The margin has been, in the last two plates, a thing the seller does with his output — making one more unit while the making pays. It is now time to turn it around and point it at the seller's inputs, at the factors of production he buys in order to produce at all: the labour, the land, the capital, the materials that go into his output and must be paid for out of its proceeds. For the same marginal logic that tells a firm how much to produce tells it also how much of each factor to employ, and the answer it gives has consequences that reach far beyond the firm — all the way to the great question of how the total product of an economy comes to be divided among those who have a hand in making it, and to the oldest and most charged of all the prices this table treats, the wage of labour. This plate follows the margin from the output it has governed so far to the inputs it governs no less, and in doing so it opens the most ideologically loaded chapter in the whole of marginal theory.
The central quantity is the marginal product of a factor: the additional output that results from employing one more unit of it, with the quantities of the other factors held unchanged. One more worker, the same machines and land, yields some increment of output — that is the marginal product of labour; one more acre, the same labour and capital, yields another — the marginal product of land; and so for each factor in turn. Multiply that physical increment by the price at which the output sells, and one has the value of the marginal product, the addition to the firm's revenue that the extra unit of the factor brings in. And now the firm's decision about how much of the factor to hire takes exactly the marginal form of all the others. So long as one more worker adds more to revenue than he adds to cost — so long as the value of his marginal product exceeds the wage — it pays to hire him; and the firm goes on hiring until the value of the marginal product has fallen to equal the wage, at which point the last worker exactly pays for himself and no further hiring can improve matters. The demand for a factor, in short, is the schedule of the value of its marginal product, and in equilibrium each factor is paid the marginal product of its last employed unit.
That the value of the marginal product falls as more of the factor is hired — that the firm reaches a stopping point at all — is the same downward slope met three plates ago, now appearing on the side of production rather than of consumption. As successive workers are added to a fixed stock of machines and land, each has less of the other factors to work with than the last, and each therefore adds less to output than the one before: the marginal product of labour diminishes as labour increases, exactly as the marginal utility of bread diminishes as bread is consumed. It is the gradient again, the same law of the falling increment, governing the factory floor as it governs the dinner table. And it carries the same consequence it carried for utility — that what sets the factor's price is the marginal unit and not the average, the contribution of the last worker hired and not the typical contribution of all of them, the typical being always the larger, since the earlier workers, with more of the other factors to themselves, were more productive than the last. The wage answers to the final hand, not the average hand, and answers to it because the final hand is where the hiring decision is made.
The same falling slope that fixes the worth of one more loaf to the one who eats it fixes the worth of one more worker to the one who hires him; and the wage, like the price, is set by the last unit, not the average — by what the final hand adds, not the typical one.
It was the American economist John Bates Clark who, in his Distribution of Wealth of 1899, raised this marginal account of factor pricing into a complete theory of how a society's income is shared out. If every factor is paid the value of its marginal product, Clark observed, then the wage bill is the marginal product of labour times the quantity of labour, the return to capital is the marginal product of capital times the quantity of capital, the rent of land is the marginal product of land times its quantity, and so on for every factor that enters production. The striking result — proved with full rigour by Wicksteed and Wicksell in those same years — is that under conditions of constant returns to scale these marginal-product shares add up exactly to the whole of the product, leaving nothing over and falling nothing short. The total output of an economy is precisely accounted for by the marginal products of the factors that produced it; the division of the pie among labour, capital, and land is determined, to the last crumb, by the slopes of the production process at the margin. It was a theory of distribution of remarkable elegance and completeness, and it remains the backbone of how economists analyse the demand for labour and the other factors to this day.
