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## Section4.6Elasticity of Demand

We are very thankful to Dr. Petra Menz and Nicola Mulberry of Simon Fraser University for allowing us to adapt their section on the Elasticity of Demand from Calculus Early Transcendentals for this section.

###### Motivating Questions
• What is the Elasticity of Demand?

• Given the Elasticity of Demand, when should the price be raised or lowered?

• How can we use the Elasticity of Demand to maximize total revenue?

### Subsection

The Elasticity of Demand measures the extent to which a change in price for a commodity will affect people's willingness to buy it. If the purchase is a luxury item, something you want but do not necessarily need (for example a new car), you may not purchase until theres a sale. On the other hand, if you consider a purchase a need (say, gas or food), you will still buy the product if the price goes up a reasonable amount. This is why its important for companies to know where to set the price of their product, since the ultimate goal is to maximize revenue. Large values of the elasticity correspond to a situation in which people will respond strongly to a change in price. Low values of elasticity correspond to situations in which a change in price has little effect on buying decisions, such as necessities.

To begin, let's look at a real example from the air travel industry and look at how the cost of air plane tickets impact the revenue of tickets sold. A simplistic view may lead one to believe that a decrease in the cost for an airplane ticket would cause the revenue to increase and vice versa. In economics, this particular relationship between unit price and revenue is referred to as elastic demand as we will learn later. Particularly in Canada, start-up airlines can collapse more readily under this condition. The Department of Finance in Canada studied the aforementioned relationship and published the research results in http://www.fin.gc.ca/consultresp/airtravel/airtravstdy_1-eng.asp by differentiating between six types of air travel that are associated pairwise: business and leisure, long-haul and short-haul, and international long-haul and North American long-haul air travel. The results of the study corroborate that the demand for business air travel is less elastic than that for leisure air travel. This finding does not come as a surprise, since even a costly booked vacation can be more readily moved to different dates than business travels. The other two results of the study are that the demand for long-haul flights is less elastic than that for short-haul flights, and similarly, the demand for international flights is less elastic than that for North American flights. This make sense because the further the destination, the less likely it is that an alternative mode of transport can be found as a substitute for an expensive flight.

We now derive the mathematical model that helps us to analyze the relationship between unit price and revenue, and determines the elasticity of demand of a particular economic situation when the demand function is given.

### SubsectionDerivation of the Elasticity

Here we will derive the equation for the elasticity of demand. Let us start by considering the function $q=D(p)\text{,}$ the quantity of an item in demand at a price $p\text{.}$ Consider the effect of a small increase of $h$ dollars in the unit price $p$ for some product to a unit price of $p+h$ dollars. Then the associated quantity demanded changes from $D(p)$ units to $D(p+h)$ units with an overall decrease of $D(p+h)-D(p)$ units. We can now calculate the percentage change in the unit price to be

\begin{equation*} \frac{\text{Change in unit price}}{\text{Price } p} \times 100 = \dfrac{h}{p}(100), \end{equation*}

and the corresponding percentage change in the quantity demanded to be

\begin{equation*} \frac{\text{Change in quantity demanded}}{\text{Quantity demanded at price } p} \times 100 = \dfrac{D(p+h)-D(p)}{D(p)}(100). \end{equation*}

By calculating the ratio of the percentage change in the quantity demanded to the percentage change in price, we can determine the effect the latter has on the former:

\begin{equation*} \frac{\text{Percentage change in the quantity demanded}}{\text{Percentage change in the unit price}} = \frac{100\dfrac{D(p+h)-D(p)}{D(p)}}{100\frac{h}{p}} = \frac{\dfrac{D(p+h)-D(p)}{h}}{\frac{D(p)}{p}} \end{equation*}

We now recognize the difference quotient in this fraction. So, if $D(p)$ is differentiable at $p\text{,}$ we can deduce for small $h$ that

\begin{equation*} \frac{D(p+h)-D(p)}{h} \approx D'(p). \end{equation*}

In other words,

\begin{equation*} \frac{\text{Percentage change in the quantity demanded}}{\text{Percentage change in the unit price}} \approx \frac{D'(p)}{\frac{D(p)}{p}} = \frac{pD'(p)}{D(p)}, \end{equation*}

when $h$ is small.

