A sale can fail without anybody losing a coin. Imagine a book worth ten pounds to its prospective reader and six pounds to the person who would supply it. Somewhere between those figures is a price both could accept, leaving four pounds of gain to divide. Now put a five-pound charge between what the reader hands over and what the supplier receives. No price can give the supplier six while keeping the reader’s payment at ten. The sale disappears, the collector receives nothing, and the four pounds of possible gain never belongs to either party. This is the loss the accounts cannot record: the benefit of an exchange which leaves no receipt because it never occurs.
The missing gain belongs to a comparison between feasible alternatives. It is not the book’s entire value, because supplying the book would have cost something; nor is it the tax printed on a transaction that never took place. It is what the reader would have gained above the supplier’s opportunity cost. Who would have received which part depends on the price they might have agreed. A blocked bargain destroys that available sum without determining its division. To call the parties robbed is to describe a foregone opportunity, not to establish that either already owned a particular amount or that the bargain could have been completed without any further cost.
Dupuit’s account of tolls already included the people who turned away without paying. A toll could transfer some of the benefit of crossing a bridge from the traveller to the collector, while extinguishing other journeys whose value fell short of the charge. The second loss could not be discovered from the receipts alone. People who continued to cross surrendered money; people deterred by the toll surrendered the advantage of crossing. The two groups left different traces, and a ledger confined to the collector missed the latter altogether. Deadweight loss gives a name to the gain which neither side nor the treasury receives.
A small invented market lets the two amounts be separated exactly. Suppose willingness to pay for the next unit is twenty pounds minus the number of units, while its marginal cost is four pounds plus that number. Competitive trade reaches eight units at a price of twelve. The buyers’ surplus is thirty-two pounds, the sellers’ surplus another thirty-two. Impose a tax of four pounds per unit and trade contracts to six units: buyers pay fourteen, sellers retain ten, and the treasury receives twenty-four. Buyers and sellers now have eighteen pounds of surplus apiece. Their combined loss is twenty-eight pounds; twenty-four has changed hands and four has vanished. The tax receipt and the deadweight loss are different entries.
The lost four pounds can be found without inspecting anybody’s payment. On the units between six and eight, buyers’ willingness to pay still exceeds marginal cost. The gap starts at four pounds and narrows to zero; its area is half a base of two times a height of four. The four pounds describes the value of those units above the resources required to supply them. Those resources remain available for other uses when output contracts, and their opportunity cost has already been deducted. Counting all the cancelled expenditure as a loss would ignore that saving. Counting the treasury’s twenty-four as destroyed would ignore its new owner.
Monopoly gives the same distinction a different destination for the money. A seller who restricts output to raise the price can take more from the buyers who remain while preventing other purchases that would have been worth their cost. The extra receipts on continuing sales are a transfer to the seller; the gain on the excluded sales is foregone. Arnold Harberger made this allocative loss the subject of an empirical study of American manufacturing, using 1924–1928 profit data and assumptions about costs, normal returns and demand. His 1954 estimate was strikingly small, but its object was specific: the resource misallocation his method could measure in that historical setting. A small estimate of one cost does not make all the other consequences of power disappear.
The familiar rule that the loss grows with the square of the tax comes from the way two margins move together. In the invented market, a tax of eight pounds reduces trade by four units instead of two and opens a gap of eight pounds instead of four. The loss becomes sixteen pounds, four times the previous loss. The rule is exact for these straight curves over this range. Near an undistorted competitive position, it also explains why the loss from a small wedge can initially be small: the first trades displaced have benefits only slightly above their costs. But increasing a wedge that already exists starts from a different place. Moving our tax from four to five pounds raises the loss from four to six and a quarter. That extra pound sacrifices trades whose gains were already appreciably above cost.
The missing trade leaves no payment to follow, only a gain that the remaining receipts cannot explain.
The resources spent acquiring a transfer create a further cost outside the elementary triangle. A licence may promise its holder an income without requiring an equivalent addition to production; rivals can then spend time, expertise and money trying to obtain or defend it. Tullock’s criticism of the narrow welfare calculation followed those efforts. The transfer itself and the resources devoted to securing it must be distinguished, including the efforts of unsuccessful contestants. This does not license counting every pound of a monopoly profit a second time as a social loss. How much is dissipated, whether an activity produces useful information, and what the resources would otherwise have done are additional questions. The prize makes waste possible; its face value does not establish the amount.
The sign of the loss depends on whose costs have entered the starting picture. Return to the trade worth ten pounds to the buyer and costing six to the supplier, but suppose it also imposes seven pounds of uncompensated damage on a neighbour. The private gain is four; the gain after that damage is counted is minus three. Preventing the trade now avoids a net loss. Pigou’s distinction between private and social net product makes room for precisely this discrepancy: the contracting parties can benefit while costs fall on people outside their bargain. A corrective charge can improve the allocation when it brings that damage into the decision. The quantity of trade by itself gives no verdict.
The use of the revenue must also enter a decision about whether a tax is worth levying. The elementary account treats the money received by the treasury as retained purchasing power. A hospital, a transfer or a wasteful project will put that purchasing power to different uses, requiring a further comparison of benefits and real costs. Likewise, the burdens of administering the tax and complying with it are resources to be counted where they arise. A triangle measuring the disturbance in one market cannot by itself judge the whole fiscal arrangement. It remains useful because it separates a particular cost from the transfer and from what is done with the proceeds.
Distribution gives the unmade sale its human meaning even after the net loss has been measured. The lost opportunity may have mattered most to a buyer with little money, to a worker whose services would have supplied the product, or to an owner whose alternative use was poor. The unweighted sum records their monetary gains under specified circumstances; it does not decide how much each loss should matter in a common judgment. Nor does a policy with a positive total gain ensure that its losers are compensated. The earlier plates’ difficulties of measurement and division survive inside this apparently simple figure.
The person deprived by a blocked trade is whoever would have shared its net benefit, but the benefit must first be shown to have been available. Once genuine costs, outside effects and feasible alternatives are included, the missing gain can be described without inventing a recipient for it. That is the power of deadweight loss: it makes an absence answerable to an account. It is also the limit of the account. Knowing which gains have been left unrealised leads to the next question—whether a condition in which no further mutual gain is available deserves to be called the best.