The Margin

Begin with a single employer deciding how many workers to take on. He will reason, if he reasons at all, at the margin — not are workers worth having? but is one more worth having? The first worker he hires may add a great deal to what the firm produces; the second, set to work beside the first, adds somewhat less, for the most valuable tasks are already taken; the tenth, less again. This is the law of diminishing returns, and it means that the value each successive worker adds — the value of their marginal product — falls as the workforce grows.

The rule for the employer is then exact. He should keep hiring as long as the next worker brings in more than they cost in wages, and stop at the one whose contribution just equals the wage. Hire fewer, and he passes up a worker who would have more than paid for themselves; hire more, and the last one costs more than they fetch. At that stopping point — and this is the whole of the theory in a single sentence — the wage equals the value of the marginal product. Not the average worker's product, which is higher; not the first worker's, higher still; but the last one's, the marginal one's. Pay is set at the margin.¹

From the worker's side the same logic runs in reverse, and competition is what closes it. Were some employer paying a worker less than the value they add, a rival could profit by luring that worker away for a few cents more; the bidding ends only when the wage has been driven up to meet the marginal product. So under competition — and the qualification will come to matter enormously — the wage is pressed to the value of the last hour from both sides at once.²

That this now seems obvious is the measure of how completely the idea won. It was not obvious for most of the history of the subject. The classical economists — Smith, Ricardo, Mill — had no satisfactory theory of the wage at all; they reached for the cost of producing a worker, for subsistence, for a “wages fund” of capital to be divided among the available hands, and none of it quite held together. The wage, like every price in their system, was anchored to cost of production, and the cost of producing a human being was a murky and circular thing to reckon.

The break came in the 1870s, almost at once and in three countries: Jevons in England, Menger in Austria, and Walras in Switzerland, working independently, relocated the source of value from the cost of making a thing to the usefulness of the last unit of it — its marginal utility.³ Within a generation the same marginal logic was carried from consumption to production, and from production to distribution itself. John Bates Clark, in America, completed the journey: if each factor of production — labor, land, capital — is paid the value of its marginal product, then the ancient question of how the national product divides among them has at last a single, unified answer. One principle, applied at the margin, fixed every factor's price at one stroke.⁴

One principle, applied at the margin, set the price of labour, of land, and of capital together. It looked, for a while, like a law of nature.

It is worth dwelling on how much this explains, because the rest of the plate will be spent on its limits, and the limits should not be mistaken for a verdict against it. Marginal productivity tells us why a surgeon is paid more than a hospital porter — the value added by the marginal surgeon is higher; why wages climbed through the twentieth century — workers gained the tools and skills that lifted their marginal product; why an ordinary worker in a rich country out-earns an identical worker in a poor one — more capital and better organization stand behind them, raising what their last hour is worth. It is the correct account of the demand for labor, and the demand for labor is one half of everything. The trouble begins only when the theory is asked to be the other half as well.

Clark did not stop at explaining the wage. He went on to bless it. If under competition each factor receives the value of what it adds at the margin, then — he argued, in prose of remarkable confidence — each gets precisely what it creates, and the distribution of income is, at its foundation, just. What a social class gets is, under natural law, what it contributes; the market, properly working, hands every man his own product and no more.⁵ The theory thus entered the world already wearing a halo: not merely an account of how wages come to be set, but a demonstration that they are, in the main, deserved.

This was a category error, and a momentous one. That the wage tends, under competition, toward the marginal product is a claim about what is — the description of an equilibrium. That the marginal product is what a worker deserves is a claim about what ought to be — a judgment of justice. The second does not follow from the first, and no quantity of mathematics will carry it across the gap. One may grant the entire positive theory and deny the ethics without the least contradiction. And the marginal product is in any case a poor candidate for a measure of personal desert, because it depends on everything standing around the worker: how much capital is at hand, how many colleagues, what technology, how the firm is run — none of it the worker's own doing.⁶ Move the same worker from a poorly equipped firm to a well-equipped one and their marginal product, and so their “deserved” wage, may double, though the worker has not changed in any particular. A quantity so dependent on circumstance is a strange thing to call what a person has earned. The positive theory survives the objection untouched. The moral gloss does not.

There was, from the very start, a worry buried in the theory, and it took a theorem to settle it — or rather, to reveal exactly when it could be settled. If every factor is paid the value of its marginal product, do the payments add up? Do wages and rent and interest, each fixed at the margin, exactly exhaust the total product — leaving neither a surplus undistributed nor a deficit to be conjured from nowhere?

