Product Exhaustion Theorem
The proof that, under constant returns to scale, paying every factor its marginal product exactly exhausts total output with no residual — Wicksteed's 1894 theorem, later tied to Euler's theorem and to the Henry George Theorem's rent-as-residual logic.
Overview
The product exhaustion theorem (also called the "adding-up problem") holds that, under constant returns to scale, if every factor of production is paid a return equal to the value of its marginal product, those payments sum exactly to the total product — no surplus and no deficit remain. Philip Wicksteed gave the theorem its first general statement in his 1894 An Essay on the Co-ordination of the Laws of Distribution, though he did not initially connect it to its underlying mathematics; that connection — that the result is a direct application of Euler's theorem for functions homogeneous of the first degree — was drawn out the same year by the reviewer A. W. Flux. Knut Wicksell offered an independent proof shortly after.[1][2]
Why It Matters for Georgism
The theorem sits directly upstream of the Henry George Theorem's logic. If labour and capital are paid exactly their marginal products under constant returns, and land is counted as a third, genuinely distinct factor, land rent becomes precisely the residual left over once labour and capital have been paid — the same conclusion the classical law of rent reaches by a different route. Historian of economics Mark Blaug records that Wicksteed himself demonstrated this equivalence: the rent computed as the "mixtilinear" area left under the marginal-product curves of labour and capital is identical to the rent computed as the ordinary rectangular area under land's own marginal-product curve.[1] The theorem is one of the technical bridges between the older, land-centred classical account of rent and the newer, symmetrical marginal productivity framework — a framework John Bates Clark used to argue the opposite conclusion, that land's return is simply "earned" like any other factor's.
Limits
The theorem's force depends entirely on the constant-returns-to-scale assumption; marginal-product payments will not exhaust the product exactly under increasing or decreasing returns. Blaug also notes that the economist Enrico Barone accused Wicksteed of reproducing an earlier, unacknowledged proof — a priority dispute that does not bear on the theorem's mathematics but is part of its historical record.[1] [CITATION NEEDED: a primary or independent secondary source for the Barone priority dispute, beyond Blaug's summary]
See Also
- Philip Wicksteed — the economist who first stated the theorem, and the bridge from George's economics into professional marginalism
- Marginal Productivity — the broader factor-payment framework the theorem underwrites
- Henry George Theorem — the modern welfare-economics result the product-exhaustion logic anticipates
- Law of Rent — the classical statement of rent as a residual, which the theorem reaches by marginalist means
- John Bates Clark — used the same marginal apparatus to argue land's return is "earned"
Sources
- Mark Blaug, Economic Theory in Retrospect, 5th ed. (Cambridge University Press, 1997), Ch. 11 §13 — used for the theorem's attribution to Wicksteed and Wicksell, the Barone priority dispute, and the equivalence Wicksteed demonstrated between rent-as-residual and rent-as-land's-marginal-product. Book page
- Philip H. Wicksteed, An Essay on the Co-ordination of the Laws of Distribution (London: Macmillan, 1894), full text via the McMaster University History of Economic Thought Archive. PDF — used for the primary 1894 statement of the theorem and its original context.