Arc Elasticity

What is Arc Elasticity?

Arc elasticity refers to the measure of elasticity between two specific points in relation to two variables. It compares the percentage changes in each variable between two specific points, making it useful when there is no explicit mathematical function defining their relationship.

Unlike point elasticity, which assesses elasticity at a specific point on the curve, arc elasticity calculates the elasticity between distinct points.

This concept is applied in both economics and mathematics. This concept is frequently employed in economic applications to assess the fluctuations between the quantity of goods demanded and their corresponding prices.

The formula for the y arc elasticity of x is expressed as:

Arc E_x,y = (percentage change in x) / (percentage change in y)

Where:

To accurately determine the percentage changes between two points, the midpoint formula is utilized:

% change in x = (x₂−x₁)/{(x₂+x₁)/2} 

% change in y = (y₂−y₁)/{(y₂+y₁)/2} 

Here, x represents the quantity of a good demanded or supplied, while y pertains to its corresponding price.

Understanding Arc Elasticity

Arc elasticity measures the percentage change of one variable in relation to the percentage change of another variable between two specific points. This method differs from point elasticity, calculated at a single point, as the distance between two points approaches zero.

It is commonly applied in the field of economics, particularly in the context of the law of demand. Instead of examining changes at a specific point, this method involves dividing the magnitude of price and demand changes by their midpoint.

This approach provides a more precise representation of elasticity across an entire curve.

The formula for arc elasticity of demand is:

Arc E_d = { % change in quantity demanded( or supplied)}/{ % change in price}

Mathematically, the formula can be expressed as:

Arc E_d = [( Q₂− Q₁)/{( Q₂+ Q₁)/2}]/ [(P₂ - P₁​)/{(P₁+ P₂)/2}​​]

Where:

• Q₁  and Q₂ present the initial and final quantities demanded (or supplied)
• P₁  and P₂ represent the initial and final prices

Interpreting Arc Elasticity

The Arc elasticity can be interpreted as follows:

1. Elastic (|E∣>1): If the absolute value is greater than 1, it indicates elastic demand or supply. This means that consumers or producers are highly responsive to price changes, leading to a significant change in quantity demanded or supplied when prices fluctuate.
2. Inelastic (|E∣<1): If the absolute value is less than 1, the demand or supply is inelastic. It indicates that the quantity demanded or supplied is not very responsive to price changes. In this scenario, consumers or producers are relatively insensitive to price fluctuations.
3. Unit Elastic (E=1): If the absolute value is exactly 1, it signifies inelastic demand or supply. This suggests that consumers or producers are not very sensitive to price changes, resulting in minor fluctuations in quantity demanded or supplied despite price changes.

Price elasticity of demand

Price elasticity of demand gauges how responsive the quantity demanded of a product or service is to a price change. This metric is calculated by dividing the percentage change in quantity demanded by the percentage change in price. 

Arc elasticity of demand uses a midpoint between the two points.

The formula for Price (Point) Elasticity of Demand is:

PE_d = % change in quantity demanded/% change in price

PE_d = {(Q₂− Q₁)/Q₁}/ {(P₂ - P_1​)/P₁​}

Where:

• Q₁  = Initial quantity demanded
• Q₂ = New quantity demanded
• P₁  = Initial price
• P₂ = New price

Calculation of Price Elasticity of Demand

Suppose the price of a product rises from $500 to $600, resulting in a decrease in quantity demanded from 200 to 150 units.

Using the price elasticity of demand formula, the calculation is as follows:

% change in quantity demanded= = (Q₂− Q₁) /Q₁ = (150– 200) / 200 = -0.25

% change in price = (P₂ - P₁) /P₁ = (600 – 500) / 500 = 0.2

Thus, PE_d = % change in quantity demanded/% change in price= -0.25 / 0.2 = -1.25

Considering the absolute value in price elasticity, the negative sign is disregarded. Thus, the price elasticity of demand for that good when the price increases from $500 to $600 is 1.25.

If we consider different start and end points in the example, the calculation of price elasticity of demand (PE_d ) will vary.

Suppose the price initially decreased from $600 to $500, and the quantity demanded increased from 150 to 200 units.

% change in quantity demanded= = (Q₂− Q₁) / Q₁ = (200– 150) / 150 = 33.33%

% change in price = (P₂ - P₁) / P₁ = (500-600) / 600 = -16.67

Thus, PE_d = % change in quantity demanded/% change in price= 33.33 / -16.67 = 2

In this scenario, the price elasticity of demand (PE_d ) is −2, which is different from 1.25.

