The Equilibrium

The marginal idea has been traced from a single good to a single choice — the value of the last unit, the laws of diminishing utility and rational allocation, the subjective source of value, the cost of the alternative forgone. Each plate has concerned one valuer, one good, one decision at the margin. Léon Walras, working in Lausanne in the same decade as Jevons and Menger and arriving independently at the same marginal insight, took the idea to its limit. He asked what happens when every valuer is equalizing margins across every good at once, when the cost of each thing is the alternative its making forgoes, and when the markets for all goods and all the means of producing them are linked, so that no single price can be understood in isolation. His answer was general equilibrium: the conception of the entire economy as one system in which the prices and quantities of all goods are determined together, simultaneously, at the margin. It is the grandest idea in economics, the flagship of the volume — and it is the idea that gives this whole series its name.

Walras's starting point was a fact the earlier marginalists had not fully confronted: that the markets of an economy are not separate but interlocked. The price of bread depends on the price of flour, which depends on the price of wheat, which depends on the price of the land and the labor that grow it — and the price of that labor depends, in turn, on the prices of all the goods the worker buys, bread among them. There is no end to the chain and no place to cut it, for every price is at once a cause and an effect of every other. To analyze a single market in isolation, holding all other prices fixed, is a useful approximation, but it is only that; in truth a disturbance anywhere ripples everywhere, and the price of any one good can be correctly understood only as part of the simultaneous determination of all prices together. Walras set himself the task the interdependence demanded: to determine all the prices of all the goods at once, within a single system.¹

His method was to write the economy as a system of equations. For every consumer, the marginal conditions of the earlier plates — utility maximized when the marginal utility per unit of money is equal across all goods — yield a set of demands, one for each good, depending on all the prices and on the consumer's income. For every producer, the conditions of cost yield a set of supplies, again depending on all the prices. And for every good, a market-clearing condition requires that the total quantity demanded equal the total supplied. Walras counted his equations and his unknowns — the unknowns being the prices — and found that, with one qualification, their numbers matched: there were exactly as many independent equations as there were prices to determine. From this equality he argued, informally, and not yet with the rigor a later generation would demand, that the system has in general a determinate solution — a single set of prices at which every market, all of them at once, stands in balance. The whole economy, he had shown, could in principle be solved.²

One qualification gave the system its characteristic form, and gave this series its name. Walras saw that his equations could determine only relative prices — how much bread exchanges for cloth, how much labor for bread — and not the absolute level of prices, for nothing in the logic of exchange fixes the scale. If every price doubled at once, every exchange ratio would be unchanged and every market would clear exactly as before; the absolute numbers are, in a sense, arbitrary. To pin them down, Walras chose one good as the unit of account, set its price equal to one, and expressed every other price in terms of it. This good he called the numéraire. It is not money in the full sense — it need do none of money's other work, neither storing value nor passing from hand to hand — but simply the measuring rod of the system, the single good in whose units all value is reckoned, so that to know a price is to know how much of the numéraire a thing commands. The device is more than a convenience. It expresses a deep truth of the marginal theory: that what exchange determines is never the absolute worth of things, which has no meaning, but only their worth relative to one another — and that to speak of value at all, one must first choose something against which to measure it.³

A system of equations may possess a solution, but a market does not solve equations; it gropes. Walras asked how an economy, with no central calculator, could actually arrive at the prices that clear all markets at once, and he answered with the process he called tâtonnement, a groping or feeling-out. Imagine, he said, that prices are called aloud — he later pictured an auctioneer crying them to the assembled market — and that at the announced prices each participant declares how much he would buy or sell. Where a good stands in excess demand, its price is raised; where in excess supply, lowered; and the adjusted prices are called again, and again, with no exchange permitted until a set of prices is reached at which every market clears. The market, in this picture, computes the equilibrium it cannot calculate, by trial and correction, each round bringing the called prices nearer to the ones that balance all markets together. Whether a real economy converges as neatly as the parable suggests is a question that has occupied economists ever since; but the parable captures something true — that a price system is a vast, decentralized procedure for finding, without anyone intending it, the numbers at which all the plans of all the agents can be made mutually consistent.⁴

