There was always something awkward about the way the early plates spoke of utility. They treated it as a quantity — so many units of satisfaction from this glass of water, so many from that, the units mounting up and falling off at the margin as neatly as pounds on a scale. The trouble is that no one has ever seen a unit of satisfaction, or found an instrument that measures one, or said with any clear meaning that the pleasure of a meal is precisely twice the pleasure of a song. The cardinal conception of utility, for all the elegant results it yielded, rested on a magnitude that could not be observed and perhaps did not exist, and this was a real embarrassment to a discipline with scientific ambitions. The achievement of the plate now before us was to show that the magnitude was never needed in the first place — that the whole apparatus of marginal choice could be rebuilt on a far humbler foundation, requiring of a person not that he measure his happiness but only that he rank his options, and that this humbler foundation gave back every result the grander one had claimed, and a few besides.
The instrument of this rebuilding is the indifference curve, and the idea behind it is disarmingly simple. Suppose a person is choosing between bundles of two goods — say food and clothing — in various combinations. Some bundles he will prefer to others; but among certain bundles he will be indifferent, content to have any one of them and caring not at all which, because each affords him the same satisfaction as the rest. Plot all the combinations of food and clothing among which he is thus indifferent, and they trace out a curve: a great deal of food with little clothing at one end, a great deal of clothing with little food at the other, and in between every blend he holds to be just as good. That curve is an indifference curve, and a person's whole structure of preference can be drawn as a family of them, curve stacked above curve, each higher one standing for a level of satisfaction he prefers to all the curves below it. The curves slope downward, since to give up some clothing and remain equally content he must be handed more food; they bow inward toward the origin, for reasons we shall come to; and no two of them ever cross, since a crossing would leave him equally satisfied by two bundles he has already ranked one above the other. Edgeworth drew the first of them in 1881; Pareto, a generation later, saw that they could carry the entire weight of the theory of choice.
Everything turns on the slope of the curve, and the slope has a name: the marginal rate of substitution. It is the rate at which the person will trade one good for the other while remaining on the same curve — the amount of clothing he will give up to get one more unit of food without becoming any better or worse off than before. And here the convexity of the curve, its inward bow, shows its meaning: as the person moves along the curve toward more food and less clothing, the rate at which he will trade clothing for food declines, because the more food he already has the less an additional unit of it is worth to him relative to the clothing he is now short of. The marginal rate of substitution diminishes as one slides down the curve — the same falling gradient that governed utility, now expressed not as a quantity of satisfaction but as a ratio of exchange. Indeed the marginal rate of substitution is exactly the ratio of the two goods' marginal utilities, the worth of one more unit of food set against the worth of one more unit of clothing; but where the marginal utilities are unobservable magnitudes, their ratio is a rate of trade-off that reveals itself plainly the moment the person actually chooses. The cardinal quantity has been replaced by an ordinal rate, and nothing essential has been lost.
The slope of the indifference curve asks not how much happiness a thing yields — a quantity no one can name — but only how much of one good a person will give up for another and feel neither better nor worse; and that he shows the world in every exchange he makes.
With the curves in hand, the consumer's problem solves itself in a single picture. Across the field of indifference curves runs the budget line, the boundary of what the person can afford: given his income and the prices of the two goods, it marks every bundle that exactly exhausts his money, and its slope is the ratio of the two prices. He wishes to reach the highest indifference curve he can — the most preferred bundle his budget allows — and the highest curve his budget can touch is the one to which the budget line is tangent. At that point of tangency the slope of the indifference curve equals the slope of the budget line; which is to say that the marginal rate of substitution equals the ratio of the prices. And this condition is no stranger: it is the equimarginal rule of the fourth plate, returned in ordinal dress. For to set the rate at which one will trade food for clothing equal to the rate at which the market trades them is precisely to equalize the satisfaction wrung from the last penny spent on each — the master rule of allocation, recovered in full, but recovered now without any appeal to a measurable utility, resting on nothing more than the slopes of curves that a person's own choices reveal. The cardinal crutch has been thrown away, and the optimum still stands.
