Understanding Price Elasticity of Demand Using the Midpoint Formula

For example, if a price rises from $$10$to$$20$, the percentage increase is $100%$. However, if the price falls from $$20$ back to $$10$, the percentage decrease is only $50%$. This means that calculating elasticity from point A to point B would yield a different result than calculating it from point B to point A. To ensure that the elasticity remains the same regardless of the direction of the price change, economists use the Midpoint Formula.

The Midpoint Formula Explained

The advantage of the midpoint method is that it uses the same base for both increases and decreases, providing a consistent value between any two points. Instead of dividing by the initial value, we divide by the average value of the price and the quantity.

To calculate the percentage change for either price ($P$) or quantity ($Q$), use the following mathematics:

$\% \text{ change in } X = \frac{\text{Final Value} - \text{Starting Value}}{\left( \frac{\text{Final Value} + \text{Starting Value}}{2} \right)} \times 100$

When applying this to the full elasticity formula, we use the average price and the average quantity as our denominators:

$\text{Price Elasticity of Demand} = \frac{\frac{Q_2 - Q_1}{(Q_2 + Q_1)/2}}{\frac{P_2 - P_1}{(P_2 + P_1)/2}}$

Worked Example: Mathematics of Demand Elasticity

Let's apply this to a real-world scenario. Suppose at an initial price of $$10$, the quantity demanded is $100$ units. When the price rises to $$20$, the quantity demanded falls to $90$ units.

Step 1: Calculate the percentage change in quantity demanded $\text{Change in } Q = 90 - 100 = -10$ $\text{Average } Q = \frac{90 + 100}{2} = 95$ $\% \Delta Q = \frac{-10}{95} \times 100 \approx -10.5\%$

Step 2: Calculate the percentage change in price $\text{Change in } P = 20 - 10 = 10$ $\text{Average } P = \frac{20 + 10}{2} = 15$ $\% \Delta P = \frac{10}{15} \times 100 \approx 66.6\%$

Step 3: Calculate Elasticity $\text{Elasticity} = \frac{-10.5\%}{66.6\%} \approx -0.15$

Since $|-0.15| < 1$, this demand is inelastic.

Elasticity and Total Revenue

Understanding elasticity is not just a mathematical exercise; it is a crucial tool for business strategy. Total Revenue ($TR$) is calculated as Price multiplied by Quantity ($P \times Q$). The relationship between price changes and total revenue depends entirely on the elasticity of the good:

• Inelastic Demand: The "price effect" dominates the "quantity effect." If you raise the price, the drop in quantity is so small that total revenue actually increases. (Example: Salt)
• Elastic Demand: The "quantity effect" dominates. If you raise the price, the drop in quantity is so significant that total revenue falls. (Example: Airfare)
• Unit Elastic Demand: The price and quantity effects perfectly balance each other out, leaving total revenue unchanged.

Summary Table for Decision Making

By mastering the midpoint formula, you can accurately determine whether a price change will help or hurt a company's bottom line.