The Externality

Begin with the homeliest of cases. A mill burns coal to drive its looms, and the burning sends a plume of smoke over the rooftops to a laundry, where it settles on the linen drying in the yard and must be washed away a second time. The mill and its customers have struck a fair bargain — the cloth is worth its price to them, and the price covers the mill's costs — and by the lights of the classical account that bargain is wholly to the good. Yet a cost has been incurred that no party to it bore: the labor and water the laundry now spends to undo the grey. That cost is as real as any in the mill's accounts. It is simply borne by someone who was neither asked nor paid, and who has no place in the reckoning that fixed the price of cloth. This is an externality, and the word, for all its modern ubiquity, names a precise thing: an effect of an economic action — a cost or a benefit — that falls upon a third party outside the transaction, and that the price of the action therefore does not carry.

The mischief such effects work is best seen as a wedge. To the mill, the cost of weaving one more bolt of cloth is what that bolt consumes — the coal, the wage, the wear on the loom — and the mill, sensibly, weaves cloth up to the point where this private cost just matches what the cloth will fetch. But the cost to society of that same bolt is larger: it is everything the mill pays plus the harm the attendant smoke does to the laundry and to every other party downwind. Between the private cost that the maker counts and the social cost that the world actually bears stands a gap, and the market, which sees only the private figure, behaves as though the gap did not exist. Where the spillover is a harm, as with the smoke, social cost exceeds private cost and the wedge points one way; where it is a benefit — the householder who vaccinates herself and in doing so shields her neighbors, or who restores a handsome façade that every passer-by is the richer for seeing — social benefit exceeds private benefit and the wedge points the other. In both, the decision is made on the private figure while the consequences land on the social one.

The result follows with a melancholy regularity. Whatever harms a bystander, the market makes too much of, for the harm is free to the one who causes it and enters none of his sums; the smoke costs the mill nothing, so the mill is lavish with it. Whatever helps a bystander, the market makes too little of, for the benefit escapes to others and the maker cannot bill for it; the vaccinated woman is paid nothing by the neighbors she protects, so she weighs only her own small risk against the cost and may well decline. This is not an error confined to the last unit, a triangle of waste at the margin of an otherwise sound calculation. It is a systematic mispricing of an entire activity, and through it the market produces the wrong quantity of everything whose effects reach past the two who struck the deal.

Why this happens repays a closer look, because the answer dictates every cure. No villain stalks the story. The mill-owner is not poisoning the air from malice; he is making cloth, which people want, and the smoke is the unregarded by-product of the making. What has failed is not his conscience but the market's reach. There exists no market in clean air, none in quiet, none in the herd protection that a vaccinated population confers — and where there is no market there is no price, and what bears no price is invisible to the arithmetic of gain and loss by which every firm and household steers. The externality is, at its root, the failure of a missing market. The cost of the smoke goes unpriced not because anyone wills it so but because the market that would have priced it — a market in which the laundry could sell the mill the right to foul its air, or charge it for the fouling — was never brought into being. Were that market to exist, the harm would enter the mill's accounts as surely as the coal does, and the mill would temper its smoke to the point worth tempering. The failure is an absence, and the remedies, as the plates will show, are all contrivances for supplying what is absent.

The phenomenon is old, though its name is not quite a century old. Alfred Marshall spoke of external economies and diseconomies — the ways one firm's activity could lower or raise the costs of others outside any contract between them — and Pigou, his successor at Cambridge, gave the idea its sharp modern form and its policy edge, dividing the world's effects into the private and the social and proposing to close the gap between them. The range of the thing is vast. On the side of harm: the smoke and the effluent, the noise of the airport, the congestion each driver adds to the road, the antibiotic overused until resistance spreads to all. On the side of benefit: the vaccine, the education whose graduate enriches more than himself, the research whose discoveries others freely build upon, the orchard and the beehive that feed each other. A single discipline of analysis runs through them all. One distinction, however, must be flagged at the outset and will occupy a later plate in full — that between a true externality, a real effect operating outside the price system, and a merely pecuniary one, transmitted through prices, which looks like a spillover and is nothing of the kind.

Let us put the harm into curves, for the diagram is where the failure becomes exact. The mill faces a private marginal cost — the rising expense of each additional bolt of cloth, given the coal and labor it must buy — and this is the supply curve a competitive market draws upon. But each bolt also sends up its measure of smoke, and that smoke inflicts a marginal external cost on the bystanders, a harm that rises, let us say, with the quantity produced. Add the external cost to the private cost and one obtains the marginal social cost: the true cost to the world of one more bolt, the mill's own outlay together with the damage done to those who never bought the cloth. The social cost curve lies above the private one, and the vertical distance between them — the harm per unit — is the heart of the matter.

Now the market does what markets do. Buyers value successive bolts along the demand curve, which we may take to register the social benefit of each; the mill supplies along its private cost curve; and trade settles where the two meet, at the quantity the figure marks as the market's. The efficient quantity, however — the one a benevolent calculator would choose, weighing the cloth's worth against its full social cost — lies where demand meets the social cost curve, and that point sits to the left of the market's. The market overproduces. It cannot help but overproduce, for it consults the mill's private cost, which omits the smoke, and so it carries the activity past the point where the world's gain from another bolt has dwindled to less than the world's cost of making it.