What is achieved when the groping comes to rest is worth pausing over, for it is the marginal idea in its fullest flower. At the equilibrium prices, every consumer has equalized the marginal utility per unit of money across all the goods he buys; every producer has pushed each activity to the margin where its cost is just covered; and every market clears, the quantity each good's makers wish to sell exactly meeting the quantity its users wish to buy. The marginal conditions of all the earlier plates hold at once, throughout the whole economy, knit together by the single web of prices. And there is more: such an equilibrium, a later generation would show with precision, is in a definite sense efficient — no one can be made better off without another being made worse off, for all the gains from reshuffling goods and resources have been exhausted at the margin. Here, made exact, is the old intuition of the invisible hand met in the founding volumes: that a multitude of individuals, each pursuing only his own advantage at his own margin, and coordinated by nothing but prices, can be led to an outcome coherent for the whole and wasteful of nothing. Walras had given the intuition a mathematical body.⁵

Walras's argument had a gap, and the closing of it is part of the plate's story. To count equations and unknowns and find their numbers equal is not to prove that a solution exists, still less that it is unique or that the groping would reach it; a system of equations can have the right count and yet no economically meaningful solution at all. Walras knew his demonstration was incomplete, and for half a century the existence of a general equilibrium remained an article of faith rather than a theorem. It was settled only in the middle of the twentieth century, when economists armed with new mathematics — the topology of fixed points — proved rigorously that, under stated conditions on tastes and technology, a set of equilibrium prices does exist. The proof vindicated Walras's vision and became one of the central results of modern economic theory; but it also revealed how demanding the conditions are, and how far a real economy, with its increasing returns and its missing markets and its imperfect competition, may stand from the world in which the theorem holds. The vision was sound; its domain was narrower than its author had hoped.⁶

With general equilibrium the volume can answer the question the author set it — why the last unit, and not the whole, sets the price of everything — and answer it now at the scale of an entire economy. The price of every good is set at the margin: by the marginal utility it yields to those who buy it, and the marginal cost, itself an opportunity forgone, of producing it. But because all markets are interlocked, these margins cannot be settled one good at a time; they are settled all at once, in a single system, each price adjusting until every margin in the economy is brought into balance with every other. The last unit sets the price of everything not in isolation but in concert: the marginal glass of water, the marginal hour of labor, the marginal loaf of bread, each valued at its margin, and all these margins reconciled simultaneously, through the web of prices, into one coherent equilibrium. The whole economy, in Walras's vision, is a single immense exercise in marginal valuation, every price a marginal price, all of them determined together. The margin does not govern one market; it governs the system entire.⁷

General equilibrium became the framework within which much of modern economics is conducted — the master model against which questions of efficiency, of taxation, of trade, of the gains and losses from any disturbance are posed, for only a general reckoning can trace how a change in one market propagates through all the others. Yet the vision is an idealization, and an austere one. It assumes that all markets exist and clear, that competition is perfect, that agents possess the information and the rationality to find their best positions, that the groping converges; and where these assumptions fail — as they do wherever there is monopoly power, met in the volume of power, or missing markets and hidden information, met in the volumes of the externality and the asymmetry, or the strategic interdependence met in the volume of the game — the smooth determinacy of the Walrasian system gives way to the frictions and failures that occupy much of the rest of this table. General equilibrium is thus at once the high point of the marginal idea and the benchmark of its incompleteness: it shows what a perfectly marginal economy would look like, and in the very perfection of the picture it marks off, by contrast, all the ways the real economy departs from it. The model is a measuring rod for the world's imperfections as much as a description of its order.⁸

Why does the last unit, and not the whole, set the price of everything? Because the economy is one system, and in it every price is a marginal price, settled together with all the rest. Léon Walras took the marginal idea — that value lies in the last unit, that cost is the alternative forgone — and showed that when every agent reckons at his own margin and every market is linked to every other, the prices of all goods are determined simultaneously, in a single equilibrium that clears every market at once and wastes nothing at the margin. He gave the invisible hand a mathematical form, conceived the groping by which a price system finds, without calculation, the numbers that reconcile every plan, and chose the one good against which all the others are measured — the numéraire that gives this series its name. The marginal revolution, which began by dissolving the paradox of a cheap glass of water, ends in the vision of the whole economy as a single act of marginal valuation, every price the price of one more, all of them reckoned together. So the quantities close on the idea that remade them: that to understand a price — any price, every price — one must look not to the total or the average, not to the labor sunk or the cost already incurred, but to the margin, the value of one more, weighed against the one more of everything else it might have been.