The Surplus

In 1844 a French engineer of bridges and roads sat down to a question his profession could no longer avoid. The state was building public works — a bridge here, a canal there, a road across a province — and someone had to say whether each was worth its cost. The difficulty was that the benefit of a bridge is not the toll it collects. A traveler who would gladly have paid ten francs to cross pays only the posted two and goes on his way richer by eight, and that private enrichment, invisible in any ledger of receipts, is precisely the public good the bridge was built to produce. Jules Dupuit saw that to value the work one had first to value the benefit it conferred beyond what it charged — and he saw, with a clarity the discipline took another half-century to absorb, how the thing might be measured.

His reasoning ran as follows. Lower the toll by a sou and a few more travelers cross who would not have crossed before; lower it again and a few more still. Each reduction reveals the most that some traveler was willing to pay — his private valuation of the crossing — and the schedule of all such valuations, ranged from the most eager to the most indifferent, is what we have since learned to call the demand curve. The benefit of the bridge to its users is then the sum of what each would have paid, less the sum of what each did pay. On the diagram it is the area beneath the demand curve and above the price: a region Dupuit was the first to read as a quantity of welfare, and which Alfred Marshall, formalizing it in his Principles of 1890, would name the consumer's surplus.

The conceptual step is easy to state and surprisingly hard to keep hold of. A price is one number. Value is a schedule. The market quotes a single figure — the going rate, the same for the eager and the indifferent alike — but the buyers behind the demand curve are not a single figure; they are a queue of distinct valuations, and the one going price serves the whole queue. The buyer at its head, who would have paid a fortune, is charged the same as the buyer at its tail, who would scarcely have bought at all. The first pockets a great surplus; the last, almost none. Sum the pocketed differences and you have the triangle: the value the exchange confers on its buyers over and above the sum it extracts from them.

The figure repays a moment's attention, for the two regions answer different questions. The rectangle is what the buyers surrender: their outlay, the seller's takings, the part of value that is visible because it moves. The triangle is what they keep: the gain that never changes hands, the part of the good's worth that the price does not reach. We do not buy bread in order to pay for bread. We buy it because it is worth more to us than its price, and the measure of that worth-over-price — that excess of value above expenditure — is the surplus. Revenue is counted, taxed, reported. Surplus is silent, and is, from the buyer's side, the whole of the point.

Marshall, who was scrupulous, hung two warnings on the measure that the textbooks have not always carried with equal care. The first is that the triangle is reckoned in money, and money is a proxy for the satisfaction it commands; to read an area of francs as a quantity of welfare is to assume that a franc stands for the same increment of well-being all along the curve — that the marginal worth of money is held fixed while the price moves. The second follows from it. A poor man's intense need and a rich man's idle preference enter the diagram on identical terms, each as a willingness to pay, and willingness to pay is bounded by ability to pay. The demand curve is drawn against an existing distribution of income, and it reads the desires of those who hold money as larger than the desires of those who do not. The surplus is an honest measure of gains as the prevailing distribution of purchasing power prices them. Whether that is an honest measure of gains as such is a question the triangle cannot answer — and the third plate, and the tenth, will press it hard.

What Dupuit did for the buyer, the same logic does for the seller, and the symmetry is clean enough to draw in a single stroke. Sellers, like buyers, are a queue and not a number. Behind the supply curve stands an array of marginal costs — the cost of producing the first unit, the second, the hundredth — rising as the cheap sources of supply give way to the dear. At the going price, the maker of the cheap first unit earns far more than it cost him to produce; the maker of the last, barely more than its cost. The gap between the price received and the marginal cost of each unit, summed, is the producer surplus: the area above the supply curve and below the price, the exact mirror of the consumer's triangle.

A caution belongs here, because the producer's surplus is more easily misread than the buyer's. It is not the same thing as profit. It is revenue in excess of variable cost — what is left once the cost of actually making each unit has been met, but before the fixed costs of being in business at all are reckoned. In the short run it is the contribution production makes toward those fixed costs and toward profit; in the long run, as fresh entry competes ordinary profit away, what survives of it is the return to whatever cannot be copied — a uniquely fertile field, a patent, a place on the busiest corner, a scarce and particular skill. Producer surplus is, at bottom, the rent of the scarce; the volume that bears that name, The Rent, takes up its anatomy. Here it is enough to see that the seller, no less than the buyer, walks away from the market holding something the price did not capture.

Set the two curves on the same axes and they cross, and the point at which they cross is the equilibrium: the price at which the quantity buyers wish to take equals the quantity sellers wish to give. At the crossing the total gain from trade stands fully on view — the consumer's triangle above the price, the producer's below it, and their sum, the whole region enclosed between demand and supply up to the traded quantity, the total surplus the market has called into being. Now comes the result that makes the diagram more than a picture. Of all the quantities that might be exchanged, the equilibrium quantity is the one that makes that total as large as it can possibly be. Trade less, and worthwhile units go unmade — gains left lying on the table. Trade more, and units are made that cost more than anyone will pay — losses incurred to no purpose. Only at the crossing is every worthwhile trade made and no wasteful one; only there is the area filled to the brim. This is the geometric seed of the most consequential proposition in welfare economics, that a competitive market wastes none of the gains from trade — a proposition we shall state with care, and qualify with more, in the eleventh plate.

