Elasticity of Demand

Section 4.1 Elasticity of Demand

Learning Objectives.

Define and interpret elasticity of demand
Understand what the elasticity of demand tells us about an objects demand function
Understand what it means for demand to be elastic, in-elastic and unit elastic.
Learn how to use elasticity of demand to maximize revenue.

Note: This section is a custom section added to OpenStax Calculus. It was written by Kim Savinon, Katie Hall and Andrew Miller.

In previous sections, we've learned about the demand function. Sometimes, when the price of a object changes a little bit, we see a huge change in demand. But other times, we can have a large change in prince and still not see a big change in demand. The elasticity of demand helps us quantify these relative changes.

Subsection 4.1.1 Definition and Examples

We want to observe how sentitive the demand for a product is to changes in price. Elasticity of demand will help us quantify this sensitivity. To help get a feel for how we should define this quanitity, we will work through a motivating example.

Let's study how the changes in price effect changes in demand for furniture versus gas. As we increase the price of furniture, we expect to get less demand. This is because furniture is not a necessity. On the other hand, when we increase the price of gas, we expect demand to be about the same since gas is a necessity.

To analyze this effectively, we want to compare these different effects on demand. However, an increase of $\$1$ to the price of gas is very different than increase the price of furniture by $\$1$ since, for example, the price of a couch is (generally) much more than the price of a gallon of gas. For this reason, when we consider price increases, we want to consider the percent change in price. We can find the percent change in price by doing $\dfrac{\Delta P}{p}.$ We will compare this to the percent change in demand $\left(\dfrac{\Delta x}{x}\right)$ by taking the ratio or quotient.

We also notice that when we increase price, we generally expect demand to decrease. Since one quanitity is going up while the other is going down, the ratio of their percent changes would give a negative quantity. Since we prefer to deal with positive quantities, we will multiply by $-1\text{.}$ Putting these ideas together leads to the following quotient:

\begin{equation*} -\dfrac{\frac{\Delta x}{x}}{\frac{\Delta p}{p}} = -\dfrac{p \Delta x}{x \Delta p}. \end{equation*}

Using our knowledge of calculus, we can use the derivative $\frac{dx}{dp}$ in place of the ratio of changes. This leads us to the following definition.

Definition 4.2.

If the demand equation is given by $x=f(p)$ where $p$ is the price per item and $x$ is the number of items sold at that price, then elasticity of demand $E$ is defined as

\begin{equation} E(p)= \dfrac{-p}{x}\cdot\dfrac{dx}{dp} \tag{4.1.1} \end{equation}

An important thing to note here is that, previously, we often wrote the demand function as $p=g(x)$ where $x$ (items demanded) was the input and price was the output. Here, we want to make sure the function is written as $x=f(p)$ where $p$ is the input. This will mean we sometimes need to solve for $x\text{.}$

When analyzing elasticity of demand, we often was to know if something like a 10% change in price will lead to more than or less than a 10% change in demand. The following terms capture this idea.

Definition 4.3.

If the percent change in demand is less than the percent change in price, which leads to $E(p)\lt 1\text{,}$ we say that demand is inelastic.

If the percent change in demand is more than the percent change in price, which leads to $E(p)>1\text{,}$ we say that demand is elastic.

If the percent change in demand is eqaul to the percent change in price, which leads to $E(p)=1\text{,}$ we say that demand is unit elastic.

Products like milk and gas have inelastic demand since they are necessities. For these products, the percent change in price is more than the percent change in demand so $E(p)\lt 1\text{.}$ For products like furniture or vehicles, the demand is elastic. When price changes by a small percentage, demand changes by a bigger percentage and thus $E(p)>1.$ In many situations, the elasticity of demand may be greater then one from some $p$ values and less than one for others. Thus demand is not always either elastic or inelastic for a given product.

Example 4.4. Elasticity of Demand of Soda at Different Prices.

