The price of the book is visible; the amount its reader would have paid is hidden. The shop records three pounds, though the reader might have paid five, and the two pounds of gain appear in neither the till nor the accounts. The preceding plates have followed that unrecorded gain into the bargain and its division. To add it across a market, economics gives it a shape: the space beneath the demand curve and above the price, narrowing as the buyers’ willingness to pay approaches what they actually pay. With a straight demand curve the space is a triangle. Its clean edges can make the concealed information look as though it has already been found.
The geometry began with a practical difficulty: a public work could be useful without collecting anything like the value it created. In 1844 Jules Dupuit asked how an engineer should value roads, canals and bridges. The receipts could not answer him. A free bridge might be valuable; a costly bridge might earn substantial tolls while offering little improvement over the crossing already available. He therefore followed the journeys that people would still make as a hypothetical toll rose, adding what each journey was worth before deducting what it cost. The stopping points disclosed something the observed traffic count concealed: how much of the traffic depended on the crossing being cheap. The method made the absent transaction part of the evidence about the transactions that occurred.
Marshall made this surplus a familiar part of the theory of demand, while keeping a warning beside the measure. The buyer’s gain was the excess of what he would have paid over what he had to pay; successive units could be valued differently even when bought at the same price. A household would give much for its first supply of tea and less for further supplies, yet pay the market price for all of them. Adding the differences gave the surplus. But the money with which the differences were measured had other uses, and its importance depended on the household’s means. Marshall’s discussion of the measure includes both the change in the value of money to one person and the difficulty of comparing it across people. The triangle came into economics with its qualifications attached.
A triangle can be exact inside an economy constructed to fit it. Imagine a buyer whose willingness to pay for the next unit starts at twelve pounds and falls by one pound for each additional unit, with no income effect over the purchases in question. At a price of four pounds he buys eight units. His total willingness to pay for those units is sixty-four pounds, his payment thirty-two, and his surplus thirty-two: half a base of eight multiplied by a height of eight. Raise the price to six and he buys six units, leaving a surplus of eighteen. The benefit of bringing the price back down is fourteen pounds. It is the difference between two areas, including a saving on units he would have bought anyway and a gain on the additional units he now buys. Measuring a change requires its starting point.
The arithmetic does not tell an observer which demand curve belongs in the picture. One purchase at four pounds is compatible with many valuations of the units before it and many responses to a different price. Even a sequence of purchases requires interpretation. Income may have changed, a competing shop may have opened, or the quality of the book may differ; quantities observed at different prices need not trace one unchanged demand schedule. The useful question is therefore often local: how much would purchases change under this price change, with the other relevant circumstances held fixed? Marshall distinguished such questions from extravagant guesses about what people would pay under conditions far removed from their experience. An estimate can be useful without pretending to know the whole curve up to the price at which the last buyer disappears.
The ordinary demand curve also carries an income effect into the area beneath it. It holds money income fixed as the price changes. A lower price then leaves the buyer able to afford more of other things, and that improvement can itself alter how much of the good he wants. The curve records both the relative attractiveness of the cheaper good and the consequences of being better off. If we keep adding willingness to pay along it as though every pound retained the same significance, those changes enter the sum. The problem is especially plain when the purchase occupies a large part of the purse. A small saving on a minor purchase and a transformation of the household budget do not demand the same approximation.
The compensated measures associated with Hicks give the monetary question two precise forms. After a price reduction, how much money could be taken away at the new prices while leaving the buyer as well off as before? That is the compensating measure of the gain. Alternatively, how much extra money at the old prices would make him as well off as the price reduction does? That is the equivalent measure. Each compares attainable ways of living, holding a specified level of well-being fixed. Each can be represented by an area under a compensated demand curve, but the level held fixed differs. The discrepancy between them is intelligible: the money is valued at different sets of prices.
Those two questions can produce substantially different answers even for a perfectly consistent buyer. Imagine a household with one hundred pounds which divides its expenditure equally between a particular good and everything else. The good’s price falls from four pounds to one, while the other price stays fixed. At the cheaper prices, fifty pounds would buy the household its former level of well-being: it could surrender fifty of its original hundred. At the old prices, two hundred pounds would be needed to give it the new level: it would require an extra hundred. The ordinary area between the two prices is about sixty-nine pounds, sitting between the two answers in this example. No contradiction has appeared. Three calculations have valued the same change under different income conditions.
The area can measure a gain only after the question has supplied its prices, its income and its people.
The difference need not make ordinary surplus unusable. Willig’s contribution was to give the approximation a test: bounds on the income elasticity of demand, together with the size of the surplus change relative to income, can bound its error against the compensated measures. Small income effects and modest changes can make the ordinary area a good approximation; a large change calls for checking rather than reassurance. With no income effect over the relevant range, the three measures coincide. This is a reason to improve the evidence supporting the triangle, rather than to abandon every calculation that has drawn one. The size and character of the possible error belong beside the estimated gain.
Even exact compensation for each buyer leaves a further question when the buyers are added together. Ten pounds is ten pounds in the money ledger, whether its owner is choosing a second holiday or deciding which meal to miss. A calculation can faithfully report what each person would sacrifice under his existing resources without establishing that their satisfactions are equal or that those resources were fairly distributed. Improving the treatment of one person’s income effect does not settle the valuation of gains across people. A project’s monetary surplus and the judgment that it ought to be built therefore require different work, even when the monetary calculation is faultless.
The triangle bears considerable weight when its load is specified. It can expose benefits that receipts conceal, compare a proposed change with an actual alternative, and show where better evidence might alter a decision. Dupuit himself allowed that the available information might give only a bound, yet that the bound could be enough to decide whether a bridge was worthwhile. The danger lies in letting the elegance of the area substitute for the demand evidence or the judgment about whose gains count. Once those are supplied, the same geometry can ask a darker question: what happens to the gain when a trade worth making is prevented? The next plate follows the empty space left behind.