The Order — The Utility, Plate iii

The substance of the first plate had a fatal pair of properties: it could not be measured, and it did not need to be. Bentham's quantity of pleasure was posited as a real magnitude, addable within a person and, he hoped, across a crowd, and for a century the marginalists built the theory of price on the supposition that each additional unit of a good yields a definite, diminishing increment of it. But no one had ever produced the instrument that would read the increment off, and by the opening of the twentieth century a suspicion had hardened into a program: that the unmeasurable magnitude was also a superfluous one, that everything the theory of demand actually required could be had from something far thinner and far less metaphysical — the bare order in which a person ranks the things before her. This is the ordinalist revolution, and its claim is exactly that a great deal can be kept after the magnitude is thrown away. The plate's question is the one that revolution leaves on the table: if all we are ever entitled to is the ranking, what, precisely, was lost when the size of a want was discarded — and was it nothing, or was it more than the victors admitted?

The first to see that the magnitude was dispensable was Vilfredo Pareto, who realized that the whole apparatus of choice could be drawn using only curves of indifference — the loci of bundles a person regards as equally good — without ever assigning a number to the satisfaction those bundles supply. A map of such curves records which bundle is preferred to which, and nothing else; it knows the order of the contours but not their height. And Pareto's point was that the order is all the theory of demand consults. To predict what a person buys when prices and income are given, one needs to know only the shape of her indifference map and the budget she can reach; the cardinal labels, if they ever existed, do no work in the prediction and could be stripped off without the slightest loss to it. Utility, on this view, was never a quantity that explained choice. It was a name for the ranking that choice expresses, and a ranking is an ordinal thing — it tells you the sequence, not the spacing.¹

1. Pareto saw the apparatus could be drawn with curves of indifference alone — loci of bundles regarded as equally good — without assigning a number to the satisfaction they supply. Such a map records which bundle is preferred to which and nothing else: it knows the order of the contours but not their height. To predict what a person buys at given prices and income one needs only the shape of her map and her budget; the cardinal labels do no work and strip off without loss. Utility was never a quantity that explained choice, but a name for the ranking choice expresses.

What Pareto glimpsed, John Hicks and Roy Allen made into a thoroughgoing reconstruction. In a paper of 1934 pointedly titled a reconsideration of the theory of value, they showed that the central machinery of demand could be rebuilt with no cardinal magnitude anywhere in it, by replacing the suspect notion of marginal utility with one that asks only about the order: the marginal rate of substitution, the quantity of one good a person would just be willing to give up to obtain a little more of another while staying on the same indifference curve. This rate is a slope on the ordinal map; it can be read off the ranking directly, with no appeal to how much pleasure anything affords. And from it the whole familiar structure follows. The old law that marginal utility diminishes is replaced by the condition that the rate of substitution diminishes — that the indifference curves bow toward the origin — which is a statement purely about the shape of the order. The response of demand to a change in price, which Eugen Slutsky had already in 1915 decomposed by formula into a movement along an indifference curve and a movement to a new one, falls out in ordinal terms, the substitution part signed and certain, the income part isolated and named. Everything the cardinalist could derive, the ordinalist could derive; and the ordinalist derived it while assuming strictly less.²

2. Hicks and Allen (1934) rebuilt demand with no cardinal magnitude, replacing marginal utility with the marginal rate of substitution — the amount of one good a person would just give up for a little more of another while staying on the same curve, a slope read off the ranking directly. The law that marginal utility diminishes becomes the condition that this rate diminishes (the curves bow inward), a statement purely about the order. Slutsky's 1915 decomposition of demand into substitution and income effects falls out in ordinal terms. Everything the cardinalist derived, the ordinalist derived — assuming strictly less.