Philip Wicksteed posed the question sharply in 1894, and the answer proves to be a piece of pure mathematics. Factor payments exhaust the product precisely when production shows constant returns to scale — when doubling every input doubles the output, no more and no less. This is Euler's theorem on homogeneous functions pressed into economic service: under constant returns, and only then, do marginal-product payments sum to exactly the whole.⁷

The qualification is not a mere technicality to be waved through. If a firm enjoys increasing returns — if doubling its inputs more than doubles its output, as happens wherever fixed costs are large or networks compound — then paying every factor its marginal product would promise away more than the firm has made; the sums refuse to close, and something must give, usually the assumption of perfect competition itself. If returns decrease, marginal-product payments fall short, and a residual is left over, which is one of the ways economists account for profit. So the elegant image of a product neatly carved into portions, each factor carrying off exactly what it added and nothing remaining, holds in its clean form under one demanding condition. Economies thick with fixed costs and increasing returns — which is to say, the actual modern ones — are not obviously among the cases where it does.

Grant the theory its logic and its condition; there remains the question of whether its central quantity can be found at all. The marginal product of a worker is the extra output the firm gets from that worker's last hour. For a lone craftsman this is plain enough. But most work is not done alone. It is done in teams, where the output is the joint result of many hands and no single share can be cleanly excised and laid at one person's feet.

Alchian and Demsetz put it memorably in 1972: when two men together lift a crate onto a truck that neither could lift alone, what is the marginal product of one of them?⁸ The output is genuinely joint; the separate contributions are not lying there waiting to be read off. This is no quibble. It means that for a great part of modern work — collaborative, organized, folded into a process — “the value this worker adds” is not an observed fact but an inference, and frequently a contested one. The theory's master quantity goes missing exactly where most of the economy actually runs.

Where the marginal product can be glimpsed — through natural experiments, sudden influxes of workers, abrupt shifts in what a firm can sell — the wage does broadly track it, which is why the theory is not to be thrown away. But the tracking is loose, and the departures from it are large and stubborn: workers paid well below any defensible estimate of their marginal product, for years at a stretch, in ways the clean theory has no room for.⁹ Those departures are not noise to be averaged away. They are the very subject of the plates that follow, and the first firm sign that the wage is not pinned by productivity alone.

What, then, is marginal-productivity theory a theory of? Not, it turns out, of the wage. It is a theory of the demand for labor — and demand fixes one thing only: the most an employer would be willing to pay. It draws the ceiling.

Return to the range of the opening plate. There was a most the employer would pay — the value the work adds — and a least the worker would accept — their next-best option — and between them a band within which the wage was left open, to be settled by power. Marginal productivity is the exact account of that upper line, and of nothing beneath it. It tells us, with real rigor, where the ceiling sits and what lifts or lowers it; it has nothing whatever to say about where, in the space below, the wage will actually come to rest.¹⁰

The orthodoxy's long and consequential error was to take this — a theory of the ceiling — for a theory of the wage. Under one assumption the error is invisible: if competition for the worker is perfect, the wage is driven all the way up to the ceiling, the floor and the position cease to matter, and the marginal product simply is the wage. That single assumption does an enormous amount of concealing work. Relax it — admit that workers cannot move without cost, that employers are sometimes few, that the bargain is struck in Smith's unequal room — and the ceiling and the wage come apart, the band reopens, and marginal productivity stands revealed as the true theory of only its upper edge.

Set in its proper place, the theory is neither the triumph Clark proclaimed nor the fraud its fiercest critics have called it. It explains something real and important; it fails to explain something else equally real; and the whole art is to keep the two apart.

What it explains is the structure of wages — their spacing, the pattern of high and low. Why surgeons out-earn porters, why the skilled out-earn the unskilled, why pay climbs with the capital and the tools a worker commands: these are questions about relative marginal products, and to them marginal productivity gives good answers.¹¹ If you wish to know why one wage stands above another, the margin is very often the place to look.

What it does not explain is the level of wages in general, nor the share of the whole product that goes to labor rather than to capital. Those are not questions about any one worker's marginal product; they are questions about the floor beneath all workers and the balance of power across the whole economy — about how much of the surplus in the first diagram, the region above the wage line, the owners contrive to keep.¹² The margin explains the shape of the wage structure. It does not explain its height, nor how the product is divided; and the division, history is emphatic, can shift a great deal without any change in anyone's marginal product at all, moved by the very things the theory does not contain — the law, the union, the single employer, the unequal power of the two sides.¹³

Which returns us to where the opening plate left off. The marginal product is the ceiling: the most the work is worth. What the worker is actually paid — where, in the range beneath that ceiling, the bargain is forced to settle — is precisely the question the theory cannot answer, and precisely the question the rest of the volume exists to pursue.