Week 4 — Graph & Model Workshop · "Growth Rates & the Rule of 70"

Course: Principles of Macroeconomics (ECON 2) · Silver Oak University (fictional sample) · Prof. Ashford
Objective 4 — macroeconomic modeling & quantitative/graphical analysis · SLO A
Worth 50 points · Model Workshops group = 15% of the grade · Workshop 4
Format: compound a small fictional economy at two different growth rates in a spreadsheet or Desmos, 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 that a growth rate compounds — and that the rule of 70 turns an abstract percentage into a number you can feel: years to double. Today you'll build a compounding table for a small fictional economy at two different growth rates and watch the gap between them go from invisible to enormous. This is the single most important intuition in this week's material: small differences in a growth rate produce huge differences in outcomes, given enough time.

The tool: 🔗 Desmos Graphing Calculator — https://www.desmos.com/calculator (free, instant, no login) — or any spreadsheet (Google Sheets, Excel, etc.). Either works for this workshop; use whichever you're more comfortable with.

Part 2 — The Guiding Question

If two economies start at the exact same size but grow at different steady rates, how long does it take before the gap between them becomes impossible to ignore — and does the rule of 70 predict it correctly?

The scenario. The fictional economy of Meadowbrook has two possible futures, both starting from the same real GDP of $100 billion:

• Path A — Meadowbrook adopts modest, steady policies and grows at 2% per year.
• Path B — Meadowbrook adopts aggressive, steady growth-oriented policies and grows at 7% per year.

Both paths compound every year on the previous year's total (not the original $100 billion) — that's what "compounding" means.

Part 3 — Set Up the Model (spreadsheet or Desmos)

If using a spreadsheet: create three columns — Year (0 through 10), Path A ($100 growing 2%/yr), Path B ($100 growing 7%/yr). In each cell after Year 0, multiply the PREVIOUS year's value by 1.02 (Path A) or 1.07 (Path B).

If using Desmos: graph both compounding formulas at once and compare them visually:
- Path A: y = 100(1.02)^x
- Path B: y = 100(1.07)^x

Either way, you are computing: this year's value = last year's value × (1 + growth rate).

Part 4 — Solve (complete this scaffold)

Fill in the blanks by computing (or reading off) each path's value. Show at least one worked step for Year 1 of each path.

(g) Using the rule of 70, about how many years should Path B (7%/yr) take to double from $100 billion to roughly $200 billion? Show 70 ÷ rate. | __
(h) Using the rule of 70, about how many years should Path A (2%/yr) take to double? | _
(i) Compare your Year-10 answer for Path B to your rule-of-70 prediction in (g) — do they roughly agree? | ___

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

In 2–3 sentences, explain what happens to the gap between Path A and Path B as time passes (compare Year 1 to Year 10), and connect this directly to the rule of 70: why does a 7%-growing economy end up near double its starting size by Year 10, while a 2%-growing economy is nowhere close? (Hint: this is the "small differences compound huge" idea from lecture — put it in your own words.)

Part 6 — Analysis Questions

1. At Year 1, Path A is at roughly $102 and Path B is at roughly $107 — a gap of about $5 billion. At Year 10, the gap is much larger. In one sentence, explain why the gap grows over time rather than staying the same size (this is the core idea of compounding, as opposed to simple/flat growth).
2. A classmate argues: "A 2-percentage-point difference in growth rates (say, 3% vs. 5%) is basically meaningless — it's such a small number." Using the rule of 70, show your classmate why they're wrong: compute the doubling time for both 3% and 5% growth and compare.
3. Connect to policy: Meadowbrook's legislature is debating a package of growth-oriented reforms that supporters claim could raise the long-run growth rate from 2% to 4%. In 2–3 sentences, explain what that change would mean for how long it takes the economy to double (use the rule of 70 for both rates) — without taking a position on whether the legislature should adopt the reforms (that's a normative question this workshop isn't asking you to settle).