The figure above is fixed at one price; the cross it draws is not. Below, the same meeting is made operable. The market’s own equilibrium price sits at the crossing, and there the two surpluses fill the whole area between the curves — the gain from trade at its greatest. Drag the price away from it, as a ceiling or a floor would, and the traded quantity retreats to the short side of the market: the trades that no longer happen were worth making, and the surplus they would have produced is transferred to no one but simply destroyed — the deadweight loss, shown in red. Only at the crossing is the loss nothing.

The price

Consumer surplus—

Producer surplus—

Total gain—

Deadweight loss—

But the figure conceals exactly as much as it reveals, and what it conceals is the whole subject of this volume. The market fixes the size of the total surplus; it does not, of itself, fix the division between the two triangles. That division is governed by the slopes of the curves — by elasticity, the readiness of each side to walk away. Where demand is steep and supply flat, where buyers have nowhere else to turn and sellers have ready alternatives, the price settles high, the consumer's triangle is squeezed thin, and the greater share of the gain accrues to sellers. Reverse the slopes and you reverse the division. The same total, the same efficient quantity, the same brimming area, can be split in wholly different proportions according to which side can the more easily do without the other. The size is a matter of technology and taste. The split is a matter of who can walk away — which is to say, of power. Nearly everything that follows is the patient elaboration of that one sentence.

A measure is only as honest as the conditions under which it holds, and the consumer's surplus rests on a condition that is almost never quite true. Marshall's triangle reads an area of money as a quantity of welfare, and that reading is exact only if a unit of money means the same to the buyer throughout — if the marginal utility of income holds fixed while the price slides along the curve. The trouble is that a change in price is itself a change in real income. When the price of something a household buys falls, the household is, by that very fact, a little richer: it can afford the old basket and have money left over. And a richer household is a different household — it wants different quantities, values the next dollar differently, sits on a different demand curve. The curve we read the surplus from has shifted beneath us in the very act of reading it.

John Hicks, in the years around 1939, gave the difficulty its lasting shape by drawing two demand curves where Marshall had drawn one. The ordinary, or Marshallian, curve traces how quantity answers to price while money income is held fixed — and so it tangles two effects together: the substitution toward the good that has grown cheaper, and the enrichment that comes of facing lower prices at all. The compensated, or Hicksian, curve isolates the first by stripping out the second. At each price it asks how much the household would buy if its income were adjusted, in the same breath, to hold its welfare — not its money — exactly constant. Along the compensated curve the buyer is pinned to a fixed level of well-being, and the area behind it is therefore a clean measure of the welfare change, uncontaminated by the shifting worth of money.

This yields not one true measure of a price change but two, according to which level of welfare one elects to hold constant — the level before the change, or the level after. Hold the buyer at her original welfare and the resulting money measure is the compensating variation: the sum that would have to be taken from her after a price fall to return her exactly to where she began. Hold her instead at her new welfare and one has the equivalent variation: the sum that would have to be handed to her in place of the price fall to leave her just as well off as the fall itself would. These are the honest measures — the most a beneficiary would pay, the least a loser would accept — and Marshall's ordinary surplus sits between them, the rough mean of two exact figures it was long mistaken for being.

How wide is the gap between rough and exact? For most of what one might worry about, narrower than the worry. The two compensated measures and the ordinary one collapse into a single figure precisely when the income effect vanishes — when the good takes so small a share of the budget, or answers so faintly to income, that growing a little richer does not change how much of it one buys. Robert Willig showed, in a paper of 1976 whose title announced that the consumer's surplus might be used without apology, that the percentage error from putting Marshall's triangle in the place of the true measures is bounded, and that the bound is governed by the income elasticity of demand and the share of income at stake. For a good on which a household spends a small fraction of what it has, the error is a fraction of one percent. The apparatus was rescued; the diagram could stay.

It is worth being exact about what was rescued and what was not, for the rehabilitation is narrower than it is usually taken to be. Willig's result vindicates the triangle as an approximation to the right money measure of a given person's gain. It says nothing whatever about the deeper objection, which is not that the triangle mismeasures welfare in money but that money mismeasures welfare across persons. Compensating and equivalent variations are as firmly bound to ability to pay as the ordinary surplus is; all three read a pauper's desperate willingness to part with a small sum as a small gain, and a magnate's careless willingness to part with a large one as a large gain. The income effect is a technical wrinkle, and it has been technically smoothed. The distributional charge — that the whole construction prices desire by the purse — is not a wrinkle but a feature, and no refinement of the curve can iron it flat. It waits for us, in its sharpest form, at the tenth plate.

The surplus of the first three plates is a small and private thing — a buyer's gain over a price, a seller's margin over a cost, the property of the two parties to a single trade. But the word entered economics carrying a larger and graver meaning, and that older sense has never gone away; it has only gone quiet, and it returns the instant the subject turns from the efficiency of a market to the wealth and division of a whole society. To the classical economists, and to the tradition that descends from them, the surplus was not a triangle on a diagram but a magnitude in the life of a nation: the product of all its labor and land over and above what had to be consumed to produce that product and to set the same production going again the following year. It was, quite literally, the residue — what remained in the granary once the seed for next season had been set aside and the laborers fed.

