Point Elasticity and Arc Elasticity

Point elasticity evaluates elasticity at a specific point on the demand or supply curve, whereas arc elasticity measures elasticity over a range.

Difference Between Point Elasticity and Arc Elasticity

Point elasticity is the elasticity measured at a particular point on the demand curve, and for a linear demand curve, it varies depending on where it is measured. In other words, it depends on which two points on the demand curve are chosen for the calculation. On the other hand, arc elasticity of demand is the elasticity measured over a range of prices, meaning that elasticity is evaluated over a portion of the demand or supply curve rather than at a specific point. In other words, instead of taking two points, the initial and final price, an average of the two is used in the calculation.

Point Elasticity

The price elasticity of demand at a specific point on the curve is calculated using the following formula:

\[ E_p = \left( \frac{\Delta Q}{\Delta P} \right) \left( \frac{P}{Q} \right) \]

$E_p$: Price elasticity of demand at a specific point.
$\Delta Q = Q_2 - Q_1$: Change in quantity demanded between two close values ($Q_1$ and $Q_2$).
$\Delta P = P_2 - P_1$: Change in price between two close values ($P_1$ and $P_2$).
$P$: Price at the point where elasticity is evaluated (can be $P_1$ or $P_2$, depending on the context).
$Q$: Quantity demanded corresponding to price $P$ (can be $Q_1$ or $Q_2$, depending on the context).

In this method, a small change ($\Delta Q$ and $\Delta P$) around the point where elasticity is calculated is used, and the reference price and quantity are selected based on the analysis.

Arc Elasticity

Arc elasticity of demand measures the sensitivity of quantity demanded to price changes over a section of the demand curve. It is expressed using the formula:

\[ E_a = \left( \frac{\Delta Q}{\Delta P} \right) \left( \frac{\bar{P}}{\bar{Q}} \right) \]

$E_a$: Arc elasticity of demand.
$\Delta Q = Q_2 - Q_1$: Change in quantity demanded between two points on the demand curve.
$\Delta P = P_2 - P_1$: Change in price between the same two points.
$\bar{Q} = \frac{Q_1 + Q_2}{2}$: Average quantity, calculated as the arithmetic mean of the initial ($Q_1$) and final ($Q_2$) quantities.
$\bar{P} = \frac{P_1 + P_2}{2}$: Average price, calculated as the arithmetic mean of the initial ($P_1$) and final ($P_2$) prices.

The formula combines two elements:

The term $\frac{\Delta Q}{\Delta P}$, which measures the absolute change in quantity ($Q$) relative to the absolute change in price ($P$).
The term $\frac{\bar{P}}{\bar{Q}}$, which adjusts the scale of the change using average values to ensure symmetry between the two analyzed points.

This approach is more accurate than other elasticity methods when analyzing a specific section of the demand curve, as it considers relative price and quantity differences rather than focusing solely on absolute values.

Example of Price Elasticity of Demand Calculation Using Arc Elasticity

Suppose the quantity demanded of a product decreases from 150 units ($Q_1$) to 100 units ($Q_2$) when the price increases from $10 ($P_1$) to $15 ($P_2$).

Step 1: Calculate the changes ($\Delta Q$ and $\Delta P$)

\[ \Delta Q = Q_2 - Q_1 = 100 - 150 = -50 \] \[ \Delta P = P_2 - P_1 = 15 - 10 = 5 \]

Step 2: Calculate the average quantities and prices ($\bar{Q}$ and $\bar{P}$)

\[ \bar{Q} = \frac{Q_1 + Q_2}{2} = \frac{150 + 100}{2} = 125 \] \[ \bar{P} = \frac{P_1 + P_2}{2} = \frac{10 + 15}{2} = 12.5 \]

Step 3: Substitute into the arc elasticity formula

\[ E_a = \left( \frac{\Delta Q}{\Delta P} \right) \left( \frac{\bar{P}}{\bar{Q}} \right) \] \[ E_a = \left( \frac{-50}{5} \right) \left( \frac{12.5}{125} \right) \] \[ E_a = \left( -10 \right) \left( 0.1 \right) = -1 \]

The arc elasticity is $-1$, indicating unitary elasticity: a 1% increase in price results in a 1% decrease in quantity demanded.

Short-run vs long-run elasticities Price ceiling