Elasticity of Demand

The Core Limitation of the Law of Demand

In classical demand analysis, the Law of Demand establishes an important functional rule: price ($P$) and quantity demanded ($Q$) move in opposite directions. When price rises, demand contracts; when price falls, demand expands.

For example, if the price of a textbook doubles from ₹100 to ₹200, the Law of Demand accurately predicts that quantity demanded will decrease. However, it cannot tell us whether the purchase volume drops by 5%, exactly 50%, or collapses to zero. To quantify this responsiveness, we must transition to the Elasticity of Demand.

Defining Elasticity & Responsiveness

Think of elasticity as a measure of structural stretchability. If you try to stretch a rigid plastic pen, its physical dimensions show zero modification—its responsiveness is zero. Conversely, pulling a rubber band or a flexible t-shirt yields significant physical stretch. This demonstrates high responsiveness.

Similarly, market demand can be highly responsive or completely rigid. While demand is influenced by an expansive matrix of variables—including Price ($P$), Consumer Income ($Y$), and Prices of Related Commodities ($P_r$)—the primary economic analytical focus centers on Price Elasticity of Demand ($E_p$).

Mathematical Formulations & Measurement Methods

1. The Percentage Method

The standard algebraic baseline defines price elasticity as the percentage change in consumption divided by the percentage change in price.

Component Breakdowns

• $$\% \text{ Change in Quantity Demanded} = \frac{\Delta Q}{Q} \times 100$$
• $$\% \text{ Change in Price} = \frac{\Delta P}{P} \times 100$$
• $$\Delta Q = Q_1 - Q \quad (\text{Where } Q_1 = \text{New Quantity}, Q = \text{Initial Baseline Quantity})$$
• $$\Delta P = P_1 - P \quad (\text{Where } P_1 = \text{New Price}, P = \text{Initial Baseline Price})$$

2. The Proportionate Method

By substituting the component terms back into the primary ratio, the multiplier term ($100$) cancels out, yielding the Proportionate Method:

Mathematical Proof of Equivalence (Classroom Transcript Case)

Let us evaluate a structural market shift using both mathematical frameworks to prove algebraic consistency.

Step-by-Step Parameter Derivation

$$\Delta P = P_1 - P = 200 - 100 = 100$$

$$\Delta Q = Q_1 - Q = 250 - 500 = -250$$

Key Properties of Elasticity Metrics

1. Unit-Free Metric: Elasticity coefficients are pure numbers. Because the calculations evaluate relative percentage variations, all base dimensions (such as kilograms, liters, meters, or currency values) cancel out completely.
2. The Sign Convention Indicator: Due to the law of demand's inverse relationship, calculating price elasticity naturally yields a negative coefficient. In economic analysis, this negative sign is treated purely as a directional indicator of that inverse relationship, not as an algebraic value less than zero. Hence, we often state the absolute value $|E_p| = 0.5$.
3. Boundary Scale Conditions: The theoretical limits for elasticity range from an absolute global minimum of $0$ (perfect rigidity) to an upper bound of $\infty$ (infinite sensitivity).

The 5 Degrees (Kinds) of Price Elasticity

Visualizing Elasticity Slopes

Geometric / Point Elasticity Mechanics

To measure elasticity at a specific precise coordinate along a linear downward-sloping demand curve, we use the Geometric Method (or Point Elasticity framework).

Mathematical Dissection of Point Coordinates

• Vertical Price Intercept (Point A): The upper segment length is zero. Dividing any positive lower segment by zero yields an infinite value: $$E_p = \frac{\text{Lower Segment (AT)}}{\text{Upper Segment (0)}} = \infty$$
• Upper Half Segment (Point B): The segment stretching down to the horizontal axis is longer than the segment extending up to the vertical axis ($BT > BA$). Because the numerator exceeds the denominator: $$E_p = \frac{BT}{BA} > 1$$
• Exact Midpoint (Point M): The segment below the midpoint exactly equals the segment above it ($MT = MA$). This matching ratio results in a unitary elasticity coefficient: $$E_p = \frac{MT}{MA} = 1$$
• Lower Half Segment (Point C): The segment extending down to the horizontal axis is shorter than the segment stretching up to the vertical axis ($CT < CA$). Because the numerator is smaller than the denominator: $$E_p = \frac{CT}{CA} < 1$$
• Horizontal Quantity Intercept (Point T): The lower segment length drops to zero. Dividing zero by any positive upper segment value results in zero elasticity: $$E_p = \frac{\text{Lower Segment (0)}}{\text{Upper Segment (TA)}} = 0$$

Macro Determinants of Elasticity

Notes compiled from comprehensive lecture transcripts · Aligned with structural university standards · Microeconomics