This was, in the history of the subject, a rare and beautiful kind of progress: not the addition of a new result but the subtraction of an unneeded assumption, with every old result left standing. Pareto had first grasped that utility need only be ordinal, a matter of more and less rather than of how much; and in 1934 Hicks and Allen carried the insight through the whole of value theory, showing that demand curves, optima, and the response of consumption to every change in price and income could all be derived from the indifference map alone, the measurable util nowhere required. The reconstruction even yielded a tool the older theory had lacked. A change in the price of a good, Slutsky and Hicks showed, could be cleanly split into two effects: a substitution effect, the pure consequence of the altered rate of exchange, found by sliding along a single indifference curve to the point of the new tangency; and an income effect, the consequence of the change in real purchasing power, found by stepping from one indifference curve to another. The substitution effect always runs against the price — a good grown dearer is always, on this account, substituted away from — and in that fact lies the hard analytical core of the law of demand, a core the indifference-curve apparatus was the first to isolate cleanly.
It was a rare thing in the history of a science: an assumption quietly let go, the unmeasurable util sent away without ceremony, and not one of the results built upon it lost in the parting.
If Pareto and Hicks had freed the theory from the measurement of utility, Paul Samuelson asked, in 1938, whether it might be freed from utility altogether — and even from the indifference curve, which after all is itself a mental construct, a map of preferences that no one can directly observe. His proposal, revealed preference, began from the only thing about a consumer that truly is observable: the choices he actually makes. If a person, able to afford bundle B, is seen to choose bundle A instead, he has thereby revealed that he prefers A to B — not in any inner, unmeasurable sense, but in the plain operational sense that A is what he took when B was his for the taking. From a few axioms requiring only that such choices be consistent — that a person who picks A over B not, in similar circumstances, turn around and pick B over A — Samuelson and those who followed him were able to reconstruct the entire theory of demand, the downward-sloping curve and all, with no mention of utility or indifference anywhere in it. It was the ordinal revolution carried to its logical terminus: the theory of choice grounded at last in nothing but choice itself, the inner life of the chooser set aside as superfluous to a science of what people do.
The economy of assumptions was real, and it was bought, as such economies usually are, at a price that was not at first apparent. What ordinalism discarded along with the measurable util was the possibility of comparing one person's satisfaction with another's: if utility is mere ranking, with no unit and no zero, then there is simply no operation by which my gain can be weighed against your loss, and the comparisons on which any judgement about the distribution of welfare must rest lose their footing. Robbins pressed this point with great force, and the older welfare economics, which had blithely added and subtracted utilities across persons, never fully recovered from it; a later volume of this table, taking up the surplus that an exchange creates, will find the wound still open. And revealed preference, for all its rigour, rests on the assumption it makes explicit — that choices are consistent — which the behavioural economists have shown to be, in real human beings, frequently and systematically false: preferences reverse when the same options are described in different words, and the person who chose A over B yesterday may, framed differently, choose otherwise today. The apparatus is exact, but its exactness is conditional on a stability of preference that the creatures it describes do not reliably possess.
For all these qualifications, the ordinal turn stands as one of the genuine refinements in the history of economic thought. It took the margin — which had entered the subject as a measurable magnitude, a quantity of utility rising and falling — and distilled it to a pure rate of trade-off, a slope readable in the act of choosing, owing nothing to any unit of happiness and answerable to observation in a way the older quantity never was. The consumer's optimum, the demand curve, the very law of demand: all of it now rested on the humblest of foundations, a person's bare ability to choose. And yet, having followed the margin up to this height of refinement, we are obliged to notice how little, in the end, it asks of anyone who would actually use it. For beneath the indifference curves and the rates of substitution and the axioms of revealed choice, the entire teaching of the margin comes down to a single instruction — one so short it can be given in three words, and so powerful that those three words are very nearly the whole of what economics has to tell a person facing a decision. The next plate is given over to those three words. As the table says of every quantity, first to last: the argument remains open.