Week 7 — Graph & Model Workshop · "Taxing the Market: Incidence, Revenue & Deadweight Loss"

Course: Principles of Microeconomics (ECON 1) · Silver Oak University (fictional sample) · Prof. Kessler
Objective 4 — surplus, efficiency & government intervention · SLO A
Worth 50 points · Model Workshops group = 15% of the grade · Workshop 7
Format: graph supply, demand, and a tax wedge in Desmos (free, no account), solve for incidence and DWL, interpret in words, then catch the AI's mistakes.

This is the course's signature weekly component — the economics analog of a lab. Every instructional week has one workshop: you set up a model, solve it, and explain what it means.

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Part 1 — The Big Picture

Last week (Workshop 6) you computed consumer and producer surplus at the competitive equilibrium — the point where total surplus is maximized. This week the government steps in and imposes a per-unit tax. You'll see exactly how much of that maximum surplus is captured as tax revenue, how much is redistributed between buyers and sellers (incidence), and how much simply disappears — the deadweight loss.

The tool: 🔗 Desmos Graphing Calculator — https://www.desmos.com/calculator (free, instant, no login)

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Part 2 — The Guiding Question

When a $4/unit tax is imposed on this market, who really pays — and how much value is permanently lost?

The verified market:
- Demand: P = 20 − 0.5Q
- Supply: P = 4 + 0.5Q
- Free-market equilibrium: Q = 16, P = 12 (pre-computed: 20 − 0.5Q = 4 + 0.5Q → Q = 16)

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Part 3 — Set Up the Model (in Desmos)

1. Open Desmos. Enter these three equations:
   - Demand: y = 20 - 0.5x
   - Original supply: y = 4 + 0.5x
   - Tax-shifted supply: y = 8 + 0.5x (original supply + $4 tax)
2. You'll see two upward-sloping lines and one downward-sloping line. Confirm:
   - Demand meets original supply at (16, 12) — the free-market equilibrium.
   - Demand meets the tax-shifted supply at a new point.
3. The vertical gap between the two supply lines at any quantity is $4 — the tax wedge.

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Part 4 — Solve (complete this scaffold)

Fill in each blank. Show all arithmetic.

Question	Your answer
(a) Free-market Q and P (read from the Desmos intersection or solve algebraically)	Q = ____ , P = ____
(b) New equilibrium Q after the $4 tax (set demand = tax-shifted supply: 20 − 0.5Q = 8 + 0.5Q)	Q = ____
(c) Buyer price Pb (plug new Q into the demand equation: P = 20 − 0.5Q)	Pb = ____
(d) Seller price Ps (Pb minus the $4 tax; also verify using original supply: P = 4 + 0.5Q)	Ps = ____
(e) Buyer's burden (Pb minus the original P*)	____
(f) Seller's burden (original P* minus Ps)	____
(g) Tax revenue = tax × new Q	____
(h) Deadweight loss = ½ × tax × ΔQ (where ΔQ = Q* − Q_new)	____

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

In 3–4 sentences, explain:
- Who bore more of the $4 tax burden — buyers or sellers — and why the split came out the way it did. (Hint: compare the demand and supply slopes.)
- What the deadweight loss means in plain language — not just its formula, but what actually happened to those transactions.

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Part 6 — Analysis Questions

1. Incidence rule. The $4 tax split exactly 50/50 here. Suppose demand were much more inelastic (a steeper demand curve) while supply stayed the same. Would buyers bear more, less, or the same share? Explain the logic in one sentence without needing to compute anything.

2. Revenue vs. DWL. Your answer to (g) is tax revenue; your answer to (h) is the DWL. Together they make up the efficiency cost and transfer from the market. Is tax revenue a loss to society? Is DWL? Explain the difference in 2–3 sentences.

3. Policy tease. Suppose instead of taxing this market, the government paid a $4/unit subsidy to sellers. Would the new equilibrium Q be higher or lower than 16? Would the new Pb be higher or lower than $12? Just predict — no calculation needed.