The shaded triangle measures the cost of the failure. Across the units from the efficient quantity to the market's, the social cost of each exceeds the social benefit — these are bolts the world would be better off without, made only because the mill never paid for their smoke — and the accumulated excess of cost over benefit is the deadweight loss, the value that the externality simply destroys, accruing to no one. It is worth pausing on what does the damage. It is not the smoke as such; a world in which the mill bore the full cost of its smoke would still produce some smoke, and rightly. The damage is done by the smoke that is free to the mill — by the absence of a price on the harm, which lets the private calculation diverge from the social one. Restore that price, by whatever device, and the divergence closes.

From this follows a conclusion that startles the newcomer and that the figure states without apology: the efficient quantity of pollution is not zero. Efficiency does not ask that the smoke cease; it asks that the activity be carried to the point where the value of one more unit just equals the social cost of the smoke that unit brings, and no further. To abate the smoke entirely would be to forgo all the cloth whose worth exceeds its full social cost — a loss as real, and as wasteful, as the pollution it would prevent. The aim is not a pristine world but a balanced one, the smoke held at the level where the gain from the activity and the harm from its by-product are brought into equality. A policy that demands zero emissions is, by the logic of this plate, exactly as misguided as one that ignores emissions altogether; both refuse the balance that efficiency requires.

Turn the wedge the other way and a second failure appears, the perfect mirror of the first. Some activities confer benefits that spill beyond the one who undertakes them. The orchard in bloom feeds the bees of the apiary next door, which return the favor by pollinating the fruit — James Meade's textbook pair, chosen for their charm. The householder who is vaccinated lowers not only her own chance of falling ill but everyone's, by denying the disease a host through which to travel. The student who is educated enriches more than himself, in the productivity of those who work beside him and the temper of the citizen he becomes. The scholar who makes a discovery hands it, once published, to all who care to build upon it. In each, a private marginal benefit accrues to the actor, and beyond it a marginal external benefit accrues to others — and the marginal social benefit, the sum of the two, exceeds what the actor alone enjoys.

The market, consulting as ever only the private figure, now falls short the other way. The grower plants his orchard up to the point where his own benefit from one more tree just covers its cost, reckoning nothing of the bees; the woman weighs her own small protection against the bother of the clinic, counting for nothing the neighbors she would shield. So the activity is carried only to where private benefit meets cost, while efficiency — which would weigh the full social benefit against the cost — asks for more of it. The market underproduces the beneficial as surely and for the same reason as it overproduces the harmful: it is blind to the part of the effect that lands on others, and that blindness, harmless-seeming, leaves a margin of valuable trees unplanted and valuable inoculations ungiven.

The triangle now measures a gain not taken rather than a loss incurred, but it is waste all the same. Across the units from the market's quantity to the efficient one, the social benefit exceeds the cost — trees worth planting, shots worth giving, whose value to the world outruns what they consume — and yet they are not provided, because the one who would provide them captures only a part of the value and the part that escapes him does not enter his decision. And the remedy is the mirror of the smoke's. Where the harmful activity wanted a tax to lift its private cost up to the social cost, the beneficial activity wants a subsidy to lift its private benefit up to the social benefit — a negative tax, paying the actor for the good he does to others, so that he carries the activity to the efficient point. Subsidized vaccination, public support for schooling, the research grant and the patent that lets a discoverer capture more of what he creates: each is a device for paying the spillover's beneficiaries' debt to the one who served them unbidden.

The most consequential instance is the one a recent age has relearned at cost. A vaccine confers a textbook positive externality: each person immunized is a link removed from the chain of contagion, so that the unvaccinated are protected by the vaccinated around them, and at a high enough coverage the disease can find no path at all — the herd immunity that shields even those who cannot be inoculated. But the individual, weighing only her own protection against her own trouble and risk, undercounts the benefit she would confer, and a population of such individuals immunizes too few of itself, stalling short of the threshold that would protect them all. Under-vaccination is a market failure of exactly the orchard's form, and the public subsidy, the school requirement, and the campaign of persuasion are its corrective. The knowledge spillover of research is another, and a profound one, for ideas are the most escape-prone of benefits — once known, a discovery serves everyone at no further cost — which is why a later volume found the market underproducing knowledge, and why the patent and the public laboratory exist to coax more of it into being.

If the failure is a missing price, the most direct cure is to install one, and this was Pigou's proposal. Levy upon the harmful activity a tax equal to the marginal external cost it inflicts — a tax of exactly the harm the smoke does — and the mill, which before paid only its private costs, must now pay the harm as well. Its private calculation and the social calculation, which the missing price had pulled apart, are thereby reunited: the mill, confronting a cost that at last includes the smoke, weaves cloth only up to the point where the value of another bolt covers its full social cost, which is precisely the efficient point. The externality has been, in the term of art, internalized — drawn back inside the ledger of the one who creates it — and the market, now seeing the true cost, makes the right amount of smoke unbidden. The tax does not command the mill to pollute less; it merely charges it for polluting, and lets the mill's own arithmetic do the rest.

