Week 7 — Graph & Model Workshop · "The Spending Multiplier"

Course: Principles of Macroeconomics (ECON 2) · Silver Oak University (fictional sample) · Prof. Ashford
Objective 6 — fiscal policy; the spending multiplier · SLO A
Worth 50 points · Model Workshops group = 15% of the grade · Workshop 7
Format: trace the actual rounds of re-spending that make up the multiplier in a spreadsheet or by hand, sum the geometric series, then compare your sum to the formula 1/(1−MPC) — 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

Segment 3 of this week's lecture gave you the formula — multiplier = 1/(1−MPC) — and showed that a $20 billion increase in government spending, at MPC = 0.8, produces a $100 billion total change in output. But where does that number actually come from? It isn't magic, and it isn't a fixed law of nature — it's the sum of an endless chain of re-spending, round after round, each one a little smaller than the last. Today you build that chain yourself, round by round, add it up, and prove for yourself that the sum really does equal what the formula predicts. (Next week is midterm review, pulling this together with everything since Week 1; the AD–AS diagram picture of this same fiscal-policy story is what you already practiced in Weeks 5–6.)

The tool: 🛠️ A spreadsheet (Google Sheets, Excel, or any spreadsheet app) — or a calculator and paper if you prefer to trace it by hand. No special software required.

Part 2 — The Guiding Question

If the government spends an extra $20 billion, and each recipient re-spends 80% of every dollar they receive, exactly how much does that ripple add up to in total — and does the running total ever actually reach the formula's prediction?

The scenario. The government of a fictional economy increases spending by ΔG = $20 billion (say, on a highway project). The marginal propensity to consume (MPC) = 0.8 — every person who receives a dollar of new income spends 80 cents of it and saves 20 cents. That 80 cents becomes someone else's income, who spends 80% of that, and so on — the ripple this week's lecture named but didn't fully trace.

Part 3 — Set Up the Model (in a spreadsheet)

1. Open a spreadsheet. Make three columns: Round, New spending this round, Running total.
2. Round 1 is the original $20 billion the government spent directly. Enter 20 as Round 1's "new spending."
3. Each later round = the previous round's new spending × MPC (0.8) — because only 80% of each round's income gets spent again (the other 20% is saved and leaves the spending stream for this model).
4. Running total = the sum of every round's "new spending" so far (add each new round onto the previous running total).

Part 4 — Solve (complete this scaffold)

Fill in the blanks. Show every multiplication step (round n's new spending = round n−1's new spending × 0.8).

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

In 2–3 sentences, explain why the running total gets closer and closer to $100 billion but each individual round contributes less and less — what is happening to the money at each round (how much is being saved vs. re-spent), and why that leakage is exactly what makes the series converge to a finite number instead of growing forever.

Part 6 — Analysis Questions

1. Suppose the MPC had been 0.9 instead of 0.8 (people save less, spend more of each new dollar). Without redoing the whole table, use the formula to state the new multiplier and the new total ΔY for the same $20 billion of spending — and explain in one sentence why a higher MPC produces a bigger multiplier.
2. A classmate says, "By Round 10, the running total is basically already at $100 billion, so the later rounds don't matter." Using your Part 4(a) answer, is that claim accurate? Explain using your own numbers.
3. Connect to policy: A policymaker is deciding between two possible stimulus packages of the same dollar size — one aimed at low-income households (typically a higher MPC, since more of each dollar tends to be spent rather than saved) and one aimed at high-income households (typically a lower MPC). Using what you just built, explain in 2–3 sentences why the total output effect (ΔY) could differ between the two packages even though ΔG is identical — without arguing which recipient group the government should target (that's a normative call this workshop isn't asking you to make).

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: "The government increases spending by $20 billion. The MPC is 0.8. Trace the first four rounds of re-spending (Round 1 through Round 4), give the running total after 4 rounds, and state the total effect on GDP using the multiplier formula."
2. Audit every claim against your own work:
   - Did it get Round 1 = 20, Round 2 = 16, Round 3 = 12.8, Round 4 = 10.24? Chatbots sometimes multiply by the wrong fraction (using MPS = 0.2 instead of MPC = 0.8, which shrinks the rounds far too fast) or restart from 20 each round instead of shrinking the base.
   - Did it correctly compute the multiplier as 1/(1−0.8) = 5, or did it confuse this with the money multiplier (1/RR — a completely different formula about bank reserves, not taught until Week 9)?
   - Did it correctly report the total effect as multiplier × ΔG = 5 × 20 = $100 billion, or did it stop early and report just the multiplier (5) or just the running total after a few rounds as the "final answer"?
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 shrink the rounds by the wrong fraction, merge the spending multiplier with the money multiplier, or forget to multiply the multiplier by ΔG — catching it is the point.

