Elasticity of Demand

Definition

Definition

The price elasticity of demand measures how sensitive customer demand is to a small percentage change in the price of a good. Intuitively, elasticity is computed by the following ratio:

\[- \: \frac{\text{% Change in Quantity Demanded}}{\text{% Change in Price}}\]

Recall that the Law of Demand implies that a positive percent change in price will result in a negative percent change in demand. Consequently, the negative sign appears in the formula for elasticity only to make sure that the final result will be a positive value.

Mathematical Definition for Elasticity of Demand

If the price of the good is \(p\) and the corresponding quantity demanded is \(f(p)\), then the elasticity of demand at price \(p\), \(E(p)\), is defined by

\[\begin{align*} E(p) = -\frac{pf'(p)}{f(p)} %\label{elasticity} \end{align*}\]

Example 1

Elasticity of demand

A store has determined that the demand for used lamps is given by

\[ f(p) = 500 - 15p \]

where \(p\) is the price (in dollars) of a lamp. Find the price elasticity of demand, \(E(p)\).

Step 1:   Compute   \(f'(p)\).

\[\begin{align*} f'(p) &= \frac{d}{dp}(500 - 15p)\\ \\ &= -15 \end{align*}\]

Step 2:   Compute   \(E(p)\).

\[\begin{align*} E(p) &= -\frac{pf'(p)}{f(p)} \\ \\ &= -\frac{p(-15)}{500 - 15p}\\ \\ &= \frac{15p}{500 - 15p} \end{align*}\]

Inelastic, Elastic & Unitary Demand

What if elasticity is less than one?

Suppose \(E(100) = 1/2\). This means that when the price is $100, a 1% increase in price (i.e., the price increases to $101), will result in a 0.5% decrease in demand. Or similarly, a 1% decrease in price (i.e., the price decreases to $99), will result in a 0.5% increase in demand. Since the increase in price is greater than the loss of demand, producers have an incentive to increase their price.

Definition

If \(E(p) < 1\) (i.e., a percent change in price results in a smaller percent change in demand), then demand is said to be inelastic.

What if elasticity is greater than one?

Suppose \(E(100) = 3\). This means that when the price is $100, a 1% increase in price will result in a 3% decrease in demand. Or similarly, a 1% decrease in price will result in a 3% increase in demand. Since the decrease in price is less than the increase in demand, producers have an incentive to decrease their price.

Definition

If \(E(p) > 1\) (i.e., a percent change in price results in a larger percent change in demand), then demand is said to be elastic.

What if elasticity is equal to one?

Suppose \(E(100) = 1\). This means that when the price is $100, a 1% increase in price will result in a 1% decrease in demand. Or similarly, a 1% decrease in price will result in a 1% increase in demand. Since the increase in price matches the decrease in demand, there is no incentive for the producer to change their price.

Definition

If \(E(p) = 1\) (i.e., a percent change in price results in the same percent change in demand), then demand is said to be unitary.

Example 2

Elasticity of demand

A store has determined that the demand for used lamps is given by

\[ f(p) = 500 - 15p \]

where \(p\) is the price of a lamp. Compute \(E(10)\) and \(E(20)\) and interpret the results.

Step 1:   Recall the elasticity of demand.

In Example 1, we computed the elasticity of demand for \(f\) as

\[E(p) = \frac{15p}{500 - 15p}\]

Step 2:   Evaluate   \(E(10)\)   and interpret.

\[E(10) = \frac{150}{500 - 150} = \frac{150}{350} = \frac{3}{7}\]

Since \(E(10) < 1\), demand is inelastic when the price of a lamp is $10.

Step 3:   Evaluate   \(E(20)\)   and interpret.

\[E(20) = \frac{300}{500 - 300} = \frac{300}{200} = \frac{3}{2}\]

Since \(E(20) > 1\), demand is elastic when the price of a lamp is $20.