total revenue rises immediately after the fare increase, since demand over the immediate period is price inelastic. Total revenue falls after a few years, since demand changes and becomes price elastic.

1Notice that since the number of units sold of a good is the same as the number of units bought, the definition for total revenue could also be used to define total spending. Which term we use depends on the question at hand. If we are trying to determine what happens to revenues of sellers, then we are asking about total revenue. If we are trying to determine how much consumers spend, then we are asking about total spending.

2Division by zero results in an undefined solution. Saying that the price elasticity of demand is infinite requires that we say the denominator “approaches” zero.