Do students need to do the algebra, or just shift curves?

The AP exam usually asks for graphs and the size of the tax, not full algebraic solving. Teach the algebra once with the widget example so the 'tax equals external cost' rule sticks, then let most students work graphically.

What is the prerequisite for this lesson?

Students should have the market-failure overview (Topic 6.1) so they already know MSC, MPC, and deadweight loss. This lesson is the fix: taxes, subsidies, good classification, and Coase.

My students size the tax or subsidy off the price gap instead of the external cost. How do I break that habit?

Make them measure the vertical distance between MPC and MSC, or between private demand and MSB, and never a horizontal price change. Say it as a slogan: the correction equals the external cost or benefit at the optimal quantity, full stop. Once they measure the right gap, the tax-versus-subsidy choice follows from whether the market over- or under-produces.

Lesson plans · AP Micro Unit 6 · MICRO 6.2, MICRO 6.3, MICRO 6.4

Externalities and Public Goods: Pigouvian Taxes and Subsidies

Essential question: When a market over- or under-produces because of spillovers, exactly how big should the tax or subsidy be to fix it?

2 × 50-minute periods · MICRO 6.2, MICRO 6.3, MICRO 6.4 · prints clean with Cmd/Ctrl+P

Objectives

Students will be able to draw negative and positive externality graphs and correctly distinguish marginal private from marginal social cost or benefit.
Students will be able to calculate the optimal Pigouvian tax as equal to the marginal external cost and show that it shifts the private curve onto the social curve.
Students will be able to explain when a per-unit subsidy corrects a positive externality and set its size to the marginal external benefit.
Students will be able to classify goods as private, public, common-resource, or club using excludability and rivalry.
Students will be able to explain the free-rider problem and the Coase theorem as a private alternative when property rights are clear.

Materials (all free, no student accounts needed)

Warm-up (8 min)

Post: 'A factory dumps 15 dollars of pollution harm into a river for every widget it makes. What is the single number a government would tax per widget to fix it?' Students write a guess and one sentence of reasoning.
Take three guesses on the board without confirming. Many will overthink it; keep the 15-dollar answer in your pocket for the worked example.

Direct instruction (34 min)

Recap the two externality graphs: negative externality means MSC is above MPC and the market overproduces; positive externality means MSB is above MPD (private demand) and the market underproduces.
Introduce the Pigouvian tax: set the per-unit tax equal to the marginal external cost so the private supply curve shifts up onto MSC, moving the market to the efficient quantity and eliminating deadweight loss.
Work the module example live. MPC: P = 10 + 0.7Q. Demand: P = 90 - 0.8Q. External cost 15 per unit. Solve the market quantity (Q = 53.3), form MSC = 25 + 0.7Q, solve the optimal quantity (Q = 43.3), and compute DWL = 0.5 x 10 x 15 = 75.
Show that a 15-dollar tax turns supply into P = 25 + 0.7Q, exactly MSC, so the correction is precise. Then flip to positive externalities: the fix is a per-unit SUBSIDY equal to the marginal external benefit.
Classify goods with the two-by-two of excludability and rivalry (private, club, common resource, public) and connect non-excludable plus non-rival to the free-rider problem. Add the Coase theorem: with clear property rights and low bargaining costs, private parties can reach the efficient outcome without government.

Guided practice (30 min)

Project /sandbox/externality on the negative-externality setting. Have a student identify the market quantity, then drag to reveal MSC and mark the optimal quantity and the DWL triangle.
Now activate the tax (or shift the private curve up by the external cost) and cold-call: 'What happened to the DWL, and why does the tax equal exactly the external cost and not more?' Elicit that a larger tax would push output below the optimal quantity, creating a new loss.
Switch to the positive-externality case. Ask the class to predict the fix before you act, then show a subsidy shifting private demand (or supply) out to the socially optimal quantity. Cold-call: 'Tax or subsidy here, and why the difference?'
Numeric check at desks: give MPC = 4 + 0.5Q, demand P = 40 - 0.5Q, external cost 6. Partners find the market quantity, the optimal quantity, the Pigouvian tax, and the DWL. Cold-call one pair to put the numbers on the board and defend that the tax equals 6.

Independent practice (18 min)

Assign 6 items from /practice/public-goods-externalities, including at least two goods-classification questions and two externality-correction questions.
One drawing task: sketch a positive-externality graph, label MSB above private demand, and show the subsidy that reaches the optimal quantity.

Exit ticket

Item 1: a good is non-excludable and non-rival. Students name its category.
Item 2: a market has a negative externality of 8 dollars per unit; state the optimal Pigouvian tax and one sentence on why that exact size works.
Grade for 2 points: public good, and 8 dollars with a correct reason that the tax shifts MPC onto MSC.

Homework

Read the Coase Theorem section of /micro/public-goods-externalities and write two sentences on why clear property rights let neighbors solve a noise dispute without a tax.
Redo the worked Pigouvian example with a new external cost of 20 dollars and report the new optimal tax and DWL.

Differentiation

Give equation-averse students a ready-made table (columns for MPC, MSC, demand, tax) so they plug in rather than derive.
Stretch: ask advanced students to prove algebraically that a tax equal to the external cost always lands the market at the MSC-demand intersection, for any linear curves.
Provide the excludability-rivalry two-by-two grid as a printed reference so classification questions become a two-question lookup.

Misconceptions to head off

Wrong: the optimal tax equals the difference between the market price and the optimal price. Correction: the Pigouvian tax equals the marginal EXTERNAL cost, the vertical gap between MSC and MPC, not any price difference.
Wrong: a subsidy fixes a negative externality. Correction: a tax fixes a negative externality (overproduction); a subsidy fixes a positive externality (underproduction).
Wrong: a bigger tax is always better because pollution is bad. Correction: a tax above the external cost overshoots and drives output below the optimal quantity, creating a new deadweight loss.
Wrong: public goods and common resources are the same. Correction: both are non-excludable, but a common resource is RIVAL (one person's use depletes it), which is why it gets overused, while a public good is non-rival.

Teacher FAQ

Do students need to do the algebra, or just shift curves?: The AP exam usually asks for graphs and the size of the tax, not full algebraic solving. Teach the algebra once with the widget example so the 'tax equals external cost' rule sticks, then let most students work graphically.
What is the prerequisite for this lesson?: Students should have the market-failure overview (Topic 6.1) so they already know MSC, MPC, and deadweight loss. This lesson is the fix: taxes, subsidies, good classification, and Coase.
My students size the tax or subsidy off the price gap instead of the external cost. How do I break that habit?: Make them measure the vertical distance between MPC and MSC, or between private demand and MSB, and never a horizontal price change. Say it as a slogan: the correction equals the external cost or benefit at the optimal quantity, full stop. Once they measure the right gap, the tax-versus-subsidy choice follows from whether the market over- or under-produces.

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