Week 4 Model Workshop: Which Economy's Residents Are Pulling Ahead?

Part 1 — The Big Picture

This week you learned that long-run growth is an outward shift of the whole frontier, that a growth rate divides the change by the starting value, that the rule of 70 turns a growth rate into an approximate doubling time, and that per-capita growth, about total growth minus population growth, tracks living standards. In this Model Workshop you work the same tools on two illustrative economies that tell different stories: one has faster total growth, and the other may be doing more for its average resident. You will build the numbers in a free tool, shift the Week 1 frontier yourself, sort positive claims from normative ones, and then check a chatbot's account of the same economies.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: Economy A's total output grows faster than Economy B's, but its population grows faster too: which economy's average resident is pulling ahead, about how long would each economy take to double, and which claims about them are positive rather than normative?

Workshop data: Two-economy growth table (illustrative, two years)

Source: A workshop-only table constructed for this course, not real data: real GDP, nominal GDP, and population for two illustrative economies in two consecutive years, printed in full in the first scaffold table.

• Rows: 2 economies observed in 2 consecutive years, 4 observations in all; the first scaffold table shows one row per economy and per measure, 6 rows
• Columns: year 0 value, year 1 value, and a blank growth-rate cell for each measure
• Economy A: real GDP 400 and 416 billion dollars; nominal GDP 400 and 436 billion dollars; population 20 and 20.6 million
• Economy B: real GDP 250 and 255 billion dollars; nominal GDP 250 and 260 billion dollars; population 25 and 25 million

Part 3 — Materials & Procedure

Tools and materials

• Desmos (free) or Google Sheets (free)

Ways of working

• read-and-compute-from-macro-data
• build-and-shift-the-ad-as-model
• separate-positive-from-normative-claims

Procedure

1. Open Desmos at desmos.com/calculator or a blank Google Sheets spreadsheet. If you use Google Sheets, type the six rows of the first scaffold table into columns A to C: the measure in column A, the year 0 value in column B, and the year 1 value in column C, with real GDP for Economy A in row 2. Tool: Desmos (free) or Google Sheets (free). Check: Six rows of data: real GDP, nominal GDP, and population for each economy, in two years.
2. Compute every growth rate as the change divided by the starting value, times 100. In Google Sheets, type =(C2-B2)/B2*100 in D2 and copy it down the column. In Desmos, type each calculation on its own line, such as (416-400)/400*100. Copy each rate into the first scaffold table. Tool: Desmos (free) or Google Sheets (free). Check: Six growth rates, each a whole number of percent; the nominal rates are larger than the real rates.
3. For each economy, compute per-capita growth as real GDP growth minus population growth, and write it in the second scaffold table beside the total growth rate. Tool: Desmos (free) or Google Sheets (free). Check: One per-capita growth rate for each economy, written with about.
4. Use the rule of 70 to estimate doubling times: divide 70 by each economy's real GDP growth rate and by each economy's per-capita growth rate. In Google Sheets, type =70/D2 beside a rate; in Desmos, type 70/4 and the like. Record each result in the second scaffold table with the word about. Tool: Desmos (free) or Google Sheets (free). Check: Four doubling times in years, each written as an approximation.
5. In Desmos, type 3x+6y=24 to draw the Week 1 labor-hours frontier, then type 3x+6y=k on a new line and accept the slider for k. Type the points (2,2), (4,2), and (6,3) on their own lines. Drag k from 24 up to 36 and watch the line. If you are working in Google Sheets instead, compute 3 × 6 + 6 × 3 and compare it with 24 and with each value of k you test. Tool: Desmos (free) or Google Sheets (free). Check: The whole frontier moves outward as k rises, and at k = 36 the point (6, 3) sits on it, while moving from (2, 2) to (4, 2) never moves the frontier at all.
6. Sort the four claims in analysis question 4 into positive and normative, and for each positive claim note the number in your tables that tests it. Tool: Desmos (free) or Google Sheets (free). Check: Two claims labelled positive with a number from the tables beside each, and two labelled normative.
7. Paste the Part 6 prompt into one approved chatbot, or use the equivalent route instead, and compare its answers with your tables on all four checks. Tool: Desmos (free) or Google Sheets (free). Check: A short critique that quotes the chatbot or the printed analysis and corrects at least two errors.