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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: "In a market with demand P = 20 − 0.5Q and supply P = 4 + 0.5Q, a $4/unit tax is levied on sellers. What is the new equilibrium quantity, the buyer price, the seller price, the tax revenue, and the deadweight loss?"
2. Audit every claim against your own scaffold:
   - Did it get Q = 12, Pb = 14, Ps = 10? (Common error: shifting demand instead of supply, or shifting supply the wrong direction.)
   - Did it compute DWL = 8? (Common error: DWL = tax revenue = 48, or using ΔP instead of ΔQ in the formula.)
   - Did it correctly split incidence 50/50 given the equal slopes? (Common error: claiming sellers bear all of it because they "pay" the tax.)
3. Write 2–3 sentences naming what the AI got right and at least one thing you had to correct or verify carefully. If it got everything right, explain how you verified each claim — that is the skill.

Chatbots routinely shift the wrong curve, mix up Pb and Ps, or equate DWL with tax revenue. Catching these errors is the point.

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Part 8 — What to Submit

One document (or text entry) with: your Part 4 scaffold (with arithmetic), your Part 5 interpretation, your Part 6 answers, and your Part 7 AI-critique paragraph. A screenshot of your Desmos graph (all three lines) is welcome but optional. Due Sun, Oct 18, 11:59 p.m. (50 points).

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Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

Every number below is pre-computed and independently verified in Python. All pass ✓.

(a) 20 − 0.5Q = 4 + 0.5Q → Q = 16, P = 12. ✓

(b) 20 − 0.5Q = 8 + 0.5Q → Q = 12. ✓

(c) Pb = 20 − 0.5(12) = 14. ✓

(d) Ps = 14 − 4 = 10. Verify: 4 + 0.5(12) = 10. ✓

(e) Buyer's burden = 14 − 12 = 2. ✓

(f) Seller's burden = 12 − 10 = 2. ✓

(g) Tax revenue = 4 × 12 = 48. ✓

(h) ΔQ = 16 − 12 = 4. DWL = ½ × 4 × 4 = 8. ✓

Part 5: The $4 tax split exactly 50/50 (buyers pay $2 more; sellers net $2 less) because demand and supply have identical slopes (|−0.5| = |+0.5|); neither side can dodge the tax more than the other. The DWL of 8 represents the 4 transactions (Q dropped from 16 to 12) that would have been mutually beneficial — a buyer's willingness to pay exceeded a seller's willingness to accept at some price — but the tax wedge made them impossible. That surplus is not transferred to government; it simply ceases to exist.

Part 6:
1. If demand were more inelastic (steeper), buyers would bear more of the tax — they can't easily reduce quantity demanded when price rises, so sellers can pass more of the cost forward.
2. Tax revenue ($48) is a transfer from buyers and sellers to the government — it's a loss to those two parties, but the government has it. DWL ($8) is a pure efficiency loss — value that no one has; it represents the foregone gains from the missing transactions.
3. A subsidy shifts supply down by $4 → new supply P = 0 + 0.5Q → new Q > 16; Pb < 12.

Part 7: Common AI errors to watch for — shifting demand instead of supply (wrong curve); getting Q = 14 or Q = 8 (algebra error); claiming DWL = 48 (tax revenue confusion); saying sellers bear the entire burden because they "pay" the tax.

Grading rubric — 50 points

Criterion	Full	Partial	None
Scaffold (Part 4) — all 8 cells correct with arithmetic: Q=16/P=12, Q=12, Pb=14, Ps=10, burdens=2 each, revenue=48, DWL=8 (24)	24	12–20	0–10
Interpretation (Part 5) — why split was 50/50 (equal slopes) + what DWL means in words (10)	10	5–8	0–4
Analysis (Part 6) — incidence logic (inelastic → more burden); revenue vs. DWL distinction; subsidy prediction (8)	8	4–6	0–3
AI-critique (Part 7) — names a specific verified or corrected claim in the AI's answer (8)	8	4–6	0–3

Quality gate (self-checked): quantitative gate — all 8 scaffold cells Python-re-verified ✓. Graph-logic check — tax on sellers shifts supply UP (not demand, not down); ceiling below eq → shortage; floor above eq → surplus; DWL = ½ · t · ΔQ (not revenue): all correct ✓.