Arc Elasticity Calculator
Use this tool to calculate the price elasticity of demand (or supply) between two points on a demand (or supply) curve using the arc elasticity method. This method provides a measure of elasticity over a range of prices and quantities.
Enter the initial Price (P1) and Quantity (Q1), and the New Price (P2) and New Quantity (Q2). The quantities usually represent quantity demanded for price elasticity of demand. Ensure consistent units for price and quantity.
Enter Price and Quantity Points
Understanding Arc Elasticity & Formula
What is Arc Elasticity?
Arc elasticity is a method used to measure the price elasticity of demand or supply between two distinct points on a curve. It calculates elasticity over a range, rather than at a single point. This makes it useful when dealing with larger price or quantity changes, as it gives the same result whether you start from point 1 and move to point 2, or vice-versa. It uses the average (midpoint) of the initial and new prices and quantities.
Arc Elasticity Formula
The Arc Elasticity (E) formula is:
E = [(Q₂ - Q₁) / ((Q₁ + Q₂) / 2)] / [(P₂ - P₁) / ((P₁ + P₂) / 2)]
This can be simplified to:
E = [(Q₂ - Q₁) / (Q₁ + Q₂)] * [(P₁ + P₂) / (P₂ - P₁)]
Where:
- P₁ = Initial Price
- Q₁ = Initial Quantity
- P₂ = New Price
- Q₂ = New Quantity
The formula measures the percentage change in quantity divided by the percentage change in price, using the average of the start and end points as the base for the percentage calculation.
Interpreting the Arc Elasticity Value (Absolute Value)
- |E| > 1: Elastic - Quantity change is proportionally larger than the price change. Consumers (or producers) are highly responsive to price changes.
- |E| < 1: Inelastic - Quantity change is proportionally smaller than the price change. Consumers (or producers) are less responsive to price changes.
- |E| = 1: Unit Elastic - Quantity change is proportionally equal to the price change.
- |E| = ∞ (Infinity): Perfectly Elastic - Any price change causes an infinite change in quantity (occurs when P₁ = P₂ but Q₁ ≠ Q₂). The demand/supply curve is horizontal.
- |E| = 0: Perfectly Inelastic - Quantity does not change regardless of price change (occurs when Q₁ = Q₂ but P₁ ≠ P₂). The demand/supply curve is vertical.
Arc Elasticity Examples
See how different changes in price and quantity affect elasticity:
Example 1: Elastic Demand
Scenario: Price decreases from $10 to $8, Quantity demanded increases from 100 to 140 units.
1. Known Values: P₁ = 10, Q₁ = 100; P₂ = 8, Q₂ = 140.
2. Calculation:
ΔQ = 140 - 100 = 40
Avg Q = (100 + 140) / 2 = 120
ΔP = 8 - 10 = -2
Avg P = (10 + 8) / 2 = 9
E = (40 / 120) / (-2 / 9) = (0.3333...) / (-0.2222...) ≈ -1.5
3. Result: E ≈ -1.5. Absolute value |E| = 1.5.
Conclusion: Since |E| > 1, demand is Elastic between these points.
Example 2: Inelastic Demand
Scenario: Price increases from $5 to $7, Quantity demanded decreases from 200 to 180 units.
1. Known Values: P₁ = 5, Q₁ = 200; P₂ = 7, Q₂ = 180.
2. Calculation:
ΔQ = 180 - 200 = -20
Avg Q = (200 + 180) / 2 = 190
ΔP = 7 - 5 = 2
Avg P = (5 + 7) / 2 = 6
E = (-20 / 190) / (2 / 6) = (-0.1053...) / (0.3333...) ≈ -0.316
3. Result: E ≈ -0.316. Absolute value |E| = 0.316.
Conclusion: Since |E| < 1, demand is Inelastic between these points.
Example 3: Unit Elastic Demand
Scenario: Price decreases from $20 to $15, Quantity demanded increases from 50 to approximately 66.67 units.
1. Known Values: P₁ = 20, Q₁ = 50; P₂ = 15, Q₂ = 66.67.
2. Calculation:
ΔQ = 66.67 - 50 = 16.67
Avg Q = (50 + 66.67) / 2 = 58.335
ΔP = 15 - 20 = -5
Avg P = (20 + 15) / 2 = 17.5
E = (16.67 / 58.335) / (-5 / 17.5) ≈ (0.2857) / (-0.2857) ≈ -1
3. Result: E ≈ -1. Absolute value |E| = 1.
Conclusion: Since |E| ≈ 1, demand is approximately Unit Elastic between these points.
