The Margin — The Wage, Plate ii

If the bargain is indeterminate, the orthodox theory of the wage may be read as a long effort to make it determinate — to replace Smith's contest of patience with a calculation that yields, for any worker, a single right number. The instrument of that calculation is the margin, and its claim is startling in its precision: the wage is not haggled over at all, but equals the value of the marginal product of labor — the worth of what the last hour worked adds to the product. Pay a worker more than that, and the last hour loses the firm money; pay less, and a competitor bids the worker away. The wage is squeezed, from above and below, onto a single point.

The theory arrived late in the nineteenth century, worked out near-simultaneously and independently by several hands, and given its most influential and most confident statement by the American economist John Bates Clark. In Clark's hands the marginal principle applied to every factor at once: labor is paid the value of its marginal product, capital the value of its marginal product, land the value of its marginal product, each agent of production receiving the worth of what its last unit contributes. The distribution of income ceased to be a matter of custom, struggle, or law and became a matter of differential calculus — the partial derivatives of a production function, read off and paid out.¹

1. John Bates Clark, The Distribution of Wealth (1899). The marginal-productivity theory was developed independently around the same years by Philip Wicksteed (1894), Knut Wicksell, and others; the very phrase "marginal productivity" owes something to a critic, J. A. Hobson, who used it in objection before it became the orthodoxy's own banner.

Clark did not present this as a morally empty result. He drew from it a doctrine of natural justice: if each factor is paid the value of what it adds, then each receives exactly what it creates, and the wage is nothing but the worker's own product handed back to him. Exploitation, on this view, is not merely absent but impossible in a competitive market, because no one takes from another what that other has made. It is a remarkable inversion of Smith — where Smith saw an unequal contest, Clark saw an exact and impartial accounting — and it gave the orthodox theory not only determinacy but a conscience.²

2. Clark's claim that marginal-product distribution gives "to each what he creates," and is therefore just, is a normative gloss the mathematics does not require. Many economists accepted the positive theory — that competitive factor prices equal marginal products — while flatly rejecting the ethical interpretation, noting that what a factor "creates" at the margin depends on how much of every other factor it is combined with, which is hardly the worker's doing.

⁂

The elegance is real, and the theory remains the working core of how economists think about labor demand. But it rests on three supports, each of which has been shown to be less solid than it looks, and the cumulative effect is to turn an apparent law into a special case. The first difficulty is arithmetical, and was noticed almost at once. If every factor is paid the value of its marginal product, do those payments, added up, exactly exhaust the total product — no surplus left over, no deficit to be covered? They do, but only on a condition: that the production function exhibit constant returns to scale, so that doubling every input doubles the output. This is a theorem of Euler's, and it is a knife-edge. Under increasing returns the marginal payments sum to more than the product; under decreasing returns, to less; and in neither case can every factor be paid its marginal product without something — a pure profit, a pure loss — appearing from nowhere to balance the books.³

3. The product-exhaustion or "adding-up" problem, posed by Wicksteed in An Essay on the Co-ordination of the Laws of Distribution (1894). By Euler's theorem the marginal products sum to the total product exactly when the function is homogeneous of degree one — constant returns to scale. The usual rescue, that competition drives the long run to constant returns and zero profit, is an assumption about the equilibrium, not a guarantee that the marginal principle alone distributes the whole product.

The second difficulty is more corrosive, and it concerns the measurement of capital. To speak of the marginal product of labor, one holds the other factors fixed; chief among them is capital. But to do the arithmetic one must say how much capital there is, and capital — unlike hours of labor or acres of land — is a heterogeneous heap of machines and buildings and inventories that can be added together only by their value. And their value depends on the rate of profit used to discount what they will earn — which is the very thing the marginal theory sets out to determine. The reasoning runs in a circle: the marginal product of capital is supposed to fix the rate of profit, yet the quantity of capital whose marginal product is taken cannot be known until the rate of profit is already fixed. A long controversy established that this is not a technicality to be waved away.⁴

4. The Cambridge capital controversy (1950s–60s), between Cambridge, England (Joan Robinson, Piero Sraffa, Luigi Pasinetti) and Cambridge, Massachusetts (Paul Samuelson, Robert Solow). The phenomena of reswitching and capital-reversing showed that the aggregate marginal product of capital need not behave as the smooth parable requires; Samuelson himself conceded the aggregate production function was a parable rather than a theorem. The doubt extends to any clean aggregate marginal-product story, labor's included.

The third difficulty is the one this volume is built around, and it is the simplest of the three. The whole result — that the wage equals the value of the marginal product — holds only if the firm is a price-taker in the labor market, able to hire as many workers as it likes at a wage it cannot itself affect. Grant the firm any power over the wage, any need to raise its offer in order to attract more hands, and the equality breaks at once: the wage falls below the marginal product, by a margin that measures the firm's power. The competitive labor market is not a mild simplification here but the entire load-bearing assumption, and it is exactly the assumption the eighth plate will deny.⁵

5. The competitive assumption is examined and overturned in the eighth plate, The Monopsony. It is worth distinguishing two separate wedges between wage and contribution: imperfect competition in the product market already drives a gap between the marginal physical product and the marginal revenue product, before any question of the labor market arises; monopsony adds a second, larger wedge on the labor-market side.

Read with these three reservations in view, marginal-productivity theory is not refuted but resized. It is the correct account of labor demand in a competitive market, a genuine and indispensable benchmark, and the ceiling it identifies — that no firm will long pay a worker more than he adds — is real. What it is not is a law of the wage as such, still less a proof that the wage is just. It is a description of one case, the frictionless and competitive one, presented for a century as though it were the nature of things. The wage equals the marginal product where no one has power; the volume's wager is that someone almost always does.