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Humphrey, Thomas M.. "Algebraic production functions and their uses before Cobb-Douglas." Economic Quarterly. Federal Reserve Bank of Richmond. 1997. HighBeam Research. 19 Oct. 2018 <https://www.highbeam.com>.
Humphrey, Thomas M.. "Algebraic production functions and their uses before Cobb-Douglas." Economic Quarterly. 1997. HighBeam Research. (October 19, 2018). https://www.highbeam.com/doc/1G1-19656437.html
Humphrey, Thomas M.. "Algebraic production functions and their uses before Cobb-Douglas." Economic Quarterly. Federal Reserve Bank of Richmond. 1997. Retrieved October 19, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-19656437.html
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Fundamental to economic analysis is the idea of a production function. It and its allied concept, the utility function, form the twin pillars of neoclassical economics. Written
P = f(L, C, T ...),
the production function relates total product P to the labor L, capital C, land T (terrain), and other inputs that combine to produce it. The function expresses a technological relationship. It describes the maximum output obtainable, at the existing state of technological knowledge, from given amounts of factor inputs. Put differently, a production function is simply a set of recipes or techniques for combining inputs to produce output. Only efficient techniques qualify for inclusion in the function, however, namely those yielding maximum output from any given combination of inputs.
Production functions apply at the level of the individual firm and the macro economy at large. At the micro level, economists use production functions to generate cost functions and input demand schedules for the firm. The famous profit-maximizing conditions of optimal factor hire derive from such microeconomic functions. At the level of the macro economy, analysts use aggregate production functions to explain the determination of factor income shares and to specify the relative contributions of technological progress and expansion of factor supplies to economic growth.
The foregoing applications are well known. Not so well known, however, is the early history of the concept. Textbooks and survey articles largely ignore an extensive body of eighteenth and nineteenth century work on production functions. Instead, they typically start with the famous two-factor Cobb-Douglas version
P = b[L.sup.k][C.sup.1-k].
That version dates from 1927 when University of Chicago economist Paul Douglas, on a sabbatical at Amherst, asked mathematics professor Charles W. Cobb to suggest an equation describing the relationship among the time series on manufacturing output, labor input, and capital input that Douglas had assembled for the period 1889-1922.(1)
The resulting equation
p = b[L.sup.k][C.sup.1-k]
exhibited constant returns to scale, assumed unchanged technology, and omitted land and raw material inputs. With its exponents k and 1 - k summing to one, the function seemed to embody the entire marginal productivity theory of distribution. The exponents constitute the output elasticities with respect to labor and capital. These elasticities, in competitive equilibrium where inputs are paid their marginal products, represent factor income shares that just add up to unity and so exhaust the national product as the theory contends.
The function also seemed to resolve the puzzling empirical constancy of the relative shares. How could those shares remain unchanged in the face of secular changes in the labor force and the capital stock? The function supplied an answer. Increases in the quantity of one factor drive down its marginal productivity and hence its real price. That price falls in the same proportion as the increase in quantity so that the factor's income share stays constant. The resulting share terms k and 1 - k are fixed and independent of the variables P, L, and C. It follows that even massive changes in those variables and their ratios would leave the shares unchanged.
From Cobb-Douglas, textbooks and surveys then proceed to the more exotic CES, or constant elasticity of substitution, function
P = [k[L.sup.-m] + (1 - k)[C.sup.-m]][sup.-1/m].
They observe that the CES function includes Cobb-Douglas as a special case when the elasticity, or flexibility, with which capital can be substituted for labor or vice versa approaches unity.
Finally, the texts arrive at functions that allow for technological change. The simplest of these is the Tinbergen-Solow equation. It prefixes a residual term [e.sup.rt] to the simple Cobb-Douglas function to obtain
P = [e.sup.rt][L.sup.k][C.sup.1-k.]
This term captures the contribution of exogenous technological progress, occurring at trend rate r over time t, to economic growth. Should new inventions and innovations fail to materialize exogenously like manna from heaven, however, more complex functions are available to handle endogenous technical change. Of these and other post-Cobb-Douglas developments, texts and surveys have much to say. Of the history of production functions before Cobb-Douglas, however, they are largely silent.
The result is to foster the impression among the unwary that algebraic production functions are a twentieth century invention. Nothing, however, could be further from the truth. On the contrary, the idea, if not the actuality, of such functions dates back at least to 1767 when the French physiocrat A. R. J. Turgot implicitly described total product schedules possessing positive first partial derivatives, positive and then negative second partial derivatives, and positive cross-partial derivatives. Thirty years later, Parson Thomas Malthus presented his famous arithmetic and geometric ratios (1798), which imply a logarithmic production function. Likewise, a quadratic production function underlies the numerical examples that David Ricardo (1817) used to explain the trend of the relative shares as the economy approaches the classical stationary state. In roughly the same period, pioneer marginalist Johann Heinrich Von Thunen hypothesized geometrical series of declining marginal products implying an exponential production function. Before he died in 1850, Thunen wrote an equation expressing output per worker as a function of capital per worker. When rearranged, his equation yields the Cobb-Douglas function.