Example 4: Perfectly Elastic Demand
Scenario: Price stays at $10, but quantity demanded changes from 100 to 150 units (theoretical).
1. Known Values: P₁ = 10, Q₁ = 100; P₂ = 10, Q₂ = 150.
2. Calculation: ΔP = 10 - 10 = 0. Avg P = 10. ΔQ = 150 - 100 = 50. Avg Q = 125. Price change percentage = (0 / 10) * 100 = 0%. Quantity change percentage = (50 / 125) * 100 = 40%. Elasticity = 40% / 0%.
3. Result: Division by zero in the denominator (price change) results in infinite elasticity.
Conclusion: This is a case of Perfectly Elastic demand.
Example 5: Perfectly Inelastic Demand
Scenario: Price changes from $50 to $70, but quantity demanded stays at 10 units.
1. Known Values: P₁ = 50, Q₁ = 10; P₂ = 70, Q₂ = 10.
2. Calculation: ΔQ = 10 - 10 = 0. Avg Q = 10. ΔP = 70 - 50 = 20. Avg P = 60. Quantity change percentage = (0 / 10) * 100 = 0%. Price change percentage = (20 / 60) * 100 = 33.33%. Elasticity = 0% / 33.33%.
3. Result: The numerator (quantity change) is zero, resulting in an elasticity of 0.
Conclusion: This is a case of Perfectly Inelastic demand.
Example 6: Price Increase (Elastic)
Scenario: Price of gadget increases from $5 to $6, units sold fall from 1000 to 700.
1. Known Values: P₁ = 5, Q₁ = 1000; P₂ = 6, Q₂ = 700.
2. Calculation:
ΔQ = 700 - 1000 = -300
Avg Q = (1000 + 700) / 2 = 850
ΔP = 6 - 5 = 1
Avg P = (5 + 6) / 2 = 5.5
E = (-300 / 850) / (1 / 5.5) = (-0.3529...) / (0.1818...) ≈ -1.941
3. Result: E ≈ -1.941. Absolute value |E| = 1.941.
Conclusion: Since |E| > 1, demand is Elastic.
Example 7: Price Decrease (Inelastic)
Scenario: Price of a staple food decreases from $2 to $1.50, quantity demanded rises from 500 to 520 units.
1. Known Values: P₁ = 2, Q₁ = 500; P₂ = 1.5, Q₂ = 520.
2. Calculation:
ΔQ = 520 - 500 = 20
Avg Q = (500 + 520) / 2 = 510
ΔP = 1.5 - 2 = -0.5
Avg P = (2 + 1.5) / 2 = 1.75
E = (20 / 510) / (-0.5 / 1.75) = (0.0392...) / (-0.2857...) ≈ -0.137
3. Result: E ≈ -0.137. Absolute value |E| = 0.137.
Conclusion: Since |E| < 1, demand is Inelastic.
Example 8: Elasticity of Supply
Scenario: Price increases from $10 to $12, Quantity supplied increases from 500 to 700 units.
1. Known Values: P₁ = 10, Q₁ = 500; P₂ = 12, Q₂ = 700.
2. Calculation:
ΔQ = 700 - 500 = 200
Avg Q = (500 + 700) / 2 = 600
ΔP = 12 - 10 = 2
Avg P = (10 + 12) / 2 = 11
E = (200 / 600) / (2 / 11) = (0.3333...) / (0.1818...) ≈ 1.833
3. Result: E ≈ 1.833. Absolute value |E| = 1.833.
Conclusion: Since |E| > 1, supply is Elastic between these points.
Example 9: Inelastic Supply
Scenario: Price decreases from $100 to $80, Quantity supplied decreases from 1000 to 950 units.
1. Known Values: P₁ = 100, Q₁ = 1000; P₂ = 80, Q₂ = 950.
2. Calculation:
ΔQ = 950 - 1000 = -50
Avg Q = (1000 + 950) / 2 = 975
ΔP = 80 - 100 = -20
Avg P = (100 + 80) / 2 = 90
E = (-50 / 975) / (-20 / 90) = (-0.0513...) / (-0.2222...) ≈ 0.231
3. Result: E ≈ 0.231. Absolute value |E| = 0.231.
Conclusion: Since |E| < 1, supply is Inelastic between these points.
