The Function — The Share, Plate i

The Wage asked what sets the price of a single hour of labor, and answered, with Clark, that each factor earns its marginal product — the value of what one more unit of it adds. The Share asks a larger question: not the price of one factor but the division of the whole product between the two great classes of factor, labor and capital, those who work and those who own. How is a nation's income split between wages and profits? The first and most influential answer extends the marginal-productivity principle from the single wage to the aggregate division, and finds the split to be a kind of law of production — natural, stable, and set by technology rather than by power. Paul Douglas, with the mathematician Charles Cobb, gave that answer its empirical form in 1928, and the volume opens where the orthodoxy has long stood: with the claim that the division of the product is set by productivity.

Douglas's instrument was the aggregate production function — a formula relating a nation's total output to its total labor and its total capital. The form he and Cobb chose, which now carries their names, makes output the product of labor and capital each raised to a power, so that output rises with both, and each power measures how responsive output is to that factor. Two features matter for distribution. First, the two powers sum to one, so that doubling both labor and capital doubles output — constant returns to scale. Second, and crucially, under competition, where each factor is paid its marginal product, the power on labor turns out to equal labor's share of the output, and the power on capital equals capital's share. The production function does not merely describe how goods are made; its very exponents are the division of the product. The mathematics of production and the arithmetic of distribution become, in this construction, the same thing.¹

1. Douglas's tool was the aggregate production function, relating total output to total labor and capital. The Cobb-Douglas form makes output the product of labor and capital each raised to a power. The powers sum to one (constant returns), and — crucially — under competition, where each factor earns its marginal product, the power on labor equals labor's share and the power on capital equals capital's. The exponents of the production function are the division of the product; production and distribution become one mathematics.

Beneath this lies an elegant theorem that answers a worry as old as the marginal-productivity theory itself. If labor is paid its marginal product and capital is paid its marginal product, do the two payments exactly use up the total output, or is there a surplus left over, or a shortfall? The answer, under constant returns to scale, is that they add up exactly: a result of Euler guarantees that the sum of each factor's marginal product times its quantity equals the whole product. Nothing is left over and nothing is missing — the product is precisely exhausted by paying each factor its margin. This was a genuine intellectual relief, for it showed the marginal-productivity theory of distribution to be internally consistent: one could pay every factor its contribution and the books would balance, with no unexplained residual and no need to invoke power or bargaining to close the accounts. The division of the product, on this view, is fully determined by the technology of production together with the rule that each factor earns its margin.²

2. An old worry: if labor and capital are each paid their marginal product, do the payments exhaust the output or leave a surplus? Under constant returns, Euler's theorem guarantees they add up exactly — the product is precisely exhausted, nothing left over or missing. The marginal-productivity theory of distribution is thus internally consistent: pay each factor its contribution and the books balance, with no residual and no need to invoke power. Distribution is fully determined by technology plus the marginal rule.

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Douglas's fame, though, rests not on the algebra but on the empirics. Fitting his function to the data of American manufacturing, and later to many countries and periods, he found two striking things. The exponents came out near three-quarters on labor and one-quarter on capital. And — measured independently, from wage and profit data — labor's actual share of the national income was also near three-quarters, holding remarkably steady through booms, slumps, and the long march of technology. The two numbers agreed: the share the marginal-productivity theory predicted, the exponent, matched the share labor actually received. This apparent constancy of labor's share became one of the celebrated regularities of economics — often called Bowley's Law, after the statistician who first remarked on it — and it seemed to confirm that the division of the product was governed by a stable law of production, fixed by technology, and not by the shifting fortunes of class conflict or the balance of power between worker and owner.³

3. Douglas's fame rests on the empirics. Fitting his function to American manufacturing and many countries, he found the exponents near three-quarters on labor, one-quarter on capital — and, measured independently from wage and profit data, labor's actual share was also near three-quarters, steady through booms and slumps. The predicted share (the exponent) matched the received share. This constancy, often called Bowley's Law, seemed to confirm that the division was a stable law of production, fixed by technology, not by power.

The implication was profound and, to many, reassuring. If labor's share is fixed by the production function at around three-quarters, then the division of the product is not a matter of power, or politics, or bargaining, but of technology — of how output responds to labor and to capital. Each factor receives the share that corresponds to its productive weight; the split is, in this sense, natural, and the apparent constancy seems to prove that neither side can durably alter it. This is the productivity answer to the volume's question in its strongest form: the split between labor and capital is set by productivity, embodied in the production function, and it is stable because the technology of production is stable. It raises to the level of an aggregate law the normative claim met in The Wage — that each factor earns precisely what it contributes — and for the better part of a century it was the orthodox understanding of the functional distribution of income, taught as settled and treated as deep.⁴

4. The implication, reassuring to many: if labor's share is fixed at around three-quarters by the production function, the division is a matter of technology, not power or bargaining. Each factor gets the share matching its productive weight; the split is natural, and the constancy seems to prove neither side can alter it. This is the productivity answer in its strongest form, raising The Wage's normative claim — each factor earns what it adds — to an aggregate law. For a century it was orthodoxy.

But the edifice rests on assumptions that the rest of the volume will test, one by one. The exponents equal the factor shares only if each factor is paid its marginal product — that is, only under competition, with no market power on either side; let firms acquire the power to mark prices up, or employers the power to hold wages down, and the link between productivity and the share is broken. The constancy holds only if the production function has a particular shape, an elasticity of substitution near one; let capital and labor substitute more freely, and a falling price of machines can push labor's share down. And even the empirical constancy, so long taken for granted, turns out to have been the feature of an era: in recent decades labor's share has not held but fallen, across the rich world and beyond. Each of these cracks opens onto a rival answer — that the split is set not by productivity but by power, by technology of a different kind, or by the politics that writes the rules. The challenge begins with power, and with the theory that distribution is governed by the markup. The next plate is the degree.⁵

5. The edifice rests on assumptions the volume will test. The exponents equal the shares only under competition with no market power — let firms mark up, or employers hold wages down, and the productivity-share link breaks. The constancy holds only if the elasticity of substitution is near one — let capital and labor substitute freely, and cheaper machines push labor's share down. And the constancy was an era's feature: labor's share has lately fallen. Each crack opens a rival answer. Next: the degree.

The volume opens with the orthodoxy: that the division of the product is a law of production. Paul Douglas and Charles Cobb fit a nation's output to its labor and its capital and found that each factor's share matched its weight in the production function — labor's near three-quarters, capital's near one-quarter — and that this share held remarkably steady, decade upon decade, exactly as the marginal-productivity theory predicted. The product is precisely exhausted by paying each factor its margin; the split is natural, technological, and stable; each gets what it adds. This is the productivity answer in its strongest form, and for fifty years it was orthodoxy. But it stands on assumptions — perfect competition, a particular elasticity of substitution, a stable production function — and on a constancy that has, in our own time, given way. The challenge to it begins with power. The next plate is the degree.