The last plate established that an exchange creates a surplus and that the market makes that surplus as large as it can be, and then stopped pointedly at the threshold of a further question it declined to answer: once the gains exist, who gets them? This plate is about that question, and it is a harder one than it looks, for it turns out to lie at the exact boundary of what the theory of prices can and cannot do. Economics can say, with precision, how large the gains from a trade are, and it can say that competition will not let them go to waste; what it cannot always say is how they will be divided between the parties who create them. The division of the surplus is where the confident machinery of supply and demand meets a region it does not fully govern — the region of bargaining — and the story of how economists came to terms with that region is the story of this plate.
Strip the problem to its bones: one buyer and one seller, alone with each other, and a single thing to be traded. Suppose the good is worth ten to the buyer — that is the most he would give rather than go without — and that it costs the seller four to part with, that being the least he would accept. Then any price at all between four and ten leaves both of them better off than no trade: the buyer pockets the difference between ten and the price, the seller the difference between the price and four, and at every price in that range the two differences add to six, the whole surplus the trade creates. But how that six is split depends entirely on where in the range the price falls. At a price of five the buyer takes five and the seller one; at a price of nine the seller takes five and the buyer one; at seven they divide it evenly. The size of the pie is fixed at six regardless, but its division swings all the way from one party's near-total victory to the other's, and nothing in the bare fact of mutual benefit picks out a point within the range. With a single buyer and a single seller — the case the older economists called bilateral monopoly — the price, and with it the division, is simply indeterminate.
Francis Ysidro Edgeworth, in a strange and brilliant book of 1881 called Mathematical Psychics, gave this indeterminacy a geometry that economists have used ever since. Picture a rectangle whose every point represents one way of dividing two goods between two people — the more the one gets of each good, the less the other has, so that a single point fixes both parties' holdings at once. Each person has a family of indifference curves running through the box, the contours of his preferences, and the endowment they start from is one particular point within it. Through that point run two indifference curves, one for each trader, and the lens-shaped sliver caught between them is the set of allocations that both would prefer to where they began — the trades, that is, which make both better off. Bargaining drives the two into that lens; but where, within it, do they come to rest? Edgeworth's answer was the contract curve: the locus of points at which the two parties' indifference curves are tangent, where their valuations of the two goods have come exactly into line and no further trade can help the one without hurting the other. Anywhere off that curve, gains remain to be had, and rational traders will not stop there. So they must end on the contract curve — but the contract curve is not a point. It is a whole arc of efficient bargains, running from terms that heavily favour the one trader to terms that heavily favour the other, and which point along it the bargaining actually reaches, the geometry cannot say.
Bargaining will carry the two to the contract curve, for anywhere short of it some gain still lies unclaimed; but the curve is a whole stretch of points, not one, and which of them they reach — how the surplus is shared between them — the lines on the diagram leave entirely open.
Edgeworth did not leave the indeterminacy where he found it, and his way out is one of the most beautiful ideas in the subject. The trouble, he saw, was a trouble of small numbers — of having only two traders, free to haggle without alternative. So he asked what happens as the numbers grow. Add a second buyer and a second seller, then more of each, and something remarkable occurs: the range of outcomes that can survive bargaining begins to shrink. With more traders present, any proposed division that treats one party too generously can be undercut, for that party's counterpart can now defect to a rival offering better terms, and the standing threat of such defection narrows what any bargain can sustain. As the number of traders on each side rises toward the many, the arc of viable outcomes on the contract curve contracts steadily, and in the limit — the limit of a great crowd of buyers and sellers, no one of them individually significant — it collapses to a single point, and that point is exactly the competitive equilibrium, the price at which supply equals demand. This is Edgeworth's conjecture, made rigorous more than eighty years afterward by Debreu and Scarf as a theorem about the shrinking of the core. Its meaning is profound: the indeterminacy of bargaining is a disease of thin markets only, and competition is the cure. Where the traders are many, the market sets the price, and the division of the surplus along with it; the haggling has nowhere left to go.
