Week 15 Model Workshop: Which Town Shares Its Income More Evenly?

Part 1 — The Big Picture

This week you learned to measure how income is shared: rank households from poorest to richest, split them into quintiles, add their income shares as you go, and plot those cumulative shares as a Lorenz curve against the straight line of perfect equality. The Gini coefficient turns the gap between the two lines into one number from 0 to 1, and it is a positive measurement, while a judgment that a society's Gini is too high is normative. In this workshop you build two towns' Lorenz tables in a spreadsheet, compute each town's area and Gini coefficient by trapezoids, compare the towns, label claims about them as positive or normative, and then catch a chatbot's slips on the same towns. The habit you build, measuring first and labelling the judgment second, is the habit the whole course has been teaching.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: Which of the two illustrative towns shares its income more unequally, by how much, and which claims about the towns can a Gini coefficient settle?

Workshop data: the illustrative towns of Hillford and Lakemont

Source: Two illustrative towns constructed for this workshop, not real data and not any real place: two small towns each rank their households from the poorest fifth to the richest. In Hillford the five quintiles receive 5, 8, 13, 20, and 54 percent of income. In Lakemont they receive 8, 12, 17, 23, and 40 percent.

• Rows: 5 quintiles per town, 10 income shares in all (the Lorenz table has 6 rows, from 0 to 100 percent of households)
• Columns: the cumulative share of households and each town's cumulative share of income, from which the trapezoid areas and the Gini coefficients are built
• Towns compared: Hillford and Lakemont, each with five quintile shares that add to 100 percent

Part 3 — Materials & Procedure

Tools and materials

• Desmos (free) or Google Sheets (free)

Ways of working

• build-the-supply-and-demand-schedule
• shift-vs-move-along-the-curve
• compute-elasticity-by-the-midpoint-formula

Procedure

1. Open a blank Google Sheets file. Label row 1 as Households, Hillford, Lakemont, Hillford strip, and Lakemont strip in columns A to E. Type the cumulative household shares 0, 0.2, 0.4, 0.6, 0.8, and 1 in A2 to A7. Tool: Desmos (free) or Google Sheets (free). Check: Six rows of household shares, from 0 to 1.
2. Type 0 in B2 and C2. Build each town's running total as a fraction: in B3 type =B2+0.05 and continue down with each Hillford share (0.08, 0.13, 0.20, 0.54); do the same in column C with Lakemont's shares (0.08, 0.12, 0.17, 0.23, 0.40). Tool: Desmos (free) or Google Sheets (free). Check: Both cumulative columns end at 1 in row 7.
3. In D3 type =(A3-A2)*(B2+B3)/2 and in E3 type =(A3-A2)*(C2+C3)/2, then copy both down to row 7. In D8 type =SUM(D3:D7) and in E8 type =SUM(E3:E7) for the areas under the two Lorenz curves. Tool: Desmos (free) or Google Sheets (free). Check: Each area is less than 0.5, the area under the line of perfect equality.
4. In D9 type =1-2*D8 and in E9 type =1-2*E8 for the two Gini coefficients. Check each against the gap method: 0.5 minus the area, divided by 0.5. Tool: Desmos (free) or Google Sheets (free). Check: Both Gini coefficients lie between 0 and 1, and the town with the smaller area has the larger Gini.
5. Optional picture: in Desmos, open a table with the household shares in the first column and each town's cumulative shares in the next two, turn on lines between the points, and type y = x for the line of perfect equality. Tool: Desmos (free) or Google Sheets (free). Check: Both curves start at the bottom-left corner and end at the top-right corner, and one sags farther below the diagonal than the other.
6. Fill in the Part 4 tables from your sheet, answer the analysis questions in Part 5, then do the AI-critique step in Part 6 or its equivalent route.

Fallback route

When to use it: Use this route if you cannot open Google Sheets or Desmos, or your device cannot run them.

Complete both Part 4 tables by hand with any calculator, writing each running total and each trapezoid sum beside its result. Describe the picture in words instead of drawing it: what is on each axis, where the line of perfect equality runs, and how far each town's curve sags below it. Answer all six analysis questions, do the critique step, and submit the same items for equal credit.

This route earns the same credit as the primary route.

Part 4 — The Scaffold

Build each town's Lorenz table first; every comparison in this workshop reads from these running totals.

Student scaffold table 1

Share of households (poorest first)	Hillford cumulative share of income (percent)	Lakemont cumulative share of income (percent)	Line of perfect equality (percent)
0 percent	0	0	0
20 percent			20
40 percent			40
60 percent			60
80 percent			80
100 percent	100	100	100

Compute each town's trapezoid areas, the area under its Lorenz curve, and its Gini coefficient, and say in words what the number means.

