The previous plate left Dupuit's triangle drawn but disregarded, a tool forged in an engineering memoir and ignored by the economists who might have used it. It was Alfred Marshall who, half a century later and at first in apparent ignorance of his predecessor, drew the same triangle, gave it the name it has carried ever since, and set it at the centre of the new economics he was building. The consumer's surplus, in Marshall's hands, ceased to be an engineer's curiosity and became one of the half-dozen most useful instruments in the whole applied science — the standard way, to this day, of putting a number on a question that sounds as though it ought to have no number at all: how much good does a market do the people who buy in it? This plate is about that instrument: the shape it takes, the work it does, and the assumption it quietly rests upon.
Marshall's definition is a model of clarity, and it is worth giving in something close to his own terms. The consumer's surplus, he said, is the excess of the price a person would be willing to pay rather than go without a thing, over the price he actually does pay. Consider a buyer purchasing not one unit but several. The first unit is worth a great deal to him, for it satisfies his most urgent want, and he would pay handsomely for it; the second is worth somewhat less, the third less again, in the familiar descent of diminishing marginal utility that the previous volume traced. Yet in an ordinary market he pays the same price for every unit — the going price — however much each was individually worth to him. On every unit but the last, then, he gets the thing for less than he would have paid; and the sum of all these little excesses, across all the units he buys, is his surplus. Laid out against the schedule of his diminishing valuations, it is the gap between what the units were worth to him and what he paid for them all — the benefit he carries away from the transaction over and above the money he laid down.
Drawn on a diagram, this gap acquires the shape that has made it famous. Put quantity along the horizontal and price along the vertical, and the buyer's schedule of diminishing valuations becomes a downward-sloping curve — his demand curve, each point of which records what one more unit is worth to him. The price he pays is a horizontal line cutting across this curve at the quantity he buys. Everything beneath that line is what he spends: a rectangle of expenditure, price times quantity. But everything above the line and below the curve is what he gains beyond what he spends: the consumer's surplus, a roughly triangular region riding atop the rectangle. For the market as a whole the construction is identical, with the market demand curve in place of the single buyer's, and the total consumer surplus the whole area trapped between the demand curve above and the price below. It is the first figure most students of the subject ever draw, and it has a satisfying concreteness — a benefit that had seemed too diffuse to weigh, the unpaid-for value of a market to its customers, rendered as an area one could in principle measure with a planimeter.
The price line cuts across the curve of demand, and everything above the line and beneath the curve is surplus — the benefit the buyers keep, drawn at last as a definite figure with a definite area, a triangle of welfare one could measure with a ruler.
It was this concreteness that made the triangle indispensable, for it converted a vague good into a measurable one, and a measurable good can be entered into the ledgers of policy. With the consumer's surplus in hand, the economist could at last answer, in money, a whole class of questions that had previously admitted only of hand-waving. What do consumers gain when a price falls? The increase in the triangle. What do they lose when a tax is laid on, or a monopoly raises the price, or a tariff shuts out a cheaper import? The portion of the triangle that the higher price destroys. Is a public bridge or railway or reservoir worth its cost? Set the cost against the surplus it will generate — precisely Dupuit's question, now equipped with a standard tool. The consumer's surplus became, and remains, the workhorse of applied welfare economics, the device by which the benefits and harms of policies are routinely reckoned; whenever an economist tells you that a regulation will cost consumers so many millions, it is some triangle, or some change in one, that lies behind the figure.
And yet there is a difficulty folded into the measure, one that Marshall himself saw clearly and that has troubled the consumer's surplus ever since. The triangle is an area measured in money — pounds of willingness to pay, stacked up unit by unit. But the thing we actually care about is not money but welfare, the satisfaction the buyers derive; and money translates into welfare at a rate that is not constant. A pound is worth more to a man when he is poor than when he is rich — the marginal utility of money itself diminishes — and as a buyer moves down his demand curve, purchasing more and laying out more, his circumstances shift, if only slightly, and with them the worth to him of each further pound. The money-area, in other words, faithfully measures welfare only if a pound of expenditure means the same in satisfaction at every point along the curve — only, that is, if the marginal utility of money is constant. Marshall knew this, and defended his triangle with an approximation: when the good in question takes up only a small part of the buyer's budget, the money he spends on it scarcely alters his wealth, and so the marginal utility of his money holds very nearly steady across the relevant range, and the area may be trusted. For small things — and most single goods are small against a whole budget — the approximation is excellent. For large ones it is not, and the triangle measured in money begins to part company with the welfare it is meant to track.