Fallback route

When to use it: Use this route if neither Desmos nor Google Sheets is available, or if you cannot get online.

Do every calculation by hand or with any calculator, writing each step on its own line, and sketch the Week 1 frontier 3x + 6y = 24 and the shifted line 3x + 6y = 36 on graph paper with the points (2, 2), (4, 2), and (6, 3) marked. Complete both scaffold tables, every analysis question, and the critique step from your hand work; submit the same items for equal credit.

This route earns the same credit as the primary route.

Part 4 — The Scaffold

Your data and growth rates: fill in each growth rate as the change divided by the year 0 value, times 100.

Student scaffold table 1

Measure	Year 0	Year 1	Growth rate (percent)
Economy A real GDP (billions of dollars)	400	416
Economy A nominal GDP (billions of dollars)	400	436
Economy A population (millions)	20	20.6
Economy B real GDP (billions of dollars)	250	255
Economy B nominal GDP (billions of dollars)	250	260
Economy B population (millions)	25	25

Living standards and doubling: use real GDP growth, not nominal, and write every doubling time with about.

Student scaffold table 2

Economy	Real GDP growth (percent)	Per-capita growth (percent)	Years for real GDP to double	Years for output per person to double
Economy A
Economy B

Part 5 — Analysis Questions

1. Show the growth-rate calculation for each economy's real GDP, nominal GDP, and population, every step on its own line. Then compute each economy's per-capita growth and say each figure in a plain sentence.
2. Which economy's total output grew faster, and which economy's average resident gained more? Use your per-capita figures to explain why the two answers differ, and say why nominal GDP growth cannot settle either question.
3. Use the rule of 70 to estimate how long each economy's real output and each economy's output per person would take to double, and explain what the gap between those doubling times means over a working lifetime. Then describe what happened in Desmos when you raised k, and explain why moving from (2, 2) to (4, 2) is not long-run growth.
4. Positive or normative? Sort these four claims about the two illustrative economies, and for each positive claim name the number in your table that tests it: (1) Economy B's real GDP grew 2 percent. (2) Economy A's government ought to spend more on schools. (3) If Economy A's population keeps growing 3 percent a year while its real GDP grows 4 percent, its output per person grows about 1 percent a year. (4) Faster growth is worth giving up some leisure time.
5. Choose one and answer it in 3–4 sentences. (a) Per-capita growth is an average. Explain what it cannot tell you about how Economy A's extra output is shared, and what other information you would want before saying most residents are better off. (b) The rule of 70 is an approximation. Explain why your doubling times are written with about, and what you would check in your spreadsheet or calculator to see how close the shortcut is.

Part 6 — AI-Critique Moment

Choose one equal-credit critique route. Route A: ask an approved AI tool to critique your completed work, then check its response against your own evidence. Route B is the no-AI equivalent printed below.

Approved tools: Claude, Gemini, ChatGPT, Grok, Microsoft Copilot.

Prompt to paste:

I am studying two illustrative economies. Economy A's real GDP rose from 400 to 416 billion dollars in a year, its nominal GDP rose from 400 to 436 billion dollars, and its population rose from 20 million to 20.6 million. Economy B's real GDP rose from 250 to 255 billion dollars, its nominal GDP rose from 250 to 260 billion dollars, and its population stayed at 25 million. Which economy grew faster? Which economy's average resident gained more? About how many years would each economy's real output take to double at its growth rate, and what multiplier explains that growth? On a production possibilities frontier, is long-run growth a movement along the frontier or a shift of it? Finally, should Economy A's government try to slow its population growth?

Test the response for these likely failures:

• Chatbots often report Economy A's growth from nominal GDP, (436 − 400) ÷ 400 × 100 = 9 percent, which mixes higher prices with more output.
• Chatbots sometimes describe growth as the economy moving along its frontier, or call a move from an inside point to the frontier growth.
• Chatbots often reach for a multiplier, such as the spending multiplier 1 ÷ (1 − MPC) or the money multiplier 1 ÷ RR, to explain or speed up long-run growth.
• Chatbots often answer the question about slowing population growth as if the growth arithmetic settled it, presenting a value judgment as a fact.

In 3–5 sentences, quote the chatbot's answer on each of the four checks, say what it got right, and correct at least two errors using your own table and the course rule, labelling which of its statements were positive and which were normative.