The Physiocrats of eighteenth-century France gave the idea its first system. In Quesnay's Tableau of 1758 — a diagram of the economy's circular flow with a fair claim to be the first macroeconomic model — only agriculture was held to yield a true net product, a produit net, a surplus of output over the inputs consumed in raising it; manufacture and trade, they thought, merely reshaped what the land had already brought forth. The doctrine was wrong in its particulars, as we now see, for industry creates surplus too. But it was right, and lastingly so, in its central instinct: that a society produces more than it must use up to keep producing, that this excess is the fund from which everything not strictly necessary is paid, and that the deepest question in political economy is who lays claim to it.

David Ricardo, half a century on, turned the idea into an engine of distribution, and in the turning wrote the seed of another volume in this table. As a growing population drives cultivation onto poorer and poorer soils, he argued, the difference between the yield of the good land and the yield of the worst land still in use falls to the owners of the good land as rent — a surplus that comes to them not for any effort or abstinence of their own but for the bare ownership of a scarce and unproduced thing. The Rent, the table's second element, is Ricardo's question grown into a volume of its own. For him the surplus was the great stake in a three-cornered struggle among landlords, capitalists, and laborers, and the laws governing how it divided among rent, profit, and wages were, he held, the proper subject of the entire science. Distribution was not a footnote to the theory of value. It was the point.

This classical surplus differs from the marginalist triangle in nearly every respect, and the difference is no matter of vocabulary. The triangle is microscopic, subjective, a thing of exchange: it lives in the gap between a buyer's valuation and a price, and it is summoned into being by tastes. The residue is macroscopic, objective, a thing of production: it lives in the gap between what a society makes and what it must use up to make it, and it is governed by technology and by the division of the product among classes. The one is measured in the willingness of individuals to pay; the other in tons of corn, or hours of labor, over and above replacement. They are very nearly two different concepts that the accident of a shared word has persuaded generations of students to mistake for one.

In our own century Piero Sraffa revived the classical conception with a rigor the classics themselves could not command. His slender book of 1960, with its austere title — a treatise on the production of commodities by means of commodities — showed that once the technical methods of production and the real wage are given, the surplus of the economy, the rate of profit, and the relative prices of all goods are determined together, with no appeal whatever to marginal utility or marginal productivity. And to measure that surplus free of the distortion any shift in distribution imposes on ordinary prices, Sraffa built a peculiar composite good — his standard commodity — meant to serve as an invariable measure of value, a yardstick that does not itself bend when the shares of wages and profit move. The reader of this series will recognize the ambition at once, for it is the series' own: to find a numéraire that holds steady while everything priced against it shifts. Sraffa was, in the most literal sense, hunting for an honest unit of account — and the difficulty of that hunt is part of what these volumes are about.

No one pressed the classical idea of the surplus to a sharper or more disturbing point than Marx, and it is worth following the argument exactly, for its force lies in its logic and not in its indignation. Marx grants, for the sake of the argument, the most favorable assumptions the orthodoxy could ask. Suppose every exchange is voluntary. Suppose every good, labor included, sells at its full value. Suppose, in short, that no one is cheated anywhere. The question he then puts is the one the orthodoxy of his day could not comfortably answer: under these conditions of universal fair exchange, where does profit come from? If everything trades at its true value, how does the capitalist, who began the day with a sum of money, contrive to end it with a larger one?

The answer turns on a distinction the classical economists had blurred and Marx made precise — that between labor and labor-power. What the worker sells is not his labor, the work itself, but his labor-power: his capacity to work, his ability to labor for a day. And like any other commodity in the argument, labor-power has a value, namely the cost of producing and reproducing it — the food, shelter, training, and rest required to bring a worker to the gate each morning fit to work, and to raise the next generation behind him. The wage, paid at the full value of labor-power, covers exactly this cost of reproduction. Here is the hinge of the entire system. The capitalist has bought labor-power at its value — and labor-power, once bought, can be set to work for longer than it takes to reproduce that value. A worker may reproduce the value of his day's wage in, say, the first six hours; but the working day is ten, and in the remaining four he goes on producing value all the same — value that belongs, by the terms of the purchase, not to him but to the man who bought his power to work.

The day thus falls into two parts. In the necessary hours the worker reproduces his own wage; in the surplus hours he produces a value the capitalist keeps and calls profit. This unpaid surplus labor — congealed in the commodities the worker makes, realized when they are sold — is surplus value, and it is, on Marx's account, the single source of profit, of interest, and of rent alike. The ratio of the unpaid hours to the paid, of surplus labor to necessary labor, is the rate of exploitation; and there are two ways to raise it. One may lengthen the working day, so that the surplus hours grow while the necessary hours stay fixed — Marx called this absolute surplus value, and the long bitter history of the struggle over the length of the day is its political expression. Or one may raise the productivity of labor in the trades that make the worker's necessities, so that his wage is reproduced in fewer hours and more of the fixed day falls into the surplus portion — relative surplus value, the quieter engine, and in the long run the more powerful.