Note, since $q=D(p)$ is a decreasing function on a certain interval $I\text{,}$ its derivative $D'(p)$ is negative for all $p\text{.}$ But this means that the value of the quantity $pD'(p)/D(p)$ is negative. Since it is preferred to work with positive values, economists define the elasticity of demand $E(p)$ to be the negative of the quantity $pD'(p)/D(p)\text{.}$

###### Elasticity of Demand

Suppose that the demand function $q=D(p)$ is differentiable. Then the elasticity of demand, $E\text{,}$ at price $p$ is defined by

\begin{equation*} E(p) = -\frac{pD'(p)}{D(p)} \end{equation*}

###### Example4.61

The quantity in demand for a certain product as a function of the price, in dollars, is given by

\begin{equation*} q=D(P)=-50p+20000 \end{equation*}
1. Determine the elasticity of demand $E(p).$

2. Calculate $E(100)\text{.}$ What can you determine from your result?

3. Calculate $E(300)\text{.}$ What can you determine from your result?

Solution
1. The elasticity of demand is given by the equation

\begin{equation*} E(p) = -\frac{pD'(p)}{D(p)} \end{equation*}

The derivative is $D'(p)=-50\text{,}$ and thus

\begin{equation*} E(p)= \frac{-p(-50)}{-50p+20000} = \frac{50p}{-50p+20000}= \frac{p}{400-p}. \end{equation*}
2. \begin{equation*} E(100) = \frac{100}{400-100} = \frac{1}{3}. \end{equation*}

Therefore, when the unit price $p$ is $\100$ per unit, a small increase in $p$ will lead to a decrease of approximately $0.33\%$ in the quantity demanded $q\text{.}$

3. \begin{equation*} E(300) = \frac{300}{400-300} = 3. \end{equation*}

Here, we see that a small increase in $p$ from $\300$ per unit will lead to a decrease of approximately $3\%$ in the quantity demanded $q\text{.}$

The following economic terminology is useful when describing demand in terms of elasticity.

###### Elastic, Unitary, and Inelastic Demand

The demand is elastic if $E(p) \gt 1\text{.}$ That is to say, the demand is elastic if the percentage change in demand is greater than the percentage change in price.

The demand is unitary if $E(p) = 1\text{.}$ That is to say, the demand is unitary if the percentage change in demand and price are relatively equal.

The demand is inelastic if $E(p) \lt 1\text{.}$ That is to say, the demand is inelastic if the percentage change in demand is less than the percentage change in price.

In our first example, we determined that the demand for the given product is elastic when $p=300$ and inelastic when $p=100\text{.}$ These calculations illustrate that (1) a small percentage change in the unit price will result in a greater percentage change in the quantity demanded (i.e. when the demand is elastic), (2) a small percentage change in the unit price will cause a smaller percentage change in the quantity demanded (i.e. when the demand is inelastic), and (3) a small percentage change in the unit price will result in the same percentage change in the quantity demanded (i.e. when the demand is unitary).

### SubsectionElasticity and Revenue

We have developed the notion of elasticity of demand by analyzing the relationship between quantity demanded and unit price in terms of percentage change. Of course this change influences revenue, so we now have a closer look at the effects of elasticity on revenue. Again we assume that $q=D(p)$ relates the quantity $q$ demanded of a certain product to its unit price $p$ in dollars. When $q$ units of the product are sold at the price $p\text{,}$ then the revenue is given by

\begin{equation*} R(p)=pq=pD(p). \end{equation*}

We now calculate the marginal revenue with respect to $p$ and obtain

\begin{equation*} R'(p)=D(p)+pD'(p)\\ =D(p)\left[1+\frac{pD'(p)}{D(p)}\right] \\ =D(p)\left[1-E(p)\right]. \end{equation*}

