Cobb-Douglas Production Function

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Cobb-Douglas Production Function (CDPF) takes the following form:
Q = Ax₁^α. x₂^β, α, β > 0
Here, Q = output, x1 = quantity of labour, x2 = quantity of capital and A, α, β are parameters of the function.
Roughly speaking, CD-PF found that about 75% of the increase in production was due to the labour input and 25% was due to the capital input. CD-PF can be estimated by regression analysis by first converting it into the following form.
log Q = log A + α log x1 + βlogx2
=> Q' = A' + αx1' + βx2'
CD-PF in log form is a linear function.
Properties of CD-PF:
1. The sum of exponents of inputs i.e. (α + β) in CD-PF measures the return to scale.
Q (tx1, tx2) = A(tx₁)^α. (tx₂)^β
= t^(α + β) . A x₁^α. x₂^β

= if (α + β) = 1, return to scale is constant. 

If (α + β) > 1, return to scale is increasing

If (α + β) < 1, return to scale is decreasing

2. MP of  a factor depends on the amount used in production. 

3. The exponents of of inputs of CD-PF measures output elasticity of inputs respectively. 

4. CD-PF can be extended to include more than two factors. 

5. When the sum of exponent (α + β) in two input factors of CD-PF is equal to one, it become linear homogeneous production function of degree one. 

6. Elasticity of substitution between two factors (labour & capital) in CD-PF is equal to unity.