Example 10: Unit Elastic Supply
Scenario: Price increases from $10 to $12.50, Quantity supplied increases from 100 to 125 units.
1. Known Values: P₁ = 10, Q₁ = 100; P₂ = 12.5, Q₂ = 125.
2. Calculation:
ΔQ = 125 - 100 = 25
Avg Q = (100 + 125) / 2 = 112.5
ΔP = 12.5 - 10 = 2.5
Avg P = (10 + 12.5) / 2 = 11.25
E = (25 / 112.5) / (2.5 / 11.25) ≈ (0.2222...) / (0.2222...) ≈ 1
3. Result: E ≈ 1. Absolute value |E| = 1.
Conclusion: Since |E| ≈ 1, supply is approximately Unit Elastic between these points.
Frequently Asked Questions about Arc Elasticity
1. What is the main difference between arc elasticity and point elasticity?
Point elasticity measures elasticity at a single point on the curve using derivatives (or very small changes). Arc elasticity measures elasticity over a discrete range between two points, using the midpoint method for averages. Arc elasticity is generally used when the price or quantity change is relatively large.
2. Why use the average price and quantity in the arc elasticity formula?
Using the averages ((Q1 + Q2) / 2 and (P1 + P2) / 2) in the midpoint formula ensures that the calculated elasticity value is the same regardless of whether you start at point 1 and move to point 2, or start at point 2 and move back to point 1. This symmetry is a key advantage of the arc elasticity method over calculating simple percentage changes from just one point.
3. What does a negative arc elasticity value mean for demand?
For price elasticity of demand, the value is almost always negative because price and quantity demanded typically move in opposite directions (according to the Law of Demand). A price increase leads to a quantity decrease (negative ΔQ/ΔP), and a price decrease leads to a quantity increase (negative ΔQ/ΔP). Economists typically focus on the *absolute value* (|E|) of price elasticity of demand, as the sign simply confirms the inverse relationship.
4. How is arc elasticity used in business and economics?
Businesses use it to predict how a price change might affect total revenue. If demand is elastic (|E| > 1), a price increase will decrease total revenue, and a price decrease will increase total revenue. If demand is inelastic (|E| < 1), a price increase will increase total revenue, and a price decrease will decrease total revenue. Economists use it to analyze market responsiveness and inform policy decisions.
5. What happens if the initial quantity (Q1) and new quantity (Q2) are both zero?
If both Q1 and Q2 are zero, the average quantity ((Q1 + Q2)/2) is zero. The formula involves dividing by this average, making the calculation undefined. The calculator will return an error in this scenario, as there's no meaningful quantity change to measure elasticity from.
6. What if the price doesn't change (P1 = P2)?
If P1 equals P2, the change in price (P2 - P1) is zero. The denominator of the price change percentage ((P2 - P1) / ((P1 + P2) / 2)) becomes zero, leading to division by zero in the elasticity calculation. This signifies infinite elasticity (Perfectly Elastic) – any non-zero change in quantity occurs without any price change. The calculator handles this specific edge case.
7. What if the quantity doesn't change (Q1 = Q2)?
If Q1 equals Q2, the change in quantity (Q2 - Q1) is zero. The numerator of the elasticity formula becomes zero, resulting in an elasticity of 0. This signifies zero elasticity (Perfectly Inelastic) – quantity does not change regardless of price changes. The calculator handles this specific edge case.
8. Can this calculator be used for other types of elasticity?
While the formula is specifically for Price Elasticity of Demand or Supply, the *structure* of the formula (percentage change in dependent variable / percentage change in independent variable) is similar for other arc elasticity measures like Cross-Price Elasticity or Income Elasticity. However, this tool is labeled and primarily designed for Price Elasticity, where P is price and Q is quantity.
9. What are the units for the inputs and output?
The units for P1, Q1, P2, and Q2 must be consistent (e.g., Dollars and Units, Euros and Kilograms). However, the resulting elasticity value is a unitless number, as it's a ratio of two percentage changes.
10. What does an elasticity of +1.5 mean?
For elasticity of *supply*, a value of +1.5 means that for every 1% increase in price between the two points, the quantity supplied increases by approximately 1.5%. Since the absolute value (1.5) is greater than 1, supply is elastic. For demand, you would typically see a negative value; an absolute value of 1.5 (|E|=1.5) would mean demand is elastic.