The demand equation for one 20 oz soda is given by $p=3-2x\text{,}$ where $p$ is the price of one soda and $x$ is the number of hundreds of sodas sold in one week. Find the elasticity of demand when $p=1.25,\;1.50, \text{ and }2.00\text{.}$ Then determine if demand is elastic, inelastic or unit elastic in each case.

Solution.

We know that

\begin{equation} E(p)= \dfrac{-p}{x}\dfrac{dx}{dp} .\tag{4.1.2} \end{equation}

To find $\frac{dx}{dp},$ we need to rewrite the demand equation so that it is of the form $x=f(p)\text{.}$ We get

\begin{align*} p \amp = 2-x \\ 2x+p \amp = 3 \\ 2x \amp = 3-p \\ x \amp = \frac{3}{2}-\frac{1}{2}p \end{align*}

Then

\begin{equation*} \frac{dx}{dp}=\frac{d}{dp}\left(\frac{3}{2}-\frac{1}{2}p\right)=-\frac{1}{2}. \end{equation*}

Plugging in to (4.1.2), we get that

\begin{equation*} E(p)= \dfrac{-p}{x}\dfrac{dx}{dp}=-\dfrac{p}{\frac{3}{2}-\frac{1}{2}p}\left(-\frac{1}{2}\right) =\frac{p}{3-p}. \end{equation*}

Now we can evaluate $E(p)$ at each of the given $p $ values.

\begin{align*} E(1.25) \amp =\frac{1.25}{3-1.25}=\frac{5}{7} \amp \amp E\lt 1, \text{ inelastic } \\ E(1.50) \amp =\frac{1.50}{3-1.50}=1 \amp \amp E=1, \text{ unit elastic }\\ E(2.00) \amp =\frac{2.00}{3-2.00}=2\amp \amp E>1, \text{ elastic } \end{align*}

When we defined elasticity of demand, we defined it as the (negative) ratio of percent change in demand over percent change in price. Since when price goes up, (usually) demand goes down, we can think of this as measuring

\begin{equation*} E(p)=\frac{\% \text{ decrease in demand}}{\% \text{ increase in price}} \text{.} \end{equation*}

So, for example, if $E(p)=1/2 \text{,}$ we can interpret this as saying that a 1% decrease in demand comes from a 2% increase in price. If $E(p)=2=\frac{2}{1} \text{,}$ we can interpret this as saying that a 2% decrease in demand comes from a 1% increase in price (or alternatively a 1% increase in price leads to a 2% decrease in demand.).

Example 4.5. Interpretting Elasticity of Demand.

In Example 4.4, we found the elasticity of demand for soda at three different prices. Interpret each of these values in terms of relative percent changes of price and demand.

Solution.

We found that $E(0.75)=0.6=\frac{0.6}{1}=\frac{3}{5} $ so we can either say that a 1% increase in price leads to a 0.6% decrease in demand or, similarly, that a 5% increase in price leads to a 3% decrease in demand.

We found that $E(1.50)=3=\frac{3}{1} $ so we can say that a 1% increase in price leads to a 3% decrease in demand.

We found that $E(1.00)=1=\frac{1}{1} $ so we can say that a 1% increase in price leads to a 1% decrease in demand.

There's a few things to note here. First, there are lots of equivalent fractions so, for example, since we can write $E(1.50)=3=\frac{6}{2} \text{,}$ we could have also said a 2% increase in price leads to a 6% decrease in demand. Second, we can also switch which quantities are increasing and decreasing. So, for example, for $E(0.75)=\frac{3}{5} \text{,}$ we get that a 5% decrease in price leads to a $3% $ increase in demand.

Checkpoint 4.6.

The demand equation for school lunches is $10-\frac{x}{10}\text{,}$ where $x $ is the number of lunches purchased and $p $ is the price in dollars. Find $E(p) \text{.}$

Hint.

Make sure you solve for $x $ first, then plug into (4.1.1).