Set beside the marginalist account, the provocation is exact and total. To the orthodoxy that would mature after Marx, each factor of production is paid the value of its marginal product, the contribution of its last unit: the wage measures the worker's productive contribution, profit measures capital's, and the distribution is, in just this sense, a set of fair exchanges yielding no exploitation at all. To Marx, the very same wage is the value of labor-power — the cost of the worker's upkeep — and the gap between what labor produces and what its upkeep costs is not a contribution priced by the market but a surplus extracted by ownership. The two readings can describe the identical pay packet, the identical voluntary contract, the identical competitive market, and arrive at opposite verdicts on what it means. The exchange is fair, says the one; the exchange is fair and the worker is exploited all the same, says the other — for the fairness of the exchange and the appropriation of the surplus are, on Marx's logic, perfectly compatible. This is the wage's hidden complement, and it returns us with force to the unfinished question of The Wage: whether what labor receives is set by what it contributes, or by the power confronting it across the gate.

We have followed the surplus into being and watched two traditions quarrel over its nature. We turn now to the contest over its keeping, and the first thing to establish about that contest is also the strangest: that not all of the surplus at stake is kept by anyone. Some of it, when the contest is joined, is not handed from one party to another but destroyed outright — annihilated, vanished, gone to no one at all. This vanished surplus is the deadweight loss, and it is perhaps the single most important idea in the economics of policy, because it is the cost that has no beneficiary, the waste that is purely waste.

It arises whenever a wedge is driven between the price a buyer pays and the cost a seller bears. A tax does this on purpose; a tariff, a quota, a price control, and above all the markup of a monopolist do it as well. Recall from the cross that every unit worth trading is one a buyer values above its cost of production, and that the competitive quantity is precisely the quantity at which all such units, and no others, change hands. Now insert a wedge — say a monopolist who, to hold the price high, keeps supply back. The quantity traded falls below the competitive level. And on every unit no longer traded — every unit between the restricted quantity and the quantity the market would otherwise have reached — there was surplus to be had: the buyer valued it above what it would have cost to make. That surplus is now realized by no one. The buyer does not get the good; the seller does not make the sale; the state collects nothing on a transaction that does not occur. The gains from those trades simply cease to be.

On the diagram the lost surplus is a triangle, the region between the demand and supply curves over the range of trades the wedge has suppressed. It is named, for the economist who first took its measure seriously, the Harberger triangle. And here the figure teaches a lesson that surprises almost everyone who meets it. The triangle is thin. It is thin because the trades destroyed at the margin were the least valuable ones — units the marginal buyer valued only a little above their cost — so the surplus lost on each is small; and because, for the same reason, the loss grows not in proportion to the wedge but with its square. A small distortion destroys almost nothing; a wedge twice as large destroys roughly four times as much. When Arnold Harberger, in 1954, actually computed the triangle for monopoly across American manufacturing, he arrived at a figure so small — a fraction of one percent of national income — that it seemed to put in doubt whether monopoly was worth troubling over at all.

The smallness of the triangle set off one of the most fruitful arguments in the discipline, and the reply to it is essential, because the triangle is not the whole of the cost. Gordon Tullock observed, in 1967, that the monopolist's gain — the rectangle of profit he wins by holding the price high — counts in the standard accounting as a mere transfer from buyers to monopolist, and so as no social loss at all. But suppose firms must spend real resources to win that rectangle: to lobby for the protective tariff, to litigate rivals into submission, to build and defend the barriers that keep competition out. Then those resources — the lawyers, the lobbyists, the contrived obstructions — are genuinely consumed, dissipated in the bare contest over who shall pocket the transfer, and the true social cost of monopoly swells from the thin triangle toward the fat rectangle behind it. The activity earned a name a few years on — rent-seeking — and it recast the whole question. The structures that make such seeking possible are the burden of The Power and The Rule, the table's tenth and eighteenth elements. To these static costs the dynamic ones must be added: a monopoly sheltered from rivalry may simply decline to improve, and the surplus never created by the inventions never made is a loss no triangle can show.

The state now enters the contest, and it enters by the very instrument the monopolist used: a wedge. A tax is a wedge driven on purpose between the price the buyer pays and the price the seller keeps, and once it is in place the surplus is carved not into two parts but into four. A part is transferred to the state — the revenue the tax is levied to raise, the rectangle whose height is the tax and whose width is the quantity still traded. A part is destroyed — the deadweight triangle of trades the tax discourages, surplus that goes, as we have seen, to no one. And the remainder, the surplus that survives as private gain, is redistributed between buyer and seller in proportions the tax itself does not announce. The whole of the difficult and beautiful theory of taxation lives in those proportions, and in the gap between what the law intends and what the market delivers.

The first and most important result is one of the genuine surprises of the science, and it goes in a sentence: who bears a tax has nothing to do with who pays it. The party the law names as the one who must hand over the money — the shopkeeper who remits the sales tax, the employer who forwards the levy on wages — is not, save by coincidence, the party who ends the poorer for it. The burden does not stay where the statute sets it down. It slides, through the adjustment of price, until it comes to rest by a logic the legislator does not command: the logic of elasticity, of which side can the more easily walk away. Tax the sale of a good, and if buyers have nowhere else to turn while sellers have ready alternatives, the price rises by nearly the full amount of the tax and the buyers bear it, whoever signs the check; reverse the elasticities and the sellers bear it instead. The more inelastic side — the side with fewer options, less able to refuse — surrenders the larger share of the surplus, every time.

