Arc Elasticity
Arc elasticity is the elasticity of one variable with respect to another between two given points. It is used when there is no general way to define the relationship between the two variables. Arc elasticity is also defined as the elasticity between two points on a curve.
The concept is used in both economics and mathematics. In economics, is it commonly used to measure the changes between the quantity of goods demanded and their prices.
Arc Elasticity
Definition: Arc elasticity refers to the elasticity between two variables at two given points. It is used when there is no general method to define the relationship between two variables. Arc elasticity is also defined as the elasticity between two points on a curve. This concept is applied in both economics and mathematics. In economics, it is commonly used to measure changes in the quantity demanded and the price of goods.
Origin
The concept of arc elasticity originated in the fields of economics and mathematics, initially used to solve the problem of calculating elasticity in nonlinear relationships. Traditional point elasticity calculation methods are not applicable in nonlinear curves, so arc elasticity was proposed as an alternative. It gradually gained acceptance and application by economists and mathematicians in the early 20th century.
Categories and Characteristics
Arc elasticity is mainly divided into two categories: demand arc elasticity and supply arc elasticity. Demand arc elasticity measures the impact of price changes on the quantity demanded, while supply arc elasticity measures the impact of price changes on the quantity supplied. The characteristic of arc elasticity is that it considers the average rate of change between two points rather than a single instantaneous rate of change, making it more accurate in handling nonlinear relationships.
Specific Cases
Case 1: Suppose the price of a commodity rises from 10 to 15 units, and the quantity demanded decreases from 100 to 80 units. Using the arc elasticity formula to calculate demand arc elasticity:
Arc Elasticity = [(Q2 - Q1) / (Q2 + Q1)] / [(P2 - P1) / (P2 + P1)]
Substitute the data: Arc Elasticity = [(80 - 100) / (80 + 100)] / [(15 - 10) / (15 + 10)] = -0.5 / 0.2 = -2.5
This indicates that for every 1% increase in price, the quantity demanded will decrease by 2.5%.
Case 2: Suppose the price of an agricultural product rises from 5 to 7 units, and the quantity supplied increases from 200 to 250 units. Using the arc elasticity formula to calculate supply arc elasticity:
Arc Elasticity = [(Q2 - Q1) / (Q2 + Q1)] / [(P2 - P1) / (P2 + P1)]
Substitute the data: Arc Elasticity = [(250 - 200) / (250 + 200)] / [(7 - 5) / (7 + 5)] = 0.111 / 0.167 = 0.67
This indicates that for every 1% increase in price, the quantity supplied will increase by 0.67%.
Common Questions
Question 1: What is the difference between arc elasticity and point elasticity?
Answer: Point elasticity refers to the elasticity at a specific point, suitable for linear relationships; arc elasticity is suitable for nonlinear relationships and calculates the average elasticity between two points.
Question 2: Why use arc elasticity instead of point elasticity?
Answer: In nonlinear relationships, point elasticity may not accurately reflect the changes between variables, while arc elasticity, by considering the average rate of change between two points, can provide more accurate results.
