Every efficiency result so far has quietly assumed that the buyer and the seller are the only people affected by a trade.
That assumption is doing far more work than it appears to. When it holds, the demand curve is society's marginal benefit and the supply curve is society's marginal cost, so the crossing point is genuinely the right quantity. When it fails, both curves are measuring the wrong thing, and the market clears at a quantity nobody would have chosen. This is the first failure in the course that afflicts even a perfectly competitive market, and it is not repaired by adding competitors.
Private cost is not social cost
An externality is a cost or benefit falling on someone who is neither buying nor selling. A coal plant's sulphur dioxide damages crops and lungs downwind; the plant's supply curve reflects its coal and labour bills and not the damage. A neighbour's vaccination protects people who never chose it. The first is a negative externality, the second positive, and the structure is identical with a sign flipped.
Write marginal private cost for what the seller bears, marginal external cost for what falls on others, and marginal social cost for their sum. The efficient quantity equates marginal social benefit with marginal social cost. The market equates marginal private benefit with marginal private cost. With a negative externality the second condition is met at a quantity where social cost already exceeds social benefit, so the market produces too much, and every unit past the efficient one destroys value.
Example. A market has demand and marginal private cost . Each unit produced imposes £20 of damage on third parties. Find the market and efficient quantities and the deadweight loss.
The market equates , giving at a price of £55. Marginal social cost is , so the efficient quantity solves , giving . The market overproduces by 10 units. The loss on each is the gap between social cost and benefit, which runs from zero at up to £20 at , so the deadweight loss is .
Now you. A market has demand , marginal private cost , and external damage of £10 per unit. Find both quantities and the deadweight loss.
Answer
The market gives , so at £50. Marginal social cost is , so the efficient quantity is at a buyer price of £55. Overproduction is 5 units and the deadweight loss is .
Note what the efficient quantity is not. It is not zero. Pollution has a benefit, in the form of the goods produced alongside it, and the right amount of it is where the marginal damage equals the marginal value of the last unit made. A policy aimed at eliminating an externality is aiming at the wrong target, which is the single most useful thing this framework has to say about environmental argument.
The Pigouvian tax
Arthur Pigou proposed the fix in The Economics of Welfare in 1920. Charge the producer a tax equal to the marginal external damage at the efficient quantity, and the private cost curve becomes the social one. The firm then maximises its own profit and produces the efficient quantity, without needing to know or care why.
In the first example, a tax of £20 per unit turns marginal private cost into , which meets demand at exactly. Buyers pay £65, sellers net £45, the government collects , and the deadweight loss is gone. That last point is worth pausing on, because it inverts the previous lesson's arithmetic: this tax removes a deadweight loss rather than creating one, since it corrects a distortion instead of introducing one. Taxes on externalities are the one case where raising revenue and improving efficiency point the same way, which is why they are sometimes said to offer a double dividend.
Two practical difficulties are real. The tax must equal marginal damage, so the damage has to be measured, and that measurement is contested: official United States estimates of the social cost of carbon have ranged from roughly 190 a tonne of carbon dioxide depending on the discount rate chosen and how damage to future generations is weighted. And the tax must be levied per unit of the externality, not per unit of output. A tax on electricity treats a wind farm and a coal plant alike; a tax on emissions does not, and only the second gives firms a reason to abate.
Coase: it may be a property rights problem
Ronald Coase argued in 1960 that Pigou had framed the problem wrongly. Externalities are reciprocal: the factory harms the laundry only because the laundry is there, and forcing the factory to stop harms the factory. What is missing is not a tax but a clearly assigned right, and if rights are clear and bargaining is costless, the parties will trade their way to the efficient outcome regardless of who holds the right.
Example. A factory earns from output , and each unit does £10 of damage to a neighbouring laundry. Find the efficient output, and show that bargaining reaches it under either assignment of rights.
Unconstrained, the factory sets and produces 20, earning £400 while doing £200 of damage, so the joint total is £200. The efficient output maximises , giving and : the factory earns £375, damage is £150, and the joint total is £225.
If the factory has the right to pollute, it starts at 20 and the laundry offers to pay it to cut back. Moving to 15 costs the factory £25 in profit and saves the laundry £50, so any payment between £25 and £50 makes both better off, and they stop at 15 because beyond that the factory's losses exceed the laundry's savings. If instead the laundry has a right to clean air, the factory starts at zero and pays for permission. Producing 15 units is worth £375 to it and costs the laundry £150, so a payment between those figures is agreed, and again they stop at 15. Same quantity, opposite distribution.
Now you. The same factory, but the damage rises to £16 per unit. Find the efficient output and the bargaining range when the factory holds the right.
Answer
Efficiency needs , so : the factory earns £336 and damage is £192, for a joint total of £144, against £80 at . With the factory holding the right it starts at 20, and moving to 12 costs it while saving the laundry . Any payment between £64 and £128 is agreed.