Others besides ThUnen presaged modern work. In 1877 a mathematician named Hermann Amstein derived from a production function the first-order conditions of optimal factor hire. Moreover, he employed the Lagrangian multiplier technique in his derivation. And in 1882 Alfred Marshall embedded an aggregate production function in a prototypal neoclassical growth model. From the mid-1890s to the early 1900s a host of economists including Philip Wicksteed, Leon Walras, Enrico Barone, and Knut Wicksell used production functions to demonstrate that the sum of factor payments distributed according to marginal productivity exactly exhausts the total product. One of these writers, A. W. Flux, introduced economists to Leonhard Euler's mathematical theorem on homogeneous functions. Finally, exemplifying the adage that no scientific innovation is christened for its true originator, Knut Wicksell presented the Cobb-Douglas function at least 27 years before Cobb and Douglas presented it.
The following paragraphs trace this evolution and identify specific contributions to it. Besides exhuming lost or forgotten ideas, such an exercise may serve as a partial corrective to the tendency of textbooks and surveys to neglect the early history of the concept. One thing is certain. Algebraic production functions developed hand-in-hand with the theory of marginal productivity. That theory progressed from eighteenth century statements of the law of diminishing returns to late nineteenth and early twentieth century proofs, of the product-exhaustion theorem.
Each stage saw production functions applied with increasing sophistication. First came the idea of marginal productivity schedules as derivatives of a production function. Next came numerical marginal schedules whose integrals constitute particular functional forms indispensable in determining factor prices and relative shares. Third appeared the pathbreaking initial statement of the function in symbolic form. The fourth stage saw a mathematical production function employed in an aggregate neoclassical growth model. The fifth stage witnessed the flourishing of microeconomic production functions in derivations of the marginal conditions of optimal factor hire. Sixth came the demonstration that product exhaustion under marginal productivity requires production functions to exhibit constant returns to scale at the point of competitive equilibrium. Last came the proof that functions of the type later made famous by Cobb-Douglas satisfy this very requirement. In short, macro and micro production functions and their appurtenant concepts--marginal productivity, relative shares, first-order conditions of factor hire, product exhaustion, homogeneity and the like--already were well advanced when Cobb and Douglas arrived.
1. PRODUCTION FUNCTIONS IMPLICIT IN VERBAL STATEMENTS OF THE LAW OF DIMINISHING RETURNS
The notion of an algebraic production function is implicit in the earliest verbal statements of the operation of the law of diminishing returns in agriculture. A. R. J. Turgot, the French physiocratic economist who served as Louis XVI's Minister of Finance, Trade, and Public Works for a year until dismissed for enacting free-market reforms against the wishes of the king, provided the best of these early statements. In his 1767 Observations on a Paper by Saint Peravy, Turgot discusses how variations in factor proportions affect marginal productivities.(2)
Suppose, he writes, that equal increments of the variable factor capital are applied to a fixed amount of land. Each successive increment adds a positive increase to output such that capital's marginal productivity is positive. But that marginal productivity, which at first rises with increases in the capital-to-land ratio, eventually attains a peak and then falls until it reaches zero. At that latter point, the total product of capital--the sum of the marginal products--is at a maximum.
Here is the first clear articulation of the law of variable proportions, or diminishing marginal productivity. Although Turgot applied the law strictly to capital, he realized that it holds for any variable factor including labor. He also recognized a corollary proposition, namely that increases in any factor raise the marginal productivities of the other cooperating factors, which now have more of the first factor to work with. Thus additions to capital, while eventually lowering capital's own marginal productivity, raise the marginal productivities of labor and land.
Turgot's Production Function
Marginal productivity, when expressed mathematically, is the first-order partial derivative of the production function with respect to the input in question, or
[Differential]P/[Differential]C.
And the rate of change of that marginal productivity, again with respect to the associated input, is the second-order partial derivative
[Differential][Differential]P/Differential]C]/[Differential]C =
[Differential.sup.2]P/[Differential][C.sup.2].
Finally, the response of an input's marginal productivity to changes in complementary inputs is a cross-partial derivative
[Differential][Differential]P/[Differential]C]/[Differential]L =
[Differential.sup.2]P/Differential]C[Differential]L.
From what has been said above, it follows that Turgot implicitly described a production function possessing positive first partial derivatives, positive then negative second partial derivatives, and positive cross-partial derivatives. His function, with its initially rising marginal productivity of capital, differs from Cobb-Douglas. In Cobb-Douglas, of course, the marginal productivity of a variable factor declines monotonically from the outset so that the second partial derivative is always negative. Also, Turgot's function, because of the fixity of land, cannot exhibit constant returns to scale like Cobb-Douglas.
2. PRODUCTION FUNCTIONS IMPLICIT IN NUMERICAL TABLES AND SERIES
More than 30 years after Turgot, English classical economists independently rediscovered his notion of production functions obeying the law of variable proportions. Unlike him, however, they expressed the concept numerically. Thus several British classicals, though presenting no explicit mathematical production functions, nevertheless used hypothetical numerical examples and series that imply specific functional forms. A logarithmic function underlies Thomas Malthus's famous arithmetical and geometrical series, which he used to illustrate the law of diminishing returns. …
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