The Law of Demand establishes the inverse relationship between the price of a good and the quantity demanded. This is valuable, but lacks a certain depth: for example, we can anticipate that if the price of a good were to increase, the quantity demanded would drop - but by how much? In other words, along with just the direction of the relationship, it would be beneficial in many scenarios to understand the magnitude of the relationship. Will quantity demanded drop by a little? By a lot? How much is “a little” or “a lot”?
We will use the concept of an elasticity to measure the sensitivity of one variable to changes in another variable (or parameter). When it comes to demand, for example, we will frame our exploration by asking “how sensitive is quantity demanded to changes in price?” To construct the measure, we will measure changes in percentage change terms 1 : a change in price from $200 to $210 will instead be characterized as a 5% increase in the price 2 .
In general, an elasticity can give the sensitivity of any variable \(x\) in response to changes in another variable \(y\text{.}\) The formula will be given as
and, in this section, we will define several specific elasticities which depict relationships between specific pairs of variables. The most useful of these is the price elasticity of demand: it defines the sensitivity of quantity demanded to changes in price, and can be defined as
It is expressed as a ratio of percentage change in quantity demanded in response to a percentage change in price. For example, if a 10% increase in price brings about a 2% decrease in quantity demanded, the price elasticity of demand for the good would be \(\epsilon_D = \frac{\%\Delta Q_D}{\%\Delta P} = \frac{-2}{10} = -0.2\text{.}\) If the Law of Demand holds, then the inverse relationship between \(P\) and \(Q_D\) causes \(\epsilon_D\) to be negative.
Importantly, any elasticity can be expressed in an equivalent way using the definition of the percentage change:
This alternate expression can be useful if the variable changes are given in raw prices (or quantities). Rather than convert to percentage changes, they can be used directly here. 3 Price elasticity of demand can therefore be equivalently expressed as:
Consider a demand function \(Q_D = 150 - 10P\text{.}\) What is the price elasticity of demand if the price is 10?
Hint.
Here you will find \(Q_{D}\) by substituting the price into the demand function.
Answer.
The quantity demanded at \(P=10\) is \(Q_{D}=150 - 10(10)=50\text{.}\) The \(\frac{\Delta Q_D}{\Delta P}=-10\) from the demand function. (Each increase in P decreases \(Q_D\) by 10.)
So our \(\epsilon_{D}=\frac{\Delta Q_D}{\Delta P}\frac{P}{Q_D}=-10(\frac{10}{50})=-2\text{.}\)
How should we interpret \(\epsilon_D\) values? Economists generally distinguish between values greater than or less than -1 when describing the elasticity of demand. Demand is:
relatively elastic if \(\epsilon_D \lt -1\text{;}\) here, smaller % changes in price lead to larger % changes in \(Q_D\text{;}\)
relatively inelastic if \(\epsilon_D > -1\text{;}\) here, larger % changes in price lead to smaller % changes in \(Q_D\text{;}\)
unit elastic if \(\epsilon_D = -1\text{;}\) here, the % change in price and % change in \(Q_D\text{;}\) are directly proportional;
One way to interpret these categorizations is to consider a 1% price change. For a good with elastic demand, \(Q_D\) would change by more than 1%; for a good with inelastic demand, \(Q_D\) would change by less than 1%; for a good with unit elastic demand, \(Q_D\) would change by exactly 1%. When demand is relatively elastic, the good is more sensitive to price changes - this holds for goods that are luxury goods, or goods which have many substitutes. In such scenarios, consumers are more likely to, say, respond to higher prices by cutting back their quantity demand more significantly. When demand is relatively inelastic, however, the good is less sensitive to price changes. This holds true for necessities, as well as goods with few substitutes: consumers are less likely to change consumption patterns for these goods in response to price changes.
Checkpoint1.5.2.
You may have noticed that \(\epsilon_{D}\) is always negative. Why is that?
Answer.
Price elasticity of demand is always negative because it represents the relationship between quantity demanded and price. From the Law of Demand, we know that this is an inverse relation. If price rises, quantity demanded falls, and vice versa. So since these variables always move in opposite directions along a demand curve, the percent change in quantity is always negative for a positive percent change in price.
The table below highlights some empirically-calculated values for price elasticity of demand. Goods generally fall into categories as described above, with a couple of additions. First, short-run purchases tend to be more inelastic: when a consumer has to buy a plane ticket in 48 hours, she is less likely to be flexible in her demand than if she were purchasing a ticket for a flight six months into the future. Additionally, specific brands (such as Chevrolet cars) have many substitutes and tend to have more elastic demand, while broad categories (such as cars in general) have fewer substitutes and, therefore, more inelastic demand.
