Elasticity of Demand

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.

Definition

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

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

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. In this situation, the incentive is for the producer 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.

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. In this situation, the incentive is for the producer to reduce 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.

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. In this situation, 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

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.