Student scaffold table 2

Town	Sum of the five trapezoid areas	Area under the Lorenz curve	Gini coefficient	What it means in words
Hillford
Lakemont

Part 5 — Analysis Questions

1. Complete the Lorenz table for both towns. At 60 percent of households, what share of income has each town's curve reached? Say in one plain sentence what that gap from the line of perfect equality means.
2. Complete the second table. For each town, show the five trapezoid sums of heights, the area under the Lorenz curve, and the Gini coefficient, then check each Gini with the gap method. Say in words what each result means.
3. Which town shares its income more unequally? Explain using both the Gini coefficients and a description of the two curves in words, and compute how many times the poorest fifth's share the richest fifth receives in each town.
4. Label each claim as positive or normative and give your reason: (a) Hillford's Gini coefficient is higher than Lakemont's; (b) Hillford ought to tax its richest fifth more; (c) a tax on Hillford's richest fifth would lower their choices to work; (d) Lakemont's distribution is fairer. Then name one thing the Gini coefficients alone cannot tell you about either town.
5. Name one way this classroom measurement differs from a real one, such as using only five groups, counting income before taxes and transfers, or leaving out wealth, and say how it could change one of your answers.
6. Choose one reflection option and answer it: (a) describe in words a group you know, such as a team or a club, where rewards were shared very unevenly, and say which of your claims about it were positive and which normative; or (b) describe a claim about inequality you have heard, in words, and say what evidence would test its positive part. Both options carry the same weight.

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 the measurement of inequality in an introductory microeconomics course. In an illustrative example, not real data, two small towns each rank their households from the poorest fifth to the richest. In Hillford the five quintiles receive 5, 8, 13, 20, and 54 percent of income. In Lakemont they receive 8, 12, 17, 23, and 40 percent. For each town, give the cumulative shares, the area under the Lorenz curve using five trapezoids each 0.2 wide, and the Gini coefficient as 1 minus 2 times that area. Which town's income is shared more unequally? Then answer: should Hillford's government tax its richest fifth more? Does a fall in the price of food shift each town's Lorenz curve, or move along it? Is a change in Hillford's Gini a change in demand or a change in quantity demanded? Is the steepness of the Lorenz curve its elasticity, and is income demand elastic or inelastic here? Does Hillford's richest fifth earn more because it has an absolute advantage or a comparative advantage? Is a legal minimum wage in Lakemont a price ceiling or a price floor? If Lakemont taxes the richest fifth's employers, who really pays the tax? Is the richest fifth's average income the same as its marginal cost? Do the towns' firms set output where price equals marginal cost or where marginal revenue equals marginal cost? Is the richest fifth's income its accounting profit or its economic profit? Does Hillford's factory smoke, a cost to its neighbors, count in its Gini coefficient?

Test the response for these likely failures:

• Chatbots often add that Hillford should tax its richest fifth more, or that its Gini is too high, presenting a value judgment as a model result.
• Chatbots sometimes say a fall in the price of food moves along or shifts a town's Lorenz curve.
• Chatbots sometimes call a change in Hillford's Gini coefficient a change in demand or in quantity demanded.
• Chatbots sometimes call the steepness of the Lorenz curve an elasticity.
• Chatbots sometimes label one town's income distribution elastic or inelastic.
• Chatbots sometimes say Hillford's richest fifth earns more because of an absolute or a comparative advantage.
• Chatbots sometimes call a legal minimum wage a price ceiling.
• Chatbots often say employers bear a tax in full because they hand it to the government.
• Chatbots sometimes treat the richest fifth's average income as a marginal cost.
• Chatbots sometimes say the towns' firms set output where price equals marginal cost, whatever their market.
• Chatbots sometimes call a quintile's income share its accounting or economic profit.
• Chatbots sometimes add the cost of a factory's smoke to Hillford's Gini coefficient.

Write 3–5 sentences naming what the chatbot got right and at least two places where it slipped into one of the failures above, quoting its words and correcting each by hand against your own completed tables and Chapter 15. For any claim about earlier material that this workshop cannot check, say so and why.