The figure shows the correction and discloses a bonus. The tax lifts the private cost curve until it sits atop the social cost curve, and the market slides down the demand curve from its untaxed quantity to the efficient one, the overproduction wrung out. But the shaded rectangle records something Pigou's critics too often forget: the tax raises revenue, and a great deal of it. This is the celebrated double dividend. An ordinary tax — on wages, on sales — buys its revenue at the price of a distortion, discouraging the very work or trade it falls upon; the Pigovian tax, alone among taxes, buys its revenue while improving the allocation, since the behavior it discourages is behavior there was too much of to begin with. The revenue it throws off can then be used to cut the distorting taxes, so that society gains twice — once from the smoke abated, once from the wage tax lightened. Tax the bad, the slogan runs, not the good; and the corrective levy is the rare instrument that funds the state and mends a market in the same stroke. The mirror device serves the orchard: a subsidy, the negative of the tax, paid to the activity whose benefits escape, lifting its private benefit to the social one.

So the apparatus is in use, and visibly. The carbon tax is Pigou applied to the gravest externality of all. The congestion charge that London and Singapore lay on cars entering the crowded center prices the delay each driver imposes on every other. Effluent fees, landfill levies, the duties on tobacco and alcohol that economists file, not always persuasively, under the same heading — all are corrective taxes, aimed at an external cost and meant to shrink it to the efficient size. Yet the apparatus rests on a quiet and demanding precondition, and it is the same one that haunted the last two plates. The tax must equal the marginal external cost at the efficient quantity — not a penny more or less — and to know that figure one must know both the harm and the point at which the harm and the activity's value come into balance. Set the tax too low and too much smoke remains; set it too high and the cloth whose worth exceeded its true cost is needlessly forgone. The corrective tax is exactly as good as the number behind it, and the number, we have seen, is hard to come by.

In nineteen-sixty Ronald Coase published the paper that would become the most cited in all of economics, and it turned Pigou on his head. Coase's first move was not a calculation but a change of view, and everything follows from it. The harm of an externality, he insisted, is not one-sided but reciprocal. We say the mill harms the laundry; but to forbid the mill its smoke is to harm the mill, for it must then forgo the cloth the smoke would have helped produce. The cattle trample the farmer's crops; but to fence out the cattle is to harm the rancher, who loses the beef. Two parties want incompatible uses of one scarce thing — the air, the field — and to grant either his wish is to deny the other. The right question, then, is not the moral one of who is at fault, but the economic one of which use is worth more; and a smoke that does ten pounds of damage to the laundry while bringing the mill twenty is a smoke the world is richer for, however it offends the tidy intuition that the polluter is the villain.

From this reframing comes the theorem that bears his name. If property rights are clearly defined, and if the parties can bargain at no cost, then they will arrive at the efficient outcome no matter to whom the right is first given. The initial assignment of the right determines who must pay whom — and so determines the distribution of the gains — but it does not determine the quantity of the activity, which settles at the efficient level either way. The externality, on this account, is not a flaw in the market mechanism crying out for a tax. It is the absence of a market, and the absence has a precise cause: no one owns the air, so no one can sell it, so the cost of fouling it never gets priced. Define the right — declare that the mill owns the air, or that the laundry does — and a market in the smoke springs into being, and the parties trade their way to efficiency without a planner's help.

The figure makes the invariance plain. Suppose first the rancher holds the right to graze as he pleases. Unchecked, he runs the largest herd; but the farmer, watching cattle whose damage to his crops exceeds their worth to the rancher, finds it pays him to offer a bribe for each such beast removed, and the rancher, offered more than the steer is worth to him, accepts — and so they walk the herd down, steer by steer, until they reach Q*, where the next animal's value to the rancher has risen to meet its damage to the farmer and no further bribe can profit them both. Now suppose instead the farmer holds the right to a crop unmolested. The rancher starts with no cattle at all; but for each steer worth more to him than the damage it would do, he can pay the farmer enough to consent and still come out ahead, and so they walk the herd up — from none toward Q*, stopping at the very same crossing. The destination does not depend on the starting point. The right decides the direction of the payments, and thus who ends up richer; it leaves the efficient herd exactly where it was.

The implications were seismic, and they founded a discipline. If a missing market, not a market flaw, is the trouble, then the cure is not Pigou's tax but the law's clear assignment of a right, after which private bargaining does the work a planner could only guess at — and does it without needing to know the harm to the penny, since the parties, who feel the harm and the benefit directly, reveal those magnitudes in their willingness to pay. Where bargaining is possible, the corrective tax is not merely unnecessary; it may be harmful, taxing a smoke the parties would have priced more accurately between themselves. The externality dissolves into a question of property and contract, and the study of how legal rules shape economic outcomes — the law-and-economics that Coase, with Guido Calabresi, set in motion — grew up around the insight. The work was honored with a Nobel in nineteen ninety-one. And yet, as the next plate will show, the theorem's most famous use is very nearly the opposite of what its author intended.

The Coasean bargain assumed that striking a deal costs nothing, and nothing could be further from the truth. To bargain, the parties must first find one another; they must learn enough of the facts to know what a deal is worth; they must haggle over the terms, each concealing what he would settle for; they must write the agreement down and stand ready to enforce it if the other reneges. Every one of these consumes time, effort, and money, and the sum of them is what Coase called transaction costs. His deepest contribution was not the theorem of the zero-cost world but the insistence that the real world is built of these costs, and that they, not the frictionless ideal, are what an honest economics must study. The fairy tale of costless bargaining was a device — a way of showing, by its very unreality, that since bargaining is in fact dear, the things the theorem made irrelevant become, in the world we inhabit, decisive.

