## Calculating Elasticity and Percentage Changes

### Calculating Percentage Changes and Growth Rates

Before we dive deeper into solving for elasticity, let's first make sure we are comfortable calculating percentage changes, also known as a growth rates. The formula for computing a growth rate is straightforward:

$\displaystyle \text{Percentage change}=\frac{\text{Change in quantity}}{\text{Quantity}}$

Suppose that a job pays $10 per hour. At some point, the individual doing the job is given a$2-per-hour raise. The percentage change (or growth rate) in pay is

$\displaystyle \frac{\2}{\10}=0.20\text{ or }20\%​$

Now to solve for elasticity, we use the growth rate, or percentage change, of the quantity demanded as well as the percentage change in price in order to to examine how these two variables are related. The price elasticity of demand is the ratio between the percentage change in the quantity demanded (Qd) and the corresponding percent change in price:

$\displaystyle \text{Price elasticity of demand}=\frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in price}}$

There are two general methods for calculating elasticities: the point elasticity approach and the midpoint (or arc) elasticity approach. Elasticity looks at the percentage change in quantity demanded divided by the percentage change in price, but which quantity and which price should be the denominator in the percentage calculation? The point approach uses the initial price and initial quantity to measure percent change. This makes the math easier, but the more accurate approach is the midpoint approach, which uses the average price and average quantity over the price and quantity change. (These are the price and quantity halfway between the initial point and the final point.) Let's compare the two approaches. Suppose the quantity demanded of a product was 100 at one point on the demand curve, and then it moved to 103 at another point. The growth rate, or percentage change in quantity demanded, would be the change in quantity demanded $\displaystyle {(103-100)}$ divided by the average of the two quantities demanded:

$\displaystyle \frac{(103+100)}{2}​$

In other words, the growth rate:

$\displaystyle \begin{array}{r}{\frac{103-100}{(103+100)/2}}\\{=\frac{3}{101.5}}\\{=0.0296}\\{=2.96\%\text{ growth}}\end{array}$

Note that if we used the point approach, the calculation would be:

$\frac{(103-100)}{100} = \text{3% growth}$

This produces nearly the same result as the slightly more complicated midpoint method (3% vs. 2.96%). If you need a rough approximation, use the point method. If you need accuracy, use the midpoint method. Note: as the two points become closer together, the point elasticity becomes a closer approximation to the arc elasticity.

In this module you will often be asked to calculate the percentage change in the quantity. Keep in mind that this is same as the the growth rate of the quantity. As you work through the course and find other applications for calculate growth rates, you will be well prepared.