Solution.

\begin{equation*} E(p)=\frac{p}{10-p} \end{equation*}

Subsection 4.1.2 Elasticity of Demand and Revenue

We've seen that when demand is elastic, ($E>1$), when we make a change to the price, the change to demand (as a percent) is larger then the percent change to price. For example, when $E=2 \text{,}$ a 10% increase in price leads to a 20% decrease in demand. If we consider the revenue, since price only goes up 10% but demand goes down 20%, we will get less revenue overall. Therefore, if $E>1\text{,}$ increasing price decreases revenue. On the other hand, when demand is inelastic, ($E \lt 1$), when we make a change to the price, the change to demand (as a percent) is smaller then the percent change to price. For example, when $E=1/2 \text{,}$ a 20% increase in price leads to a 10% decrease in demand. If we consider the revenue, since price goes up 20% but demand only goes down 10%, we will get more revenue overall. Therefore, if $E\lt 1\text{,}$ increasing price increases revenue. Right in the middle of these cases, when $E=1 \text{,}$ we are at an equilibrium point and revenue is maximized. This is summarized below.

Note 4.7. Using Elasticity of Demand to Tell us About Revenue.

When $E=1 $ revenue is maximized.
If $E\gt 1 \text{,}$ increasing price decreases revenue.
If $E\lt 1 \text{,}$ increasing price increases revenue.

Example 4.8. Maximizing the Revenue.

The consumer demand equation for chicken tenders at "The Rent" is given by $x=(36-2p)^2\text{,}$ where $p $ is the price of the tenders and $x$ is the demand in weekly sales. Find the price they should charge in order to maximize revenue.

Solution.

We know that we need $E=1$ to maximize revenue. We already have the demand equation in terms of "$x=$", so we can take the derivative and plug it into the elasticity of demand formula.

\begin{equation*} \frac{dx}{dp}=\frac{d}{dp}(36-2p)^2=2(36-2p)(-2)=-4(36-2p) \end{equation*}

Thus,

\begin{equation*} E =-\frac{dx}{dp}\frac{p}{x} = 4(36-2p)\frac{p}{(36-2p)^2}=\frac{4p}{36-2p}. \end{equation*}

To maximize revenue, we set $E=1\text{.}$ This gives

\begin{gather*} E=\frac{4p}{36-2p}=1 \\ 4p=36-2p \\ 6p=36 \\ p=6 \end{gather*}

Therefore, they should charge $\$6$ to maximize revenue.

Checkpoint 4.9.

A small grocery store makes their own candy bats. The elasticity of demand is given by $E(p)=\frac{p}{5-p}$ where $p$ is the price of each candy bar and $x$ is the number of candy bars they sell in one week. How much should this small grocery store set as the price for their candy bars in order to maximize revenue?

Hint.

We need $E=1 $ to maximize revenue.

Solution.

$p=\$2.50$

Example 4.10. Change in Demand of Football Tickets.

Assume the price of student season tickets for football is $\$40$ and $E(40)=1.75\text{.}$ If the price increases by $\$4\text{,}$ what is the approximate change to demand?

Solution.

We know that

\begin{equation*} E(p)=\frac{\% \text{ decrease in demand}}{\% \text{ increase in price}} \text{.} \end{equation*}

Let's find the $\% \text{increase in price }$ and then we can plug that and the elasticity into the above expression. Since the price goes up $\$4$ and was originally $\$40\text{,}$ the percent change in price is

\begin{equation*} \dfrac{4}{40}=\dfrac{1}{10}=0.10 \text{ or } 10\% \end{equation*}

Thus we get,

\begin{equation*} E(p)=1.75=\frac{\% \text{ decrease in demand}}{10\%} \text{.} \end{equation*}

Solving for $\% \text{ decrease in demand}$ gives us:

\begin{equation*} \% \text{ decrease in demand}=17.5\%. \end{equation*}

Thus demand will decrease by 17.5%.