Grant the preceding plates everything they win. Grant the ordinalist his victory — that within a single person only the ranking of wants is real, and the magnitude an illusion we can do without. Grant the behaviorist his — that a preference is nothing over and above the consistent choices that reveal it. A person, on these terms, possesses a utility that is genuine but thin: a function that represents how he ranks the world, fixed only up to any transformation that preserves the order, and, where his choices among risks are concerned, fixed up to the choice of an origin and a scale.1 Each person carries such a function. Now put to the theory the one question for whose sake welfare economics exists at all, the question every tax, every transfer, every public decision silently answers: is the dollar worth more to the man who has few than to the man who has many? Is the gain to the one who wins this policy larger than the loss to the one who bears it? To answer, you must set my utility beside yours and read off which is greater. And there is no operation in the whole of economics that performs the reading.
The reason is not a shortage of cleverness. It is a fact about what the quantity has become. Each person's utility is settled only up to a free choice of zero and of unit; nothing in the choices it was distilled from says where my zero sits relative to yours, or how long my unit is against yours. To add the two is therefore to add quantities with no common scale — and that is not a hard sum but a meaningless one, like adding my Tuesday to your kilogram and reporting the total.2 The phrase total utility of the community, which Bentham wrote down without a tremor, turns out under examination to be not a large quantity difficult to measure but a string of words with no determinate referent at all. The scandal of this plate is that the most important number in welfare economics may not be a number.
It was not always felt as a scandal, because for a long while economists simply helped themselves to the comparison. The classical utilitarians compared freely: Bentham counted heads, and Edgeworth and Pigou built an entire old welfare economics on a pair of assumptions that, taken together, licensed the most consequential conclusion the comparison has ever yielded. Assume that people are broadly similar in their capacity to turn income into satisfaction; assume that the satisfaction yielded by income diminishes at the margin as income grows — that the hundredth thousand means less to its owner than the first thousand means to its. It follows at once that moving a dollar from a rich man to a poor one raises the total of satisfaction, because the dollar buys less want where it leaves than where it lands. Redistribution, on this reasoning, is not a sacrifice of efficiency to compassion; it is the efficient course, the one that maximizes the very sum the science exists to maximize.3 That argument, for half a century, was the intellectual engine of the case for the welfare state.
Robbins withdrew the licence. In a short and devastating essay, he argued that comparisons of satisfaction between persons are not the kind of thing economic science can establish, because there is no observation that could confirm or refute one. That my loss from a tax exceeds your gain from the spending is not a finding about the world; it is a verdict I have reached on grounds that lie outside the world the economist can see. Such a verdict may be wise or foolish, humane or cruel, but it is an ethical judgment, and to present it as a scientific result is to smuggle one's values into the books under the cover of arithmetic.4 Robbins did not deny that we make such judgments, or that we must; he denied only that economics, as a positive science, has any authority to make them. The effect was to cut the engine of the old welfare economics out of the machine.
What was left, once the comparison was forbidden? Only the one ranking that needs no comparison at all. To say that a change makes someone better off and no one worse off requires no weighing of one person's gain against another's loss; it is a statement entirely within each person's own ranking. This is the criterion of Pareto, and the new welfare economics retreated to it as to high ground. But the high ground is very nearly barren. Almost no real policy makes no one worse off; every tax, every tariff, every reform creates losers as well as winners, and on losers as well as winners the Pareto criterion is simply silent. It cannot rank the overwhelming majority of the choices a society actually faces, because it refuses on principle to do the one thing — compare a gain with a loss — that ranking them requires. Its rigor is purchased at the price of its relevance: it is unimpeachable and almost useless in the same breath.
The attempts to extend it only sharpened the difficulty. The compensation tests proposed that a change be counted an improvement if the winners could compensate the losers and still come out ahead — whether or not they actually did. The hope was to recover a usable criterion without an interpersonal comparison; the result was incoherence. Scitovsky showed that the test could pronounce a move from one state to a second an improvement, and then pronounce the move back from the second to the first an improvement as well — an approval running in a circle, which is to say no approval at all.5 The more honest response was to stop pretending. Bergson and Samuelson proposed that the economist simply write the society's value judgments down, explicitly, as a social welfare function, and reason from there. This is candor, and it is the right move; but it must be seen for what it is. It does not derive the interpersonal comparison; it assumes it, in the very shape of the function chosen, and then keeps the books consistently. The weighing of person against person is not performed by the science. It is performed by whoever writes the function, and the science merely works out the consequences.6
Could the comparison be earned rather than assumed? The most beautiful attempt is Harsanyi's. Imagine, he proposed, an impartial observer who must rank social states without knowing which person in them he will turn out to be, and who treats each identity as equally likely. Such an observer, if he is rational in his choice under this manufactured risk, will rank the states by the average of the utilities in them — and so a form of utilitarianism, the very summing Robbins forbade, falls out of nothing but rationality and impartiality. It is a remarkable argument, and it does real work. But it does not conjure the comparison from nothing. The assumption that the observer might with equal probability be any person already supposes that the persons' utilities are laid on a common scale, comparable enough to be averaged; the comparability that seemed to emerge at the end was placed at the beginning, in the structure of the lottery over identities.7 Sen's response is gentler and, in the end, more durable: perhaps the choice is not between full comparability and none. Perhaps we can compare partially — say with confidence that the starving man gains more from bread than the sated man loses, without pretending to a precise ratio of their utilities — and let whatever rankings survive that partial comparison be all we claim. It is less than Bentham wanted and more than Robbins allowed, and it may be the most that honesty permits.8
The temptation, having come this far, is to file the impossibility of comparison as a technical limitation — a measurement problem awaiting a better technique, perhaps the instrument of the tenth plate. That reading is a mistake, and naming the mistake is the work of this plate. The impossibility is not a gap in our measurements. It is a consequence of what utility was decided to be. When the discipline emptied utility of its psychology and rebuilt it as a bare summary of one agent's choices, it discarded precisely the thing a comparison would need: some common substance, present in you and in me, of which each of us might have more or less. A representation of my choosing contains nothing that reaches across the gap to your choosing, because choosing is something each of us does alone, and the representation was built to capture nothing else. The information required to compare us was not lost in transmission; it was thrown away on purpose, as the price of rigor, in the plates that came before. To compare us now, one must put that information back by hand — must add, from outside the positive theory, an ethical premise about how much your hunger counts against my comfort. There is no avoiding the premise. The only choice is whether to state it or to hide it.
From which the deepest consequence follows, and it is not technical but political. Welfare economics is not, and never was, value-free. Its long retreat into Pareto efficiency and potential compensation was not an escape from ethics but a particular ethics held silently — for a criterion that can bless any change making the rich richer and no one poorer, while it stands mute before every change that would take from the rich to feed the poor, is not neutral between the contented and the desperate. Its neutrality leans, and it leans toward what already is. The refusal to compare, dressed as scientific scruple, turns out to carry a thumb on the scale. The next plate will show that this is no accident of attitude but a matter of logic: that the refusal to compare utilities, joined to a few conditions that each seem beyond reproach, makes coherent social choice not merely difficult but impossible, and hands the whole of it, in the limit, to a single will. The comparison we cannot make is the one a just society most needs to make. The number does not exist. The judgment is unavoidable. Honesty begins in confessing that we are making the judgment — and have been all along — rather than reading it off the world.