# Point Elasticity of Demand

Point elasticity of demand is the ratio of percentage change in quantity demanded of a good to percentage change in its price calculated at a specific point on the demand curve.

Point elasticity of demand is actually not a new type of elasticity. It is just one of the two methods of calculation of elasticity, the other being arc elasticity of demand. All major measures of elasticity i.e. (price) elasticity of supply, income elasticity, cross elasticity of demand/supply have their point elasticity and arc elasticity versions even though point elasticity method is simpler and more popular method.

## Example

In order to understand the difference between point elasticity and arc elasticity, let’s consider the market for public transportation in Market XYZ. Let’s assume that if cost of a trip changes from $2 (P0) to$3 (P1), passenger demand per day falls from 0.5 million (Q0) to 0.4 million (Q1).

Elasticity of demand is defined as the percentage change in quantity demanded divided by percentage change in price:

$$\text{E} _ \text{d}=\frac{\Delta \text{Q%}}{\Delta \text{P%}}$$

The percentages are most commonly defined with reference to P0 and Q0 and this gives us the price elasticity of demand for public transportation of -0.4.

$$\text{E} _ \text{d}=\frac{\Delta \text{Q%}}{\Delta \text{P%}}=\frac{(\text{0.4}-\text{0.5})}{\text{0.5}}÷\frac{(\text{\3}-\text{\2})}{\text{\2}}\\=-\text{0.1}\times \frac{\text{\2}}{\text{0.5}}=-\text{0.4}$$

Now, imagine if the price movement is opposite i.e. if price decreases from $3 to$2 and passenger demand increases. In this case, the calculation of elasticity of demand would be as follows:

$$\text{E} _ \text{d}=\frac{\Delta \text{Q%}}{\Delta \text{P%}}=\frac{(\text{0.5}-\text{0.4})}{\text{0.4}}÷\frac{(\text{\2}-\text{\3})}{\text{\3}}\\=-\text{0.1}\times \frac{\text{\3}}{\text{0.4}}=-\text{0.75}$$

Just using a different starting point for the price movement has caused a roughly 100% increase in elasticity which doesn’t sound right. The formula for the price elasticity itself shows that the elasticity of demand at a point on a curve depends on the ratio of change in quantity demanded to change in price and on the ratio of initial price and quantity at the point on the curve on which we want to calculate elasticity. If the difference between P0 and P1 or Q0 and Q1 is high, our estimate for price elasticity will not be accurate.

### vs Arc Elasticity

One way to address the sensitivity of point elasticity to starting price and quantity is to calculate the arc elasticity. The arc elasticity of demand is calculated by finding percentage based on average of the starting and closing prices and quantities. The arc price elasticity of demand for the public transport in Market XYZ would be -0.55:

$$\text{E} _ \text{d}=\frac{\text{Q} _ \text{1}-\text{Q} _ \text{0}}{\frac{(\text{Q} _ \text{1}+\text{Q} _ \text{0})}{\text{2}}}\div\frac{\text{P} _ \text{1}-\text{P} _ \text{0}}{\frac{(\text{P} _ \text{1}+\text{P} _ \text{0})}{\text{2}}}\\=\frac{\text{0.4}-\text{0.5}}{\frac{(\text{0.4}+\text{0.5})}{\text{2}}}\div\frac{\text{\3}-\text{\2}}{\frac{(\text{\3}+\text{\2})}{\text{2}}}=\frac{-\text{0.1}}{\text{0.45}}\div\frac{\text{\1}}{\text{\2.5}}=-\text{0.55}$$

The arc elasticity method of elasticity calculation is also called mid-point method.

Where the change in price or quantity demanded is large, arc elasticity method is an improvement on the point method of calculation. However, where the change is small, point elasticity of demand is preferred.

Point elasticity of demand can also be calculated for any point on the demand curve using a bit of calculus as follows:

$$\text{E} _ \text{d}=\frac{\text{dQ}}{\text{dP}}\times\frac{\text{P}}{\text{Q}}$$

Where dQ/dP is the first derivative of the demand curve/function. It measures the change in quantity demanded for very small change in price at price P. Since dQ/dP can be calculated at an exact point on a curve, the above equation gives a better estimate of elasticity.