Picture, then, the two ends of the spectrum. At the low-cost end stand the rancher and his single neighbor, or you and the one neighbor whose late music disturbs your sleep. The parties are few, the right is clear, and a conversation can settle it; here the Coase theorem holds, and the law's only task is to assign the right cleanly, after which the two will bargain to the efficient outcome and the assignment will govern merely who pays whom. At the high-cost end stands the factory whose smoke descends on ten thousand households, no two of them acquainted, each with so small a stake that none will trouble to organize the rest, every one tempted to let his neighbors do the bargaining and to enjoy the cleaner air for free. Among such multitudes no bargain forms at all; the free-rider sees to that, and the holdout, and the sheer cost of gathering ten thousand voices into one negotiating party. The externality stands uncorrected, not because the gains from correcting it are small but because the cost of transacting devours them.

And here the matter that the theorem pronounced irrelevant becomes all-important: when bargaining cannot occur, the assignment of the right decides not merely the distribution but the efficiency of the outcome, for there is no subsequent trade to undo a bad first assignment. The law can no longer wash its hands after handing out the entitlement; it must hand it out well. The guiding principle of the law-and-economics that grew from Coase is that, where the market cannot operate, the law should imitate the result the market would have reached — assigning the right to the party who values it most, or arranging the rules so as to minimize the total harm the externality does. Calabresi and Melamed gave the lawyer's version of the toolkit: an entitlement may be protected by a property rule, which forbids anyone to take it without the holder's consent and sends the parties to bargain, or by a liability rule, which lets it be taken so long as the taker pays a damage fixed by a court — the first apt where bargaining is cheap, the second where it is dear and a judge's valuation must stand in for a price.

So the choice of remedy is, at bottom, a reading of transaction costs. Where they are low, the law defines a right and withdraws, trusting the parties to trade their way to efficiency. Where they are high, no such trust is warranted, and the law must do the market's work itself — by liability rules that price the harm through damages, or by the regulation and the corrective tax of the earlier plates, which for a pollutant that touches millions are not Pigou's challenge to Coase but the only instruments the situation permits, the planner's levy standing in for the market that the transaction costs forbid to form. The tax we met in the fourth plate is, seen this way, the high-transaction-cost remedy, and the bargain of the fifth the low-cost one; they were never truly rivals.

This is the synthesis the volume has been building toward, and it dissolves a quarrel that has consumed too much ink. Pigou and Coase are not antagonists but instruments suited to different frictions. Where transaction costs are low — the few, the identifiable, the well-defined — Coase governs: assign the rights and let bargaining internalize the spillover. Where they are high — the many, the diffuse, the anonymous — Pigou governs: tax or regulate, because no bargain can. To ask in the abstract whether the tax or the bargain is the right approach to externalities is to ask a malformed question; the well-formed one is always about the transaction costs of the case at hand, which decide whether a market can be summoned into being or must be simulated by the state. The single neighbor, the global atmosphere, the patent thicket in which a single useful invention requires licenses from dozens of overlapping rights-holders and so is never built — each is a different point on the same spectrum, and the friction, not the philosophy, tells us which tool to reach for.

For the great pollution problems — the smoke of a thousand factories descending on a whole region's air — the bargain is impossible and the tax is hard to set, and a third device was invented that is cleverer than either, for it manufactures the very market the externality lacked. The idea, sketched by Thomas Dales and made rigorous by William Montgomery, is to create out of thin air a property right in the harm itself. Decide how much total pollution the air can bear; print exactly that many permits, each entitling its holder to emit one ton; hand or sell them to the firms; and then — this is the stroke — let the firms buy and sell the permits among themselves. A right that did not exist now exists, is owned, and is traded, and the missing market stands supplied. The scheme goes by the name of cap-and-trade, and its two halves are a direct marriage of the volume's two great traditions: the cap is Pigou's, fixing the quantity of harm society will tolerate; the trade is Coase's, letting a market in the right allocate that harm to where it does the least damage to do.

Why the trading matters — why one does not simply order every firm to cut its emissions by the same share — is the subject of the figure, and it turns on a fact the regulator cannot see: firms differ enormously in what it costs them to abate. For one plant, a modest retrofit slashes emissions cheaply; for another, the same cut demands ruinous reengineering. A uniform rule, blind to these differences, forces the high-cost plant to abate just as much as the low-cost one, squandering effort where it is dear and sparing it where it is cheap. Trading repairs this. The firm that can cut cheaply does so and sells its surplus permits; the firm that would cut dearly buys permits instead and cuts less; and the buying and selling continues until no two firms have different marginal costs of abatement — for as long as they do, the high-cost firm can profit by buying a permit from the low-cost firm, who can profit by cutting one more ton and selling it. At rest, every firm's marginal abatement cost equals every other's, and equals the price of a permit. The same total reduction has been achieved, but at the least total cost any allocation could manage.

The merit of the scheme is not only that it minimizes cost but that it does so without anyone having to know the costs in advance. The regulator who tried to set efficient firm-by-firm standards would need each plant's abatement-cost curve, which each plant has every reason to misrepresent; the permit market extracts that private information automatically, in the only language that cannot lie — the willingness to pay. The price at which permits change hands is the marginal cost of the cap, discovered rather than decreed, and it tells the regulator, after the fact, what the chosen quantity of cleanliness actually cost. This is the same trick the whole subject keeps performing: where the planner lacks the information to act well, a market is contrived in which the parties reveal the information by acting in their own interest.