The figure shows the wedge resting on the more inelastic of the two parties. When demand is steep — when buyers, facing a necessity, cannot readily do without — the price they pay climbs steeply with the tax while the price sellers receive scarcely falls, and the great bulk of the burden lands on buyers, though the tax be gathered entirely from the firms. This is why a tax on cigarettes or on insulin falls overwhelmingly on those who buy them, and a tax on a thing in fixed supply on those who own it; the statute may say what it likes. The principle is among the most secure in economics and among the least understood beyond it, and the misunderstanding is not idle: a great deal of political argument proceeds as though naming the remitter settled the burden, when the market reopens the question the instant the law is signed. The full anatomy of this sliding is the work of The Tax, the table's seventeenth element. Here it is the surplus we are tracking, and the lesson is that the state's claim upon it, like the monopolist's, is mediated by power — by the elasticities that decide who can least afford to walk away.

A second result governs the size of the loss, and it carries the whole modern theory of how to tax. The deadweight triangle, we saw, grows with the square of the wedge: a small tax destroys very little surplus, a large one disproportionately much. Two maxims follow, and they pull against each other. The first is that it is cheaper to raise a given sum from many small taxes than from a few large ones — broad bases at low rates, the wedge spread thin across the whole economy rather than driven deep into one corner of it. The second, subtler and more troubling, is that a tax destroys least where the parties can least escape it: to spare the most surplus, one should tax the inelastic things, the necessities no one can forgo. But the inelastic necessities are exactly what the poor cannot avoid, so the rule that spares the most surplus falls hardest on those with the least. Efficiency counsels taxing bread; justice forbids it. The conflict is no flaw in the theory but the theory's honest report of a real and unavoidable tension — the very tension, between the size of the surplus and the justice of its division, that haunts every plate of this volume.

There is one wedge that escapes the dilemma, and it deserves naming for being so nearly unique. A tax on the unimproved value of land destroys no surplus at all. Because the supply of land is fixed — no tax can discourage the making of more of it, there being no more to make — the wedge suppresses no trade and kills no worthwhile transaction; it merely transfers the landlord's surplus, the pure rent of position, from owner to public, and leaves the use of the land untouched. Henry George built a movement on the observation, and the volume on The Rent will weigh it. The point here is the clean one the surplus framework makes visible: that the loss a tax inflicts is the loss of the trades it deters, so that a tax which deters nothing — falling on a surplus that owes nothing to effort or to production — can take without wasting.

We come to the seller's maximal claim — the arrangement under which one party to the trade walks away with the entire surplus, and the other with none. It is called price discrimination, and its perfect form is the cleanest demonstration in all of economics that efficiency and fairness are different questions, and can return opposite answers on the very same case.

The ordinary seller posts a single price, and we have seen the consequence: the eager buyer at the head of the demand curve, who would have paid far more, keeps a large surplus, and the seller collects only the one price from everyone. Now imagine a seller who could read each buyer's mind — who knew, for every customer, the very most she would pay, and could charge her exactly that, and the next customer something else, and so on down the curve. This is first-degree, or perfect, price discrimination, and its result is doubly remarkable, in ways that point in opposite moral directions.

The first remarkable thing is that the outcome is efficient — fully, completely efficient, with no deadweight loss whatever. The reason is exact. A single-price monopolist holds back supply because selling one more unit means cutting the price on all the others; the marginal sale cannibalizes the inframarginal ones, so the monopolist stops short of the efficient quantity and the deadweight triangle opens. But the perfect discriminator suffers no such conflict. Because he charges each buyer her own price, selling one more unit costs him nothing on the units already sold; he will therefore go on selling to every buyer who values the good above its cost of production — which is precisely the efficient quantity, the same the competitive market would reach. Every worthwhile trade is made. The triangle of lost surplus, the waste that troubled the sixth plate, vanishes entirely.

The second remarkable thing is that the buyer keeps nothing. Every scrap of the surplus that the single price would have left in buyers' hands is now extracted, converted, carried over to the seller; the consumer's triangle is emptied into the seller's pocket, unit by unit, down to the last buyer, who is charged exactly her cost and so gains nothing at all. Hold the two findings together, for their conjunction is the whole of the lesson. The perfectly discriminating monopolist runs an economy that is efficient — not a grain of surplus is wasted — and in which one party has taken everything. Efficiency has been achieved and equity annihilated in the very same stroke; the two are not merely distinct but, in this case, perfectly opposed, the one secured by the very means that destroyed the other. To the lay intuition that efficient means good, or at least fair, the perfect discriminator stands as a permanent refutation: the largest possible total surplus, divided as unequally as a division can be.

Perfect discrimination is an idealization — no seller reads minds — but its weaker forms are everywhere, and each is an attempt to capture more of the surplus than a single price would yield. Third-degree discrimination sorts buyers into observable groups and charges each its own price, more where demand is less elastic: the student fare, the senior's discount, the same drug sold dearer in the richer country. Second-degree discrimination cannot observe the groups directly and so induces buyers to sort themselves, offering a menu — the large size at a lower unit price, the two-part tariff of a fixed fee plus a charge per use, the stripped and the premium versions of one essential good — built so that each kind of buyer, choosing what suits her, reveals which kind she is and pays accordingly. Each device recovers for the seller some of the surplus a uniform price would have surrendered; each is a technology of capture.

