Understanding Arc Elasticity in Economics

Point Elasticity vs. Arc Elasticity

There are two primary ways to calculate elasticity depending on whether you are looking at a single coordinate or a segment of a curve.

Point Elasticity

Point elasticity provides the exact measurement of elasticity at a specific single point $(X, P)$ on a demand curve. It is calculated using the following formula:

$\text{Elasticity} = \frac{\Delta X}{\Delta P} \times \frac{P}{X}$

Where:

• $\Delta X$: The change in quantity demanded.
• $\Delta P$: The change in price.
• $P/X$: The ratio of the current price to the current quantity.

This formula is closely related to the slope of the demand curve. Since $\text{slope} = \frac{\Delta P}{\Delta X}$, the term $\frac{\Delta X}{\Delta P}$ is simply the reciprocal of the slope ($1/\text{slope}$). Therefore, you can also express point elasticity as:

$\text{Elasticity} = \frac{1}{\text{slope}} \times \frac{P}{X}$

Arc Elasticity (The Midpoint Method)

In practice, economists often deal with changes between two distinct points rather than an infinitesimal point. To avoid the mathematical inconsistency that occurs when calculating percentage changes from different starting points, we use arc elasticity.

The arc elasticity method uses the percent change measured as a percent of the average value of the two points. This ensures that the elasticity remains the same whether you are moving from Point A to Point B or from Point B to Point A.

The formula for arc elasticity ($\eta$) is:

$\eta = \frac{\frac{Q_2 - Q_1}{(Q_1 + Q_2)/2}}{\frac{P_2 - P_1}{(P_1 + P_2)/2}}$

By simplifying the complex fraction, we arrive at a more practical version for computation:

$\eta = \frac{Q_2 - Q_1}{Q_1 + Q_2} \times \frac{P_1 + P_2}{P_2 - P_1}$

Worked Example: Calculating Arc Elasticity

Let's apply the arc elasticity formula to a real-world scenario.

Scenario: Suppose a company observes the following demand data:

• At a price of $20, the quantity demanded is 60 units.
• At a price of $18, the quantity demanded is 64 units.

Step 1: Identify the variables

• $P_1 = 20, Q_1 = 60$
• $P_2 = 18, Q_2 = 64$

Step 2: Apply the formula $\eta = \frac{64 - 60}{64 + 60} \times \frac{20 + 18}{18 - 20}$

$\eta = \frac{4}{124} \times \frac{38}{-2}$

$\eta = \frac{1}{31} \times (-19)$

$\eta = -\frac{19}{31} \approx -0.6129$

Step 3: Interpret the result The absolute value is approximately 0.61. Since $0.61 < 1$, the demand in this price range is inelastic.

Key Conceptual Nuances

Variation Along a Linear Demand Curve

A common misconception is that a linear demand curve has a constant elasticity. While a linear demand curve has a constant slope, the elasticity is not constant.

As you move down a linear demand curve, the ratio of $P/X$ changes at every point. Consequently, the elasticity will vary: it is typically elastic at high prices/low quantities and becomes inelastic at low prices/high quantities.

Total Expenditure and Elasticity

Understanding elasticity is crucial for businesses deciding whether to raise or lower prices. This is because price changes affect Total Expenditure (Price $\times$ Quantity):

• If demand is inelastic ($|E| < 1$): A price increase will lead to an increase in total expenditure because the percentage drop in quantity is smaller than the percentage increase in price.
• If demand is elastic ($|E| > 1$): A price increase will lead to a decrease in total expenditure because the percentage drop in quantity is larger than the percentage increase in price.
• Unit Elastic ($|E| = 1$): Total expenditure is maximized at this point.