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: "An economy's real GDP grows steadily at 5% a year, starting from $100 billion. Using the rule of 70, about how many years will it take to double? And separately, if real GDP grows from $500 billion to $520 billion in one year, what is the growth rate?"
2. Audit every claim against your own work:
   - Did it get the rule-of-70 doubling time as 70 ÷ 5 = 14 years (not 70 × 5 = 350, and not some other number)? Chatbots routinely multiply instead of dividing on the rule of 70.
   - Did it compute the growth rate as ($520B − $500B) ÷ $500B (the OLD/starting value) × 100 = 4% — or did it mistakenly divide by the new value ($520B) instead, giving a slightly different (wrong) answer?
   - Did it keep total growth and per-capita growth straight, or did it wander into a population-adjusted answer when none was asked for?
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 multiply instead of divide on the rule of 70, or divide a growth rate by the wrong base — catching it is the point.

Part 8 — What to Submit

One document (or text entry, or spreadsheet export) 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 or export of your spreadsheet/Desmos graph is welcome but optional. Due Sun, Sep 27, 11:59 p.m. (50 points).

Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

Every number pre-computed and independently verified (see the Week-4 verified-numbers check, _build/logs/week-04-numbers.txt). Path A: $100 × 1.02ⁿ. Path B: $100 × 1.07ⁿ.

• Year 1: Path A = 100 × 1.02 = $102.00. Path B = 100 × 1.07 = $107.00. ✓
• Year 5: Path A = $110.41. Path B = $140.26. ✓
• Year 10: Path A = $121.90. Path B = $196.72. ✓
• (g) Rule of 70 for Path B: 70 ÷ 7 = 10 years to double (from $100B toward $200B). ✓
• (h) Rule of 70 for Path A: 70 ÷ 2 = 35 years to double. ✓
• (i) Yes — at Year 10, Path B is at $196.72B, extremely close to the $200B the rule of 70 predicted for doubling in 10 years at 7% growth. This confirms the shortcut works. Path A, by contrast, is at only $121.90B at Year 10 — nowhere near doubling, exactly as expected since the rule of 70 says it needs 35 years, not 10. ✓
• Part 5: at Year 1, the two paths look almost identical ($102 vs. $107 — a gap that seems small). By Year 10, they have pulled miles apart ($121.90 vs. $196.72 — Path B is over $74 billion ahead). This is compounding: each year's growth is applied to an already-larger base, so a persistent rate advantage snowballs rather than staying constant. The rule of 70 captures this precisely — 7% growth needs only 10 years to double, while 2% growth needs 35 — which is exactly why Path B is nearly double its start by Year 10 while Path A has barely budged past $120B.
• Part 6: (1) the gap grows because compounding applies each year's growth rate to an ALREADY-GROWN base, not the original $100B — so the dollar amount added each year keeps increasing for the faster path, while simple/flat growth would add the same fixed amount every year and the gap would stay constant. (2) 70 ÷ 3 ≈ 23.3 years to double at 3%; 70 ÷ 5 = 14 years to double at 5% — nearly a 10-year difference in doubling time from just a 2-percentage-point gap, which is very much NOT "meaningless" over a policy-relevant time horizon (a working generation). (3) at 2% growth, Meadowbrook needs 70 ÷ 2 = 35 years to double; at 4% growth, it needs 70 ÷ 4 = 17.5 years — HALF the time. Full credit for explaining this factual (positive) comparison WITHOUT taking a position on whether the legislature should adopt the reforms — that verdict depends on values (e.g., what the reforms would cost, trade off, or affect distributionally) that this workshop does not ask students to weigh.
• Part 7: full credit for a specific catch — most commonly the AI multiplying instead of dividing on the rule of 70 (giving 350 instead of 14), or dividing the growth-rate calculation by the wrong (new) base (getting a number close to but not exactly 4%, e.g., dividing 20 by 520 instead of 500).

Grading rubric — 50 points

Quality gate (self-checked): quantitative gate — Path A/B values at Years 1/5/10 (102.00/110.41/121.90 and 107.00/140.26/196.72), rule-of-70 doubling times (35 and 10 years), the 3%/5% comparison (23.3 and 14 years), and the 2%/4% policy comparison (35 and 17.5 years) all Python-re-verified ✓. Graph-logic check — N/A for this workshop (Week 4 is a pure compounding/growth-rate model, no AD–AS or other curve-shift claims; curve-shift work resumes Week 5) ✓. Quantitative gate: PASS. Graph-logic check: N/A (no curve-shift claims this week).