# Cobb-Douglas Production Function

**Definition:** The **Cobb-Douglas Production Function**, given by Charles W. Cobb and Paul H. Douglas is a linear homogeneous production function, which implies, that the factors of production can be substituted for one another up to a certain extent only.

With the proportionate increase in the input factors, the output also increases in the same proportion. Thus, there are constant returns to a scale. In Cobb-Douglas production function, only two input factors, labor, and capital are taken into the consideration, and the elasticity of substitution is equal to one. It is also assumed that, if any, of the inputs, is zero, the output is also zero.

Likewise, in the linear homogeneous production function, the expansion path generated by the cobb-Douglas function is also a straight line passing through the origin. The CD function can be expressed as follows:

**Q = AL**^{α}**K**^{β}

Where, Q = output

A = positive constant

K = capital employed

L = Labor employed

α and β = positive fractions shows the elasticity coefficients of outputs for inputs labor and capital, respectively.

Β = 1-α

This algebraic form of Cobb-Douglas function can be changed in a log linear form, with the help of regression analysis:

**Log Q = log A + α log L + β log K**

The homogeneity of the Cobb-Douglas production function can be checked by adding the values of α and β. If the sum of these parameters is equal to one, then it shows that the production function is linearly homogeneous, and there are constant returns to a scale. If the sum of these parameters is less or more than one, then there is a decreasing and increasing returns to a scale respectively.