The scheme has left the blackboard for the statute book, and its record is real. The American program of the early nineteen-nineties that capped the sulfur dioxide behind acid rain and let utilities trade allowances is the celebrated success, cutting emissions faster and far more cheaply than the engineers' command-and-control estimates had forecast. The European Union's emissions trading system put a price on carbon across a continent; the northeastern states' regional initiative and California's market did the same on smaller scales. In each, the missing market in a harm was conjured into being by the simple acts of capping a quantity and permitting a trade — Pigou setting the ceiling, Coase apportioning what fell beneath it. The instrument is, when it works, the most elegant in the whole repertoire of the failures: it fixes the environmental result with the certainty of a quota and achieves it with the thrift of a market.

The corrective tax and the cap on permits look, at first, like two routes to one place. Fix the price of pollution at the right level and firms will abate until their marginal cost meets it, settling at the efficient quantity; fix that quantity directly and the permit price will rise to the right level on its own. Under perfect knowledge the two instruments are mere images of each other — name the price and the quantity follows, name the quantity and the price follows — and an economist of Pigou's certainty would find the choice between them empty. But we do not possess perfect knowledge. The one thing the regulator most conspicuously lacks is the true cost of abating, the marginal abatement cost curve being a thing he must guess at and the firms have every reason to obscure. Martin Weitzman asked the question this uncertainty makes urgent: given that we will set the lever in the wrong place, because we have misjudged the cost, which lever leaves us less badly off — the one that fixes the price, or the one that fixes the quantity?

His answer is one of the prettiest results in the economics of policy, and it turns entirely on the shape of the marginal damage. Set the tax, and if costs turn out higher than you expected, firms abate less than you intended and the quantity of pollution drifts upward; the price is held, the quantity errs. Set the cap, and if costs turn out higher, the quantity is held fast but the marginal cost of meeting it climbs far above what you meant to impose; the quantity is held, the cost errs. Each instrument protects one variable and exposes the other, and which exposure you should fear depends on how steeply the harm rises with the quantity of pollution.

So the rule falls out of the figure. Where marginal damage is steep — where the harm climbs sharply with the quantity, as it would near a threshold beyond which some irreversible calamity waits — it is the quantity that must not be allowed to slip, and the cap is the safer instrument; a tax that let emissions drift past the threshold, because abatement proved dearer than forecast, would be a grave mistake. Where marginal damage is flat — where each additional unit does about the same modest harm, as with a pollutant that accumulates slowly so that any single year's emissions scarcely move the total — it is the cost that must not be allowed to run away, and the tax is the safer instrument; a cap that forced firms to spend wildly to hit a rigid number, for a benefit that barely changed with the number, would waste far more than it saved. Steep harm wants a quantity; flat harm wants a price; and the slope of the damage, not any prior preference for markets or for plans, settles which lever the prudent regulator pulls.

The rule has a sharp bearing on the gravest case. Carbon dioxide is a stock pollutant: it is the accumulated stock in the atmosphere, built up over a century and more, that drives the warming, and the emissions of any single year add only a sliver to that stock. The marginal damage of one year's carbon is therefore nearly flat — this year's ton does almost exactly the harm of last year's — and Weitzman's rule accordingly favors the price instrument, the carbon tax, whose virtue is to fix the cost of abatement and spare the economy the wild swings a rigid annual cap would impose. This is among the strongest theoretical arguments for taxing carbon rather than capping it, and it is why so many economists, asked to choose, choose the tax. The choice is not forced — hybrid designs that cap the quantity but fence the permit price between a floor and a ceiling try to capture the merits of both — but the flat damage of a stock pollutant tilts the balance toward the price, and the plate's lesson is that this tilt is a fact about the harm, not a matter of taste.

There is a positive externality so peculiar and so pervasive that it deserves a plate of its own, for it reshapes whole industries and breaks several of the comfortable presumptions of competition. It is the network externality, and its mark is that the value of a good to any one user rises with the number of others who use it. A single telephone is a paperweight; a telephone is worth having only because others have one too, and worth more the more others do. Each person who joins the network confers a benefit on everyone already connected — one more soul they can now reach — and receives, in turn, the accumulated benefit of all who joined before. That benefit passes between strangers uncompensated, exactly as the orchard's blossom passed to the bees, and so the network good carries a positive externality woven into its very nature: to adopt is to enrich every other adopter, and none of them pays you for it.

Once one has the eyes for it, the pattern is everywhere. The communication networks wear it plainly — the telephone, the fax in its day, electronic mail, the social platforms whose worth to each member is the multitude of other members. But so do the great standards, whose value lies precisely in being shared: the layout of a keyboard, the format of a video or a document, the operating system that attracts software because it has users and users because it has software. A language is a network good of the purest kind, worthless to speak if no one else speaks it and invaluable if all do; and money, as an earlier volume found, is the network good at the foundation of the economy, for a currency is worth holding only insofar as others will accept it, its value rising with the breadth of its acceptance. The platform that has come to dominate the modern economy is a network good doubled — a two-sided market, in which buyers are drawn by the throng of sellers and sellers by the throng of buyers, each side an externality upon the other.

