Paradox of value

Why does water, which nobody can live without, sell for less than a stone nobody needs?

Adam Smith set the puzzle in 1776 and left it unsolved: nothing is more useful than water, yet it will buy almost nothing; a diamond has hardly any use, yet a great quantity of goods can be had for it. If price tracked usefulness the ranking would be reversed, so price evidently does not track usefulness. What does it track?

The ready answer is scarcity. But test that on its own and it collapses at once. A great many things are scarce and worth nothing at all — the exact configuration of dust on my windowsill this morning is unique in the universe and nobody will pay for it. Meanwhile insulin is both scarce and indispensable, and it is expensive. Neither usefulness nor scarcity is doing the work alone, so the answer must be a relation between them rather than a property of either.

Before hunting for the relation, look at how the question is phrased, because it may be malformed. It sets "water" against "a diamond" — a whole category against a single unit. Nobody has ever been offered all the world's water in exchange for all the world's diamonds. What is offered, always, is one more glass or one more stone.

So ask about the unit actually in play. What is one additional litre worth to you? That depends on what you would do with it — and here is the step the whole subject turns on: you do not use water at random. You drink first, because dying is the worst outcome. Then you cook, then wash, then water the garden, then hose the drive. The uses arrange themselves in descending order of urgency and you satisfy them from the top down.

So what does an extra litre get spent on? The next job down the list. And what does losing one cost you? The same job. With water abundant, the value of one more unit is the value of hosing the drive, and that is small, no matter that the top of the list is life itself. Make water scarce — a siege, a drought — and you are working near the top again, and a litre is worth whatever you have. The substance did not change. Its position on the list did.

The diamond's list, by contrast, has one entry. There is no abundant supply working its way down to trivial uses, so the marginal stone is still being put to the most valued thing anyone wants to do with a stone.

This is what Jevons, Menger and Walras arrived at independently in the 1870s, and it is the reason the episode is called the marginal revolution. Smith himself went a different way, toward labour and cost of production, and how much of the scarcity side he already grasped is a live argument among historians of the subject.

Two things need adding, or the picture flatters itself. First, buyers only settle one half of a price. The other half is what it costs to produce one more unit, and price sits where the two meet — water is cheap partly because the marginal litre is cheap to deliver, and dear in places where it is not. Second, the textbook example is a slightly embarrassing one. Gem diamonds are not geologically rare in the way the folk story implies; the twentieth-century price owed a great deal to a cartel that controlled release and an advertising campaign that manufactured the demand it then satisfied. The paradox is sound; its mascot is compromised.

Three refinements, and one consequence worth carrying away.

Price does not measure worth. It measures the worth of the last unit, at present abundance, to whoever is only just willing to buy it. Total worth and marginal worth are different quantities and the paradox is entirely the gap between them.

That gap has a name and is not a defect. What you would have paid for the water you actually need, minus what you did pay, is consumer surplus — and water's enormous surplus is the very thing its low price demonstrates. The cheapness of water is evidence of how well off water makes us, not evidence that we undervalue it.

And scarcity in the useful sense is not rarity but scarcity relative to wants — how far down the list the available supply lets you go. A thing nobody wants is not scarce however few of it exist.

The consequence: any argument of the form "this is more essential, so it should command more" repeats Smith's error. Essentialness sits at the top of the list, and prices are made at the bottom.

• Consumer surplus as a measurement, and why it makes the cheapest goods the ones we gain most from.
• Why price rises during shortages are both the most resented and the most informative prices in any market.
• How the same reasoning explains what a wage is: the worth of the last worker hired, not the importance of the job.

Every term the collection defines is gathered in the glossary.