The modern frontier of the subject is its oldest dream brought within reach. Perfect, first-degree discrimination required knowing each buyer's maximum, and that was long impossible. It is impossible no longer, or less so: a seller who knows a buyer's history, her device, her location, the very search that delivered her, can estimate her willingness to pay and quote her a price built for her alone. The theory's verdict on this is uncomfortably, instructively double. Such personalized pricing can expand access — a buyer offered a low price fitted to her thin purse may be served whom a uniform price would have shut out — even as it strips the surplus from those whose willingness to pay is read and met to the cent. The same procedure that includes the marginal buyer extracts from the eager one, and it does both at once. Whether, on balance, it widens the market or merely empties the buyers' triangle into the seller's depends on which buyer one happens to be — which is only the perfect-discrimination paradox returning in modern dress. The economics of the intermediary who stands between and reads is the burden of The Spread; the rules that might govern such pricing, of The Rule.

In the thick competitive market of the cross, no one bargains. The price is simply there, a fact each trader takes as given, and the surplus splits by the elasticities, impersonally, as though by a law of nature. But thin the market to two — one buyer and one seller, a negotiation, a single deal with no ready alternative on either side — and the price ceases to be given. It becomes indeterminate, openly up for grabs, and the division of the surplus is no longer settled by the market but contested by the parties. This is the general case the volume has been building toward, for it is here that the question who walks away with it is posed most nakedly.

Francis Ysidro Edgeworth saw the structure of the problem in 1881, in a book whose strange title — a treatise of mathematical psychics — concealed one of the most penetrating results in the theory of exchange. With two traders, he showed, there is not one price but a range. Any division of the surplus that leaves both parties better off than no deal, and that cannot be bettered for one without worsening the other, is a possible settlement; the set of all such divisions is what he taught us to call the contract curve, and the modern theory of the core is its descendant. Theory alone cannot say which point on the curve the parties will reach — it can only fence the range within which they must end. And then Edgeworth showed the force that collapses the range: competition. Add traders to each side — replicate the buyers, replicate the sellers — and the range of possible bargains shrinks; in the limit, as the traders grow many, it contracts to a single point, and that point is the competitive price. The determinacy of the market price is bought with the multitude of traders. Remove the crowd, and indeterminacy floods back.

The figure is Edgeworth's box, the most compact object in the theory of exchange. Two traders, two goods, the whole of what each holds measured from opposite corners; the curving lines are the levels of satisfaction each would count indifferent; the lens between the two lines through their starting point is the set of all trades both prefer to no trade; and the contract curve threading the lens is the locus of bargains that waste nothing — the efficient divisions, every one of them, from the one that hands almost all the surplus to the first trader to the one that hands almost all to the second. The whole shaded range is efficient. The theory cannot, from within, choose among its points. Something outside the bare logic of efficiency must do the choosing — and what does the choosing is power.

If theory cannot deduce the split, perhaps reasonable axioms can select it, and John Nash, in 1950, proposed that they could. His bargaining solution picks the division that maximizes the product of the two parties' gains, each gain reckoned from what that party would have if the deal fell through — from its outside option, its disagreement point. The content of the result is sharp and memorable: your share of the surplus rises with the value of your alternative to the bargain. What you can credibly walk away to is what you can demand by staying. The party with the better outside option — the seller with other buyers, the worker with other offers, the buyer with other suppliers — commands the larger slice, not because he contributes more to the surplus but because he needs this particular deal less. This is the monopsony insight of The Wage's eighth plate raised to a general principle: power, in a bargain, is the poverty of the other side's alternatives.

Ariel Rubinstein, in 1982, made the bargaining move through time, and found a second determinant beneath the first. Let the parties trade offers back and forth, each round of delay a little costly to both, and the haggling has a unique resolution — a determinate split, where the static theory had only a range. The split is governed by patience. The party who suffers less from delay, who can the better afford to wait the other out, captures the larger share; impatience is weakness, and the capacity to outlast is a form of power as real as any outside option. As the interval between offers shrinks toward nothing, Rubinstein's division approaches a Nash-like split weighted by the parties' relative patience. Time, and the ability to wait, joins the exit and the alternative as a lever upon the surplus.

Gather the threads. The size of the surplus, we have said throughout, is a matter of technology and taste — of how much value the trade can create. Its division, wherever the market is too thin for the crowd to settle it, is a matter of outside options, of patience, of information, and of the cost each party bears from conflict — which is to say a matter of power, in the several concrete forms power takes at a bargaining table. The pie's size is one question; its slicing, another; and the second is answered not by anything intrinsic to the good exchanged but by the structure of the encounter and the alternatives each party holds in reserve. The structures that confer such power are The Power; the unwritten understandings that fix shares with no explicit bargaining at all are The Norm.

The plates so far have measured the surplus, located it, divided it, and watched it destroyed and captured. We have spoken freely of total surplus rising or falling, of policies that enlarge or shrink it, as though more total surplus were plainly better. It is time to examine that plainly, for it conceals a value judgment as consequential as any in the discipline — and one economics has spent a century trying, and failing, to do without.

