Week 14 — Graph & Model Workshop · "Find the Social Optimum and the DWL of a Negative Externality"

Course: Principles of Microeconomics (ECON 1) · Silver Oak University (fictional sample) · Prof. Kessler
Objective 8 — economic modeling & quantitative/graphical analysis · SLO A
Worth 50 points · Model Workshops group = 15% of the grade · Workshop 14
Format: plot three curves in Desmos, find the market and social equilibria, compute the Pigouvian tax and the DWL, interpret every number 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. All tools are links to free external sites — nothing to buy or download.

Part 1 — The Big Picture

Markets fail when costs or benefits spill over to people who aren't in the transaction. A negative externality drives a wedge between the marginal private cost (MPC) the producer bears and the marginal social cost (MSC) society bears. The market ignores MSC and settles where MB = MPC — overshooting the socially efficient quantity. Today you'll plot the three lines (MB, MPC, MSC), find both equilibria, compute the Pigouvian tax that restores efficiency, and calculate the deadweight loss the externality creates.

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

Part 2 — The Guiding Question

When a factory's pollution costs the neighborhood $6 for every unit produced, exactly how much does the unregulated market overshoot — and what tax precisely closes the gap?

The market (pre-computed, verified — use these numbers exactly):
- MB (demand): P = 40 − Q
- MPC (private supply): P = 4 + 0.5Q
- Marginal external cost: $6 per unit
- MSC (social cost): P = 10 + 0.5Q (= MPC + 6)

Part 3 — Set Up the Model (in Desmos)

1. Open Desmos. Type each equation on its own line:
   - y = 40 - x (MB / demand — label it "MB")
   - y = 4 + 0.5x (MPC / private supply — label it "MPC")
   - y = 10 + 0.5x (MSC / social cost — label it "MSC")
2. You'll see three lines. Two of them run parallel (MPC and MSC — same slope, different intercepts). The MB line slopes down and intersects both of them.
3. Find the two intersections visually, then verify them with algebra.

Part 4 — Solve (complete this scaffold)

Show all the algebra. Pre-verify every answer with the equations before submitting.

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

Answer each in 1–2 sentences:

1. In plain English, what does Q_mkt = 24 and Q_soc = 20 mean — what is the market doing wrong, and who is harmed?
2. What does the DWL of 12 represent? Who loses that value, and does it appear anywhere in the producer's or consumer's accounting?
3. What does the Pigouvian tax do to the supply curve on the graph — does it shift MPC up, down, or not at all? How does that change where the market settles?

Part 6 — Analysis Questions

1. Direction check: suppose instead the factory creates a positive externality (the factory's heat warms a neighboring greenhouse, saving the farmer heating costs). In that case, does the unregulated market over- or underproduce relative to the social optimum? Would the correction be a tax or a subsidy? Explain the logic in one sentence.

2. Coase check: a neighbor argues: "The factory and the neighborhood should just negotiate — if the damage is only $6 per unit, they can work it out." Under what two conditions would Coase-theorem reasoning support this suggestion? Give one reason why those conditions might fail in a real neighborhood-vs-factory dispute.

3. Real-world connect: name one real-world industry or product where you think a negative externality exists. What would a Pigouvian tax on that industry target (what is the "external cost" per unit)?

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 where MB = 40 − Q, MPC = 4 + 0.5Q, and the marginal external cost is $6 per unit (so MSC = 10 + 0.5Q), find: (a) the market equilibrium Q and P; (b) the social optimum Q and P; (c) the correct Pigouvian tax; and (d) the deadweight loss of the externality."

2. Audit every claim against your own scaffold answers (Part 4):
   - Did it get Q_mkt = 24, P_mkt = 16? (Common error: adding $6 to MB, not MPC, when computing MSC.)
   - Did it get Q_soc = 20, P_soc = 20? (Common error: reading P off MPC at Q=20, getting 14, instead of MB at Q=20, getting 20.)
   - Did it correctly set the Pigouvian tax = $6? (Common error: saying "$4" or "$10.")
   - Did it compute DWL = ½ × 4 × 6 = 12? (The most common error: using Q_mkt = 24 as the base instead of ΔQ = 4. DWL = ½ × 24 × 6 = 72 is a typical chatbot mistake.)

3. Write 2–3 sentences naming what the AI got right and at least one thing you had to correct or watch. If it got everything right, explain how you verified each step.

