What Is Arc Elasticity?

Discover the concept of arc elasticity, its importance in economics and mathematics, and learn how to calculate it using the midpoint formula with practical examples.

Arc elasticity is a method used to calculate the responsiveness or elasticity of one variable in relation to another between two distinct points. It is particularly useful in economic analysis for measuring the elasticity of demand or supply when there is a significant change in the price or other determining factors. This economic measurement allows for a more accurate reflection of elasticity over a range of values rather than at a single point.

Definition

In economic terms, arc elasticity measures the degree of responsiveness in the quantity demanded or supplied relative to changes in price. Unlike point elasticity, which measures elasticity at a specific point on the demand or supply curve, arc elasticity provides a broader perspective over a segment of the curve.

Mathematical Expression

The general formula for arc elasticity is:

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

Where:

• \(E\) is the arc elasticity.
• \(Q_1\) and \(Q_2\) are the initial and final quantities.
• \(P_1\) and \(P_2\) are the initial and final prices.

The Midpoint Formula

The midpoint formula, also known as the arc elasticity formula, is crucial in calculating arc elasticity. This method takes the average of the start and end points to minimize the distortion that can occur from large percentage changes.

Midpoint Formula

$$ E_{midpoint} = \frac{\frac{\Delta Q}{(Q_1 + Q_2)/2}}{\frac{\Delta P}{(P_1 + P_2)/2}} $$

Where:

• \(\Delta Q\) is the change in quantity (Q2 - Q1).
• \(\Delta P\) is the change in price (P2 - P1).

Example Calculation

Consider the following data:

• Initial quantity (\(Q_1\)): 100 units
• Final quantity (\(Q_2\)): 120 units
• Initial price (\(P_1\)): $10
• Final price (\(P_2\)): $8

Using the midpoint formula:

1. Calculate \(\Delta Q\) and \(\Delta P\):

   $$ \Delta Q = Q_2 - Q_1 = 120 - 100 = 20 $$
   $$ \Delta P = P_2 - P_1 = 8 - 10 = -2 $$

2. Determine the averages of quantities and prices:

   $$ \frac{(Q_1 + Q_2)}{2} = \frac{100 + 120}{2} = 110 $$
   $$ \frac{(P_1 + P_2)}{2} = \frac{10 + 8}{2} = 9 $$

3. Substitute these values into the midpoint formula:

   $$ E_{midpoint} = \frac{\frac{20}{110}}{\frac{-2}{9}} = \frac{0.1818}{-0.2222} = -0.818 $$

The arc elasticity in this example is approximately -0.818, indicating that the demand is relatively inelastic.

Historical Context

The concept of elasticity in economics, including arc elasticity, was first introduced by Alfred Marshall in his pioneering work, “Principles of Economics,” published in 1890. Marshall’s framework set the foundation for many modern economic theories on consumer behavior and market dynamics.

Applicability

Arc elasticity is widely applied in various fields of economics and business:

• Demand and Supply Analysis: Helps businesses and policymakers understand how quantity demanded or supplied reacts to price changes.
• Revenue Management: Assists companies in pricing strategies to maximize revenues based on consumer responsiveness.
• Public Policy: Ensures effective implementation of tax policies, subsidies, and price controls.

Related Terms

• Price Elasticity of Demand: Measures the responsiveness of quantity demanded to changes in price.
• Income Elasticity of Demand: Measures how the quantity demanded of a good responds to a change in consumer income.
• Cross Elasticity of Demand: Assesses the responsiveness of the demand for a good to a change in the price of another good.

FAQs

How is arc elasticity different from point elasticity?

Arc elasticity measures elasticity over a range of values, making it useful when there are significant changes. Point elasticity, on the other hand, measures elasticity at a specific point on the demand or supply curve and is more appropriate for small changes.

When should I use arc elasticity?

Use arc elasticity when analyzing substantial changes in variables over a broader interval, such as evaluating the impact of a 20% price drop on total sales volume.

Is arc elasticity always negative?

For demand, arc elasticity is generally negative because price and quantity demanded usually move in opposite directions. However, it can be positive for supply elasticity since price and quantity supplied move in the same direction.

Summary

Arc elasticity is a crucial concept in both economics and mathematics, providing insights into the responsiveness of one variable to changes in another over a specific range. Using the midpoint formula, it offers a reliable method for calculating elasticity, aiding in informed decision-making in various economic and business contexts.

References

1. Marshall, Alfred. “Principles of Economics.” Macmillan, 1890.
2. Samuelson, Paul A., and William D. Nordhaus. “Economics.” McGraw-Hill Education, 2010.
3. “Elasticity.” Investopedia. www.investopedia.com