The argument for equality appears to have been waiting inside the argument for value. The first loaf matters greatly; the hundredth matters little. Give a loaf from the abundance of one table to the emptiness of another, and the gain should exceed the loss. What began as an explanation of the falling value of successive units seems to become an instruction for distributing them. Diminishing marginal utility supplies the motion; redistribution supplies its direction; equality promises the point at which no further transfer can enlarge the sum. The mathematics is compelling. The question is which part of the conclusion the mathematics has proved, and which part it was asked to assume.
Start with a world simple enough for the argument to be exact. Two people share one hundred units of a divisible consumption good; both receive a positive amount; nothing is produced or destroyed by redistribution. Suppose their well-being is measured on a common scale by the natural logarithm of consumption, and suppose equal increments on that scale receive equal social weight. The objective is then log c₁ + log c₂, with c₁ + c₂ = 100. Because the logarithm increases at a diminishing rate, a unit taken from the larger holding and given to the smaller can increase the total. Moving from twenty and eighty to thirty and seventy raises it by about 0.272. At fifty each, the marginal gain from another unit is the same for both, and the sum reaches its maximum. Within this invented world, the conclusion is a theorem rather than a hope.
Common units, identical functions, equal weights, and a fixed resource total each help produce the equality result. Diminishing utility says that another unit means less as a particular person has more. The common scale says how an increment for one person is compared with an increment for the other. Identical functions say that, at the same consumption, the increments are equal. Equal social weights say that a comparable gain receives the same importance whoever receives it. The fixed total and costless transfer say that the act of redistribution does not change the resources available. Remove one premise and the equal division need no longer be the maximizing division. This does not discredit the result. It identifies its content. The proof is useful precisely because it shows where agreement must begin and where disagreement can alter the answer.
Arthur Pigou gave the redistribution argument a more qualified form than its familiar slogan suggests. In The Economics of Welfare, his case for a transfer from a relatively rich person to a relatively poor one considered people of similar temperament, the goods and services they consumed, and the effect on the aggregate dividend. Income received was not automatically income enjoyed: a change in saving could affect how much of a transfer became present consumption. Nor was a transfer that shrank the dividend interchangeable with a costless movement of claims over a fixed stock. His argument compared satisfactions under stated conditions; it did not make every reduction in an inequality statistic sufficient evidence that welfare had risen.
Francis Edgeworth's earlier pursuit of the greatest sum of pleasure made equality conditional on equal capacities for pleasure. His equality result applied to people assumed equally capable of pleasure; when capacities differed, his construction generally assigned more means to those he took to have greater capacity. This is the uncomfortable feature of a sum: it follows the assumed productivity of resources in producing its chosen good, wherever that productivity is placed. Edgeworth made interpersonal commensurability and differences of capacity explicit parts of the apparatus. The equations did not establish the psychological classifications that fed them. Their significance here is that diminishing utility was compatible with an unequal result from the beginning.
The same dependence appears in a small alteration of our example. Keep one hundred units, but maximize 2 log c₁ + log c₂. Both terms still rise at diminishing rates. The maximum now gives the first person two-thirds of the resources and the second one-third: roughly 66.67 and 33.33. The factor of two could represent a judgment that the first person's gain deserves twice the social weight. Alternatively, with equal social weights, it could represent an asserted difference in the conversion of consumption into comparable well-being. Those are different claims, and each would need its own defense. Neither follows from the observation that both people dislike losing resources. The curvature has remained diminishing while the equal allocation has ceased to maximize the sum.
Nor can the coefficient be recovered by choosing a more convenient ordinal scale. Multiplying one person's utility by two preserves that person's rankings. If the original sum expressed a fixed social judgment, the coefficient attached to the rescaled utility would have to change inversely to preserve that judgment. The previous plate on the gulf showed how a total can otherwise reverse without any change in anyone's choice. Concavity itself requires care: an increasing transformation can preserve an ordinal ranking while changing the curvature of its representation. A diminishing curve used to support redistribution therefore needs an interpretation beyond mere rank. The claim concerns the size of a benefit, not just the order in which someone chooses bundles.
Costs can change the answer even after the comparison has been agreed. Return to the equal log functions and the original twenty–eighty division. Suppose a transfer takes ten units from the second person but delivers only six to the first; four are lost in carrying it out. The result, twenty-six and seventy, still has a larger utility sum than twenty and eighty. If the same ten-unit loss delivered only two, the result, twenty-two and seventy, would have a smaller sum. A cost is therefore relevant without being an automatic veto. The question is its size and incidence in the particular comparison. An appeal to incentives must likewise specify what behavior changes, how the resources change with it, and who bears the consequence. Naming an effect does not measure it; ignoring it does not make it disappear.
There is a further question that no correction to the transfer cost will answer: why should the greatest sum be the whole of justice? Equal shares of a resource can leave unequal opportunities where needs differ; equal reported contentment can conceal a life adjusted to deprivation. A person who needs costly assistance to move freely may require more resources to reach an opportunity another enjoys with less. Conversely, a claim grounded in equal standing need not wait for proof that its holder will turn each dollar into more pleasure. Sen's work on the information used in social judgment makes room for these distinctions. A welfare calculation can illuminate them, but choosing what enters that calculation remains part of the judgment. Equality of resources, equality of welfare, and equality of practical freedom name different things to be made equal.
The force of the levelling argument survives this accounting of its premises. Where comparable benefits really diminish, where the worse-off gain more from the transfer than the better-off lose, and where the relevant consequences have been included, redistribution can increase the specified social sum. That is a serious argument. It becomes weaker when presented as something requiring no interpersonal or ethical premise at all. The appropriate answer to a challenged premise is evidence or a reason for adopting it. Calling the conclusion scientific cannot supply the missing reason; calling the premise ethical cannot dispose of it.
Arithmetic can carry a judgment faithfully. It cannot choose what deserves to be equal.
Justice does not follow from diminishing utility alone. Under explicit comparisons and weights, an equal allocation can follow exactly; under other comparisons and weights, an unequal one can follow just as exactly. The mathematics has returned the implications of the world and objective supplied to it. Its proper service is to make those implications visible, exposing both a neglected gain from sharing and an unsupported claim that the calculation itself ordained the shares. The next plate attempts an even greater addition: to combine not two calibrated satisfactions but the rankings of many people into a coherent choice for them all.