From this single property flow consequences that ordinary goods do not show. Networks feed upon themselves: the more who join, the more attractive joining becomes, so that growth, once it passes a critical mass, accelerates of its own accord, and a network that fails to reach that mass dwindles to nothing. Such positive feedback drives a market to tip — to swing decisively toward a single winner who takes most or all of it — rather than to settle, as the textbook market does, into a comfortable plurality of competitors. And once a network has won, those within it are locked in: to leave for a rival, however superior, one would need the rest to leave at the same moment, a feat of coordination no individual can perform, so that the established standard persists by the mere fact of its establishment. The gravest implication is that the standard which wins need not be the best, for adoption turns on expectations and on who got there first as much as on merit, and a history of small accidents can lock a society into an inferior technology — the celebrated, and as we shall see disputed, tale of the QWERTY keyboard and its supposedly superior rivals. The market in network goods does not reliably select the best; it reliably selects a winner, which is not the same thing.

The network externality binds this volume to several of its neighbors. The feedback that tips a market to one winner is, seen from the side of the loser, a barrier to entry of formidable height, for a rival with a better product cannot dislodge an incumbent whose advantage is simply that everyone is already there — which is why the dominance of the great technology platforms rests so largely on network effects, and why the antitrust questions of the volume on power return here in modern dress. The currency that gains value with acceptance is the network good of the volume on money. The platform that manufactures trust between strangers is the institution the volume on asymmetry described. And the policy questions are correspondingly tangled. If networks confer benefits that grow with their reach, there is a case for the state to compel compatibility and interoperability — to require that telephone numbers travel between carriers, that chargers be common, that one network's users be able to reach another's — and even, where a network has hardened into a monopoly, to break it up. But each remedy risks forfeiting the very benefit that made the network valuable, and sets the state the unenviable task of choosing standards a market might have chosen better.

A concept that explains everything explains nothing, and the externality, left undisciplined, explains far too much. This plate erects the fence that keeps it useful, and the fence runs between two kinds of effect that look alike and are profoundly different. The first kind, the only kind that signals a failure, is the technological externality we have studied throughout — the smoke that physically fouls the laundry's air, the vaccine that physically denies the disease a host — a real effect on a third party operating outside the price system, consuming or conferring real resources for which no one pays or is paid. The second kind, which apes the first and is forever mistaken for it, is the pecuniary externality: an effect on a third party transmitted entirely through prices.

Consider what pecuniary effects look like, for they are constant and they are real. When I develop a taste for bread and buy more of it, I nudge up the price of bread, and every other buyer pays a little more on my account — I have done them a harm. When a new firm enters an industry and undercuts the incumbents, it drives down the price and may ruin them — a harm again, and a large one. When a new technology raises the demand for one skill, it lifts that skill's wage, enriching those who possess it at the expense of every employer who must now pay more. Each of these is an effect of one party's action on others who were no part of the transaction, and each, by the loose usage, might be called a negative externality. But none is a market failure, and the reason is decisive: the effect runs through a price, and a price change is a transfer, not a destruction. What the bread-buyers lose, the bread-sellers gain; what the incumbents lose to the entrant, consumers gain in lower prices and then some. No resources are consumed, no value annihilated; the gains and losses net to zero, or to a social gain, and the price has simply done its work of reallocating who gets what. This is not the market failing. It is the market functioning — competition, redistribution through prices, the mechanism operating exactly as it should.

The distinction is the load-bearing wall of the whole subject, because without it the concept devours economics entire. Every successful firm takes custom from a rival; every innovation strands an old technology and those who lived by it; every shift in taste or supply moves a price and so helps some and harms others. If each of these counted as an externality to be corrected, then competition itself — the engine of the system — would stand condemned as a vast machine of uncompensated harms, and the logic of externality would license the state to shield every incumbent from every rival in the name of sparing third parties their losses. The pecuniary-technological line is precisely what forbids this absurdity. It says that the harms competition inflicts through prices are not failures but the system working, and that only the harms inflicted outside prices, in the physical world the price system never reached, are the failures this volume has anatomized.

Other apparent externalities fail other tests, and the careful analyst keeps all of them to hand. Some effects are inframarginal — they alter no one's decision at the margin, and so distort no choice and cause no inefficiency, however real they feel. Some are already internalized, handled by a market or an institution the theorist failed to notice: Cheung's bees and orchards, which contracted away the spillover the textbook thought impossible; the countless effects among a firm's own workers and divisions that the firm, by drawing them inside its boundary, internalizes without any market at all. And some claimed externalities are not effects on resources but effects on feelings — the displeasure I take in a stranger's lawful private conduct, in whom he marries or what he reads or how he lives. To dignify such moral disapproval as an externality, a harm to be priced and corrected, would be to hand the majority a warrant to regulate every private life that offends it, and most economists have drawn the line firmly against counting mere preferences about others' conduct, though the line, as the edge will confess, is among the most contested in the field. Before reaching for tax or permit or rule, then, the honest question is threefold: is the effect technological, a real spillover outside the price system, or merely pecuniary, a transfer through prices? Is it already internalized by some market or institution? Does it actually distort a decision at the margin? Most of what is loosely called an externality fails at least one of these tests, and dissolves on examination into the market doing its work.

