Cobb-Douglas

From Wikipedia, the free encyclopedia

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 Y = AL^αK^β

Where:

Y = output
L = labor input
K = capital input
A, α and β are constants determined by technology.

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

The exponents α and β are output elasticities with respect to labor and capital, respectively. 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 = 1.5, a 1% increase in labor would lead to approximitely a 1.5% increase in 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 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:

$\prod_{i=1}^N x_i^{\alpha_{i}}$

where $x i$ are the quantities consumed of each good i and $α i$ are the demand elasticites of utility.

[edit] Various representations of the production function

The Cobb-Douglas function form can be estimated as a linear relationship using the following expression:

log_e(O) = a₀ +	∑	a_ilog_e(I_i)
	i

Where:

O = Output
I_i = Inputs
a_i = model coefficients

The model can also be written as

$O = (I_1)^{a_1} * (I_2)^{a_2} \cdots$

A common Cobb-Douglas function used in macroeconomic modeling is

O = K α L 1 - α

where K is capital and L is labor. When the model coefficients sum to one, as in this example, the production function is first-order homogeneous, which implies constant returns to scale, that is, if all inputs are doubled that output will double.

It has been generalized in the translog functional form.