# $\S$ 3.7 Demand Elasticity

The Demand Elasticity of a good is denoted $E$ and takes on non-negative values. A range of values of $E$ and examples of goods with that $E$ are given in this table:

| Elasticity Value | Type of Demand | Examples | Detailed Explanation |
|------------------|------------------------|-----------------------------------------------------------|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| $E = 0$ | Perfectly Inelastic | Life-saving drugs, Essential medical treatments, Air (for breathing) | In a perfectly inelastic demand scenario, the quantity demanded does not change, regardless of any price changes. This applies to essential goods such as specific life-saving medications and even air for breathing in controlled environments (e.g., space stations), where consumers have no substitutes and will pay any price. |
| $0 < E < 1$ | Inelastic | Gasoline, Basic groceries, Cigarettes, Pet rocks | Goods with inelastic demand experience less than proportional changes in quantity demanded to changes in price. This category includes daily essentials like gasoline and groceries, addictive products like cigarettes, and even pet rocks, where despite minimal utility, a niche market remains consistently interested regardless of price adjustments. |
| $E = 1$ | Unit Elastic | Coffee, Certain types of clothing, Collectible spoons | Unit elastic demand indicates that the percentage change in quantity demanded is exactly equal to the percentage change in price. This equilibrium is seen in products like coffee, specific clothing items, and unexpectedly, in collectible spoons, where enthusiasts match their purchasing behavior proportionately to price changes. |
| $E > 1$ | Elastic | Luxury cars, Electronics, Fashion apparel, Novelty ice cube molds | Elastic demand occurs when the quantity demanded responds more than proportionately to changes in price. This sensitivity is typical for luxury items and non-essential electronics, but also extends to novelty items like themed ice cube molds, where a slight decrease in price can significantly increase demand among niche buyers. |
| $E = \infty$ | Perfectly Elastic | Digital products with free substitutes, Instant noodles in a college town | Perfectly elastic demand is an extreme scenario where consumers are infinitely sensitive to price changes. A tiny increase in price can cause demand to drop to zero, especially in markets like digital products with free alternatives and, humorously, for instant noodles within a college town, where students will flock to any store offering them for less. |

How do we calculate the Demand Elasticity of a good given its price-demand function $p(q)$

we use calculus to find the elasticity of demand, which is represented as:

$$ E(q) = -\frac{dp}{dq} \cdot \frac{q}{p} $$

Here, $\frac{dp}{dq}$ is the derivative of the price with respect to the quantity (the slope of the price-demand function), and $\frac{q}{p}$ represents the ratio of quantity to price at the point where we're calculating the elasticity.

### Steps to Calculate Demand Elasticity from $p(q)$:

1. **Differentiate $p(q)$ with Respect to $q$**: Find the first derivative of the price-demand function $p(q)$ with respect to $q$. This derivative, $\frac{dp}{dq}$, represents the rate of change of price with respect to quantity.

2. **Substitute Values into the Elasticity Formula**: Substitute $\frac{dp}{dq}$ and the values of $q$ and $p$ into the elasticity formula $E(q) =- \frac{dp}{dq} \cdot \frac{q}{p}$ to calculate the elasticity at a specific point.

3. **Interpret the Result**: The value of $E$ indicates how responsive the quantity demanded is to price changes:
- $E < 1$: Inelastic demand.
- $E = 1$: Unit elastic demand.
- $E > 1$: Elastic demand.
- $E = 0$: Perfectly inelastic demand (in practice, this result is highly unlikely from a continuous function).
- $E = \infty$: Perfectly elastic demand (also an extreme and unlikely outcome for most real-world functions).

### Example Calculation:

Given a price-demand function $p(q) = 100 - 2q$, let's calculate the elasticity of demand when $q = 30$.

1. Differentiate $p(q)$ with respect to $q$: $\frac{dp}{dq} = -2$.
2. Substitute into the elasticity formula:
$$ E(30) = -(-2) \cdot \frac{30}{p(30)} $$
$$ p(30) = 100 - 2(30) = 40 $$
$$ E(30) = -(-2) \cdot \frac{30}{40} = 1.5 $$

The magnitude $1.5$ tells us the demand is elastic at $q = 30$, meaning a 1% decrease in price would lead to a 1.5% increase in quantity demanded.