Chapter 5: Elasticity and Its Application – Comprehensive Notes (Mankiw, Principles of Microeconomics)

The Price Elasticity of Demand

Elasticity measures how much buyers/sellers respond to changes in market conditions.
Specifically, price elasticity of demand (PED) measures how much the quantity demanded (Q_d) responds to a change in price (P).
Key idea: along a demand curve, price and quantity move in opposite directions, so the elasticity is negative in the traditional calculation. We report elasticities as positive numbers (absolute values).
Definition:
Ed = \frac{\%\Delta Qd}{\%\Delta P}
Important note: there are multiple ways to compute the percentage changes. The midpoint method is preferred because it avoids dependence on the direction of change.

The Percentage Changes: Standard Method vs Midpoint Method

Standard method (arc from start to end):
\%\Delta P{std} = \frac{P2 - P1}{P1} \times 100
\%\Delta Q{std} = \frac{Q2 - Q1}{Q1} \times 100
Ed^{std} = \frac{\%\Delta Q{std}}{\%\Delta P_{std}}
Midpoint method (arc between start and end, using the midpoint):
- The midpoint is the average of start and end values: (\text{mid} = (\text{start} + \text{end})/2).
  \%\Delta P{mid} = \frac{P2 - P1}{(P2 + P1)/2} \times 100 \%\Delta Q{mid} = \frac{Q2 - Q1}{(Q2 + Q1)/2} \times 100
  Ed = \frac{\%\Delta Q{mid}}{\%\Delta P{mid}} = \frac{ (Q2 - Q1)/((Q2+Q1)/2) }{ (P2 - P1)/((P2+P_1)/2) }
Why midpoint? It avoids asymmetry and gives a single elasticity value for a price move, unlike the start/end method which yields different values depending on the direction of the change.

Our Scenario: Social Media Accounts Pricing

Current situation: charge \$P = 2000$ per business and maintain Q = 12 accounts/year.
Proposed price: \$P = 2500$.
Question: how many fewer accounts (Q) would you have? how would revenue change?
Observed data from the example:
- With a price increase to \$2500, Q falls to 8 (i.e., 4 fewer accounts).
Revenue check:
- At \$2000$ and 12 accounts: revenue R = (P\times Q = 2000 \times 12 = 24{,}000) (in dollars).
- At \$2500$ and 8 accounts: revenue R = (2500 \times 8 = 20{,}000).
- Revenue falls by 4{,}000 dollars.
Midpoint elasticity calculation (P: 2000 -> 2500, Q: 12 -> 8):
\%\Delta P{mid} = \frac{2500 - 2000}{(2500 + 2000)/2} \times 100 = \frac{500}{2250} \times 100 \approx 22.2\% \%\Delta Q{mid} = \frac{8 - 12}{(8 + 12)/2} \times 100 = \frac{-4}{10} \times 100 = -40.0\%
Ed = \frac{-40.0\%}{22.2\%} \approx -1.80 \Rightarrow |Ed| \approx 1.80
Interpretation:
- Elasticity > 1 in absolute value (elastic demand) implies that raising price reduces total revenue, which is consistent with the revenue decline observed here.
Additional checks using the standard method (start vs end values) show different results depending on direction:
- From B (P=2500, Q=8) to A (P=2000, Q=12):
  \%\Delta P{std} = \frac{2000 - 2500}{2500} \times 100 = -20\% \%\Delta Q{std} = \frac{12 - 8}{8} \times 100 = 50\%
  E_d^{std,B\to A} = \frac{50\%}{-20\%} = -2.50
- From A (P=2000, Q=12) to B (P=2500, Q=8):
  \%\Delta P{std} = \frac{2500 - 2000}{2000} \times 100 = 25\% \%\Delta Q{std} = \frac{8 - 12}{12} \times 100 = -33.3\%
  E_d^{std,A\to B} = \frac{-33.3\%}{25\%} = -1.333
- These illustrate why the midpoint method is preferred for a consistent elasticity measure.

Active Learning 1: Elasticity for iPhones (Midpoint Method)

Problem: Calculate the price elasticity of demand for iPhones given:
- Case 1: (P = 400, Q_d = 10{,}600)
- Case 2: (P = 600, Q_d = 8{,}400)
Midpoint changes:
\%\Delta P = \frac{600 - 400}{(600 + 400)/2} \times 100 = \frac{200}{500} \times 100 = 40\%
\%\Delta Qd = \frac{8400 - 10600}{(8400 + 10600)/2} \times 100 = \frac{-2200}{9500} \times 100 \approx -23.16\% Ed = \frac{-23.16\%}{40\%} \approx -0.579 \Rightarrow |E_d| \approx 0.58
Answer (as given in the notes):
- A. \% change in P = 40\%
- B. \% change in Q_d = -23.16\%
- C. Price elasticity of demand = (0.58) (taking absolute value)

Determinants of Price Elasticity of Demand

The elasticity of demand varies across goods depending on several factors. Three illustrative examples are discussed in the notes:
EXAMPLE 1: Cheerios vs Airfare
- Prices rise 20% for both goods.
- Cheerios has many close substitutes; airfare has few close substitutes.
- Conclusion: price elasticity is higher for Cheerios (more substitution options).
EXAMPLE 2: Insulin vs Rolex watches
- Insulin is a necessity for diabetics; a price rise causes little or no decrease in quantity demanded.
- A Rolex is a luxury; a price rise causes some people to forego it.
- Conclusion: elasticity higher for luxuries than for necessities.
EXAMPLE 3: Gasoline, Short Run vs Long Run
- A price rise in gasoline leads to limited short-run adjustments (no quick substitutes).
- In the long run, consumers can switch to smaller cars, carpool, etc.
- Conclusion: elasticity is higher in the long run than in the short run.