Table1.5.3.Price elasticity of demand for a selection of goods. 4
Good or service
\(\epsilon_D\)
Salt
-0.1
Airline travel, short-run
-0.1
Gasoline
-0.2
Coffee
-0.25
Tobacco products, short-run
-0.45
Movies
-0.9
Automobiles
-1.2
Restaurant meals
-2.3
Airline travel, long-run
-2.4
Fresh green peas
-2.8
Chevrolet automobiles
-4.0
Fresh tomatoes
-4.6
There are many reasons we might care about the price elasticity of demand in a given market. If the government is considering taxing carbon products like gasoline to curb carbon emissions, it might be concerned how sharply increasing the price of gasoline will decrease consumption. Similarly, a firm considering changing the price of its product might want to consider how sensitive its consumers are to those price changes. We will formalize this second example later this chapter.
There are two special cases for the shape of the demand curve worth considering. Demand could be perfectly inelastic. As the name implies, this is demand where the quantity demanded is so inelastic it is not responsive to changes in price at all. Consider an absolute necessity, such as a life-saving medication, where a consumer would buy the same number of units at any price. Perfectly inelastic demand creates a demand curve which is completely vertical, as you can see below. In this case, \(\epsilon_D = 0\text{.}\) Demand can also be perfectly elastic: so responsive to price, that any price increase makes \(Q_D = 0\text{,}\) while any price decrease makes \(Q_D = \infty\text{.}\) Perfectly elastic demand is horizontal, and has \(\epsilon_D = \infty\text{.}\)
Figure1.5.4.Left: Perfectly inelastic demand. As the price increases from \(P_1\) to \(P_2\text{,}\) quantity demanded stays the same at \(\bar{Q}\text{.}\) Right: Perfectly elastic demand.
We can use the concept of elasticity to also measure the sensitivity of quantity demand to changes in demand parameters. Two elasticities in particular are of interest:
Cross-price elasticity of demand measures the sensitivity of the quantity demanded of good \(x\) to changes in the price of a related good\(y\text{:}\)\(\epsilon_{xy} = \frac{\%\Delta Q_{D, x}}{\%\Delta P_y} = \frac{\Delta Q_{D, x}}{\Delta P_y}\frac{P_y}{Q_{D, x}}\text{;}\)
Income elasticity of demand measures the sensitivity of the quantity demanded of good \(x\) to changes in consumer income\(I\text{:}\)\(\epsilon_{I} = \frac{\%\Delta Q_{D}}{\%\Delta I} = \frac{\Delta Q_{D}}{\Delta I}\frac{I}{Q_{D}}\text{.}\)
Each of these elasticities can be positive or negative, depending on the demand parameter relationship. For example, \(\epsilon_I > 0\) for normal goods, while \(\epsilon_I \lt 0\) for inferior goods. Similarly, \(\epsilon_{xy} > 0\) for a related good which is a substitute, while \(\epsilon_{xy} \lt 0\) for a related good which is a complement. Measurements of the magnitude of these elasticities have different interpretations as well. For price elasticity of income, a larger value (in absolute value) indicates greater sensitivity of demand to changes in income: a good which responds to a 10% income increase with a 40% decrease in demand (\(\epsilon_I = -4\)) could be described as more inferior than a good which responds with a 6% decrease in demand (\(\epsilon_I = 0.6\text{.}\)) Magnitude of cross-price elasticity can describe the degree of complementarity or degree of substitutability between the two goods. For example, a value of \(\epsilon_{xy} = 3\) suggests the two goods are stronger substitutes (and demand for one good is more greatly impacted by changes in the the price of the related good) than two goods between which \(\epsilon_{xy} = \frac{1}{3}\text{.}\)
Checkpoint1.5.5.
1. If income elasticity \(\epsilon_I = 0.5\) for a good is it normal or inferior? How do we read \(\epsilon_I = 0.5\) in terms of actual changes?
2. If cross-price elasticity between bagels(\(x\)) and coffee (\(y\)) is \(\epsilon_{xy} = -0.75\text{,}\) are these goods complements, substitutes, or neither? How do we interpret \(\epsilon_{xy} = -0.75\text{?}\)
Answer.
1. Because income elasticity is positive, this means that income and quantity demanded change in the same direction, so this is a normal good.
We interpret the value to mean that for each 1% change in income, quantity demanded for this good will increase by 0.5%, ceteris paribus.