Part 8 — What to Submit

One document (or text entry) with: your Part 4 scaffold (the full round-by-round table through at least Round 10, with all three sub-answers a/b/c), your Part 5 interpretation, your Part 6 answers, and your Part 7 AI-critique paragraph. A spreadsheet screenshot is welcome but optional. Due Sun, Oct 18, 11:59 p.m. (50 points).

Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

Every number pre-computed and independently verified (see the Week-7 verified-numbers check). ΔG = 20, MPC = 0.8.

Full round-by-round table (Part 4):

• (a) Running total after 10 rounds = $89.2626 billion. ✓ (approaching but not yet at $100 billion — the rounds keep contributing, just smaller and smaller amounts).
• (b) Sum of the infinite geometric series: ΔG / (1 − MPC) = 20 / (1 − 0.8) = 20 / 0.2 = $100 billion. ✓
• (c) Multiplier = 1/(1−0.8) = 1/0.2 = 5; multiplier × ΔG = 5 × 20 = $100 billion. Yes — it matches (b) exactly, because the multiplier formula 1/(1−MPC) is the closed-form sum of this exact geometric series (each round is the previous round times MPC — a geometric series with first term ΔG and common ratio MPC, whose infinite sum is ΔG/(1−MPC), algebraically identical to ΔG × [1/(1−MPC)]). ✓
• Part 5: at each round, 80% of the new income gets spent again (continuing the ripple) while 20% leaves the spending stream as saving — that 20% leakage is exactly why each round is smaller than the last (0.8 times the previous), and why an endless chain of shrinking rounds nonetheless adds up to a finite total rather than growing forever: the rounds shrink geometrically fast enough that their sum converges.
• Part 6: (1) MPC = 0.9 → multiplier = 1/(1−0.9) = 1/0.10 = 10; ΔY = 10 × 20 = $200 billion — a HIGHER MPC means MORE of each round gets re-spent (less leaks to saving), so the ripple runs longer and the total is bigger. (2) The claim is not accurate — the running total after 10 rounds is $89.2626 billion, still $10.7374 billion short of the full $100 billion; the remaining (smaller and smaller) rounds still matter and, taken together, close that gap. (3) A package aimed at a higher-MPC group (often lower-income households) will tend to generate a larger multiplier and thus a larger ΔY for the same ΔG, because more of each dollar re-enters the spending stream at every round rather than being saved; a package aimed at a lower-MPC group produces a smaller ripple for the identical dollar amount. Full credit for stating this positive, mechanical difference clearly WITHOUT recommending which group the government should target — that recommendation is a normative call this workshop does not ask students to make (evenhandedness).
• Part 7: full credit for a specific catch — most commonly the AI using MPS (0.2) instead of MPC (0.8) to shrink the rounds (which would make Round 2 = 4, not 16 — a very common and very wrong chatbot error), confusing the spending multiplier with the money multiplier, or reporting the multiplier value (5) alone instead of multiplier × ΔG = $100 billion.

Grading rubric — 50 points

Quality gate (self-checked): quantitative gate — round-by-round values (20, 16, 12.8, 10.24, 8.192, 6.5536, 5.2429, 4.1943, 3.3554, 2.6844), running total after 10 rounds ($89.2626B), infinite-series sum ($100B), and the multiplier cross-check (5 × 20 = $100B) all Python-re-verified ✓. Graph-logic check — this workshop is purely quantitative (no curve-shift claim); the multiplier-formula/geometric-series equivalence and the MPC-vs-multiplier direction (higher MPC → bigger multiplier) both verified against the NUMBERS_PACK canon ✓. Quantitative gate: PASS. Graph-logic check: PASS (N/A curve-shift; formula-equivalence and MPC-direction both verified).