Every thread of this volume gathers at the changing climate, the externality beside which all the others are miniatures. When a ton of carbon dioxide rises from a chimney or a tailpipe, it joins an atmosphere shared by every person alive and every person yet to be born, and the warming it abets will visit its costs — the floods and droughts, the failed harvests, the drowned coasts — on all of them, across continents and across centuries, with no regard for who burned the carbon or who profited by it. Nicholas Stern called it the greatest market failure the world has ever seen, and the description is not rhetorical but exact: it is the smoke of the entire species, settling on the entire future, and unpriced. Three features lift it beyond the reach of the tidy diagrams. It is global, so that no single government has the standing to correct it and each nation is tempted to let the others bear the cost of abating while it emits freely — the free-rider problem raised to a planetary scale, and the reason the treaties of Kyoto and Paris bind so weakly. It is intergenerational, so that those who will suffer most cannot bargain, cannot vote, cannot so much as be consulted, and their interests hang entirely on how much weight the living choose to give them. And it is uncertain, shadowed by the possibility of catastrophe — the tipping points, the runaway feedbacks, the fat tail of outcomes too dreadful and too ill-understood to insure against.

The instrument the theory recommends is, in principle, the one the fourth plate built: a price on carbon equal to the harm a ton does, the social cost of carbon, which is simply the marginal external cost of this particular smoke. To compute it, economists join a model of the climate to a model of the economy — the integrated assessment models, of which William Nordhaus's was the pioneer — and trace a ton of carbon through warming into damage into dollars. The figure shows the logic. Abatement should proceed up to the point where its rising marginal cost meets the marginal damage a ton does, and the carbon price is the height of that meeting. But the damage from a ton, being a stock pollutant's, is nearly flat — and where that flat line of damage sits is the entire question, fixed less by the physics than by an ethical choice the next paragraphs must expose.

The ethical choice hides inside a single parameter: the discount rate, the factor by which a harm falling a century hence is reckoned against a cost borne today. The disagreement it produced is the most famous in the economics of climate, and it is not a disagreement about the science. Nicholas Stern, in his review for the British government, adopted a very low rate of discount — declining, on grounds that are frankly ethical, to count a future child's suffering for much less than a present adult's — and arrived thereby at a high social cost of carbon and a call for swift and heavy abatement. William Nordhaus, building the model that would win him a Nobel, used a higher rate drawn from the observed returns on capital, counted distant harms for considerably less, and arrived at a lower social cost and a gentler, rising ramp of action. The gulf between their prescriptions is vast, and yet it issues almost entirely from that one number — a number that expresses how a society weighs the welfare of the unborn against its own, which is a question of ethics that no econometrician can settle and that the discount-rate volume of this series examined in its own right. A shift in the rate of a single percentage point can multiply or divide the recommended carbon price several times over. The physics fixes the warming; the ethics fixes the price.

The instruments are the volume's, applied to the largest case. Most economists favor a carbon tax, and the eighth plate explained why: carbon is a stock pollutant whose marginal damage is nearly flat, and flat damage calls for a price rather than a quantity, the tax fixing the cost of abatement and sparing the economy the lurching of a rigid cap. Others prefer cap-and-trade, which fixes the environmental result and has the political advantage that its permits can be handed out to soften the opposition; still others reach for direct regulation of vehicles and power plants. On the merits the profession has reached a rare near-consensus that carbon should be priced, by tax or by cap, rather than left free. The politics have been merciless to that consensus. Carbon taxes have been voted down, repealed, and revolted against — the French fuel-tax rising that filled the streets with yellow vests is the emblem — because a diffuse and future benefit is a poor match for a concrete and present cost at the pump. And the global structure compounds the trouble: a nation that prices its carbon may simply drive the emitting industry to a neighbor that does not, so that the emissions leak abroad and the planet is no better off, which is why economists speak of border carbon adjustments to tax the carbon embodied in imports, and why the deepest obstacle of all is the absence of any global authority to do what no single nation can.

We began with smoke drifting over a rooftop and a cost that appeared in no one's accounts, and we have followed that small omission to the edge of the gravest problem the species faces. The first plate named the thing: an effect on a third party the price never consulted, a cost or a benefit that falls outside the transaction and so bends no decision. The second and third sized the two faces of it — the harmful activity the market overdoes, because the harm is free to its maker, and the beneficial one the market underdoes, because the benefit escapes him. Then came the cures, in the order in which the discipline discovered them. Pigou's corrective tax, which installs the missing price by the state's hand and raises a double dividend in the doing. Coase's bargain, which answers that the trouble is a missing right rather than a missing price, that the harm is reciprocal, and that parties who can deal will reach efficiency from any assignment of rights. The transaction costs that decide between these remedies, and reconcile them, by telling us when bargaining can occur and when it cannot. The tradable permit, which manufactures the missing market outright, marrying Pigou's cap to Coase's trade. The choice between a price and a quantity, which the shape of the harm settles. And the special cases that test the concept's edges — the network whose worth swells with its users, and the mirage of the merely pecuniary effect that wears a failure's face and is none.

Beneath all of it lies a single idea, and it is worth stating once more in its plainest form, for it is the contribution of the volume. The externality is the failure of a missing market. A cost or a benefit goes unpriced not because anyone is wicked but because the market that would have priced it was never built — there is no market in clean air, none in quiet, none in the protection a vaccine spreads, none, above all, in a stable climate — and what bears no price is invisible to the arithmetic by which every firm and household steers. State the failure so, and the unity of the cures becomes visible, for every one of them is a way of supplying the market that was missing. The tax puts a price on the unpriced. The assignment of a right lets a market in the harm spring up and the parties trade. The permit conjures, out of a cap and a trade, a market in a thing that had none. The externality and its remedies are a single thought turned over and examined from every side: complete the account; price what the price ignored; bring the bystander, at last, into the ledger.

