Week 1 — Graph & Model Workshop · "Build an Economy's PPF"

Course: Principles of Macroeconomics (ECON 2) · Silver Oak University (fictional sample) · Prof. Ashford
Objective 1 — macroeconomic modeling & quantitative/graphical analysis · SLO A
Worth 50 points · Model Workshops group = 15% of the grade · Workshop 1
Format: build and read a production possibilities frontier for a whole economy in Desmos (free, no account), complete a short scaffold, interpret the result in words, 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 — nothing to buy or download.

Part 1 — The Big Picture

This week you learned macro's first model — the production possibilities frontier (PPF) — which shows, on one graph, scarcity, trade-offs, opportunity cost, and efficiency, all at the scale of an entire economy. The PPF's slope literally is the opportunity cost, and a point inside the frontier is your first picture of what a recession looks like. Today you'll build one for a fictional economy, drag points around, and read the trade-off straight off the line. (Next week you'll use the same measuring instinct to add up GDP — the number economists use to size up the whole economy.)

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

Part 2 — The Guiding Question

If a whole economy's labor is scarce, exactly how many units of one good must it give up to produce one more unit of another — and what does the graph of that trade-off look like?

The scenario. The fictional economy of Isla Verde has 24 million labor-hours available this period. Producing one unit of consumer goods (x) takes 3 hours; producing one unit of capital goods (y) takes 6 hours. Isla Verde can split its labor-hours any way it chooses.

Part 3 — Set Up the Model (in Desmos)

1. Open Desmos. The PPF is every combination that uses all 24 million labor-hours:
   3·(consumer goods) + 6·(capital goods) = 24.
2. Let x = consumer goods and y = capital goods. Type into Desmos: 3x + 6y = 24
3. You'll see a straight line. Find where it hits each axis (the intercepts) — those are the "all of one good" points.

Part 4 — Solve (complete this scaffold)

Fill in the blanks from your Desmos graph and a little arithmetic. Show the steps.

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

In 2–3 sentences, explain what the slope of this PPF means in plain English (not just the number), and what a point inside the frontier would mean for the whole Isla Verde economy — not just "a wasted resource," but the specific macro problem it represents. (Hint: at a national scale, "inside" means idle labor and capital — in other words…?)

Part 6 — Analysis Questions

1. Isla Verde's PPF is a straight line, so the opportunity cost is constant. Real economies' PPFs bow outward. In one sentence, why — what's different about a whole economy's resources compared to this simplified example?
2. A policymaker says, "Isla Verde should just produce 6 units of consumer goods and 3 units of capital goods — maximize both!" Use the model to explain why that's unattainable with 24 million labor-hours.
3. Connect to policy: Isla Verde's government is debating whether to devote more of its labor-hours to capital goods (factories, machinery, infrastructure) or consumer goods (things people use and enjoy right now). In 2–3 sentences, explain the trade-off involved in that choice — what does the economy gain and give up either way? (You are not being asked which choice is "right" — just to explain the trade-off fairly.)

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 an economy with 24 million labor-hours, where consumer goods take 3 hours each to produce and capital goods take 6 hours each, what is the opportunity cost of one capital good, in consumer goods? Is the point (6 consumer goods, 3 capital goods) attainable?"
2. Audit every claim against your own work:
   - Did it get the opportunity cost of 1 capital good = 2 consumer goods (not ½)? Chatbots routinely flip the ratio or state it in the wrong good.
   - Did it correctly say (6, 3) is unattainable (needs 36 million labor-hours, since 3·6 + 6·3 = 36 > 24), or did it wave it through as attainable without checking the equation?
   - Did it keep "inside = idle resources/unemployment" vs. "outside = unattainable" straight, rather than confusing the two?
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 claim — that's the skill.)

The habit all term: the tool drafts, you judge. A chatbot will confidently flip an opportunity-cost ratio or wave an unattainable point through as fine — catching it is the point.

Part 8 — What to Submit

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

Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

Every number pre-computed and independently verified (see the Week-1 verified-numbers check). The frontier is 3x + 6y = 24.

• (a) 24 ÷ 3 = 8 units of consumer goods (x-intercept (8, 0)). ✓
• (b) 24 ÷ 6 = 4 units of capital goods (y-intercept (0, 4)). ✓
• (c) Opportunity cost of 1 capital good = 8 consumer goods ÷ 4 capital goods = 2 consumer goods. ✓
• (d) Opportunity cost of 1 consumer good = 4 capital goods ÷ 8 consumer goods = ½ capital good. ✓
• (e) 3·4 + 6·2 = 12 + 12 = 24 = 24 → on the frontier → efficient. ✓
• (f) 3·2 + 6·2 = 6 + 12 = 18 < 24 → inside the frontier → inefficient (6 million labor-hours idle). ✓
• (g) 3·6 + 6·3 = 18 + 18 = 36 > 24 → outside the frontier → unattainable (would need 36 million labor-hours, 12 million more than exist). ✓
• Part 5: the slope's magnitude (2 consumer goods per capital good) is the opportunity cost — each capital good costs Isla Verde 2 consumer goods. A point inside the frontier means some of the 24 million labor-hours sit idle — at the scale of a whole economy, that is the macro preview of unemployment / a recession: the resources exist to produce more of both goods, but aren't being used.
• Part 6: (1) a whole economy's resources are specialized (not equally good at both goods), so pushing all-in on one raises its opportunity cost → the curve bows outward (Isla Verde's straight line is the simplified constant-cost case). (2) (6, 3) needs 3·6 + 6·3 = 36 million labor-hours > 24 million available → unattainable; scarcity binds. (3) Producing more capital goods now (factories, machinery, infrastructure) trades away consumer goods today but can push the ENTIRE frontier outward faster later (the Week 4 growth preview); producing more consumer goods now favors current living standards but slower future growth. Both are legitimate choices — full credit for any answer that states the trade-off fairly without declaring one option objectively "right" (evenhandedness).
• Part 7: full credit for a specific catch — most commonly the AI flipping the ratio (saying ½ a consumer good), mislabeling (6, 3) as attainable, or confusing "inside = idle/unemployment" with "outside = unattainable."

Grading rubric — 50 points

Quality gate (self-checked): quantitative gate — intercepts (8, 4), ratios (2 consumer goods; ½ capital good), point tests (24=24 on; 18<24 inside; 36>24 outside) all Python-re-verified ✓. Graph-logic check — interior point = idle resources/unemployment, exterior = unattainable, slope = opportunity cost, all correct ✓. Quantitative gate: PASS. Graph-logic check: PASS.