Week 5 — Graph & Model Workshop · "Computing and Interpreting Elasticity"

Course: Principles of Microeconomics (ECON 1) · Silver Oak University (fictional sample) · Prof. Kessler
Objective 3 — Elasticity (price, income, cross) & total revenue · SLO A
Worth 50 points · Model Workshops group = 15% of the grade · Workshop 5
Format: use the midpoint formula to compute and interpret all four elasticities on the verified week numbers, set up the demand curve in Desmos (free, no account), then catch the AI's mistakes.

This is the course's signature weekly component. Every instructional week has one workshop: you set up a model, solve it, and explain what it means. All tools are links to free external sites.

---

Part 1 — The Big Picture

This week's toolkit is elasticity — a measure of how sensitive quantity is to price, income, or the price of a related good. Elasticity drives pricing strategy (should a firm raise or lower price to earn more revenue?), tax policy (which goods produce stable revenue and who bears the burden?), and the interpretation of market data.

Today you will compute all four elasticities from the week's verified numbers, apply the total revenue test, and explain every result in plain English. Then you'll have the AI attempt the same problems — and catch its errors.

Tools:
- Desmos Graphing Calculator: https://www.desmos.com/calculator (free, no login)
- Spreadsheet (Google Sheets or Excel): optional, for organizing your calculations

---

Part 2 — The Guiding Question

When a price, an income, or a related good's price changes — how much does quantity demanded actually move, and what does that tell a firm or a government about revenue and incidence?

The verified numbers for this workshop:
- Inelastic segment: P rises $4 → $6; Q falls 80 → 60 (demand curve Qd = 120 − 10P).
- Elastic segment: P rises $10 → $12; Q falls 100 → 70.
- Income change: income rises $40,000 → $60,000; Q demanded rises 50 → 70.
- Cross-price change: price of coffee rises $2 → $3; Q of tea demanded rises 100 → 120.

---

Part 3 — Set Up the Model (in Desmos)

Plot the demand curve Qd = 120 − 10P in Desmos. Enter it as y = 120 − 10x (x = price, y = quantity demanded).

1. Find Q when P = 4: read y when x = 4. It should equal 80.
2. Find Q when P = 6: it should equal 60.
3. Find Q when P = 10 on this curve: Q = 120 − 10(10) = 20. (Heads up: the elastic-case example in Part 6 uses a separate demand scenario — keep the two straight; here we work the curve Qd = 120 − 10P.)
4. Mark the two points (4, 80) and (6, 60) on your Desmos graph.

---

Part 4 — Solve (complete this scaffold)

Show every midpoint-formula step. Do not skip the numerator and denominator.

Section A — Price Elasticity of Demand (inelastic segment)

Step	Your work
Q₁ and Q₂	80 and 60
P₁ and P₂	$4 and $6
%ΔQ = (Q₂−Q₁) / average(Q₁,Q₂)	(60−80) / ((80+60)/2) = −20 / 70 = ___
%ΔP = (P₂−P₁) / average(P₁,P₂)	(6−4) / ((4+6)/2) = 2 / 5 = ___
PED = %ΔQ ÷ %ΔP	___ ÷ ___ = ___
	PED
Classification	elastic / inelastic / unit-elastic?
TR at P₁	$4 × 80 = ___
TR at P₂	$6 × 60 = ___
TR test	P ↑ and TR ↑/↓ → confirms elastic/inelastic?

Section B — Price Elasticity of Demand (elastic segment)

Step	Your work
%ΔQ	(70−100) / ((100+70)/2) = −30 / 85 = ___
%ΔP	(12−10) / ((10+12)/2) = 2 / 11 = ___
PED	___
	PED
TR at P=$10	$10 × 100 = ___
TR at P=$12	$12 × 70 = ___
TR test	___

Section C — Income Elasticity of Demand

Step	Your work
%ΔQ	(70−50) / ((50+70)/2) = 20 / 60 = ___
%ΔY (income)	(60,000−40,000) / ((40,000+60,000)/2) = 20,000 / 50,000 = ___
YED	___ ÷ ___ = ___
Classification	normal (necessity / luxury) or inferior?

