Week 11 — Graph & Model Workshop · "Monopoly: MR = MC, Read Price off Demand, Compute DWL"

Course: Principles of Microeconomics (ECON 1) · Silver Oak University (fictional sample) · Prof. Kessler
Objective 6 — perfect competition & monopoly (MR = MC, market power) · SLO A
Worth 50 points · Model Workshops group = 15% of the grade · Workshop 11
Format: derive MR from demand, find the monopoly optimum, graph it in Desmos, compute deadweight loss, and then catch the AI's mistake.

This is the course's signature weekly component. 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

Last week you found a perfectly competitive firm's output by setting P = MC. This week a single seller faces the whole market's demand curve — so it can choose the price, not just accept it. The catch: to sell more, the monopolist must lower the price on every unit, which makes marginal revenue less than price and leads to a smaller-than-competitive output and a higher-than-competitive price. The gap in between is the deadweight loss — real value that evaporates.

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

Part 2 — The Guiding Question

When a single seller controls a market, where does it produce, what price does it charge, and how much does society lose compared to a competitive market?

The scenario. A monopolist faces demand P = 100 − 2Q and has a constant marginal cost MC = 20. (ATC = 20 as well, for the profit calculation.)

Part 3 — Set Up the Model (in Desmos)

1. Open Desmos (link above).
2. Type in three equations:
   - Demand: y = 100 - 2x
   - MR (marginal revenue): y = 100 - 4x
   - MC: y = 20
3. You will see two downward-sloping lines (demand is shallower; MR is steeper and hits the x-axis at half the demand x-intercept) and a horizontal MC line.
4. Find where MR and MC cross — that x-value is the monopoly quantity (Qm). Then trace straight up from Qm to the demand line — that y-value is the monopoly price (Pm).

Part 4 — Solve (complete this scaffold)

Fill in the blanks. Show every arithmetic step.

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

In 3–4 sentences, answer: What does the gap between Qm and Qc mean for real people in this market? Why does the DWL represent "lost" value — who loses it, and why doesn't it just transfer to the monopolist?

Part 6 — Analysis Questions

1. The key graph. In words, describe what the Desmos diagram looks like: where do the three lines sit relative to each other? What is the shape of the deadweight loss "triangle" — what are its three corners (in Q, P terms)?
2. Price discrimination. Suppose the monopolist can perfectly identify each buyer's willingness to pay and charge each buyer exactly that amount (first-degree price discrimination). Would Qm rise, fall, or stay the same? What happens to the DWL? Why?
3. Connect to last week. In perfect competition, the firm produces at the point where P = MC, and P = 20 in this market. The monopolist produces Qm = 20 and charges Pm = 60. Name one way the monopoly outcome is different from the competitive outcome and explain whether that difference is "bad" from an economics standpoint — and what makes it normative vs. positive.

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 a monopolist facing demand P = 100 − 2Q and constant MC = 20, find the profit-maximizing quantity and price. Then compute the deadweight loss compared to the competitive outcome."
2. Audit every claim against your own verified work:
   - Did it correctly derive MR = 100 − 4Q?
   - Did it solve Qm = 20 from MR = MC?
   - Critical: did it read Pm = 60 off the DEMAND curve? Or did it mistakenly say the price is 20 (reading off MR)? This is the most common AI error on monopoly problems.
   - Did it compute DWL = 400 using the triangle formula?
3. Write 3–4 sentences naming what the AI got right and identifying at least one thing you had to verify or correct. If it got every step right, explain how you verified the price-reading step — that is the key habit.

The price-off-demand step is where chatbots stumble most. Even when they state the rule correctly, they sometimes plug Qm into MR instead of demand. Catching that slip is the whole point of this AI-critique moment.

Part 8 — What to Submit

One document (or text entry) with: your Part 4 scaffold (all steps shown), your Part 5 interpretation, your Part 6 analysis answers, and your Part 7 AI-critique paragraph. A screenshot of your Desmos graph is welcome but optional. Due Sun, Nov 15, 11:59 p.m. (50 points).

Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

Every number pre-computed and independently Python-verified. Demand P = 100 − 2Q; MR = 100 − 4Q; MC = 20.

• (a) MR = 100 − 4Q. (Same intercept 100; slope doubles: 2Q → 4Q.) ✓
• (b) MR = MC: 100 − 4Q = 20 → 4Q = 80 → Qm = 20. ✓
• (c) Pm = 100 − 2(20) = 100 − 40 = 60. (READ OFF DEMAND.) ✓
• (d) Plugging Qm = 20 into MR: 100 − 4(20) = 20. That number is MC (= 20), NOT the price. This is the classic trap — MR = MC told us where to stop; it did not tell us what consumers pay. ✓
• (e) P = MC: 100 − 2Q = 20 → 2Q = 80 → Qc = 40; Pc = 20. ✓
• (f) DWL = ½ · (40 − 20) · (60 − 20) = ½ · 20 · 40 = 400. ✓
• (g) Profit = (60 − 20) · 20 = 40 · 20 = 800. ✓

Part 5 model answer: Between Qm = 20 and Qc = 40, every unit would be valued by a buyer at more than $20 (their demand price lies above the $20 MC line) — so both buyer and seller would gain from the trade. The monopolist chooses not to produce these units because selling them at the lower price would reduce profit. As a result, neither party gets the gain from those trades: the buyer goes without and the monopolist doesn't serve them. That missing surplus is the DWL — it doesn't go to the monopolist; it simply disappears.

Part 6 model answers:
1. In Desmos: demand (y = 100 − 2x) is a line from (0, 100) to (50, 0). MR (y = 100 − 4x) is steeper, from (0, 100) to (25, 0), sitting below demand for positive Q. The MC line is horizontal at y = 20. The DWL triangle has corners at (20, 20) [MC at Qm], (40, 20) [competitive equilibrium where demand meets MC], and (20, 60) [monopoly price on the demand curve] — i.e., it is bounded by the demand curve above and the MC line below, between Qm = 20 and Qc = 40. Area = ½ × 20 × 40 = 400.
2. Under first-degree price discrimination, Qm rises to Qc = 40 (the competitive quantity). DWL falls to zero because the monopolist now captures all consumer surplus as profit — no mutually beneficial trade is missed. Output is allocatively efficient; only the distribution of surplus changes.
3. In perfect competition: P = 20, Q = 40. Under monopoly: P = 60, Q = 20. The difference that is "bad" in an efficiency sense (positive): 20 fewer units are produced, creating DWL = 400. Whether antitrust intervention is warranted is normative — it requires weighing the efficiency cost (DWL) against the potential loss of innovation incentives or scale economies, which is a values question.

Part 7 target: the most common AI error is reading the price off MR (getting Pm = 20 instead of 60). A good critique notes: "The AI correctly derived MR = 100 − 4Q and set MR = MC to find Q = 20. However, it then stated the monopoly price is $20 — which is just MC, not the price. The price is read from the demand curve: P = 100 − 2(20) = $60. I had to correct this step."

Grading rubric — 50 points

Quality gate (self-checked): quantitative gate — Qm = 20, Pm = 60 (demand, not MR), Qc = 40, DWL = 400, profit = 800 — all Python-verified ✓. Graph-logic check — MR = 100 − 4Q is steeper than demand, crosses MC at Qm = 20, price read from demand at 60, DWL triangle between Qm and Qc bounded by demand above and MC below ✓. Trap identified: plugging Qm into MR gives 20 = MC, not price ✓.