Cobb-Douglas
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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, and tested against statistical evidence by Paul Douglas and Richard Cobb in 1928.
For production, the function is Y = ALαKβ
Where:
If α and β = 1, the production function has constant returns to scale (if L and K are increased by 20%, Y increases by 20%). If α and β are less than 1, returns to scale are decreasing, and if they are greater than 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.
The Cobb-Douglas function can be applied to utility.
The Cobb-Douglas function form can also be estimated as a linear relationship using the following expression:
- <math> \log_e(O) = a_0 + \sum_i{a_i \log_e(I_i)} <math>
Where:
- O = Output
- Ii = Inputs
- ai = model coefficients
The model can also be written as
- <math> O = (I_1)^{a_1} * (I_2)^{a_2} \cdots <math>
A common Cobb-Douglas function used in macroeconomic modeling is
- <math> O = K^\alpha L^{1-\alpha} <math>
where K is capital and L is labor. When the model coefficents sum to one, as in this example, the production function is first-order homogeneous, which implies that if all inputs are doubled that output will double.
It has been generalized in the translog functional form.