Section D — Cross-Price Elasticity of Demand

Step	Your work
%ΔQ_tea	(120−100) / ((100+120)/2) = 20 / 110 = ___
%ΔP_coffee	(3−2) / ((2+3)/2) = 1 / 2.5 = ___
XED	___ ÷ ___ = ___
Classification	substitutes / complements / unrelated?

---

Part 5 — Interpret in Words (the SLO-A skill)

Write 1–2 sentences for each of the following interpretations:

1. Section A (inelastic): What does PED ≈ −0.71 mean for a seller who is deciding whether to raise price? What does the TR test confirm?
2. Section B (elastic): What does PED ≈ −1.94 mean for the same question? How does the TR test answer differ?
3. Elasticity ≠ slope: Explain in your own words why these two segments of a demand curve can have different |PED| values even if the demand curve has a constant slope. (Hint: what changes when price and quantity are at different levels?)
4. Section C (YED): What does YED ≈ +0.83 tell us about this good? Is it a normal or inferior good — and is it closer to a necessity or a luxury?
5. Section D (XED): What does XED ≈ +0.45 tell us about coffee and tea? What would a negative XED have meant instead?

---

Part 6 — Analysis Questions

1. A city wants to tax either coffee (|PED| = 1.9) or gasoline (|PED| = 0.3) to fund road repairs. Based only on revenue stability, which should the city tax? Which group of buyers bears the larger share of the burden, and why?

2. A transit authority is considering raising fares. Using the inelastic segment verified numbers ($4 → $6, Q 80 → 60, PED = −5/7), compute the change in total revenue and explain why the authority's revenue rises even though it sells fewer rides.

3. Connect it: think of a good you buy regularly. Based on SLAT (Substitutes, Luxury vs. necessity, Adjustment time, budget share), estimate whether your own demand for it is elastic or inelastic — and what that implies for how a seller should price it.

---

Part 7 — AI-Critique Moment (required — the BYOAI step)

Bring in your approved chatbot (Gemini, Claude, or ChatGPT) and be the economist who checks its work.

1. Paste this to the chatbot:
   "Using the midpoint formula, compute the price elasticity of demand when price rises from $4 to $6 and quantity falls from 80 to 60. Then compute the income elasticity when income rises from $40,000 to $60,000 and quantity rises from 50 to 70. Classify each result."

2. Audit every claim against your own work:
   - Did it use the midpoint formula correctly — or the simpler (asymmetric) percent-change formula?
   - Did it get PED = −5/7 ≈ −0.71 (inelastic)? Or did it make a sign error (positive PED)?
   - Did it get YED = +5/6 ≈ +0.83 (normal good, necessity range)?
   - Did it correctly classify: |PED| < 1 → inelastic; YED > 0 → normal good?
   - Did it confuse elasticity with slope anywhere?

3. Write 2–3 sentences naming what the AI got right and at least one error or near-miss you caught (or, if it was perfect, explain exactly how you verified each step — that's the skill too).

The habit all term: the tool drafts, you judge. Chatbots often forget the midpoint average denominator, flip the sign of PED, or call YED < 1 an "inferior" good when it's actually a normal necessity. Catching these is the point.

---

Part 8 — What to Submit

One document (or text entry) with: your Part 4 scaffold (with every arithmetic step), your Part 5 interpretations (five short paragraphs), your Part 6 analysis (three questions), and your Part 7 AI-critique paragraph. A screenshot of your Desmos graph is welcome but not required. Due Sun, Oct 4, 11:59 p.m. (50 points).

---

Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

Every number pre-computed and independently verified in Python. The midpoint formula produces the following results.