The difficulty announces itself the moment a policy helps some and harms others, which is to say the moment any real policy is proposed. To call such a change an improvement, one must somehow weigh the gainers' gains against the losers' losses, and to weigh them one must set them on a common scale. But here the science had, by the 1930s, talked itself into a corner. Lionel Robbins had argued, with a rigor that proved hard to answer, that comparisons of satisfaction across persons are not scientific propositions at all: there is no instrument that reads how much a loaf means to you against how much it means to me, no fact of the matter the science can establish. If that is so, the economist may not say that the loaf does more good in the starving man's hands than in the glutton's — not because it is false, but because it is not the kind of thing the science can know. And if the economist may not compare persons, how shall he add one person's gain to another's loss and pronounce upon the sum?

The answer the profession reached for, in the hands of Kaldor and of Hicks at the close of the 1930s, was the compensation principle, and it is an ingenious evasion. A change is to be counted an improvement, they proposed, if the winners gain enough that they could fully compensate the losers and still remain better off — whether or not they actually do. This merely potential improvement seems to slip the Robbins noose: it asserts not that the winners' gains outweigh the losers' losses in any felt sense, but only that the winners could buy the losers' consent and have something left over, which is a fact about sums of money and not, apparently, a comparison of satisfactions. On this principle rests the great practical machine of policy appraisal, cost-benefit analysis, which totes up the willingness to pay of everyone affected and recommends whatever yields the largest net sum. It is the working meaning of maximize total surplus, and it governs the appraisal of dams, drugs, highways, and regulations the world over.

But the evasion does not hold, and it matters to see exactly where it fails, for the failure is no technicality. To add my willingness to pay to yours and compare the total to some alternative is to treat a dollar of willingness to pay as worth the same whoever holds it — to set the marginal social value of a dollar equal for the pauper and the prince. That is the buried judgment, and it is a powerful one. Willingness to pay is bounded by ability to pay: the rich man can voice a casual preference as a large number of dollars, the poor woman an urgent need as only a small one, and cost-benefit analysis, faithfully summing, will weigh his idle convenience above her necessity because his purse speaks the louder. The procedure meant to avoid comparing persons compares them after all, and in the most questionable way imaginable — by the depth of their pockets. Distributional neutrality, the refusal to weight a poor man's dollar above a rich man's, presents itself as the absence of a distributional position. It is in truth a distributional position, and a strong one: that the existing distribution of purchasing power is the right set of weights to use.

Nor is the principle even internally secure. Tibor Scitovsky pointed out, in 1941, that the compensation tests can contradict themselves. A move from one state to another may pass the test — the winners could compensate the losers — while the reverse move, back again, also passes it, because the set of feasible redistributions differs at the two states, and what the winners can afford to pay depends on where they stand. The test can thus pronounce one state an improvement on a second, and the second an improvement on the first, in the same breath. Scitovsky proposed a double criterion to patch the contradiction; but no patch restores the original dream of a ranking that takes no stand on distribution. The dream was incoherent from the start.

None of this is cause to discard the surplus. It remains the most powerful instrument we possess for sizing the gains and losses of a policy, and to know the size of what is at stake is no small thing. The lesson is rather to read its verdicts in their true and narrow currency. This policy raises total surplus is a true and useful statement about the size of the pie. It is not, and cannot be made into, a statement that the policy is good — unless one is prepared to defend the unweighted dollar aloud, to say in plain words that a dollar of the magnate's convenience counts equally with a dollar of the pauper's need. The tools to weight the dollars differently exist; distributional weights can be written into a cost-benefit analysis as readily as left out. That they are almost always left out, and the unweighted sum reported as though it were the neutral truth, is itself a verdict — the quiet, recurring verdict of a discipline that finds the size of the surplus so much easier to discuss than the justice of its keeping.

We arrive at the summit of the classical theory of the surplus: the two propositions that crown welfare economics and are, between them, the most overread results in the discipline. They are called the fundamental theorems, and to state them precisely is already to begin to qualify them, for nearly all the trouble comes of carrying them past what they say.

The first theorem makes rigorous the intuition we read off the cross in the second plate. Under a definite set of conditions — many traders, each too small to sway the price; markets for everything; no cost or benefit spilling onto third parties; buyers who always prefer a little more to a little less — every competitive equilibrium is Pareto efficient. In the language of this volume: a competitive market, left to settle, wastes none of the gains from trade; it fills the area between the curves to the brim, exhausts the surplus, leaves no mutually beneficial trade unmade. This is the precise and defensible core of the case for markets, the marble pedestal beneath the phrase the market is efficient, and it is a genuine and beautiful result. One must only attend to what efficiency means here — and to the word the theorem does not contain.

The word it does not contain is fair, and the silence is total. Pareto efficiency asks only that no one can be made better off without making another worse off; it says nothing whatever about how well or how badly anyone is doing to begin with. An allocation in which one man owns everything and the rest own nothing is Pareto efficient — you cannot improve the lot of the starving without taking something from the man who has it all, and so the brutal arrangement passes the test undisturbed. The first theorem certifies that the competitive market reaches some efficient point; it offers not a syllable on which, and the efficient points include the monstrous alongside the humane. Maximizing the surplus and achieving a decent society are simply different objectives. The theorem speaks to the first and is mute on the second, and to hear in it a vindication of market outcomes as such is to hear a sentence that was never spoken.

