Negative externalities — too much of a bad thing — Economics

Before this

This leads to

Too much paint

The last lecture built a market with a pollution problem. Marginal private benefit (MPB, the demand curve) is $P = 30 - 0.2 Q$ . Marginal private cost (MPC, the supply curve) is $P = 6 + 0.1 Q$ . Each unit also does $6 of damage downstream, so marginal social cost (MSC) lies $6 above MPC. The market makes $Q^{*} = 80$ units a week, but the socially optimal quantity is $Q_{opt} = 60$ . The 20 extra units cost society $60 a week in deadweight loss.

That lecture promised to look at the tools that fix the problem. There are two main ones: put a price on the damage with a tax, or set a rule that limits it. This lecture works through both with the same numbers, and compares them.

There is also an older promise to keep. The tax incidence lecture showed that a $5 tax in an ordinary market creates a deadweight loss. It ended by saying that when a product harms people outside the market, a tax can make the market better rather than worse. Here is that case.

A tax equal to the damage

In 1920, the Cambridge economist Arthur Pigou argued in The Economics of Welfare that negative externalities should be offset by a tax. A tax designed for this job is called a Pigouvian tax (or a corrective tax). The rule is simple to state:

The Pigouvian tax

Set the tax per unit equal to the external cost per unit at the socially optimal quantity. The tax lifts the supply curve (MPC) onto the marginal social cost curve (MSC), so the market settles at $Q_{opt}$ by itself.

In our market, the external cost is $6 per unit, so the tax is $t = $6$ . Sellers now need $6 more per unit, so supply becomes $P = 12 + 0.1 Q$ . That is exactly the MSC line. Setting it equal to demand:

30 - 0.2 Q 18 Q = 12 + 0.1 Q = 0.3 Q = 60

The market now stops at 60 units, the social optimum. Buyers pay $P_{c} = 30 - 0.2 \times 60 = $18$ , sellers keep $P_{p} = 18 - 6 = $12$ , and the government collects $6 \times 60 = $360$ a week.

A $6 Pigouvian tax lifts supply (MPC) onto the MSC line. The market moves from Q* = 80 to Q_opt = 60. Buyers pay Pc = $18, sellers keep Pp = $12, and the tinted rectangle is the tax revenue, $6 × 60 = $360 a week. No deadweight-loss triangle remains.

Compare with the old price of $14. Buyers pay $4 more and sellers keep $2 less, the same two-thirds-to-one-third split the incidence lecture would predict for these slopes. The tax lecture's rules still apply. What has changed is the verdict.

Why this tax removes a loss instead of creating one

Count everything. Without the tax, buyers and sellers together gain $960 a week (the area between MPB and MPC up to 80 units), and the river suffers $6 \times 80 = $480$ of damage. Society is left with $960 - 480 = $480$ .

With the tax, buyers and sellers gain $900 from 60 units, but $360 of that goes to the government as tax, leaving them $540. The government has $360 to spend. The damage falls to $6 \times 60 = $360$ . Society now has $540 + 360 - 360 = $540$ .

Society is $60 a week better off: exactly the deadweight loss that the externality caused. In an ordinary market, the 20 units that the tax removes would be worth making, so removing them is a loss. Here, each of those units cost society more than it was worth. Removing them is a gain. Same tax, same diagram, opposite verdict, because the damage is now counted.

Stop and think

A factory owner complains: "This tax punishes us just for doing our job." Is that a fair description of a Pigouvian tax?

Show the answer

The economic answer is no. Before the tax, the factory paid for its materials and workers but not for the damage to the river. People downstream were, in effect, paying part of its costs. The tax asks the factory to pay the full cost of what it makes, including the part it was passing on. This idea is often called "the polluter pays". The owner is right about one thing, though: the tax is only fair if it is close to the real damage. A tax far above the damage would punish production that is genuinely worth having.

What if the tax is the wrong size?

The rule needs the external cost per unit, and in real life nobody knows it exactly. How much does it matter? Try different taxes below. The external cost stays at $6.

Graph builder — choosing the size of a pollution tax

The true external cost is $6 per unit, so the best quantity is 60. Choose a tax. The grey dotted line is supply plus your tax. Watch where the market ends up, the tax revenue and the deadweight loss that remains. Try $3, $9 and $12.

