Week 5 — Lecture Outline · Elasticity

Course: Principles of Microeconomics (ECON 1) · Silver Oak University (fictional sample) · Prof. Kessler
Objective 3 — Elasticity (price, income, cross) & total revenue · SLO A & B
Meeting pattern: two 75-min sessions (≈150 min). Segments below total ~150 min.

The deck (E), the tutorial (C), and the workshop (P) all teach from this outline. Every number here is pre-computed and independently verified.

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Week at a glance

Big question	How responsive is quantity demanded to a price change — and does that responsiveness make a seller richer or poorer when price rises?
By week's end students can	(1) Compute PED via the midpoint formula and classify it; (2) use the TR test to confirm; (3) compute YED and XED and interpret each; (4) explain why elasticity ≠ slope; (5) name the four determinants that make demand more elastic.
Key vocabulary	price elasticity of demand (PED), midpoint formula, elastic / inelastic / unit-elastic, total revenue (TR) test, income elasticity (YED), cross-price elasticity (XED), normal / inferior / luxury good, substitutes / complements, determinants of elasticity
Materials	whiteboard; Desmos; calculator; approved chatbot
Timing note	8 segments across two sessions. Segment 7 (interaction) can be trimmed.

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Segment 1 — HOOK: "Should a transit authority raise fares?" (10 min)

Open with a real-world puzzle: a city's transit authority is running a deficit. Someone proposes raising the bus fare from $2 to $3. Will that raise more revenue — or less?

The answer depends entirely on how much ridership falls. If people have no good alternatives, they keep riding and revenue goes up. If they easily switch to biking or rideshare, ridership crumbles and revenue falls. The concept that captures this: elasticity — the responsiveness of quantity to a price change.

Promise: "By the end of this week, you'll be able to compute exactly how elastic or inelastic any demand segment is, and you'll predict which way total revenue moves without running the numbers."

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Segment 2 — PLAIN-LANGUAGE IDEA: what elasticity means (15 min)

Price elasticity of demand (PED) = the percentage change in quantity demanded ÷ the percentage change in price.

Plain-language first: PED tells us how sensitive buyers are. A large |PED| means buyers are very responsive — they cut purchases a lot when price rises. A small |PED| means buyers barely react.

Classifications:
- |PED| > 1 → elastic (quantity % change exceeds price % change; buyers very responsive)
- |PED| < 1 → inelastic (quantity % change less than price % change; buyers barely react)
- |PED| = 1 → unit elastic (both percentages equal; TR unchanged)

PED is almost always negative (law of demand — price up, quantity down). We often work with |PED| and note the sign separately.

Four determinants of elasticity (help students remember with "SLAT"):
1. Substitutes — more substitutes → more elastic (easy to switch)
2. Luxury vs. necessity — luxuries more elastic, necessities inelastic
3. Adjustment time — longer time horizon → more elastic (find alternatives)
4. Budget share — bigger budget share → more elastic (price change really hurts)

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Segment 3 — THE MIDPOINT FORMULA (20 min)

✅ VERIFIED NUMBERS (pre-computed; do not recompute live)

Inelastic case: P rises $4 → $6; Q falls 80 → 60.
- %ΔQ = (60 − 80)/((80 + 60)/2) = −20/70 = −2/7
- %ΔP = (6 − 4)/((4 + 6)/2) = 2/5
- PED = (−2/7) ÷ (2/5) = (−2/7) × (5/2) = −10/14 = −5/7 ≈ −0.71
- |PED| = 0.71 < 1 → inelastic

TR check: 4 × 80 = $320 → 6 × 60 = $360 (TR rises when P rises → confirms inelastic) ✓

Elastic case: P rises $10 → $12; Q falls 100 → 70.
- %ΔQ = (70 − 100)/((100 + 70)/2) = −30/85 = −6/17
- %ΔP = (12 − 10)/((10 + 12)/2) = 2/11
- PED = (−6/17) ÷ (2/11) = (−6/17) × (11/2) = −66/34 = −33/17 ≈ −1.94
- |PED| = 1.94 > 1 → elastic

TR check: 10 × 100 = $1,000 → 12 × 70 = $840 (TR falls when P rises → confirms elastic) ✓

Why the midpoint formula? Using the average of the two values as the base gives the same percentage regardless of direction (going from $4 to $6 gives the same |%ΔP| as $6 to $4). The simple formula gives different answers depending on direction — inconsistent and misleading.

Do both examples on the board, step by step. Have students track the arithmetic, then interpret: "So moving from $4 to $6 on this demand segment, the seller gains revenue. Moving from $10 to $12 on the elastic segment, the seller loses revenue."

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Segment 4 — THE TR TEST (12 min)

The total revenue test is a quick shortcut:

Demand is...	Price ↑ → TR	Price ↓ → TR
Inelastic	↑ (same direction)	↓ (same direction)
Elastic	↓ (opposite direction)	↑ (opposite direction)
Unit elastic	unchanged	unchanged

Memory hook: "Inelastic = price and TR travel together. Elastic = price and TR travel in opposite directions."

Intuition: when demand is inelastic, buyers barely cut quantity, so the higher price is multiplied by nearly the same quantity — TR rises. When demand is elastic, buyers cut quantity sharply, so the revenue lost from fewer units outweighs the revenue gained from the higher price.

Confirm with the verified numbers: inelastic $4→$6, TR $320→$360 (same direction ✓); elastic $10→$12, TR $1,000→$840 (opposite direction ✓).