The two great traditions that this volume set against each other turn out, seen from here, to be partners rather than rivals. Pigou and Coase are not competing philosophies of the externality but instruments suited to different frictions, and the friction that decides between them is the cost of transacting. Where the parties are few and the rights are clear and a bargain is cheap to strike, Coase governs: define the entitlement and let the market that appears do the work, more accurately than any planner could, because the parties reveal in their dealing the very magnitudes the planner would have to guess. Where the parties are a multitude and anonymous and no bargain can form — the smoke over a city, the carbon over a planet — Coase's own logic hands the problem to Pigou, and the tax or the cap or the rule must stand in for the market that the transaction costs forbid to exist. The famous quarrel was always malformed. The real question, in any case at hand, is the structure of its transaction costs, which tells us whether a market can be summoned or must be simulated.

From this center the volume sends its lines across the table. To The Tax it is nearest of all, for the corrective levy and its grandest instance, the price of carbon, are Pigou's device delivered into the hands of the public finance that volume studies. To The Commons, the volume that comes next, it is bound as a twin: a shared pasture or fishery or atmosphere is precisely a field of externalities among its many users, each of whom imposes on all the others the cost of his own use, so that the tragedy of the commons is the externality written in the plural, and the two volumes are one subject seen at two scales. To The Power it reaches through the network effect, which is at once a positive externality and a barrier that builds the modern monopoly, and through the patent, which prices the knowledge externality at the cost of a temporary power over price. To The Rule it gives the whole program of law-and-economics, the design of legal entitlements to mend the failures that markets cannot. To The Rate it owes the discount on which the price of carbon turns. To The Asymmetry it is kin, for a discovery is a positive externality of the purest kind and information a good the market underprovides by the same missing-market logic. To The Bias it lends the vexed idea of an internality, the cost a person imposes on his own future self, and the paternalist taxes that idea is summoned to justify. And to The Surplus it returns the deadweight triangle, the welfare accounting in which every one of its failures was measured.

Within its own family the Externality takes the third place. The Failures are the ways a market may betray us with no villain anywhere in view — no monopolist, no liar, no fool — and each names a different structural treachery. The Power was the failure of number, of too few sellers. The Asymmetry was the failure of knowledge, of the fact one party holds and the other cannot see. The Externality is the failure of the missing market, of the cost or benefit that falls outside the price because no market carries it. And the Commons, the volume still to come, is the failure of the unowned shared resource, which all may use and none will tend. With the Externality finished, the group stands three of four complete, and only the commons remains — the externality's own twin, the same logic of the uncounted spillover deepened into the tragedy of a resource that belongs to everyone and so to no one. To build it will complete the Failures, and make of this family the third fully finished group in the table.

The achievement is real and it is large. Externality theory is the backbone of environmental economics, of urban economics, of the economics of the public sector; Pigou and Coase between them founded two whole research programs, the welfare economics that prices the unpriced and the law-and-economics that designs the rights; and the working apparatus of climate and pollution policy the world over — the carbon tax, the emissions market, the social cost of carbon, the congestion charge — is this theory in applied form. It is, by the measure of consequence, the most practically weighty of all the failures, for the gravest problem of the age is one of its instances. The concept illuminates an astonishing range, from the smoke of a single mill to the warming of a whole planet, and it does so with the same handful of curves and the same clarifying question: what does this price leave out, and who bears what it omits?

And yet the volume has insisted at every turn on the limits of its instrument, and the insistence is part of its honesty. The theory demands numbers it cannot reliably supply — the marginal external cost, the social cost of carbon — and those numbers are unmeasurable, or uncertain across orders of magnitude, or, at the climate's heart, an ethical choice about the unborn dressed in the clothing of a technical parameter. Its remedies require the very information the failure denies us, for one cannot set the right tax without knowing the harm one cannot measure. The concept is chronically over-applied, swelling until the discipline of the pecuniary line and the moralized preference must be invoked to keep it from condemning competition itself. Apparent externalities prove, with humbling frequency, to be effects some overlooked market or institution already internalizes, as the bees and the orchards taught. Coase's true lesson is not the textbook tax but the harder truth that transaction costs and the assignment of rights are usually what matter. And the greatest externality of all may exceed the marginalist tools entirely, demanding precaution against catastrophe where the curves call for the fine-tuning of a price.

So the volume ends where it began, with a verdict and its honest qualification. The externality is the failure of the missing market — the cost or benefit that falls outside the price because no market exists to carry it — and its cures, the tax and the bargain and the permit, are so many attempts to supply that market and complete the account the price left short. The theory names this failure with rare clarity, and prices it only with great difficulty, for the figure it must put on the harm is the one figure no market ever revealed. It leaves to judgment, case by case, whether the missing market can be built at all, by whom, and at what cost — and whether, in the gravest case, the building of it lies within the power of any institution we have. A price that takes no notice of the bystander is a verdict that the bystander does not count; this volume has labored to bring him into the ledger, and to be candid about how hard, how uncertain, and how contested that reckoning remains. It hands the unfinished work onward — to The Tax and The Commons, to The Rule and The Rate. The argument remains open.