2. Because cross-price elasticity is negative, this means that the quantity demanded for bagels (\(x\)) moves in the opposite direction when the price for coffee (\(y\)) changes, so these goods are complements. Higher prices for coffee means lower quantity demanded of coffee (Law of Demand). Since people consume bagels and coffee together, Less coffee demanded means fewer bagels demanded.
We interpret the value to mean that for each 1% change in the price of coffee, quantity demanded for bagels will fall by 0.75%, ceteris paribus. This is an indicator of the strength of the complementarity.
In Depth1.5.6.Slope of demand versus \(\epsilon_D\).
Question: what is the relationship between the slope of a demand curve and the price elasticity of demand for that demand curve?
The relationship between demand slope and \(\epsilon_D\) is an often-confused one. In many applications, economists use slope to provide a short-hand reference to the relative elasticity (or inelasticity) of a demand curve: a “flat demand curve” suggests more relatively elastic demand, while a “steep demand curve” suggests more relatively inelastic demand. Despite this rule of thumb, price elasticity of demand and slope of demand are not equivalent. Here, we aim to show: (1) how slope relates to demand; (2) with a linear demand curve, price elasticity changes at different points along the curve; and (3) this is precisely by design!
Figure1.5.7.Suggested depiction of “relatively elastic” or “inelastic” demand curves.
Is this the right way to think about elasticity?
(1) By definition, price elasticity of demand can be expressed as
The term \(\frac{\Delta Q_D}{\Delta P}\) captures the change in quantity demand given a change in price. It looks like a slope in the rise-over-run format, except that price is on the vertical axis and quantity on the {horizontal} axis. You might call it a run-over-rise expression! Therefore, the term \(\frac{\Delta Q_D}{\Delta P}\) is the inverse of the slope of the demand curve.
This means slope of demand - the rate of change between price and quantity demanded in absolute terms - plays an important role in determining the price elasticity of demand. But, since elasticity measures the relationship between \(P\) and \(Q_D\) in percentage change terms, the rate of change needs to be scaled by the initial values (before a price change) of each.
\begin{equation*}
\epsilon_D = \underbrace{\frac{\Delta Q_D}{\Delta P}}_{\text{ inverse of slope of } D}\times\underbrace{\frac{P}{Q_D}}_{\text{ initial values } }
\end{equation*}
(2) The consequence of defining by percentage change is that the same price-\(Q_D\) change has different implications depending on initial values. If demand is defined by the function \(Q_D = 200 - 10P\text{,}\) then a decrease in price by one always increases quantity demanded by 10 units. Since the demand curve is linear, slope must be the same everywhere! However, a change in price from 19 to 18 is a 5.2% decrease in price, while a change in price from 3 to 2 is a 33.3% decrease in price. Both price changes decrease \(Q_D\) by 10, but as percentage changes, the impacts are very different.
As a result, calculating price elasticity of demand at different points on the curve give different values for \(\epsilon_D\) at each point. In the graph below, notice the range of values for \(\epsilon_D\) at prices of 16, 12, 8, and 4.
Figure1.5.8.Price elasticity of demand at points along the curve \(Q_D = 200 - 10P\text{.}\)
The left side of the demand curve, where prices are high and quantity demanded is low, is the elastic part of the demand curve. The right side of the demand curve, where prices are low and quantity demanded is high, is the inelastic part of the demand curve. This pattern holds for any linear demand curve. Moreover, the midpoint of the demand curve (here, at a price of 10 and quantity demanded of 100) corresponds to unit elastic demand.
(3) The variation in \(\epsilon_D\) should not be surprising. At a high price of 16, for example, price changes are less consequential (smaller in % terms) and quantity changes are more consequential (larger in % terms). This, by definition, leads to a higher price elasticity of demand! Likewise, at a low price of 4, price changes are larger in % terms and quantity changes are smaller in % terms: this implies demand must be relatively inelastic when prices are low.
(4) Linear demand curves may have constant slope, but they do not have constant price elasticity of demand. So saying that a linear demand “curve” is elastic or inelastic is misleading. It would be more accurate to say that the point of interest on the curve or the observed outcome falls on the elastic part (or inelastic part) of the demand curve. The graph below highlights what is really happening when we draw a flatter or steeper demand curve: we are merely drawing attention to the relevant and corresponding section of the demand curve.
In some applications, equating slope of demand with \(\epsilon_D\) seems like a reasonable rule of thumb. It seems innocent enough, but can lead to critical misunderstandings in many models of both decision makers and markets. We should proceed with care!
More accurately, with linear demand curves, a flat curve draws attention to the elastic part of the demand curve, while a steep curve draws attention to the inelastic part of the demand curve.