The Variety of Demand Curves

Elasticity and the categories of response:
- Elastic demand: (E_d > 1)
- Inelastic demand: (E_d < 1)
- Unit elastic: (E_d = 1)
A more detailed map of the demand curve:
- Perfectly inelastic demand: (E_d = 0); vertical demand curve (quantity demanded does not respond to price).
- Inelastic demand: (0 < E_d < 1); relatively steep demand curve.
- Unit elastic: (E_d = 1); intermediate slope.
- Elastic demand: (E_d > 1); relatively flat demand curve; quantity responds a lot to price changes.
- Perfectly elastic demand: (E_d = \infty); horizontal demand curve; quantity responds by any amount at a given price.
Key intuition: the flatter the demand curve, the greater the price elasticity of demand.

Perfectly Inelastic, Inelastic, Unit Elastic, Elastic, and Perfectly Elastic Demand (Visual Concepts)

Perfectly Inelastic: E_d = 0; % change in Q = 0 regardless of % change in P; vertical demand curve.
Inelastic: E_d < 1; quantity responds little to price changes.
Unit Elastic: E_d = 1; percentage changes in Q and P are equal in magnitude.
Elastic: E_d > 1; quantity responds strongly to price changes; demand is relatively flat.
Perfectly Elastic: E_d = ∞; at a given price, quantity demanded can be any amount; horizontal demand curve.

Real-World Elasticities (Selected Examples)

Eggs: 0.1
Healthcare: 0.2
Cigarettes: 0.4
Rice: 0.5
Housing: 0.7
Beef: 1.6
Peanut butter: 1.7
Restaurant meals: 2.3
Cheerios: 3.7
Mountain Dew: 4.4
Interpretation:
- Lower values (0.1–0.5): very inelastic; quantity responds very little to price changes (e.g., eggs, healthcare).
- Higher values (above 1): elastic; quantity responds strongly to price changes (e.g., restaurant meals, Cheerios, Mountain Dew).

Elasticity Along a Linear Demand Curve

A linear demand curve has a constant slope, but elasticity varies along the curve.
General formula for elasticity at a point on a demand curve:
E_d = \frac{dQ}{dP} \cdot \frac{P}{Q}
If the demand is linear: Q = a - bP, then dQ/dP = -b, so
E_d = (-b) \cdot \frac{P}{Q}
Implication:
- At higher prices (and lower quantities), elasticity is larger in magnitude (more elastic).
- At lower prices (and higher quantities), elasticity is smaller in magnitude (more inelastic).
The illustration notes that elasticity can take values such as 5.0, 1.0, or 0.2 along a single linear demand curve depending on the price and quantity point.

Quick Reference: Key Formulas

Price elasticity of demand (definition):
Ed = \frac{\%\Delta Qd}{\%\Delta P}
Percentage changes (standard):
\%\Delta P{std} = \frac{P2 - P1}{P1} \times 100, \quad \%\Delta Q{std} = \frac{Q2 - Q1}{Q1} \times 100
Percentage changes (midpoint):
\%\Delta P{mid} = \frac{P2 - P1}{(P2 + P1)/2} \times 100, \quad \%\Delta Q{mid} = \frac{Q2 - Q1}{(Q2 + Q1)/2} \times 100
Midpoint elasticity:
Ed = \frac{\%\Delta Q{mid}}{\%\Delta P{mid}} = \frac{ (Q2 - Q1)/((Q2+Q1)/2) }{ (P2 - P1)/((P2+P_1)/2) }
Elasticity for a linear demand curve: if Q = a - bP, then
E_d = (-b) \cdot \frac{P}{Q}

Connections to Revenue and Real-World Relevance

Revenue implications depend on elasticity:
- If elasticity in absolute value > 1 (elastic), a price increase tends to reduce total revenue.
- If elasticity in absolute value < 1 (inelastic), a price increase tends to raise total revenue.
- If elasticity in absolute value = 1 (unit elastic), a price increase leaves total revenue unchanged.
The social-media pricing example shows a revenue decline when price increases from 2000 to 2500 because the demand is elastic (|E_d| > 1).
Real-world elasticity data helps firms price goods, forecast consumer response, and assess tax incidence and policy effects.

Foundational Takeaways for Exam Prep

Always determine whether you should report elasticity as an absolute value (practice in the notes).
Use the midpoint method to compute percentage changes for a robust elasticity estimate.
Remember the relationship between elasticity and revenue: elastic demand means price hikes can reduce revenue; inelastic means price hikes can increase revenue.
Distinguish between short-run and long-run elasticity (elasticity tends to be higher in the long run due to more available adjustments).
Be able to identify the type of demand curve (elastic, inelastic, unit, perfect) from given elasticity values, and know how the slope relates to elasticity along a linear curve.

Quick Worked Example (Recap)

Given a move from P1 = 2000 to P2 = 2500 and Q1 = 12 to Q2 = 8:
- Midpoint percent changes:
  \%\Delta P{mid} \approx 22.2\% ,\; \%\Delta Q{mid} = -40.0\%
- Elasticity: |E_d| \approx 1.80 (elastic)
- Revenue change: from \$24{,}000 to \$20{,}000 (revenue falls).

Notes Summary

Elasticity is a central tool for understanding how price changes affect quantity demanded, revenue, and market behavior.
The midpoint method provides a stable way to calculate elasticity and avoids inconsistencies tied to direction of change.
The type of good (necessity vs luxury), availability of substitutes, and the time horizon all shape elasticity.
Real-world elasticity values vary widely across goods, reinforcing the idea that pricing strategies must account for consumer responsiveness.
On a linear demand curve, elasticity is not constant; it depends on where you are on the curve, via the relationship E_d = (-b) \cdot (P/Q).