Week 7 Model Workshop: Who Really Pays for the Firewood?

Part 1 — The Big Picture

This week you learned what happens when a rule changes a market's price: a binding ceiling creates a shortage, a binding floor a surplus, and a per-unit tax drives a wedge between what buyers pay and what sellers keep. You also saw that the side that responds less to price bears more of a tax, whoever writes the check. In this workshop you build a mountain town's firewood market in a spreadsheet and in Desmos, add a tax and then a price ceiling, measure who pays and what is lost, and then catch a chatbot's slips on the same market. The habit you build here, compare prices before and after and keep money that changes hands apart from value that disappears, carries into every policy question the course asks.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: When the town taxes each firewood bundle or caps its price, who really pays, how many bundles stop being traded, and how much value is lost?

Workshop data: the illustrative mountain-town firewood market

Source: An illustrative market constructed for this workshop, not real data: a mountain town's weekly market for firewood bundles, with demand Qd = 150 − 10P and supply Qs = 5P, P in dollars per bundle and Q in bundles a week. Every value you need comes from those two equations and is checked in Part 4.

• Rows: 5 listed prices on one demand line and one supply line (the scaffold tables have 5, 7, and 7 rows)
• Columns: price in dollars per bundle, quantity demanded, and quantity supplied in bundles a week
• Rules: demand Qd = 150 − 10P and supply Qs = 5P, with a 3-dollar tax per bundle and an 8-dollar price ceiling as the two interventions

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 Price, Quantity demanded, and Quantity supplied in columns A to C. Type the prices 6, 8, 10, 11, and 12 in A2 to A6. Tool: Desmos (free) or Google Sheets (free). Check: Five prices run down column A, lowest at the top.
2. In B2 type =150-10*A2 and in C2 type =5*A2, then copy both down to row 6. Tool: Desmos (free) or Google Sheets (free). Check: Quantity demanded falls and quantity supplied rises down the table, and one row shows the two columns equal.
3. Open Desmos. Quantity goes on the x-axis and price on the y-axis, so type y = 15 - x/10 for demand and y = x/5 for supply, and click the point where the two lines cross. Tool: Desmos (free) or Google Sheets (free). Check: Demand falls from 15 dollars on the price axis, supply rises from the origin, and Desmos labels the crossing point.
4. Add the tax: type y = 3 + x/5, the supply line moved up by the 3-dollar tax, and click where it crosses demand. Then click the original supply line at the same x value to read the price sellers keep. Tool: Desmos (free) or Google Sheets (free). Check: The new crossing sits up and to the left of the first one, and the vertical gap between the two supply lines at that quantity equals the tax.
5. Add the ceiling: type y = 8 and click where it meets each original line. Use the polygon command with three corners to shade each deadweight-loss triangle: for the tax, the new crossing, the point straight below it on the original supply line, and the first crossing; for the ceiling, the two points on the demand and supply lines at the quantity that still trades, and the first crossing. Tool: Desmos (free) or Google Sheets (free). Check: Two shaded triangles each point right toward the first crossing; the ceiling line meets supply to the left of where it meets demand.
6. Fill in the Part 4 tables from your sheet and graph, 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 all three Part 4 tables by hand with any calculator, writing each substitution or area formula in words beside its result. Describe the graph in words instead of drawing it: what is on each axis, where each line sits, where the original and taxed supply lines meet demand, where the 8-dollar line meets each curve, and which triangle is each deadweight loss. 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 the firewood market's schedule from the two equations; this table shows where the market clears before any rule.

Student scaffold table 1

Price (dollars per bundle)	Quantity demanded: 150 − 10P	Quantity supplied: 5P	Shortage or surplus (bundles)
6
8
10
11
12

Record what a 3-dollar tax per bundle, collected from sellers, does to the market, piece by piece.

Student scaffold table 2

Quantity	With no tax	With the 3-dollar tax
Price buyers pay (dollars)
Price sellers keep (dollars)
Bundles traded a week
Tax revenue (dollars a week)
Deadweight loss (dollars a week)
Buyers' share of the tax (dollars)
Sellers' share of the tax (dollars)

Record what a price ceiling of 8 dollars per bundle does to the same market.

