Cobb–Douglas
In economics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs, particularly physical capital and labor, and the amount of output that can be produced by those inputs. The Cobb-Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas during 1927–1947.
In its most standard form for production of a single good with two factors, the function is
where:
Output elasticity measures the responsiveness of output to a change in levels of either labor or capital used in production, ceteris paribus. For example, if α = 0.45, a 1% increase in capital usage would lead to approximately a 0.45% increase in output.
Sometimes the term has a more restricted meaning, requiring that the function display constant returns to scale, meaning that doubling the usage of capital K and labor L will also double output Y. This holds if
If
returns to scale are decreasing, and if
returns to scale are increasing. Assuming perfect competition and α + β = 1, α and β can be shown to be capital's and labor's shares of output.
In its generalized form, the Cobb-Douglas function models more than two goods. The Cobb–Douglas function may be written as:
where:
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 = ALβK1−β, previously used by Knut Wicksell. Estimating this using least squares, he obtained a result for the exponent of labour of 0.75—which was subsequently confirmed by the National Bureau of Economic Research to be 0.741. Later work in the 1940s prompted them to allow for the exponents on K and L to vary, resulting in estimates that subsequently proved to be very close to improved measure of productivity developed at that time.
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