Arc elasticity is a measure used in economics to quantify the responsiveness of one variable to a change in another variable. Specifically, arc elasticity calculates the percentage change in one variable relative to the percentage change in another variable, considering the average or midpoint values of the two variables.

The formula for calculating arc elasticity (E) is as follows:

\[ E = \frac{\left(\frac{\text{Change in Quantity (Q)}}{\text{Average Quantity}}\right)}{\left(\frac{\text{Change in Price (P)}}{\text{Average Price}}\right)} \]

Where:

– \(\text{Change in Quantity (Q)}\) is the absolute change in the quantity of a good or service.

– \(\text{Average Quantity}\) is the average of the initial and final quantities.

– \(\text{Change in Price (P)}\) is the absolute change in the price of the good or service.

– \(\text{Average Price}\) is the average of the initial and final prices.

Key points about arc elasticity:

1. **Midpoint Method:**

– The use of averages in the denominator (average quantity and average price) makes arc elasticity less sensitive to the choice of starting and ending points. This is sometimes referred to as the midpoint method.

2. **Percentage Changes:**

– Arc elasticity expresses the percentage change in quantity relative to the percentage change in price. It provides a measure of the responsiveness of quantity demanded or supplied to changes in price.

3. **Interpretation:**

– The sign of the elasticity coefficient indicates the direction of the relationship between price and quantity. If \(E\) is positive, it suggests a direct or positive relationship (as price increases, quantity increases), while a negative \(E\) suggests an inverse or negative relationship.

4. **Elasticity Values:**

– The magnitude of the elasticity coefficient (\(|E|\)) indicates the degree of responsiveness. If \(|E| > 1\), the good is considered elastic (responsive to price changes); if \(|E| < 1\), it is inelastic (less responsive); and if \(|E| = 1\), it is unit elastic.
5. **Calculation Example:**
- If the arc elasticity coefficient is -2, it implies that a 1% increase in price is associated with a 2% decrease in quantity demanded.
6. **Application:**
- Arc elasticity is commonly used in the analysis of demand and supply in economics. It helps economists and businesses understand how sensitive the quantity demanded or supplied is to changes in price.
It's important to note that arc elasticity is just one method for measuring responsiveness. Other elasticity measures, such as point elasticity, may be used in different contexts or when specific starting or ending points are relevant to the analysis. The choice of elasticity measure depends on the specific requirements of the analysis and the characteristics of the data.