Week 13 — Graph & Model Workshop · "The VMPL Table: How Many Workers Should a Firm Hire?"

Course: Principles of Microeconomics (ECON 1) · Silver Oak University (fictional sample) · Prof. Kessler
Objective 7 — factor/labor markets; derived demand; the hiring rule · SLO A
Worth 50 points · Model Workshops group = 15% of the grade · Workshop 13
Format: build a VMPL schedule from an MPL table in a spreadsheet (free), apply the hiring rule at two different output prices, interpret the results in words, 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.

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Part 1 — The Big Picture

This week's central model is the value of the marginal product of labor (VMPL). A profit-maximizing firm hires as long as VMPL ≥ wage — the same marginal logic as MR = MC, now applied to the labor market. The VMPL curve is the firm's labor-demand curve, and it slopes downward because of diminishing MPL: each additional worker adds less output than the one before. In this workshop you will build the VMPL table from scratch, find the optimal number of workers at a given wage, change the output price, re-solve, and interpret what changed.

The tool: a spreadsheet — Google Sheets (🔗 https://sheets.google.com, free with a Google account) or Microsoft Excel. A spreadsheet is the natural home for a table with arithmetic in every cell.

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Part 2 — The Guiding Question

If a firm knows how much each additional worker produces and what the product sells for, exactly how many workers should it hire — and what happens to that number when the product becomes more valuable?

The scenario (verified; use these numbers exactly).

A firm sells output at P = $5 per unit. The daily wage is W = $50. The firm's MPL schedule — units of output added by each worker — is:

Worker	MPL (extra units of output)
1st	20
2nd	18
3rd	14
4th	10
5th	6

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

1. Open a spreadsheet. Create five columns: Worker, MPL, VMPL (MPL × $5), Wage ($50), and Hire? (VMPL ≥ Wage).
2. Enter the five workers in rows, the MPL values from the table, and in the VMPL column write a formula: = B2 * 5 (replacing B2 with your MPL cell). Copy it down for all five workers.
3. Fill the Wage column with $50 for all five rows.
4. In the Hire? column write: = IF(C2 >= D2, "Hire", "Do Not Hire") and copy it down.
5. You now have the firm's labor-demand schedule.

🔗 Google Sheets: https://sheets.google.com
🔗 Desmos (optional — for plotting the VMPL as a step function): https://www.desmos.com/calculator

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Part 4 — Solve (complete this scaffold)

Fill in the blanks from your spreadsheet. Show the arithmetic.

Worker	MPL	VMPL = MPL × $5	Wage = $50	Hire? (VMPL ≥ $50)
1st	20	______	$50	______
2nd	18	______	$50	______
3rd	14	______	$50	______
4th	10	______	$50	______
5th	6	______	$50	______

(a) How many workers does the firm hire at P = $5 and W = $50? ______

(b) Why doesn't the firm hire the 5th worker? (Compare VMPL[5] to the wage and say what hiring the 5th worker would do to profit.) ______

Now change the output price to P = $10 and add a new VMPL column in your spreadsheet (= MPL × $10). Keep the wage at $50.

Worker	MPL	VMPL = MPL × $10	Wage = $50	Hire?
1st	20	______	$50	______
2nd	18	______	$50	______
3rd	14	______	$50	______
4th	10	______	$50	______
5th	6	______	$50	______

(c) How many workers does the firm hire at P = $10 and W = $50? ______

(d) What happened to the number of workers demanded when the output price doubled? By how many workers did demand increase? ______

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

In 3–4 sentences, explain in plain English:
- why the VMPL column slopes downward as you move from worker 1 to worker 5 (what economic principle is at work?);
- why the firm hired more workers when the output price rose from $5 to $10, even though productivity and the wage stayed the same;
- what this result shows about the concept of derived demand (the connection between the product market and the labor market).

(Hint: think about what VMPL = MPL × P means when P changes. The math is clear; the interpretation is the skill.)

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Part 6 — Analysis Questions

1. In your spreadsheet at P = $5, the 4th worker's VMPL exactly equals the wage ($50 = $50). Should the firm hire the 4th worker or not? Explain the reasoning — what happens to profit if the firm hires vs. does not hire that worker?
2. Suppose a new machine makes every worker 50% more productive (all MPL values rise by 50%). Even without changing the output price, what happens to the VMPL of each worker? Does the firm want to hire more, fewer, or the same number at the old wage? This illustrates what kind of shift in the labor-demand curve?
3. Connect it: name one industry or occupation where you think a rise in consumer demand for the product has recently increased the demand for workers in that field. Explain the connection through the VMPL logic.

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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: "A firm has five workers with MPL values 20, 18, 14, 10, and 6. Output sells at $5 per unit and the daily wage is $50. How many workers should the firm hire to maximize profit? Show the VMPL for each worker."
2. Audit every claim against your own spreadsheet:
   - Did it compute VMPL = MPL × $5 correctly for all five workers: $100, $90, $70, $50, $30?
   - Did it apply the hiring rule correctly — hire workers 1–4 (VMPL ≥ $50), not the 5th (VMPL = $30 < $50)?
   - Common chatbot errors: (a) comparing MPL directly to the wage without multiplying by the output price; (b) stopping at worker 3 instead of worker 4 because it mistakenly applies VMPL > wage rather than VMPL ≥ wage; (c) including the 5th worker because $30 is "close enough to $50."
3. Write 2–3 sentences naming at least one thing the AI got right and at least one error or questionable step you caught — and state the correct reasoning.