This demonstrates the sensitivity of the PE_d calculation to the specific points chosen on the demand curve, resulting in a different elasticity value when different start and end points are considered. It gives different values depending on whether the price rises or falls.

Arc Elasticity of Demand

To solve the above-mentioned inconsistency, the arc elasticity of demand can be used.

Arc elasticity of demand calculates elasticity at the midpoint between two chosen points on the demand curve. This is done by using the midpoint between the quantities and prices of the two points.

The formula is as follows: 

Arc E_d= {(% change in quantity demanded)/(Midpoint Quantity Demanded)}/{( % change in price)/(Midpoint Price)}

Let’s calculate the arc elasticity following the example presented above:

Midpoint Qd = (Q₂+ Q₁) / 2 = (200 + 150) / 2 = 175

Midpoint Price = (P₂+ P₁) / 2 = (500 + 600) / 2 = 550% 

Change in qty demanded = (150-200) / 175= 0.2857% and Change in price = (600– 500) / 550 

= -0.1818

Arc E_d = 0.2857 / -0.1818 = -1.5702

When using arc elasticity of demand, you don't have to worry about which prices come first or last. It is not necessary to be concerned about identifying the starting and ending points because this method yields consistent elasticity values regardless of whether prices increase or decrease.

Using the midpoint has the following properties:

• it is symmetric with regard to the two prices and quantities, 
• it is independent of the units of measurement and 
• it produces a value of unity if the total revenues (price times quantity) at both points are identical.

Significance of Arc Elasticity

Below are a few reasons why arc elasticity is important:

1. Handling Price Changes Over a Range: It allows economists and businesses to measure the responsiveness of quantity demanded to price changes over a range, which is often more relevant and practical for decision-making as in real-world scenarios, prices seldom change only at a single point.
2. Accurate Measurement for Nonlinear Demand Curves: Useful when dealing with curved or nonlinear demand curves. It provides a better approximation of the true elasticity between two points on the curve, giving a more accurate representation of consumer responsiveness to price changes.
3. Policy Analysis: It helps predict the impact of consumer behavior when governments consider implementing indirect taxes (such as excise taxes) or subsidies that affect prices. This information is vital for designing tax policies that generate revenue without significantly reducing consumption.
4. Optimal Pricing Strategies: It is useful in analyzing how price changes might affect the firm's total revenue. By understanding the price elasticity of demand over a specific range, businesses can make informed decisions about setting optimal prices to maximize their revenue.
5. Market Segmentation: It is useful for firms to identify different price elasticities for different market segments. Understanding how different customer segments respond to price changes can help in market segmentation and targeted pricing strategies.
6. Infrastructure Planning: It helps forecast changes in demand with changes in prices for industries such as utilities (electricity, water, etc.). This information is crucial for long-term infrastructure planning and investment.
7. Cross-Price Elasticity: It is utilized for determining cross-price elasticity, which assesses how the demand for one product adjusts in reaction to alterations in the price of another associated product.
8. Elasticity of Supply: Arc elasticity principles can also be applied to supply curves, providing insights into how price changes affect the quantity of goods or services that producers are willing to supply.

Limitations of Arc Elasticity

While arc elasticity provides valuable insights, it does have limitations.

1. Dependence on Interval Choice: The arc elasticity value can vary depending on the specific interval (the two points chosen) on the demand curve. Different intervals may yield different elasticity values, leading to potential ambiguity in interpreting the demand responsiveness.
2. Not Suitable for Linear Demand Curves: For linear demand curves, arc and point elasticity will yield the same result. Using arc elasticity in such cases is unnecessary and might lead to confusion, especially for those who are new to the concept.
3. Less Suitable for Highly Variable Demand Curves: In markets where demand fluctuates significantly, calculating arc elasticity over a broad range might not accurately represent consumer responsiveness, as demand elasticity can change at different points along the curve.
4. Complexity in Interpretation: Interpreting arc elasticity values can be more complex than interpreting point elasticity. This complexity arises due to the averaging effect of quantities and prices over a range, making it harder to draw straightforward conclusions about consumer behavior.

Arc Elasticity vs. Point Elasticity

Let's comprehensively compare the differences between Arc Elasticity and Point Elasticity:

While both elasticities have their specific applications, the choice between them depends on the context and the level of precision required for the analysis.

Authored and researched by Rani Thakur⎮LinkedIn

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