The bridge is either built or absent. The formula may divide its span into a thousand increments, but half a bridge does not carry half the traffic across the river. The last loaf presents another difficulty. A person who needs it to stay alive may offer less than a well-fed person offers for a delicacy, because the first person's purse is nearly empty. Neither case makes comparison useless. Each asks what is being compared, which alternatives can actually be chosen, and what a monetary answer is capable of telling us. The margin reaches its edge when a good question about another unit is made to stand in for all three.
The ordinary marginal rule has earned its authority. If another feasible unit yields more benefit than cost, stopping too soon leaves an opportunity unused. If it costs more than it yields, going further destroys a gain that stopping would preserve. But the equality between marginal benefit and marginal cost describes a particular kind of optimum: an interior stopping point in a sufficiently regular problem, with conditions making that point a maximum. At a boundary, an inequality may be the right condition. With several local peaks, a zero slope does not identify the highest. Where the choice comes in lumps, the relevant comparison may be between complete feasible alternatives. The discipline is to examine the consequence of changing the decision; the derivative is one powerful way of doing that when the change admits it.
Arrow's account of competitive efficiency makes room for such distinctions. At a corner, the appropriate price conditions need not take the form of an equality between every pair of derivatives. The role of convexity in finding prices that support efficient allocations is another question again. It is a mistake to roll the different results into a single promise that every optimum is smooth, every market has an equilibrium, and every efficient arrangement can be reached by the same price adjustment. The assumptions belong to the conclusions individually. A limit on one theorem does not erase every useful comparison that economics can make.
Consider an illustrative crossing that costs sixty to establish and two for each use. There are ten potential users, each willing to pay ten for one crossing, and no other benefits or harms. The fixed cost is avoidable before construction. Serving all ten yields benefits of one hundred against costs of eighty, leaving twenty. Charging the marginal cost of two admits every use worth making once the crossing exists, but its twenty of receipts pay only the operating costs. The establishment cost remains sixty short. Charging eight would cover the whole cost with these stipulated users, all of whom still prefer to cross. The operating margin and the financing requirement can be reconciled here because the example has supplied a favorable distribution of willingness to pay. That is a fact about the case, not a theorem about every bridge.
Now leave the technology unchanged and reduce the number of prospective users to five. Each crossing still has a benefit of ten and a marginal operating cost of two. Yet the whole project yields fifty against a cost of seventy. Before construction it should be rejected under the example's net-benefit criterion, despite the attractiveness of every additional use considered in isolation. If the bridge has already been built and its establishment cost cannot be recovered, admitting those five users produces a further forty of benefit net of operating costs. Building it and using it are different decisions. The previous plate's treatment of The Sunk supplies the distinction: the relevant cost depends on what can still be changed.
This fixed-cost example also keeps two often-confused slopes apart. Average cost falls as the establishment cost is spread over more users; the marginal operating cost remains two throughout. Declining average cost does not require a declining marginal cost. A single facility can cost less than duplicate facilities because duplication repeats the establishment expense, even while the cost of each further use stays unchanged. Joskow's account of natural monopoly sets out these distinctions through the costs of serving a specified output. What follows for ownership, entry or regulation requires additional questions about information, demand, incentives and the alternatives. A cost advantage alone cannot answer them all.
An idea pushes the financing question further. Once a design exists, one person's use need not prevent another from using the same design. The labor required to discover it may nevertheless be substantial, and applying it can require rival materials, equipment and attention. Romer's account of technological change separates that nonrival element from the resources used alongside it. In his model, producing new designs and using existing ones cannot be coordinated by simply paying every input its marginal product under ordinary price-taking conditions. The economics must account for how creation is financed as well as how the resulting idea is used. This does not mean that every new idea creates a monopoly, or that public support, prizes and other arrangements are conceptually excluded. It means that a negligible additional use cost is an incomplete account of the undertaking.
The indivisible project similarly asks for a comparison that respects its actual shape. Suppose a public body can choose one of two complete works, with ten available to spend. Work A costs eight and yields twelve; work B costs five and yields eight. Each can be undertaken at most once, and together they exceed the budget. B has the higher ratio of benefit to cost, but A yields the higher net benefit: four rather than three. A is therefore preferred under that criterion. There is no divisible strip of B that can be combined with a fraction of A to implement the ratio ranking. Listing the feasible packages solves the small problem cleanly. No derivative has failed; the decision never offered the infinitesimal rearrangement that the derivative would describe.
The other boundary is distributive. Suppose a scarce unit of clean water can serve a person's basic need, but that person has only three to spend. Another person can offer twenty to put the same unit to a decorative use. If the seller assigns it to the highest bidder, the second use wins. That establishes the ranking of these bids under these budgets. It does not establish that the second person's welfare gain is greater, or that the first person's need is less important. The offer is a relation among preference, resources and the available alternatives. Treating it as a direct common measure of human urgency adds an interpretation the transaction has not supplied.
The opening Paradox explained why plentiful water can command little even though life depends on water as a whole. This final case alters the conditions: the unit is scarce, the need urgent, and the capacity to bid unequal. The explanation of a price remains a useful explanation. It may reveal exactly why a person is excluded. Whether that exclusion should stand is a question about rights, distribution and the standards by which access is judged. A transfer of purchasing power changes the budget entering the choice; a direct allocation changes the rule. Neither creates extra water by itself. The resource constraint survives the choice of a fairer arrangement, and marginal reasoning can help identify its consequences once the social objective has been stated.
The same caution applies to adding monetary gains. Ten dollars of willingness to pay is ten dollars in an account of willingness to pay; it need not be the same change in well-being for every person. A total can be valuable for comparing resource uses while remaining insufficient to decide who ought to gain and who ought to bear the cost. The welfare theorems accordingly distinguish efficiency from the selection among efficient allocations. They do not say that every market outcome is efficient, nor that an efficient allocation is fair. Information, external effects and market power can disturb the first claim; the distribution of rights and resources remains relevant even where those difficulties have been set aside.
The margin can price the next use without deciding whose need should come first.
The answer it gives is therefore worth having, provided the question is kept visible. It can expose the cost of an additional use, the alternative sacrificed by a choice, and the assumptions needed to make a local equality govern a whole undertaking. It can also show where a budget, a threshold or an indivisible object prevents the proposed adjustment. What it cannot do alone is turn the purchasing power behind a preference into a verdict on that preference's moral weight. The Surplus now gathers the gains left around the price. This volume leaves it both an instrument and a warning: count the changes carefully, name the feasible alternatives, and do not ask a measured gain to conceal the judgment by which it is valued. The argument remains open.