The habit all term: the tool drafts, you judge. Chatbots routinely skip the ×P step, botch the ≥ vs > inequality, or add workers that reduce profit.

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Part 8 — What to Submit

One document (or text entry) with: your Part 4 scaffold (filled in with arithmetic), your Part 5 interpretation, your Part 6 answers, and your Part 7 AI-critique paragraph. A screenshot of your spreadsheet is welcome but optional. Due Sun, Nov 29, 11:59 p.m. (50 points).

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Instructor answer key — REMOVE BEFORE PUBLISHING TO STUDENTS

Every number pre-computed and independently verified. VMPL = MPL × P.

Part 4 — P = $5, W = $50:

Worker	MPL	VMPL	Wage	Hire?
1st	20	$100	$50	Hire (100 ≥ 50)
2nd	18	$90	$50	Hire (90 ≥ 50)
3rd	14	$70	$50	Hire (70 ≥ 50)
4th	10	$50	$50	Hire (50 = 50, break even — still hire)
5th	6	$30	$50	Do Not Hire (30 < 50)

(a) Optimal employment = 4 workers. ✓
(b) The 5th worker's VMPL = $30 < wage = $50. Hiring the 5th worker would cost $50 and add only $30 in revenue — a $20 reduction in profit. ✓

Part 4 — P = $10, W = $50:

Worker	MPL	VMPL	Wage	Hire?
1st	20	$200	$50	Hire
2nd	18	$180	$50	Hire
3rd	14	$140	$50	Hire
4th	10	$100	$50	Hire
5th	6	$60	$50	Hire (60 ≥ 50)

(c) Optimal employment = 5 workers. ✓
(d) Demand increased from 4 to 5 workers — an increase of 1 worker — when the price doubled from $5 to $10. The entire VMPL curve shifted upward, making all workers more valuable. ✓

Part 5 (model interpretation): VMPL slopes downward because MPL falls (diminishing marginal product) — each additional worker produces fewer extra units as the fixed inputs are shared more widely. When the output price rose from $5 to $10, each unit of output became more valuable, so each worker's output was worth more in dollars — VMPL rose at every employment level, and the firm found it profitable to hire all five workers. This is derived demand: the firm's demand for workers is derived from the demand for the product; when the product market became more lucrative, the labor market followed.

Part 6:
1. Hire the 4th worker. VMPL = $50 = wage → the firm breaks exactly even on this worker. Hiring adds $0 to profit — but not hiring would forgo the 4th worker's contribution without saving any cost (the firm would still be paying for that capacity elsewhere). Not hiring costs nothing directly but leaves potential output and revenue uncaptured. Standard rule: hire at equality. ✓
2. If MPL rises 50%: new MPL = 30, 27, 21, 15, 9. New VMPL at P=$5: $150, $135, $105, $75, $45. At wage $50: hire 4 workers (VMPL[4]=$75≥$50; VMPL[5]=$45<$50 — same count as before, though the firm is now better off at each employment level). This is a rightward shift of the labor-demand curve — at any given wage, the firm wants more workers (or the same number at a higher wage). ✓
3. Any reasonable industry with a named VMPL chain earns credit — examples: streaming → writers and directors; electric-vehicle adoption → battery engineers; post-pandemic travel surge → pilots and hospitality workers.

Part 7 (AI-critique): full credit for a specific catch — most commonly: the bot skips the MPL × P step and directly compares MPL to the wage, or it uses strict inequality (VMPL > wage) and excludes worker 4 (VMPL = $50 = wage = $50 → should hire), or it includes the 5th worker because $30 is "near" $50.

Grading rubric — 50 points

Criterion	Full	Partial	None
VMPL table at P=$5 — all five VMPL values correct with arithmetic shown (12)	12	6–10	0–5
Hiring rule at P=$5 — correct count (4), correct reasoning for stopping before worker 5, handles the equality case (8)	8	4–6	0–3
VMPL table and count at P=$10 — correct new VMPL column, correct count (5) (10)	10	5–8	0–4
Interpretation (Part 5) — diminishing MPL, derived demand, price-to-demand connection, in plain English (12)	12	6–10	0–5
AI-critique (Part 7) — names at least one specific error caught or verified (8)	8	4–6	0–3

Quality gate (self-checked):
- Quantitative gate: VMPL at P=$5 = {100,90,70,50,30} ✓; hire 4 (5th VMPL=30<50) ✓; VMPL at P=$10 = {200,180,140,100,60} ✓; hire 5 (5th VMPL=60≥50) ✓. All Python-verified.
- Graph-logic check: VMPL slopes downward due to diminishing MPL ✓; hiring rule VMPL≥wage correct ✓; output-price rise shifts VMPL curve right (derived demand) ✓; wage change = movement along, not a shift ✓.