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Segment 5 — ELASTICITY ≠ SLOPE (15 min)

This is the most important conceptual point of the week. Reinforce it explicitly.

Slope = ΔP/ΔQ (or ΔQ/ΔP) — a ratio of absolute changes; constant along a linear demand curve.

PED = (%ΔQ)/(%ΔP) — a ratio of percentage changes; changes at every point along a linear demand curve.

Why they differ: as you move down a linear demand curve, P falls and Q rises, so the same absolute changes become a smaller percentage of P and a larger percentage of Q — PED moves from elastic (upper segment) toward inelastic (lower segment) even though slope is constant.

Illustration (draw on board): a single linear demand curve has elastic demand near the top (high P, low Q → small price change is a big %) and inelastic demand near the bottom (low P, high Q → same absolute change is a small %). There is exactly one point in the middle where PED = −1 (unit elastic).

Two demand curves with the same slope can have different elasticities at any given price. And two demand curves with different slopes can have the same elasticity at some price. Slope is a geometric property; elasticity is a behavioral measure.

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Segment 6 — INCOME & CROSS-PRICE ELASTICITY (20 min)

✅ VERIFIED NUMBERS (pre-computed)

YED (income elasticity): income rises $40k → $60k; Q demanded rises 50 → 70.
- %ΔQ = (70 − 50)/((50 + 70)/2) = 20/60 = 1/3
- %ΔY = (60,000 − 40,000)/((40,000 + 60,000)/2) = 20,000/50,000 = 2/5
- YED = (1/3) ÷ (2/5) = (1/3) × (5/2) = 5/6 ≈ +0.83
- YED > 0 → normal good; 0 < YED < 1 → necessity (quantity rises with income, but less than proportionally) ✓

XED (cross-price elasticity): price of coffee rises $2 → $3; Q of tea demanded rises 100 → 120.
- %ΔQ_tea = (120 − 100)/((100 + 120)/2) = 20/110 = 2/11
- %ΔP_coffee = (3 − 2)/((2 + 3)/2) = 1/2.5 = 2/5
- XED = (2/11) ÷ (2/5) = (2/11) × (5/2) = 10/22 = 5/11 ≈ +0.45
- XED > 0 → substitutes (coffee price ↑ → more tea demanded) ✓

Income elasticity of demand (YED) = %ΔQ ÷ %ΔIncome:
- YED > 0 → normal good (Q rises when income rises)
- 0 < YED < 1 → necessity (Q rises, but slowly)
- YED > 1 → luxury (Q rises faster than income)
- YED < 0 → inferior good (Q falls when income rises — people trade up)
- Examples: name brands vs. store brands; organic food; bus rides vs. car ownership

Cross-price elasticity of demand (XED) = %ΔQᵢ ÷ %ΔPⱼ (good i in response to price of good j):
- XED > 0 → substitutes (Pⱼ↑ → Qᵢ↑, buyers switch to i)
- XED < 0 → complements (Pⱼ↑ → Qᵢ↓, buyers want less of both)
- XED ≈ 0 → unrelated goods

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Segment 7 — TECHNOLOGY WORKFLOW + AI-CRITIQUE (15 min)

Desmos demo: plot Qd = 120 − 10P (enter as y = 120 − 10x, with x = price and y = quantity). Read Q at P = 4 (Q = 80) and P = 6 (Q = 60). Use these to recompute PED — confirm −5/7.

AI-critique moment: Ask a chatbot to compute PED between (P=4, Q=80) and (P=6, Q=60) using the midpoint formula. Common chatbot errors to catch:
- Using the simple percent change formula (different answer depending on direction) instead of the midpoint formula
- Forgetting the negative sign and calling the result positive
- Confusing elasticity with the slope of the demand curve
- Getting the TR test backward (saying TR falls when demand is inelastic)

Have the class catch at least one error and state the correct reasoning.

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Segment 8 — CALLBACKS, TEASE & THE WEEK'S WORK (13 min)

• Callback: elasticity tells us how much quantity responds — a skill we built on top of Weeks 3–4's demand, supply, and equilibrium mechanics.
• Tease next week: "Now that we know the size of the market — how much surplus do buyers and sellers capture? That's consumer and producer surplus, the topic that lets us measure the welfare gains from a competitive market."
• The week's work: Lecture Tutorial (midpoint formula → classify → TR test → YED → XED), Practice, Quiz 5 (closed to AI), Discussion 5, Assignment 5, and Workshop 5 (compute all four elasticities and catch the chatbot's mistakes).

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Instructor FAQ — common stumbles

• "Why is PED negative?" The law of demand: P↑ → Qd↓, so %ΔQ and %ΔP always have opposite signs → ratio is negative. We often work with |PED| for classification but keep track of the sign.
• "Elastic vs. inelastic — which way does TR go?" Inelastic = P and TR move together. Elastic = P and TR move opposite. Unit elastic = TR doesn't change. Drill until automatic.
• "Why doesn't elasticity equal slope?" Slope uses absolute changes; elasticity uses percentage changes. The percentages shift as you move along the curve even when absolute changes don't.
• "Is gasoline elastic or inelastic?" In the short run: inelastic (few alternatives immediately). In the long run: more elastic (buy a more fuel-efficient car, move closer to work). Determinant: time horizon.
• "YED vs. XED — how do I keep them straight?" YED responds to income (normal/inferior/luxury). XED responds to the price of another good (substitutes/complements). Different question, different elasticity.
• "Can YED be negative?" Yes — an inferior good's quantity falls when income rises (people trade up). Examples: instant noodles, bus passes (in some contexts).