Clark, however, did not stop at describing how the product is in fact divided; he went on to pronounce upon whether the division is right, and it is here that the theory acquired its lasting charge and its lasting controversy. The marginal-productivity result, he argued, shows the distribution of income under competition to be not merely determinate but just — for each factor receives exactly the value of what it contributes to the product, no more and no less, so that labour gets what labour makes, capital what capital makes, and no one is robbed. This was aimed squarely at the socialist charge, sharpest in Marx, that labour is exploited, made to produce more than it is paid and to yield up the difference to the owners of capital. Not so, Clark replied: each factor, labour included, is paid the full value of its own marginal contribution, and a distribution in which everyone receives what he produces is a distribution that does justice to all. The proposition is seductive, and it has done enormous ideological work, for it appears to certify the existing distribution of income as the natural and equitable reward of contribution. But it does not survive a close look, and the place where it fails is precisely the seam between what a factor is paid and what its owner can be said to deserve.
To each what he makes, Clark believed his theory had proved, and in the proof he heard the distribution of wealth pronounced just; but the theory shows only what each factor is paid, and the step from what a man is paid to what he deserves is one the mathematics will not take.
Three difficulties, at least, stand between the marginal-productivity theory and the ethical conclusion Clark drew from it. The first is that the marginal product of a factor is not, in any pure sense, that factor's own contribution, for it depends entirely on how much of the other factors stand beside it: the marginal product of a worker is large when he has abundant capital and land to work with and small when he has little, so that what is credited to labour is in truth a joint result of labour and everything else, artificially imputed to the one. The second is that the theory is a positive account, not a normative one — it describes what factors tend to be paid under competition, and from a description of what is paid no conclusion about what ought to be paid can validly be drawn; the wage that equals the marginal product is a fact about markets, and its justice is a separate question on which the marginal equation is silent. And the third is that even the positive claim holds only under its assumed conditions: where an employer has power over the wage, as the monopsony plate of the first volume describes, workers are paid less than their marginal product and the tidy identity fails; while the very notion of the marginal product of capital, on which the symmetry of the whole depends, was later thrown into deep doubt by the discovery that capital cannot in general be measured independently of the rate of return it is meant to explain — a circularity at the theory's heart that a later volume of this table must confront.
None of this is to dismiss the theory, which would be as great a mistake in the other direction as Clark's overreach was in his. As an account of the demand for factors of production — of how a firm decides how much labour and capital and land to employ, and of why those decisions make the factor markets behave as they do — the marginal-productivity theory is powerful, well-confirmed, and indispensable; it is simply true that a firm hires up to the point where the value of the marginal product meets the factor's price, and the consequences of that truth run through the whole of applied economics. What overreaches is not the theory of factor demand but the moral gloss laid upon it, the slide from the firm pays each factor its marginal product to each factor receives its just deserts. Strip the gloss away and a sound and useful proposition remains: under competition, and within the limits competition imposes, each factor tends to be paid the value of what one more unit of it adds to output. Whether that distribution is fair, whether the ownership of the factors that command those payments is itself just, whether the marginal products would even be the same under a different arrangement of property — these are real and pressing questions, but they lie beyond what the margin can answer. The margin tells us how the wage is set. It does not tell us whether the wage is right.
Applied to labour, all of this becomes a theory of the wage, and with it the margin closes a long circle. The marginal-productivity doctrine holds that a worker tends to be paid the value of his marginal product — the worth of what one more worker of his kind adds to the output of the firm that hires him — and in that proposition the table returns to the very first price it set down, in the first volume of its first family. The wage was where this whole work began; and what the margin brings back to it, on its return, is a particular and forceful account of that price: the wage as the marginal product of labour, the value of the last hand hired. It is an account that carries back into the wage the entire contention this plate has raised — for to say that a worker is paid his marginal product is one thing, and to say with Clark that he therefore gets exactly what he deserves is quite another, and the gulf between them is the ground on which the hardest arguments over the fairness of wages are still waged. That return is the bridge this plate throws back to the table's first volume. The volume in hand, though, is far from done: there remains the curve along which one want is traded for another, the whole of the economist's counsel compressed into three words, the lump that will not be sliced, and the quarrel over whether anyone computes the margin at all. The next plate takes up the first of these. As the table says of every quantity, first to last: the argument remains open.