Student scaffold table 3

Quantity	Value at the ceiling
Quantity demanded at 8 dollars
Quantity supplied at 8 dollars
Shortage (bundles a week)
Bundles traded a week
Value of the last traded bundle to its buyer (dollars)
Cost of the last traded bundle to its seller (dollars)
Deadweight loss (dollars a week)

Part 5 — Analysis Questions

1. Complete the schedule table. At what price and quantity does the firewood market clear, and what is the shortage or surplus at each other listed price? Say in one plain sentence what the clearing point means for buyers and sellers.
2. With the 3-dollar tax collected from sellers, write the new supply equation, solve for the price buyers pay, the price sellers keep, and the quantity, then compute the tax revenue and the deadweight loss. Say in words why the revenue is not part of the loss.
3. How much of the 3-dollar tax do buyers bear, and how much do sellers bear? Explain the split using how many bundles each side changes per dollar, and say whether the answer would change if buyers handed the tax to the town instead. Name which curve moved, which way, and where the new price and quantity land.
4. With the 8-dollar ceiling, find quantity demanded, quantity supplied, the shortage, the bundles that still trade, and the deadweight loss. Say why a ceiling at 12 dollars would change nothing.
5. Name one limitation of this model, such as the straight lines, the assumption that the bundles under the ceiling go to the buyers who value them most, or the fact that real responsiveness is estimated rather than known, and explain how it could change one of your conclusions.
6. Choose one reflection option, (a) or (b), and answer it in 3–4 sentences.

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 price controls and taxes in an introductory microeconomics course. In an illustrative example, not real data, a mountain town's weekly market for firewood bundles has demand Qd = 150 − 10P and supply Qs = 5P, with P in dollars per bundle and Q in bundles a week. Find where the market clears. Then the town collects a tax of 3 dollars per bundle from sellers: find the price buyers pay, the price sellers keep, the quantity, the tax revenue, the deadweight loss, and how much of the tax each side bears. Separately, the town instead sets a legal maximum price of 8 dollars per bundle: is that a price ceiling or a price floor, and what are the shortage or surplus, the quantity traded, and the deadweight loss? Then answer: does the tax change demand for firewood? What is the price elasticity of demand here, and are buyers elastic or inelastic? Should the town adopt the tax or the price limit? Should the town cut its own firewood because its crews are the fastest cutters? What price should a firewood seller charge to maximize profit, and how much profit do sellers earn after the tax? And does the tax make sellers pay for the smoke from burning firewood?

Test the response for these likely failures:

• Chatbots often say sellers bear the whole 3-dollar tax because sellers hand it to the town.
• Chatbots sometimes call an 8-dollar legal maximum a price floor, or say it leaves a surplus of unsold bundles.
• Chatbots often add that the tax or the ceiling is a bad idea, presenting a value judgment as a model result.
• Chatbots sometimes say the tax shifts the demand curve, or that the ceiling shifts supply.
• Chatbots often say demand for firewood falls when the buyer price rises to 11 dollars.
• Chatbots often call the 10 bundles per dollar that buyers give up the price elasticity of demand.
• Chatbots sometimes call firewood buyers inelastic because they bear part of the tax, with no calculation.
• Chatbots sometimes say the town should cut its own firewood because its crews are the fastest cutters around.
• Chatbots sometimes invent an average cost per bundle and treat it as the cost of one more bundle.
• Chatbots often say a firewood seller should set its price equal to marginal cost to maximize profit.
• Chatbots often call producer surplus after the tax the sellers' profit.
• Chatbots sometimes claim the tax is good because it makes sellers pay for the smoke from burning firewood.