Tax per unit: $6

The market's quantity and the deadweight loss left for different taxes, when the true external cost is $6 a unit (per week):

Tax	Quantity	Revenue	DWL left
$0	80	$0	$60
$3	70	$210	$15
$6	60	$360	$0
$9	50	$450	$15
$12	40	$480	$60

Two lessons come out of this. First, a tax that is roughly right does most of the good: a $3 tax, half the correct size, removes three quarters of the loss. Second, there are limits. In this market, a tax of $12, double the damage, leaves a loss as big as doing nothing, and anything larger is worse. "Some tax is always better than none" is true only if the tax is in the right range.

The other tool: rules and limits

Instead of putting a price on pollution, a government can simply set a rule. This is called regulation, or "command and control". Examples include limits on how much pollution a car or a factory may emit, a requirement to fit filters, bans on smoking in enclosed public places, and limits on noise at night. In our market, a rule could cap output at 60 units, or require each factory to cut its pollution to a set level.

Regulation has real strengths. The outcome is more certain, because the limit is written into law. It is easy to understand. And for very dangerous substances, where even a little too much is harmful, a firm limit or a ban may be the safest choice. But it also has weaknesses. The regulator must know the right limit. The rule gives no reason to cut pollution below the limit. And it treats every firm the same, even when some could cut pollution much more cheaply than others. Make a prediction about that last point.

Predict, then reveal — a rule or a tax?

Two firms pollute a river. The government wants 20 units of pollution cut in total. Firm A can cut up to 20 units at $2 per unit. Firm B can cut at $8 per unit. Option 1: a rule that each firm must cut 10 units. Option 2: a tax of $5 on every unit of pollution. Which option achieves the 20-unit cut more cheaply?

The tax is cheaper. Under the rule, A cuts 10 units at $2 and B cuts 10 units at $8: a total cost of $20 + $80 = $100. Under the $5 tax, A finds it cheaper to cut all 20 units ($2 each) than to pay $5 each in tax. B finds it cheaper to pay the tax than to cut at $8. So A cuts all 20, at a total cost of $40. The same 20-unit cut costs $60 less.

This is the main economic argument for taxes over rules: a tax lets each firm decide how much to cut, so the cuts are made where they are cheapest. Rules have the edge when the quantity must be certain, when pollution is very dangerous, or when it is hard to measure each firm's emissions well enough to tax them.

Feature	Pigouvian tax	Regulation
What is certain	The price of polluting	The amount of pollution
Cost of the cut	Lowest: cheapest cuts first	Often higher: same rule for all
Revenue	Yes, for the government	No
Information needed	The external cost per unit	The right limit
Reason to cut further	Yes: every unit cut saves tax	No, once the limit is met

A real case: British Columbia's carbon tax

Worked Example — A carbon tax (Canada, from 2008)

The policy. On 1 July 2008, the Canadian province of British Columbia introduced a tax on the carbon dioxide released by burning fossil fuels. It started at $10 per tonne (Canadian dollars) and rose in steps to $30 per tonne by 2012. It was designed to be revenue neutral: the money raised was meant to be returned through cuts in personal and business income taxes.

The prediction. Carbon dioxide adds to climate change, an external cost that the price of fuel did not include. A carbon tax is a Pigouvian tax on that cost. The model predicts that fuel becomes more expensive, people and firms use less of it, and emissions fall below what they would otherwise have been.

What happened. A review by Canada's Smart Prosperity Institute reports that modelling studies attribute emission reductions of between 5% and 15% to the tax. It adds that the evidence so far finds no measurable effect on the province's overall economic output. It also notes that, by more recent accounts, the tax has not stayed fully revenue neutral.

The verdict. The tax moved behaviour in the direction the model predicts, at little visible cost to the economy. Whether $30 a tonne was the "right" size is much harder to say. It depends on how large the external cost of a tonne of carbon dioxide really is, and estimates of that vary widely. This is the information problem from the graph builder, in real life.

Cigarettes: a harder case

Smoking creates a clear negative externality: second-hand smoke harms people nearby who did not choose to smoke. Many countries tax cigarettes heavily and ban smoking in enclosed public places, using both tools at once. The bans reduce exposure to second-hand smoke directly; the taxes raise the price and reduce smoking.

Some argue that smokers also impose costs on public health systems, which everyone pays for. That claim is debated. Smokers do use more health care, but on average they also live shorter lives, which reduces some other public costs. Economists disagree about the net figure. And much of the harm of smoking falls on smokers themselves, which, as the last lecture showed, is a private cost, not an externality. So cigarette taxes are partly Pigouvian and partly about raising revenue and discouraging a harmful habit for other reasons.

Three classic mistakes

Trap 1 — "The aim is zero pollution"

The Pigouvian tax aims for $Q_{opt}$ , not zero. In the example, 60 units are still made, and they still do $360 of damage, because those 60 units are worth more than their full social cost.