Section A — PED (inelastic):
- %ΔQ = −20/70 = −2/7
- %ΔP = 2/5
- PED = (−2/7) ÷ (2/5) = (−2/7) × (5/2) = −10/14 = −5/7 ≈ −0.71 ✓
- |PED| = 0.71 < 1 → inelastic ✓
- TR₁ = $4 × 80 = $320; TR₂ = $6 × 60 = $360. TR ↑ when P ↑ → same direction → inelastic confirmed ✓

Section B — PED (elastic):
- %ΔQ = −30/85 = −6/17
- %ΔP = 2/11
- PED = (−6/17) ÷ (2/11) = (−6/17) × (11/2) = −66/34 = −33/17 ≈ −1.94 ✓
- |PED| = 1.94 > 1 → elastic ✓
- TR₁ = $10 × 100 = $1,000; TR₂ = $12 × 70 = $840. TR ↓ when P ↑ → opposite → elastic confirmed ✓

Section C — YED:
- %ΔQ = 20/60 = 1/3
- %ΔY = 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 range ✓

Section D — XED:
- %ΔQ_tea = 20/110 = 2/11
- %ΔP_coffee = 1/2.5 = 2/5
- XED = (2/11) ÷ (2/5) = (2/11) × (5/2) = 10/22 = 5/11 ≈ +0.45 ✓
- XED > 0 → substitutes ✓

Part 5 model interpretations:
1. PED ≈ −0.71 means a 1% price increase leads to only a 0.71% drop in quantity demanded — buyers barely react. The seller gains revenue by raising price (TR $320→$360).
2. PED ≈ −1.94 means a 1% price increase leads to a 1.94% drop in quantity demanded — buyers are very responsive. TR falls ($1,000→$840); the seller loses revenue by raising price.
3. Elasticity ≠ slope: slope uses absolute changes (ΔP/ΔQ), which are constant on a linear curve. But elasticity uses percentages. At high-P/low-Q, the same ΔP is a smaller % of a large P, so the percentage changes are large → elastic. At low-P/high-Q, the same ΔP is a large % of a small P → inelastic.
4. YED ≈ +0.83: quantity demanded rises when income rises → normal good. YED < 1 → necessity (income goes up more than quantity does proportionally). Examples: utilities, basic food.
5. XED ≈ +0.45 > 0: when coffee price rises, tea demand rises — buyers switch from coffee to tea → substitutes. Negative XED would mean the goods are complements (price of one up → quantity of other down).

Part 6 model answers:
1. Tax gasoline (|PED|=0.3, inelastic): quantity barely falls → tax base stable → more reliable revenue. Buyers (the inelastic side) bear most of the burden because they can't easily reduce their purchases.
2. TR change: $320 → $360, an increase of $40. Even though they sell 20 fewer rides, the higher price more than compensates because demand is inelastic — each remaining rider pays $2 more, and the quantity drop is modest.
3. Any reasonable personal example with SLAT reasoning earns credit.

Part 7 AI-critique checklist (common errors to catch):
- Using simple % change (e.g., %ΔQ = −20/80 = −25% and %ΔP = 2/4 = 50%) instead of midpoint → gets PED = −0.5 (wrong)
- Positive PED (sign error — forgot the sign convention of the law of demand)
- Calling YED = 0.83 an "inferior" good (wrong — it's positive, so normal)
- Confusing elasticity with slope ("the curve is steep, so demand is inelastic")

Grading rubric — 50 points

Criterion	Full	Partial	None
Scaffold (Part 4) — all four elasticity calculations with every arithmetic step, correct results, and TR test applied to both PED cases (20)	20	10–16	0–8
Interpretations (Part 5) — five short paragraphs that translate each result into plain English, including the elasticity ≠ slope explanation (14)	14	7–11	0–6
Analysis (Part 6) — three questions answered with elasticity reasoning; TR change computed (10)	10	5–8	0–4
AI-critique (Part 7) — names a specific error or verification step in the AI's answer (6)	6	3–5	0–2

Quality gate (self-checked): quantitative gate — PED −5/7 ≈ −0.71 (inelastic, TR $320→$360 ✓); PED −33/17 ≈ −1.94 (elastic, TR $1,000→$840 ✓); YED 5/6 ≈ +0.83 (normal, necessity ✓); XED 5/11 ≈ +0.45 (substitutes ✓) — all Python-re-verified. Graph-logic check: TR test directions correct (inelastic → same direction; elastic → opposite direction); elasticity ≠ slope explicitly stated; YED sign correct; XED sign correct. PASS ✓