Write 3–5 sentences naming what the chatbot got right and at least two places where it slipped into one of the failures above, or counted the tax revenue as the deadweight loss, quoting its words and correcting each by hand against your own completed tables and Chapter 7. For any claim about earlier or later 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 4 (measure consumer surplus, producer surplus, and market efficiency, and analyze price ceilings, price floors, taxes, and subsidies, including who bears a tax and the deadweight loss) and Objective 3 (compute and interpret price, income, and cross-price elasticities of demand by the midpoint formula, and relate price elasticity to total revenue), 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 firewood market, printed here in full: "The market clears at 10 dollars and 50 bundles. The 3-dollar tax is paid by sellers, so sellers bear all 3 dollars and buyers bear nothing. The tax shifts the demand curve left, so demand for firewood falls. The deadweight loss is 3 × 40 = 120 dollars a week. Buyers give up 10 bundles per dollar, so their price elasticity is 10 and demand is elastic. The 8-dollar limit is a price floor, so it leaves a surplus of unsold firewood. Both rules are bad ideas, so the town should reject them. The town should cut its own firewood because its crews are the fastest. A seller should charge a price equal to marginal cost, and producer surplus after the tax is the sellers' profit. The tax is good because it makes sellers pay for the smoke." Mark every slip, name the failure each shows, correct each one by hand against your own tables and Chapter 7, 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 three 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 firewood market: three completed tables (the schedule, the 3-dollar tax, and the 8-dollar ceiling), written answers to six analysis questions, and a critique of a chatbot's answers about the same market, 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 diagram that describes the wedge and the two loss triangles 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
Schedule built and market solved	The five-row schedule is complete, the clearing price and quantity are found by solving Qd = Qs, every gap is labelled a shortage below the clearing price or a surplus above it, and the clearing point is read in a plain sentence.	The schedule is mostly complete, but one or two quantities or gap labels are wrong, or the plain sentence is missing.	No schedule, or no clearing point found.
Tax wedge, revenue, and loss computed	The taxed supply equation is written, the buyer price, seller price, and quantity are solved with steps shown, revenue is the tax times the quantity still traded, and the deadweight loss is the triangle, with revenue kept out of the loss in words.	The tax outcome is set up, but one value is wrong, or revenue and loss are computed without explaining that revenue is a transfer.	No tax outcome, or the loss computed as tax times quantity with no triangle.
Who bears the tax explained	Each side's share is the change in its own price, the shares add to the tax, the split is explained by bundles per dollar on each side, the buyer-collected case is shown to give the same prices, and the shifted curve, its direction, and the new price and quantity are named.	The split is computed but the responsiveness reason or the buyer-collected comparison is missing or unclear.	No split, or the whole tax assigned to the side that writes the check.
Ceiling shortage and loss computed	Quantity demanded and supplied at 8 dollars are computed, the gap is named a shortage, the traded quantity and the last bundle's value and cost are read, the loss triangle is computed, and the 12-dollar ceiling is explained as nonbinding.	The ceiling is set up, but one value is wrong, the gap is mislabelled, or the nonbinding case is missing.	No ceiling analysis, or the 8-dollar maximum treated as a floor throughout.
Plain-sentence analysis, limitation, and reflection	All six analysis questions are answered from the market's own numbers, the limitation is named with its effect on a conclusion, and the chosen reflection option is answered with a specific example.	Most questions are answered, but the limitation is missing or vague, or the reflection is missing.	No analysis answers.
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 or later 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.	No critique on either route, or the chatbot's reply is pasted without any check.

Quality Gate

• Every buyer price, seller price, and quantity comes from solving the two equations, shown by formula or by hand, never from memory or a chatbot.
• The tax is drawn as supply moving up by 3 dollars, and the ceiling moves no curve.
• Each side's share of the tax is the change in its own price, and the shares add up to the tax.
• Tax revenue is reported as a transfer and never counted inside the deadweight loss.
• A ceiling is called a ceiling and creates a shortage; a floor is never confused with it.
• Every computed result ends with one plain sentence about what it means for buyers, sellers, or the town.
• Every judgment about whether the town should adopt a rule is labelled normative.
• No real-world price, tax rate, 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.