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The Cobb-Douglas functional form of production functions in economics is widely used to represent the relationship of an output to inputs.
The form is 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 (see Homogeneous (mathematics)), which implies
that if all inputs are doubled that output will double.
The Cobb-Douglas function was developed Paul Douglas and Richard Cobb.
It has been generalized in the translog functional form.
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