The second theorem seems to come to the rescue, and is the more subtly treacherous for it. It runs the first in reverse: under some further conditions — chiefly that tastes and technologies be suitably convex — any Pareto-efficient allocation one likes, including a just one, an equal one, any a society might choose, can be reached as a competitive equilibrium, provided the initial endowments are first redistributed to suit. This is the theorem that underwrites the reassuring division of labor one hears everywhere: let the market do what it does best and find the efficient frontier; handle distribution separately, by redistributing the starting points and then letting prices work. Efficiency and justice, on this reading, are cleanly separable concerns, to be pursued with separate instruments and never confused.

The treachery lies in the instrument the theorem quietly requires. The redistribution it calls for must be by lump-sum transfers — taxes and grants that depend on nothing a person can alter: not on income, not on consumption, not on effort, since a tax on any of those would distort the very prices the theorem leans upon. The transfer must fall on what a person simply is, never on anything he does. But lump-sum taxes of that kind, in any actual society, are very nearly impossible: we cannot observe a person's earning capacity directly, only its traces in what he earns and buys and chooses, and to tax the traces is to tax the choices, and to tax the choices is to drive a wedge and lose surplus after all. The clean separation — grow the pie freely, slice it justly with costless transfers — rests on a tool that does not exist. Real redistribution distorts; the wedge of the seventh plate returns; and efficiency and equity, which the second theorem promised to divorce, turn out to be married after all — their union the whole subject of the modern theory of optimal taxation, which is nothing but the careful accounting of the trade-off the second theorem assumed away.

Read exactly, then, the two theorems establish something at once more modest and more pointed than the uses to which they are put. They establish that the competitive market is an extraordinary engine for making the surplus as large as it can be, and an instrument wholly indifferent to whose surplus it becomes; that efficiency can be had at any level of justice or injustice; and that the hope of buying efficiency now and justice later, at no cost of the one for the other, founders on the non-existence of the painless transfer. This is, once more and at the highest generality, the single finding of the volume: economics can tell us, with theorems, how to keep the surplus whole, and cannot tell us, with theorems, to whom it should belong.

We return, at the last, to the question the volume set out from, and we are now in a position to say why it cannot be answered quite as it was asked. An exchange creates value beyond its price: who walks away with it? The plates have shown this to be two questions wearing the dress of one, and that economics answers them with wildly unequal success.

The first question is how much. How large is the surplus an exchange or an economy creates — the buyer's gain over price, the seller's margin over cost, the social product over what reproduction consumes? This question economics answers, and answers well, and answers better with each decade. It can draw the triangle and, with Hicks's and Willig's corrections, defend the drawing; it can sum the two surpluses at the cross and prove the competitive quantity makes the sum the greatest possible; it can measure the deadweight loss a wedge destroys and show it growing with the wedge's square; it can trace, in theorems, the efficiency of the whole. The size of the surplus is a question about technology, about tastes, about the efficiency of exchange — and on that terrain the discipline is powerful, precise, and largely right.

The second question is whose. Of the surplus an exchange creates, who keeps it — the buyer or the seller, labor or capital, the owner or the public, this party to the bargain or that? And here, the plates have shown, economics from its own resources can say remarkably little. The split at the cross is fixed by elasticity, by who can walk away. The split in a bargain is fixed by outside options and by patience, by who needs the deal the less. The seller who can discriminate keeps the whole of it, with no loss of efficiency to reproach him. The state's share, and who truly bears it, are fixed by elasticity again, and not by the statute. And the grand theorems that crown the subject prove only that the surplus can be kept whole at any division whatever, the just and the monstrous alike. The keeping of the surplus is a question about power — about exits and alternatives, about who owns the scarce thing, about who can outlast whom — and on that terrain economics describes the mechanisms with care and pronounces upon the outcome not at all.

This is why a price is a verdict and not a fact — the conviction the whole series is built to vindicate. A wage, a rent, a rate, a markup, a platform's take: each looks like a number read off the world, a datum as neutral as a temperature. Each is in truth a settlement of the keeping — a particular answer, struck by a particular arrangement of power, to the question of who shall walk away with the surplus the exchange creates. To present such a number as a fact of nature is to naturalize the arrangement of power that produced it, to make a contingent victory wear the face of a law. The work of denaturalizing it, of restoring the verdict to the contest that issued it, is the work this table exists to do.

The Surplus does not adjudicate that contest; it sizes the prize and names the parties, and hands the adjudication on. It hands it, above all, to The Share — the table's twenty-second element, where the keeping is fought out at the scale of a whole economy, between those who work and those who own, and the volume's question becomes the great question of political economy: whether labor's falling portion is the reading of its marginal product, or the mark of its declining power. It hands the structures that let one side keep the more to The Power and to The Rent; the state's formal entry into the contest to The Tax; and the unwritten settlement — the share fixed by custom with no bargain struck at all — to The Norm. The surplus is the value at stake in every one of them. This volume has insisted only upon three things: that the value is real, that the contest over its keeping is real, and that no price has ever been the neutral fact it is forever taken to be.

The series sets itself to fix a numéraire of honesty against which such verdicts may at last be read. The Surplus offers that measure its hardest case — a quantity the discipline can total to the penny, and cannot, unaided, assign to a single soul. The measure holds. The assignment remains, as in honesty it must, an open argument.