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Paul Douglas explained that his first formulation of the Cobb–Douglas production function was developed in 1927; when seeking a functional form to relate estimates he had calculated for workers and capital, he spoke with mathematician and colleague Charles Cobb, who suggested a function of the form Y = AL β K 1−β, previously used by Knut Wicksell, Philip Wicksteed, and Léon Walras ...
With this assumption, we can re-express aggregate output in per capita terms (,) = (,) = For example, if we use the Cobb–Douglas production function with =, =, then () =. To obtain the first key equation of the Ramsey–Cass–Koopmans model, the dynamic equation for the capital stock needs to be expressed in per capita terms.
The equation below (in Cobb–Douglas form) is often used to represent total output (Y) as a function of total-factor productivity (A), capital input (K), labour input (L), and the two inputs' respective shares of output (α and β are the share of contribution for K and L respectively). As usual for equations of this form, an increase in ...
The AK model production function is a special case of a Cobb–Douglas production function: Y = A K a L 1 − a {\displaystyle Y=AK^{a}L^{1-a}\,} This equation shows a Cobb–Douglas function where Y represents the total production in an economy.
A Cobb-Douglas-type function satisfies the Inada conditions when used as a utility or production function.. In macroeconomics, the Inada conditions are assumptions about the shape of a function that ensure well-behaved properties in economic models, such as diminishing marginal returns and proper boundary behavior, which are essential for the stability and convergence of several macroeconomic ...
The production functions listed below, and their properties are shown for the case of two factors of production, capital (K), and labor (L), mostly for heuristic purposes. These functions and their properties are easily generalizable to include additional factors of production (like land, natural resources, entrepreneurship, etc.)
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These functions are commonly used as examples in consumer theory. The functions are ordinal utility functions, which means that their properties are invariant under positive monotone transformation. For example, the Cobb–Douglas function could also be written as: + . Such functions only become interesting when there are two or more ...