Equivalent route (no AI required)

No AI account is required for this Model Workshop. Instead of the chatbot, critique this flawed analysis, printed here in full: "Economy A grew 9 percent, because its GDP went from 400 to 436, so it grew more than twice as fast as Economy B. Since Economy A grew faster, its average resident must be gaining more too. Using the spending multiplier of 5, Economy A's output will double in about 70 ÷ (9 × 5) years, well under two years. Moving from the inside point (2, 2) to the frontier point (4, 2) is long-run growth, because the economy is producing more. The numbers prove that Economy A's government should slow its population growth." Mark it up: quote each flawed sentence, name what is wrong (nominal growth used in place of real growth, total growth read as per-capita growth, a multiplier brought into a growth calculation, a movement toward the frontier called a shift, and a normative judgment presented as a fact), cite the evidence from your tables or the course rule, and write a corrected sentence beneath it. This route assesses the same objectives (Objective 1 and Objective 4) as the chatbot critique, takes comparable effort (about 30 minutes and 200–300 words), is scored by the same scoring criteria for the same credit, and its marked-up analysis is the submission artifact that replaces the chatbot report and share link.

Part 7 — What to Submit

• Both completed scaffold tables
• A screenshot of your Desmos frontier at two values of k, or a labelled hand sketch of both frontier lines with the three points
• Written answers to all five analysis questions
• The chatbot critique report with its share link, or the equivalent route's marked-up flawed analysis

Deliverable:

A short Model Workshop report on two illustrative economies: two completed tables (six growth rates, then per-capita growth and four doubling times), a screenshot or sketch of the Week 1 frontier before and after it shifts, written answers to five analysis questions including the positive-or-normative sort, and a critique of a chatbot's account of the same economies, or, on the equivalent route, a marked-up correction of the printed flawed analysis. Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text. Choose a written, recorded, or visual form for your answers; every form is graded by the same scoring criteria.

Rubric

The rubric below shows how each submitted part is evaluated; bands are full, partial, or none.

Workshop evaluation criteria

Criterion	Full	Partial	None
Computing growth from the data	All six growth rates and both per-capita figures are computed by dividing the change by the year 0 value, with real and nominal kept apart and every step shown.	Most rates are right, but one or two divide by the year 1 value, mix real with nominal, or skip their steps.	No growth rates are computed, or the numbers are not drawn from the printed table.
Doubling times and living standards	All four doubling times come from 70 divided by the right growth rate, each is written as an approximation, and the answer explains which economy's average resident gains more and why.	Some doubling times are right but one multiplies by 70, uses nominal growth, or is stated as exact, or the living-standards comparison lacks its reason.	No doubling times or living-standards comparison appear.
Building and shifting the frontier	The Week 1 frontier is drawn and shifted outward, the point (6, 3) is shown reaching the shifted frontier, and the answer separates an outward shift from a move toward the frontier.	The frontier is drawn or shifted, but the shift-versus-movement distinction is missing or reversed.	No frontier work is submitted.
Positive versus normative	All four claims are sorted correctly, and each positive claim is tied to a number in the tables that tests it.	Two or three claims are sorted correctly, or the positive claims are not tied to evidence.	The sort is missing or fewer than two claims are sorted correctly.
Critique of the chatbot or the printed analysis	On either route, the chatbot critique or the equivalent route's marked-up analysis quotes the text it checks, corrects at least two errors with evidence from the tables or the course rule, and labels its positive and normative statements; both routes earn the same credit.	On either route, the critique names an error but does not correct it with evidence, or corrects only one.	On either route, no critique is submitted, or pasted output appears without any checking.

Quality Gate

• Every growth rate in the tables comes from your own calculation on the printed two-economy table, never from memory or a chatbot.
• Every growth rate divides the change by the year 0 value, and growth is judged with real GDP, not nominal GDP.
• Every doubling time is written with about and comes from 70 divided by a growth rate in percent.
• The frontier answer separates a shift of the whole frontier from a movement toward it.
• Each of the four claims is labelled positive or normative, with a number from the tables beside each positive claim.
• No number, rule, or source is invented anywhere in the submission.
• The critique quotes the chatbot or the flawed analysis and corrects at least two errors with evidence or the course rule.