Cobb–Douglas production function

The Cobb–Douglas form was developed and tested against statistical evidence by Charles Cobb and Paul Douglas between 1927 and 1947;[1] according to Douglas, the functional form itself was developed earlier by Philip Wicksteed.

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, Philip Wicksteed, and Léon Walras, although Douglas only acknowledges Wicksteed and Walras for their contributions.

[3] Not long after Knut Wicksell's death in 1926, Paul Douglas and Charles Cobb implemented the Cobb–Douglas function in their work covering the subject manner of producer theory for the first time.

[4] 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.

[5] A major criticism at the time was that estimates of the production function, although seemingly accurate, were based on such sparse data that it was hard to give them much credibility.

Douglas remarked "I must admit I was discouraged by this criticism and thought of giving up the effort, but there was something which told me I should hold on.

"[5] The breakthrough came in using US census data, which was cross-sectional and provided a large number of observations.

Douglas presented the results of these findings, along with those for other countries, at his 1947 address as president of the American Economic Association.

Shortly afterwards, Douglas went into politics and was stricken by ill health—resulting in little further development on his side.

However, two decades later, his production function was widely used, being adopted by economists such as Paul Samuelson and Robert Solow.

[5] The Cobb–Douglas production function is especially notable for being the first time an aggregate or economy-wide production function had been developed, estimated, and then presented to the profession for analysis; it marked a landmark change in how economists approached macroeconomics from a microeconomics perspective.

Thus, this function satisfies the law of "diminishing returns"; that is, the marginal product of capital, while always positive, is declining.

Sometimes the term has a more restricted meaning, requiring that the function display constant returns to scale, meaning that increasing capital K and labor L by a factor k also increases output Y by the same factor, that is,

Cobb and Douglas were influenced by statistical evidence that appeared to show that labor and capital shares of total output were constant over time in developed countries; they explained this by statistical fitting least-squares regression of their production function.

It is now widely accepted that labor share is declining in industrialized economies.

[11] Another issue within the fundamental composition the Cobb–Douglas production function is the presence of simultaneous equation bias.

In some cases this simultaneous equation bias doesn't appear.

A building is composed of commodities, labor and risks and general conditions.

It was instead developed because it had attractive mathematical characteristics[citation needed], such as diminishing marginal returns to either factor of production and the property that the optimal expenditure shares on any given input of a firm operating a Cobb–Douglas technology are constant.

In the modern era, some economists try to build models up from individual agents acting, rather than imposing a functional form on an entire economy[citation needed].

Similarly, it is not necessarily the case that a macro Cobb–Douglas applies at the disaggregated level.

An early microfoundation of the aggregate Cobb–Douglas technology based on linear activities is derived in Houthakker (1955).

[14] The Cobb–Douglas production function is inconsistent with modern empirical estimates of the elasticity of substitution between capital and labor, which suggest that capital and labor are gross complements.

A 2021 meta-analysis of 3186 estimates concludes that "the weight of evidence accumulated in the empirical literature emphatically rejects the Cobb–Douglas specification.

As the result, a monotonic transformation of a utility function represents the same preferences.

denote the goods' prices, she solves: It turns out that the solution for Cobb–Douglas demand is: Since

When the model exponents sum to one, the production function is first-order homogeneous, which implies constant returns to scale—that is, if all inputs are scaled by a common factor greater than zero, output will be scaled by the same factor.

[18] To see this, the log of the CES function: can be taken to the limit by applying L'Hôpital's rule: Therefore,

It is often used in econometrics for the fact that it is linear in the parameters, which means ordinary least squares could be used if inputs could be assumed exogenous.