Week 6 — Graph & Model Workshop · "Draw the Surplus Triangles and Measure the Pie"

Course: Principles of Microeconomics (ECON 1) · Silver Oak University (fictional sample) · Prof. Kessler
Objective 4 — economic modeling & quantitative/graphical analysis · SLO A
Worth 50 points · Model Workshops group = 15% of the grade · Workshop 6
Format: graph supply and demand in Desmos (free, no account), identify the surplus triangles, compute CS and PS, interpret the result, 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

This week you saw that markets do something remarkable: when buyers and sellers meet at the competitive equilibrium, they create the maximum possible total economic value — the largest "pie" of consumer and producer surplus combined. That's not just a claim; it's something you can see on the diagram as two triangles, and measure with ½ × base × height. Today you'll build the market in Desmos, draw those triangles, compute every number, and then turn an AI loose on the same problem — and grade its work.

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

Part 2 — The Guiding Question

When demand and supply intersect at the competitive equilibrium, how large is the total "surplus pie" — and who gets which slice?

The market. Demand: P = 20 − 0.5Q. Supply: P = 4 + 0.5Q. Buyers and sellers meet in this market every day.

Part 3 — Set Up the Model (in Desmos)

1. Open Desmos. In the expression bar, type: y = 20 - 0.5x (demand).
2. On the next line, type: y = 4 + 0.5x (supply).
3. You'll see two lines intersecting. That intersection is the competitive equilibrium — the one price where the quantity demanded exactly equals the quantity supplied.
4. Find the intersection coordinates (you can click the dot in Desmos). Record the equilibrium point.

Part 4 — Solve (complete this scaffold)

Fill in the blanks. Show every step — algebra for equilibrium, formula for each surplus.

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

Write 3–4 sentences answering these:
1. What does CS = __ mean in plain English — what does it represent for buyers in this market?
2. What does PS = ____ mean for sellers?
3. What does "allocative efficiency" mean in the context of this market's equilibrium?

(Hint: don't just repeat the formula — explain the idea as if to someone who has never seen an economics diagram.)

Part 6 — Analysis Questions

1. CS vs. PS: In this market, CS = PS. Is that always true? In one sentence, what determines how the total surplus is split between buyers and sellers in general? (Think about which curve is steeper.)

2. Off-equilibrium loss: Suppose a price of P = 16 is enforced by law (above the equilibrium). Use your diagram to explain: (a) what happens to quantity traded (which side is the constraint?), and (b) whether CS goes up or down compared to equilibrium. (No computation required — reason from the diagram.)

3. Connect it: The discussion this week asks whether an "efficient" outcome is also a "fair" one. Based on what you've computed today, write one sentence that clearly separates the positive claim ("the equilibrium maximizes TS") from a normative question ("is the split of CS and PS fair?").

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: "For the market with demand P = 20 − 0.5Q and supply P = 4 + 0.5Q, find the equilibrium, then compute consumer surplus and producer surplus. Show your work."
2. Audit every claim against your Part 4 scaffold:
   - Did it find Q* = 16 and P* = 12? (Watch for arithmetic errors or skipping the algebra.)
   - Did it correctly identify CS as ½ × 16 × (20 − 12) = 64? Chatbots frequently forget the ½ (getting 128 instead), use the wrong height (using Q* instead of the intercept gap), or swap the CS and PS triangles.
   - Did it get PS = 64 from the correct formula, PS = ½ × Q* × (P* − supply intercept)?
   - Did it state TS = 128?
3. Write 2–3 sentences naming what the AI got right and at least one error or risk you caught (or, if it was correct on everything, explain exactly how you verified each step — that's still the skill).

The habit all term: the tool drafts, you judge. A chatbot will confidently use the rectangle area (½ missing) or flip the surplus triangles — catching it is the point.

Part 8 — What to Submit

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

Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

Every number pre-computed and independently verified. Demand: P = 20 − 0.5Q; Supply: P = 4 + 0.5Q.

• (a) Set equal: 20 − 0.5Q = 4 + 0.5Q → 16 = Q → Q* = 16. ✓
• (b) P* = 20 − 0.5 × 16 = 20 − 8 = 12. ✓
• (c) Verify: supply gives 4 + 0.5 × 16 = 4 + 8 = 12. ✓ (matches)
• (d) Demand intercept: Q = 0 → P = 20. Demand intercept = 20. ✓
• (e) Supply intercept: Q = 0 → P = 4. Supply intercept = 4. ✓
• (f) CS = ½ × 16 × (20 − 12) = ½ × 16 × 8 = 64. ✓
• (g) PS = ½ × 16 × (12 − 4) = ½ × 16 × 8 = 64. ✓
• (h) TS = 64 + 64 = 128. ✓
• Part 5: CS = 64 means buyers as a group gain $64 of value beyond what they pay at P* = 12 — it's the accumulated "I would have paid more" across all 16 units traded. PS = 64 means sellers gain $64 above their minimum acceptable price — the accumulated "I would have accepted less." Allocative efficiency means Q* = 16 is exactly the right output level: every unit where the buyer's value (demand) exceeds the seller's cost (supply) gets produced; no unit where cost exceeds value is produced. Total surplus of 128 cannot be increased.
• Part 6: (1) CS = PS here is a coincidence of equal slope magnitudes (both ½). In general, the side with the more inelastic curve (steeper slope) captures a larger share of total surplus. (2) At P = 16: buyers demand only (20 − 16) / 0.5 = 8 units — demand is the binding constraint. Compared to equilibrium (CS = 64), CS at Q = 8, P = 16 is ½ × 8 × (20 − 16) = 16 — CS falls sharply. (3) Example: "The equilibrium maximizes TS = 128 [positive]; whether the 64/64 split is fair [normative] is a separate question that economics alone cannot answer."
• Part 7: Most common AI errors to look for: (i) forgetting the ½ and reporting CS = 128 or PS = 128; (ii) flipping CS and PS; (iii) using Q* as the height instead of (demand intercept − P*); (iv) rounding the equilibrium. If the AI is correct, credit goes to the student for specifying how each step was verified.

Grading rubric — 50 points

Quality gate (self-checked): quantitative gate — Q*=16, P*=12 (set 20−0.5Q=4+0.5Q, verified supply=12), intercepts (20 and 4), CS=½×16×8=64, PS=½×16×8=64, TS=128 — all Python-re-verified ✓. Graph-logic check — CS is the triangle UNDER demand ABOVE P*=12; PS is the triangle ABOVE supply BELOW P*=12; both triangles share base Q*=16; efficiency = TS maximized at equilibrium; CS=PS here because slopes are equal in magnitude (±0.5) — all correct ✓.