This last equation tells us that elasticity influences revenue. In order to determine what the effects are, we analyze the sign of the marginal revenue. We first note that $D(p)$ is positive for all values of $p$ and consider three cases:

$1)$ Suppose the demand is elastic when the unit price is set at $p$ dollars. Then

\begin{equation*} E(p) \gt 1 \implies 1-E(p) \lt 0, \end{equation*}

and therefore

\begin{equation*} R'(p) = D(p)\left[1-E(p)\right] \lt 0, \end{equation*}

which means that revenue $R$ is decreasing at $p\text{.}$ In other words, a small increase/decrease in the unit price results in a decrease/increase respectively in the revenue.

$2)$ Suppose the demand is inelastic when the unit price is set at $p$ dollars.

\begin{equation*} E(p)\lt 1 \implies 1-E(p)\gt0, \end{equation*}

and therefore

\begin{equation*} R'(p) = D(p)\left[1-E(p)\right] \gt 0, \end{equation*}

which necessitates that revenue $R$ is increasing at $p\text{.}$ This implies that a small increase/decrease in the unit price results in an increase/decrease respectively in the revenue.

$3)$Lastly, suppose the demand is unitary when the unit price is set at $p$ dollars. Then

\begin{equation*} E(p)=1 \implies 1-E(p)=0, \end{equation*}

and therefore

\begin{equation*} R'(p) = D(p)\left[1-E(p)\right]=0, \end{equation*}

which causes revenue $R$ to be stationary at $p\text{,}$ i.e. neither increasing nor decreasing. This means that a small increase/decrease in the unit price does not affect a change in the revenue. More importantly, the revenue is maximized here.

The results of this analysis are summarized below:

###### Effects of Unit Price Changes to Revenue

If the demand is elastic at $p\text{,}$ i.e. $E(p)\gt1\text{,}$ then a small increase/decrease in the unit price results in a decrease/increase respectively in the revenue.

If the demand is unitary at $p\text{,}$ i.e. $E(p)=1\text{,}$ then a small increase/decrease in the unit price does not affect a change in the revenue. What's more, the total revenue is maximized.

If the demand is inelastic at $p\text{,}$ i.e. $E(p)\lt1\text{,}$ then a small increase/decrease in the unit price results in an increase/decrease respectively in the revenue.

Note, the goal then is to find the price $p$ for which the demand is unitary $E(p)=1$ in order to maximize the total revenue from.

###### Example4.62

The daily demand for game rentals from Suluclac Games is a function of the rental price $p$ (in dollars), given by

\begin{equation*} D(p)=200-40p. \end{equation*}
1. Compute the elasticity of demand $E(p)\text{.}$

2. If the price of game rentals is $\1.50\text{,}$ should the price be raised or lowered to maximize total revenue?

3. At what price is the revenue maximized?

Solution
1. The elasticity of demand is given by the equation

\begin{equation*} E(p) = -\frac{pD'(p)}{D(p)} \end{equation*}

First find the derivative: $D'(p)=-40\text{.}$ Then

\begin{equation*} E(p)= \frac{-p(-40)}{200-40p} = \frac{40p}{200-40p}= \frac{p}{5-p}. \end{equation*}
2. \begin{equation*} E(1.5) = \frac{1.5}{5-1.5} = \frac{1.5}{3.5}. \end{equation*}

Since $E(1.5)\lt1\text{,}$ the price is inelastic. Therefore, when the unit price $p$ is $\1.50$ per unit, a small increase in $p$ will lead to a increase in total revenue.

