Week 14 — Who Pays for the Smoke? — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

Assignment 13 of the term — find the efficient quantity when a market's costs spill onto bystanders, measure the deadweight loss, show how a corrective tax fixes it, value a public good, and explain to a non-expert why free riders can sink a project worth doing.

This week you learned what happens when part of a market's cost lands on people outside the trade: sellers pay only their private cost, so the market makes more than the efficient quantity; the extra units are a deadweight loss; a corrective tax equal to the harm per unit moves the market back; and public goods, which no one can be kept from, invite free riders. In Week 6 you measured surplus as triangles, and in Week 7 you put a 2-dollar tax on this same market. In this assignment you work four short problems, interpret each one, and explain the free-rider problem to someone who has never studied economics. This week's lecture tutorial, quiz, discussion, and Model Workshop practice the same ideas from other angles.

The task

You will use the running market with smoke from this week's chapter, Dataset P, and the fireworks show from Dataset Q, with illustrative numbers built for this course (not real data):

• Demand is Qd = 90 − 10P, or P = 9 − Q ⁄ 10, and supply is Qs = 10P − 30, or P = 3 + Q ⁄ 10, with P in dollars per unit and Q in units per day.
• Each unit does 2 dollars of harm to neighbors who neither buy nor sell the good, so marginal social cost is P = 5 + Q ⁄ 10.
• Ignoring the harm, the market trades 30 units a day at 6 dollars.
• In the neighborhood, 100 residents each value a fireworks show at 10 dollars, the show costs 600 dollars, and no one can be kept from watching it.

Work these four problems, showing your arithmetic:

1. In the course's illustrative running market, demand is P = 9 − Q ⁄ 10 and each unit does 2 dollars of harm to neighbors, so marginal social cost is P = 5 + Q ⁄ 10. Setting marginal social benefit 9 − Q ⁄ 10 equal to marginal social cost 5 + Q ⁄ 10, what is the efficient quantity, in units per day?
2. The illustrative market trades 30 units a day, the efficient quantity is 20, and at the 30th unit marginal social cost is 8 dollars while buyers value it at 6. Using the triangle one half times the gap times the extra units, what is the deadweight loss of the externality, in dollars per day?
3. A 2-dollar corrective tax per unit on sellers shifts supply in the illustrative market to Qs = 10P − 50, while demand stays Qd = 90 − 10P. Setting Qd = Qs, what price do buyers pay, in dollars per unit?
4. In the course's illustrative neighborhood, 100 residents each value a fireworks show at 10 dollars, and the show costs 600 dollars. Using total value minus cost, what is the net gain from putting on the show, in dollars?

Beside each answer, add one brief interpretation. For problem 1, say what the efficient quantity means, compare it with the market's 30 units at 6 dollars, and explain why the market makes too much. For problem 2, explain what the deadweight loss is and why it is not the 60 dollars of total harm on all 30 units. For problem 3, say what sellers keep and how many units trade, and explain why the same 2-dollar tax that caused a deadweight loss of 10 in Week 7 removes one of 10 here. For problem 4, explain why the show is a public good and why it may not happen even though it is worth more than it costs.

Explain it to a non-expert

The treasurer of the neighborhood association says: "If the show is really worth 1,000 dollars to us, people will chip in the 600 on their own." Write a 3–5 sentence message to the treasurer that explains in everyday words why voluntary gifts can fall short even for a show worth more than it costs, and one limit, such as that the numbers are a classroom model or that neighbors who know each other may give out of goodwill. Explain any technical term in everyday words. More than one accurate wording can be defensible, and the message does not need to say how the association should raise the money.

For a different, ungraded example, imagine a friend who says: "Everyone on our street loves the new speed bumps, so the residents will gladly split the cost." Model answer: "Everyone on the street gets the quieter, safer road whether or not they pay, so each person is tempted to let the neighbors cover it. When enough people think that way, the collection comes up short even though the street as a whole would gladly pay the full price. That is why shared things like this are often paid for through dues or a city budget instead of a hat passed around. One limit: on a small street where everyone knows each other, people may chip in anyway, so I would want to see how the collection actually goes."

Use that pattern for your message to the treasurer, with your own explanation and evidence.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

Deliverables

Submit a single report containing:

• the four answers, each with the arithmetic shown and a brief interpretation as the task describes;
• your message to the treasurer, explaining why voluntary gifts can fall short and naming one limit; and
• a one-line restatement of the values you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes with a transcript, or as a labelled sketch that describes the demand line, the supply line, and the marginal-social-cost line in words with a short note. Every form is graded by the same scoring criteria. 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.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

This version is designed to be completed without AI: do the computations, the interpretations, and the writing yourself, without any AI tool. You may use a calculator or a spreadsheet for the arithmetic. Submitting AI-generated work as your own, or copying another student's work, is an integrity violation under the course policy.

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