Equivalent route (no AI required)

No AI account is required for this workshop. This route assesses the same objectives, Objective 8 (analyze market failures — externalities, public goods, common resources, and asymmetric information — evaluate responses such as corrective taxes, cap-and-trade, and private bargaining, and survey behavioral economics and the measurement of inequality evenhandedly) and Objective 1 (use the tools of economic modeling — scarcity, opportunity cost, the production possibilities frontier, comparative advantage, and the gains from trade — and distinguish positive statements (what is) from normative statements (what ought to be)), takes comparable effort of about the same 90 minutes, uses the same rubric, and earns the same credit. Instead of the chatbot, critique this flawed analysis of the two towns, printed here in full: "Hillford's Lorenz curve passes through 5, 8, 13, 20, and 54 percent, so its area is 0.1 × (0.05 + 0.08 + 0.13 + 0.20 + 0.54) = 0.10 and its Gini is 1 − 2 × 0.10 = 0.80. Lakemont's Gini is lower, so its income is shared more unequally, because a Gini near 0 means one household gets almost everything. Hillford's government should tax its richest fifth more, because the model proves its Gini is too high. A fall in food prices shifts both Lorenz curves, which is a change in quantity demanded, and the steep last segment of Hillford's curve is its elasticity, so its income is elastic. Hillford's richest fifth earns more because of its absolute advantage, a minimum wage there would be a price ceiling, and employers would bear any payroll tax in full. The richest fifth's average income equals its marginal cost, its income is its economic profit, the towns' firms all set price equal to marginal cost, and Hillford's Gini should include the cost of its factory smoke." Mark every slip, name the failure each shows, correct each one by hand against your own tables and Chapter 15, and flag each claim about earlier or later material that this workshop cannot check. Submit the marked-up analysis with your corrections in place of the chatbot report and share link.

Part 7 — What to Submit

• The two completed scaffold tables, from a spreadsheet, Desmos, or by hand
• Written answers to all six analysis questions
• The chatbot critique report with its share link, or the equivalent route's marked-up correction of the printed flawed analysis

Deliverable:

A short Model Workshop report on the two towns: two completed tables (the Lorenz table and the area-and-Gini table), written answers to six analysis questions, and a critique of a chatbot's answers about the same towns, 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 how you show it: written answers, a recorded explanation with a transcript, or a labelled sketch that describes the two curves and the diagonal in words with a short note; every form is graded by the same 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
Lorenz tables built	Both towns' cumulative shares are built as running totals with each step shown, and the share reached at 60 percent of households is read for each town with what the gap from equality means.	The tables are mostly complete, but one or two running totals are wrong, or the 60 percent reading is missing or unexplained.	No Lorenz table, or each fifth's own share plotted instead of the running total.
Areas and Gini coefficients	For each town the five trapezoid sums, the area under the curve, and the Gini coefficient are computed with each step shown, each Gini is checked by the gap method, and each result is read in a plain sentence.	Most steps are worked, but one area or one Gini is wrong, the gap check is missing, or the plain sentences are missing.	No areas or Gini coefficients, or the area taken as the Gini itself throughout.
Comparing the towns	The town with the higher Gini is named as the more unequal one with 0 read as perfect equality, both curves are described in words, and the richest-to-poorest ratio is computed for each town.	The comparison is right but one of the word description or the ratios is missing, or one ratio is wrong.	No comparison, or the lower Gini read as the more unequal town.
Positive and normative labels	All four claims are labelled with a reason that ties positive to what evidence can test and normative to values, and one thing the Gini alone cannot show is named.	Two or three claims are labelled correctly with reasons, or the limit of the Gini is missing.	No labels, or the claims that a town ought to act or is fairer labelled positive.
Limitation and reflection	A real limitation of the measurement is named with its effect on an answer, and the chosen reflection option is answered with a specific example described in words.	The limitation is vague or not linked to an answer, or the reflection is missing.	No limitation and no reflection.
AI critique or equivalent correction	The chatbot's reply, or the printed flawed analysis on the equivalent route, is quoted, at least two slips are named with the failure each shows, each is corrected by hand, and claims about earlier material are flagged; either route earns this band equally.	One slip is named and corrected on either route, or slips are named without a correction; either route earns this band equally.	No critique on either route, or the chatbot's reply is pasted without any check.

Quality Gate

• Every cumulative share is a running total of the quintile shares, shown by formula or by hand, never from memory or a chatbot.
• Every area is a sum of five trapezoids, each 0.2 wide, and every Gini coefficient is 1 minus 2 times its area, checked by the gap method.
• The comparison names the town with the higher Gini as the more unequal one and reads 0 as perfect equality, never as perfect inequality.
• Each Lorenz curve is described in words: what is on each axis and how far it sags below the line of perfect equality.
• Every judgment about what a town ought to do, or about which town is fairer, is labelled normative.
• No real-world income figure, Gini value, statistic, or study is invented anywhere in the submission.
• The critique quotes the chatbot, or the printed flawed analysis, and corrects each slip by hand.