Trap 2 — "Every tax creates a deadweight loss"

That is true in a market with no externality, as the tax incidence lecture showed. With a negative externality, a tax close to the external cost removes a deadweight loss. Always ask whether the market was efficient before the tax.

Trap 3 — "A rule and a tax that cut the same amount are equally good"

They can reach the same quantity at very different costs. A tax lets the firms that can cut cheaply do most of the cutting. In the prediction above, the same 20-unit cut cost $40 with a tax and $100 with a rule.

Try it yourself

Questions 1 and 2 use MPB: P = 30 − 0.2Q and MPC: P = 6 + 0.1Q.

The external cost is $3 per unit. Find the Pigouvian tax, the new quantity, the price buyers pay, the price sellers keep and the revenue.

Hint 1
The tax equals the external cost per unit. Supply becomes $P = 9 + 0.1 Q$ .

Hint 2
Set $30 - 0.2 Q = 9 + 0.1 Q$ .

Full answer
Tax $= $3$ . $21 = 0.3 Q$ , so $Q = 70$ . Buyers pay $30 - 14 = $16$ , sellers keep $16 - 3 = $13$ , and revenue is $3 \times 70 = $210$ a week.
The external cost is $6, but the government sets a tax of $9. Find the quantity and the deadweight loss that remains. Is the market now making too much or too little?

Hint 1
Supply becomes $P = 15 + 0.1 Q$ , so $15 = 0.3 Q$ .

Hint 2
The triangle has base $60 - Q$ and height $9 - 6$ .

Full answer
$Q = 50$ , which is 10 below the optimum of 60, so the market now makes too little. Deadweight loss $= \frac{1}{2} \times 10 \times 3 = $15$ a week. That is a quarter of the $60 loss with no tax, so this too-large tax is still a big improvement.
Firm A can cut pollution at $3 per unit (up to 30 units) and Firm B at $7 per unit. The government wants 30 units cut. Compare the cost of a rule that each firm cuts 15 units with a tax of $5 per unit of pollution.

Hint 1
Under the tax, a firm cuts a unit if cutting costs less than the $5 tax.

Hint 2
Rule: $15 \times 3 + 15 \times 7$ . Tax: only A cuts.

Full answer
Rule: $15 \times 3 + 15 \times 7 = 45 + 105 = $150$ . Tax: A cuts all 30 units (cutting costs $3, less than the $5 tax); B pays the tax instead ($7 is more than $5). Cost $= 30 \times 3 = $90$ . The tax achieves the same 30-unit cut for $60 less.
A chemical is harmless in small amounts but causes severe harm to people's health once the amount in a river passes a certain level. Explain why a government might choose a strict legal limit rather than a Pigouvian tax for this chemical.

Hint 1
Which tool makes the quantity certain, and which makes the price certain?

Hint 2
What happens if the tax is set a little too low, when the damage jumps sharply past a threshold?

Full answer
A tax fixes the price of polluting, but the quantity depends on how firms respond, which the government cannot know exactly. If the tax turns out slightly too low, pollution could pass the danger level, and the harm would be severe. A legal limit fixes the quantity directly, so the danger level is never crossed. When the external cost rises steeply past a threshold, certainty about the quantity matters more than getting the cheapest cuts, so a strict limit is the safer tool.

The bigger idea

A negative externality is a cost with no price on it. A Pigouvian tax puts the price back. A rule removes the choice instead. Both can reach the socially optimal quantity, but in different ways. The tax changes incentives and lets firms find the cheapest cuts; the rule sets the outcome directly. Neither works well without good information about the damage, and that is the hardest part in real policy.

A third approach combines the certainty of a rule with the flexibility of a tax. The government fixes the total amount of pollution and lets firms buy and sell permits to emit it. That is the subject of the lecture on tradeable permits. First, though, the next lecture turns the picture round, to goods whose benefits spill over to others and which the market makes too little of.

Key concepts in this lesson

A negative externality makes MSC lie above MPC, so the market overproduces: Q* is above Q_opt, with a deadweight loss.
A Pigouvian tax equal to the external cost per unit at Q_opt lifts supply onto MSC, so the market reaches Q_opt by itself.
With an externality, such a tax removes a deadweight loss instead of creating one; a tax far too large can do more harm than none.
Regulation fixes the quantity directly; it gives more certainty but usually costs more, because it does not let the cheapest cuts be made first.
Both tools need information about the size of the damage, which is often uncertain.