Cobb-Douglas Totally Explained

Everything about Cobb-douglas totally explained

In economics, the Cobb-Douglas functional form of production functions is widely used to represent the relationship of an output to inputs. It was proposed by Knut Wicksell (1851-1926), and tested against statistical evidence by Paul Douglas and Charles Cobb in 1928.
For production, the function is ť Y = AL^αK^β,

where:

Y = total production (the monetary value of all goods produced in a year)
L = labor input
K = capital input
A, α and β are the output elasticities of labor and capital, respectively. These values are constants determined by available technology.

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 = 0.15, a 1% increase in labor would lead to approximately a 0.15% increase in output.
Further, if:
ť α + β = 1,

the production function has constant returns to scale. That is, if L and K are each increased by 20%, Y increases by 20%. If ť α + β < 1,

returns to scale are decreasing, and if ť α + β > 1

returns to scale are increasing. Assuming perfect competition, α and β can be shown to be labour and capital's share of output.
Cobb and Douglas were influenced by statistical evidence that appeared to show that labour 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. There is now doubt over whether constancy over time exists.

Difficulties

Further, neither Cobb nor Douglas provided any theoretical reason why the coefficients α and β should be constant over time or be the same between sectors of the economy. Remember that the nature of the machinery and other capital goods (the K) differs between time-periods and according to what is being produced. So do the skills of labor (the L).
   The Cobb-Douglas production function wasn't developed on the basis of any knowledge of engineering, technology, or management of the production process. It was instead developed because it had attractive mathematical characteristics, such as diminishing marginal returns to either factor of production.
   Crucially, there are no microfoundations for it. In the modern era, economists have insisted that the micro-logic of any larger-scale process should be explained. The C-D production function fails this test.
   For example, consider the example of two sectors which have the exactly same Cobb-Douglas technologies:
   if, for sector 1, ť Y₁ = AL₁^αK₁^β

and, for sector 2, ť Y₂ = AL₂^αK₂^β,

that, in general, does not imply that ť Y₁ + Y₂ = A(L₁ + L₂)^α(K₁ + K₂)^β

This holds only if L₁ / L₂ = K₁ / K₂ and α+β = 1, for example for constant returns to scale technology.
   It is thus a mathematical mistake to assume that just because the Cobb-Douglas function applies at the micro-level, it also applies at the macro-level. Similarly, there's no reason that a macro Cobb-Douglas applies at the disaggregated level.

Some applications

Nonetheless, the Cobb-Douglas function has been applied to a lot of other contexts besides production. It can be applied to utility as follows:
U(x₁,x₂)=x₁^αx₂^β; where x₁ and x₂ are the quantities consumed of good #1 and good #2.
On its generalized form, the Cobb-Douglas utility function is written as: ť

prod_

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