But thin markets are everywhere — a single firm bargaining with a single union, a homeowner with the one builder in the valley, two nations negotiating a treaty — and for these the question of division does not dissolve into a crowd; it must be faced head on. The tool that faces it is the theory of games, which set out to do precisely what price theory could not, and predict the outcome of bargaining among the few. John Nash, in 1950, attacked the problem axiomatically: he asked what properties a reasonable division ought to have — that it waste no gains, that it treat symmetric parties symmetrically, that it not behave perversely under mere changes of scale — and proved that exactly one division satisfies them all, the one that maximises the product of the two parties' gains. Later theorists made the bargaining concrete, modelling it as a sequence of offers and counter-offers; Rubinstein showed in 1982 that when two parties alternate proposals, each impatient to settle, the division they reach is pinned down by their relative patience — the party who can better afford to wait extracts the larger share. The indeterminacy, in these treatments, is at last resolved; but notice what resolves it. It is no longer supply and demand but the parties' patience, their outside options, the credibility of their threats to walk away — the very elements of bargaining power that the theory of competitive markets had been constructed so as never to need.
What the market cannot settle, power settles; and the bargains of game theory turn on the things the price-taker never had to weigh — who can afford to wait, who has somewhere else to go, who may credibly walk away — so that the surplus is divided, in the end, by the strength of the parties and not the logic of the market.
Stand back, and a clean and slightly unsettling picture emerges of what the discipline can and cannot deliver. Where a market is competitive — many buyers, many sellers, easy alternatives on every side — the price is determined by the impersonal forces of supply and demand, and the division of the surplus falls out of it automatically; one can even read the shares from the slopes of the curves, the more inelastic side, the one with the poorer alternatives, ending with a particular portion of the gains and bearing a particular share of any tax that is laid on. Here economics is at its most powerful, deriving the division from the data of the market without ever once inquiring into the wills of the traders. But where a market is thin — where the parties are few and face one another without ready substitutes — that machinery falls silent, and the division becomes a matter of bargaining power: of who is patient and who is desperate, who is informed and who is bluffing, what each can credibly threaten and what each can be brought to believe. The surplus, the very quantity this volume has been measuring with such confidence, splits along a line that the theory of prices can locate only when the market is thick enough to locate it for them, and that in the hard and human cases is fixed by power rather than by price.
And the hard and human cases are not the exception but much of the substance of economic life. The wage a worker is paid is a division of the surplus that his employment creates, fought over between him and his employer; the rent a tenant pays divides the surplus of the tenancy; the terms a supplier wrings from a dominant buyer, the price two merging firms agree upon, the concessions struck in a trade negotiation between states — all are divisions of a surplus, and all are settled, to the degree the parties are few, by relative power rather than by market equilibrium. This places a quiet but serious limit on the reach of price theory, and it is worth stating plainly: the part of economics that is most rigorous and most determinate — the size of the gains, the competitive price — is very often not the part that people care about most, which is who ends up with the gains. Distribution, the question that drives so much of politics and so many of the bitterest disputes between classes and between nations, lives precisely in the region where the theory is least determinate, where it must hand the problem over to game theory, or to bargaining, or in the last resort to power. The surplus can be measured to the penny and still divided by force.
The volume has now watched the surplus come into being, in the gains a trade creates, and watched it parcelled out, by price where markets are thick and by power where they are thin. There remains a third thing that can happen to a surplus, and it is the darkest of the three, for it is neither creation nor division but destruction. A surplus that exists can be made not to exist — not transferred from one party to another, which leaves the total untouched, but annihilated outright, vanishing from the sum of human benefit as though it had never been possible. A tax, a monopoly, a quota, a tariff: each of them, by pushing the quantity traded away from the point where the gains are greatest, leaves some portion of the attainable surplus simply unrealised, a benefit that fails to occur and accrues to no one at all. Economists call this vanished benefit the deadweight loss, and it is the subject of the next plate. As the table says of every quantity, first to last: the argument remains open.