⁂

It is worth dwelling on how powerful an argument from parsimony this is, because it is what made the revolution irreversible. If two theories make identical predictions, and one of them rests on an entity that can never be observed while the other does not, the second is not merely tidier; it is, by the only standards a science recognizes, better. Cardinal utility had been carrying a metaphysical passenger — a private magnitude of feeling — and earning nothing for the fare. The ordinalist showed that the passenger could be put off at no cost to the journey, and once that was shown, keeping it on looked less like rigor than like sentiment. The triumph was real and it was deserved. Demand theory became a discipline that committed itself only to what could in principle be checked against behavior: a person's ranking, expressed in what she chooses, with no claim about an inner quantity that no experiment could reach. The ghost of the first plate had not been exorcised, exactly, but it had been made to do without a body. Only its outline remained — the order of its wants — and the order, the ordinalists insisted, was all anyone had ever really been using.³

3. The argument from parsimony made the revolution irreversible. If two theories predict identically and one rests on an entity that can never be observed while the other does not, the second is better by the only standards a science recognizes. Cardinal utility carried a metaphysical passenger — a private magnitude of feeling — and earned nothing for the fare; the ordinalist put it off at no cost to the journey. Demand theory became a discipline committed only to what could be checked against behavior: a ranking, expressed in choice, with no claim about an inner quantity no experiment could reach.

And yet the question of the plate is what was lost, and the honest answer is: not nothing. To keep only the order is to lose the spacing, and the spacing was carrying meaning the order cannot. A cardinal magnitude lets one say that the step from poverty to comfort is a greater gain than the step from comfort to luxury — not merely that each is a step up, but that one is larger; the ordinalist can rank the rungs but can no longer say that they are unevenly spaced, for evenness of spacing is a cardinal fact and there is no cardinal scale to carry it. Diminishing marginal utility, which had been a claim about a real and lessening increment of pleasure, survives only in the pale ordinal form of curves that bow the right way — a statement about shapes, not about a fading intensity actually felt. These losses might be borne; the discipline judged them a fair price. But beneath them lay a third that could not be shrugged off, and it is the hinge on which the rest of this volume turns. If a person's utility is only an order, private to her and calibrated to nothing outside her, then there is no scale on which my ranking and yours can be laid side by side. The magnitude, whatever its faults, had at least held out the hope of a common unit — Bentham's dream of adding satisfaction across a crowd. Ordinalism, in winning its rigor, abolished that hope at the root: two orders, each real, share no zero and no unit, and cannot be summed. The flagship of this volume is the working-out of that abolition; the ordinalist revolution is what made it inevitable.⁴

4. To keep only the order is to lose the spacing, which carried meaning the order cannot. A cardinal scale lets one say the step from poverty to comfort is a greater gain than comfort to luxury — not merely another step up, but a larger one; the ordinalist ranks the rungs but cannot say they are unevenly spaced. Diminishing marginal utility survives only as curves that bow the right way — shapes, not a fading intensity felt. And worst: if utility is only an order, private and calibrated to nothing outside, no scale lays my ranking beside yours. The magnitude had held out the hope of a common unit; ordinalism abolished it at the root.

So the revolution was a genuine advance bought at a genuine cost, and the cost was concealed by the very success that justified it. Because the ordinal theory predicted everything the cardinal theory had predicted, it was easy to conclude that nothing of substance had been given up — that the magnitude had been pure surplus, a wheel turning nothing. But prediction of market demand was never the whole of what the older quantity was wanted for. It was wanted, too, for the weighing of one person's gain against another's loss, for the judgment of whether a change made the world better and not merely different, for the entire province of welfare. In that province the lost spacing was not surplus at all; it was the thing itself. The ordinalists had not shown that intensity and comparability were meaningless, only that demand theory could proceed without them — and the discipline, hearing the first, quietly assumed the second. What was thrown away when the magnitude went was the ability to say how much, and how much is exactly what one needs in order to say better.⁵ The ranking is enough to explain a purchase. It is not enough to ground a politics, and the want had always been asked to do both.

5. The cost was concealed by the success that justified it: because the ordinal theory predicted everything the cardinal had, it seemed nothing of substance was given up. But the older quantity was wanted not only for market demand but for weighing one person's gain against another's, for judging whether a change made the world better and not merely different — the whole province of welfare. There the lost spacing was not surplus but the thing itself. The ranking is enough to explain a purchase; it is not enough to ground a politics, and the want was always asked to do both.