The habit all term: the tool drafts, you judge. The DWL-base error (using Q_mkt instead of ΔQ) is one of the most consistent chatbot mistakes in this material — catching it here means you'll never make it yourself.

Part 8 — What to Submit

One document (or text entry) with: your Part 4 scaffold (algebra shown), your Part 5 interpretations, your Part 6 analysis answers, and your Part 7 AI-critique paragraph. A screenshot of your Desmos three-line graph is welcome but optional. Due Sun, Dec 6, 11:59 p.m. (50 points).

Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

Every number pre-computed and independently verified in Python. Use these for grading.

Part 4 scaffold — verified answers:

• (a) Market equilibrium (MB = MPC):
  40 − Q = 4 + 0.5Q → 36 = 1.5Q → Q_mkt = 24; P = 40 − 24 = 16. ✓

• (b) Social optimum (MB = MSC):
  40 − Q = 10 + 0.5Q → 30 = 1.5Q → Q_soc = 20; P = 40 − 20 = 20. ✓
  (Cross-check: MSC at Q=20 = 10 + 0.5·20 = 20 = MB at Q=20. ✓)

• (c) Overproduction: 24 − 20 = 4 units beyond the social optimum. ✓

• (d) Pigouvian tax: = marginal external cost = $6/unit. After the tax, effective supply = (4 + 6) + 0.5Q = 10 + 0.5Q = MSC → market settles at Q=20, P=20. ✓

• (e) DWL: ½ × (24 − 20) × 6 = ½ × 4 × 6 = $12. ✓
  (Triangle base = ΔQ = 4; height = external cost = $6. At the market Q=24, MSC=22 and MB=16, so MSC−MB=6. Confirm: ½ × 4 × 6 = 12. ✓)

Part 5 interpretations (accept equivalent wording):
1. The unregulated market produces 4 units (24 vs. 20) beyond what is efficient — for each of those extra units, the social cost (MSC) exceeds the social benefit (MB). The factory and buyers each gain surplus from those units, but the neighborhood absorbs uncompensated pollution damage.
2. The DWL of $12 is the net social value destroyed by overproduction — the surplus that would exist at the social optimum but is wiped out by the 4 extra units (MSC > MB). It does not appear in the factory's accounts or consumers' bills; it falls entirely on the uninvolved third parties.
3. The Pigouvian tax shifts the MPC curve up by exactly $6 (the tax is added to producers' cost per unit). The new effective supply curve = MSC. The market now settles at MB = new MPC = MSC → Q=20, P=20. The market equilibrium IS the social optimum.

Part 6 analysis:
1. Positive externality → market underproduces (MSB > MPB, so MB curve understates value) → correction = subsidy (not tax). The subsidy shifts demand up (or supply down) to the social optimum.
2. Coase conditions: (i) clear, enforceable property rights (who has the right to the clean air?) AND (ii) low transaction costs (few parties, easy negotiation). They fail when: many affected parties (many neighbors with heterogeneous damages), high negotiation/coordination costs, difficulty of assigning rights to diffuse pollution, or strategic holdout behavior.
3. Any reasonable real-world externality earns credit — e.g., vehicle emissions (external cost = health and climate damage per mile driven; Pigouvian approach = a fuel tax or per-mile tax targeting emissions). Accept any industry with a plausible external cost per unit.

Part 7 AI-critique (most common errors to flag):
- DWL computed as ½ × 24 × 6 = 72 (uses Q_mkt instead of ΔQ = 4). Correct: ½ × 4 × 6 = 12.
- P at social optimum read off MPC (14) instead of MB (20). Correct: P_soc = 40 − 20 = 20.
- Pigouvian tax set as $4 (e.g., "difference between the two prices P_mkt=16 and P_soc=20 minus something") instead of = the external cost = $6.
- MSC derived incorrectly (e.g., adding $6 to MB rather than MPC).

Grading rubric — 50 points

Quality gate (self-checked):
- Quantitative gate: Q_mkt=24, P_mkt=16, Q_soc=20, P_soc=20, Pigouvian tax=$6, DWL=12 — all Python-re-verified ✓.
- Graph-logic check: MSC = MPC + 6 (parallel, shifted up) ✓; market overshoots (24 > 20) ✓; Pigouvian tax shifts MPC up to MSC ✓; DWL triangle base = ΔQ=4, height=EMC=6 ✓; positive externality → underproduction → subsidy (not tax) ✓.