The triangle is drawn in money, and money is not welfare at a fixed rate — a pound means less the more of them one holds — so the surplus is true to the satisfaction beneath it only where the spending is too slight to make the spender any richer or poorer.
Worse was to come, for the ordinal revolution the previous volume described seemed for a time to pull the ground from under the consumer's surplus altogether. Marshall's measure leaned on cardinal utility and on the very notion of a marginal utility of money, and these the ordinalists had declared meaningless: one could speak of preferring more to less, but not of how much utility a pound conferred. If utility could not be measured, how could the area under a demand curve be a measure of it? The rescue came from John Hicks, in a paper of 1941 pointedly titled "The Rehabilitation of Consumer's Surplus." Hicks showed that one could define exact money measures of a person's gain or loss from a change — a fall in price, say — without any appeal to cardinal utility at all, using only the ordinal apparatus of indifference curves. Two such measures present themselves. The compensating variation is the sum of money that, taken from the buyer after the price has fallen, would leave him exactly as well off as he was before it fell — the most he would give to have the change. The equivalent variation is the sum that, handed to him while prices stay as they were, would make him exactly as well off as the fall in price would have — the least he would accept to forgo it. Both are defined purely by his preferences, both are exact, and both are unimpeachably ordinal. And Marshall's rough triangle, Hicks demonstrated, lies always between these two, and coincides with them precisely when the income effect vanishes — when, that is, the marginal utility of money is constant after all.
This was a rehabilitation in the fullest sense: the consumer's surplus was shown to be not a confused relic of cardinal utility but a serviceable approximation to two perfectly rigorous ordinal measures, exact in the limiting case and close to exact wherever income effects are small. A later result sealed the reprieve. Robert Willig, in a paper of 1976 whose title — "Consumer's Surplus Without Apology" — captures its spirit exactly, showed that for the great majority of practical problems the gap between Marshall's triangle and the exact Hicksian measures is tiny, as a rule smaller than the error involved in estimating the demand curve in the first place. The economist may therefore go on using the simple area under the demand curve, as Marshall and Dupuit did, with a clear conscience. But the subtleties do not vanish for being small. When a change is large, or falls upon a good that looms large in the budget, the compensating and the equivalent variation diverge, and can return different answers to the same question, and one must then decide which exact measure the question calls for — a decision the rough triangle silently papers over. And beneath all of this lies a deeper reservation that none of these refinements touches: every measure here is of one person's gain, reckoned in that one person's money; the moment one wishes to speak of the surplus a policy confers upon a whole population, one must add one person's pounds to another's, and that addition, as the coming reservations of this volume will insist, smuggles back the comparison of different people's welfare that the ordinal turn had forbidden.
The triangle, then, is at once the most used and the most quietly qualified instrument in the welfare economist's kit — a money measure of benefit that works beautifully in the small and rests, in the large, on a choice of conventions and an act of addition it cannot itself justify. Marshall gave it its shape and its centrality; Hicks gave it foundations the ordinalists could accept; Willig gave it leave to be used without apology. What none of them altered is that it measures the benefit of an exchange to one side of it only — the buyer's. But every exchange has two sides, and the seller, too, ordinarily comes away with more than the bargain cost him: the price he receives exceeds, on all but the marginal unit, what he would have been willing to accept for it. There is a surplus on the seller's side of the market as surely as on the buyer's, a second triangle drawn against the curve of supply, and the gains from a trade are the two surpluses summed. To that mirror-image quantity — the producer's surplus, and the rent that hides within it — the next plate turns. As the table says of every quantity, first to last: the argument remains open.