The Coase theorem is often quoted as showing that externalities need no government action, which reverses its actual point. Coase's argument was that the outcome depends on transaction costs, and that these are almost never zero. Two parties can bargain; ten thousand households downwind of a power station cannot, because organising them costs more than the agreement is worth and every one of them has an incentive to hold out for a larger share. What Coase established is that the choice between a tax, a court, a regulation and a negotiation is an empirical comparison of transaction costs, not a matter of principle.
Quantity instruments
A third route sets the quantity rather than the price. A regulator issues permits totalling the efficient level of emissions and lets firms trade them. Firms that can abate cheaply sell permits to firms that cannot, so abatement ends up concentrated where it is cheapest, and the permit price settles at the marginal abatement cost of the last firm that needs one.
The largest test was the United States sulphur dioxide allowance market created by the 1990 Clean Air Act Amendments. Emissions from covered power plants fell by roughly half over the following two decades, and the cost of compliance came in far below the pre-programme forecasts, largely because trading let plants substitute towards low-sulphur coal instead of installing scrubbers everywhere. The European Union's carbon market has had a rougher history, with an initial over-allocation of permits that collapsed the price, which is the characteristic failure mode of a quantity instrument: the regulator has to guess the abatement cost curve to set the cap, and a wrong guess shows up as a wild price.
That is the general tradeoff. A tax fixes the price of the externality and lets the quantity fall where it will; a permit scheme fixes the quantity and lets the price fall where it will. Which is safer depends on which mistake is more costly, an argument Martin Weitzman formalised in 1974.
Rivalry, excludability, and public goods
A different failure comes from the nature of the good itself. Two properties classify it. A good is rival if one person's use leaves less for others, and excludable if non-payers can be kept out.
Ordinary private goods are both: a sandwich, a haircut. Club goods are excludable but not rival, like a subscription streaming service, where an extra viewer costs nothing but can be shut out. Common resources are rival but not excludable, like an ocean fishery. And public goods are neither: national defence, a lighthouse beam, the result of basic research. One person's consumption does not diminish anyone else's, and nobody can be prevented from consuming it.
Non-rivalry changes the efficiency condition itself. For a private good, efficiency requires each person's marginal benefit to equal marginal cost, and quantities are added across people. For a public good, everyone consumes the same quantity, so it is the benefits that add. Paul Samuelson stated the condition in 1954:
Marginal benefits are summed vertically rather than quantities horizontally, which is a genuine change of arithmetic and not a technicality.
Example. Three households value street lighting at , and per unit, and lighting costs £45 per unit to provide. Find the efficient quantity, and what private provision would deliver.
Summing, . Setting that equal to 45 gives , where the three marginal benefits are £25, £15 and £5, summing to £45 as required. Left to themselves, each household compares its own benefit with the full cost. Household 1 provides until , that is 5 units, and the others then have marginal benefits of £35 and £25 at that point, both below £45, so they add nothing and enjoy the lighting for free. Private provision delivers 5 units against an efficient 25.
Now you. The same three households, but lighting now costs £36 per unit. Find the efficient quantity and what private provision gives.
Answer
gives , with marginal benefits of £22, £12 and £2 summing to £36. Household 1 acting alone provides until , that is 14 units, and neither of the others will add to it. Private provision reaches exactly half the efficient level, and the household that values the good most bears the entire cost.
That gap is the free rider problem, and it is not a claim that people are selfish. Contributing is individually irrational even for someone who wants the good badly, because their own contribution is small and they get the benefit either way. This is why defence, street lighting and basic research are tax-financed almost everywhere.
The commons, and two honest corrections
Common resources fail the other way: everyone uses them and nobody bears the full cost of use, so they are overused. Each extra fishing boat catches fish that other boats would have caught, and that loss is external to the owner. Garrett Hardin named this the tragedy of the commons in 1968 and concluded that the only remedies were private ownership or state control.
Elinor Ostrom spent decades showing that conclusion was wrong as a matter of fact. Working through Swiss alpine pastures, Japanese forest commons, Spanish irrigation systems and Philippine water associations, she documented communities that had managed shared resources sustainably for centuries under neither private property nor central authority, using monitoring, graduated sanctions and local dispute resolution that they designed themselves. She was awarded the Nobel prize in 2009 for it. The theory identifies a real incentive problem; it does not license the conclusion that only two institutions can solve it.
Coase supplied the other correction, and pointedly. The lighthouse had been economists' standard example of a good the market could not supply, repeated from Mill through Sidgwick to Samuelson. In 1974 Coase went and looked, and found that British lighthouses had for centuries been built and operated by private parties, financed by dues collected at ports where ships docked, since ships that use lighthouses also have to land somewhere. Excludability turned out to be a fact about available institutions rather than a fact about photons.
Both corrections point the same way. The categories in this lesson identify where the incentives push, and they do not settle what any particular society can achieve. What they do settle is that the competitive equilibrium is no longer automatically efficient once costs escape the trade. The next lesson finds the other route to the same conclusion, where the trade is between two parties only but one of them knows something the other does not.