3. To find the price at which the revenue is maximized, set $E(p)=1$ and solve for $p\text{:}$

\begin{equation*} E(p) = \frac{p}{5-p}=1 \implies p=5-p\\ 2p=5\\ p=5/2 \end{equation*}

Thus the price at which revenue is maximized is $\2.50\text{.}$

###### Example4.63

Carla's Cookies learns from experience that demand for boxes of cookies is well modeled by

\begin{equation*} q=D(p)=15\sqrt{5-p}, \end{equation*}

where $p$ is the price of a box of cookies in dollars.

1. Determine the elasticity of demand $E(p)\text{.}$

2. If the price of a box of cookies is $\4.00\text{,}$ should the price be raised or lowered to maximize total revenue?

3. At what price is the revenue maximized?

Solution
1. The elasticity of demand is given by the equation

\begin{equation*} E(p) = -\frac{pD'(p)}{D(p)}. \end{equation*}

First find the derivative using the chain rule: $D'(p)=\frac{15}{2}(5-p)^{-1/2}(-1)\text{.}$ Therefore,

\begin{equation*} E(p)= \frac{-p\frac{15}{2}(5-p)^{-1/2}(-1)}{15(5-p)^{1/2}} = \frac{0.5p}{(5-p)^{1/2}(5-p)^{1/2}}= \frac{0.5p}{5-p}. \end{equation*}
2. \begin{equation*} E(4) = \frac{0.5(4)}{5-4} = 2. \end{equation*}

Since $E(4)\gt 1\text{,}$ the price is elastic. Therefore, when the unit price $p$ is $\4$ per unit, a small increase in $p$ will lead to a decrease in total revenue.

3. To find the price at which the revenue is maximized, set $E(p)=1$ and solve for $p\text{:}$

\begin{equation*} E(p) = \frac{0.5p}{5-p}=1 \implies 0.5p=5-p\\ 1.5p=5\\ p=5/1.5=3.33 \end{equation*}

Thus the price at which revenue is maximized is $\3.33\text{.}$

###### Example4.64

Sunshine Nursery determines that the daily demand for tomato plants in early summer is the following function of their price, $p$ (measured in dollars):

\begin{equation*} D(p) = 50e^{-0.1p} \end{equation*}
1. Determine the elasticity of demand $E(p)\text{.}$

2. If the price is currently $\3$ per plant and we raise the price slightly, do we expect our revenue to increase or decrease?

3. At what price is the revenue maximized?

Solution
1. The elasticity of demand is given by the equation

\begin{equation*} E(p) = -\frac{pD'(p)}{D(p)} \end{equation*}

First find the derivative: $D'(p)=50e^{-0.1p}(-0.1)\text{.}$ Therefore,

\begin{equation*} E(p)= \frac{-p50e^{-0.1p}(-0.1)}{50e^{-0.1p}} = 0.1p. \end{equation*}
2. \begin{equation*} E(3) = 0.1(3)=0.3. \end{equation*}

Since $E(3)\lt1\text{,}$ the price is inelastic. Therefore, when the unit price $p$ is $\3$ per unit, a small increase in $p$ will lead to a increase in total revenue.

3. To find the price at which the revenue is maximized, set $E(p)=1$ and solve for $p\text{.}$

\begin{equation*} E(p) = 0.1p=1 \implies p=10 \end{equation*}

Thus the price at which revenue is maximized is $\10\text{.}$

### SubsectionSummary

• The Elasticity of Demand measures the extent to which a change in price for a commodity will affect people's willingness to buy it. Given the demand function $q = D(p) \text{,}$ and given that this function is differentiable, then the elasticity of demand at price $p$ is given by $E(p) = -\frac{pD'(p)}{D(p)}$ .

• If $E(p) \gt 1 \text{,}$ then the demand is called elastic. In this case, a small increase in the unit price results in a decrease in the revenue. If $E(p) = 1 \text{,}$ then the demand is called unitary. In this case, a small increase in the unit price does not change the revenue, and the total revenue is maximized. Finally, if $E(p) \lt 1 \text{,}$ then the demand is called inelastic. In this case, a small increase in the unit price results in an increase in revenue.