Extension of the Cobb-Douglas Production Function—The Cobb-Douglas production function can be shown to be a special case of a larger class of linear homogeneous production functions having the following mathematical form:
Q = γ[∂K−ρ + (1 − ∂)L−ρ]−ν/ρ
where γ is an efficiency parameter that shows the output resulting from given quantities of inputs; ∂ is a distribution parameter (0 ≤ ∂ ≤ 1) that indicates the division of factor income between capital and labor; ρ is a substitution parameter that is a measure of substitutability of capital for labor (or vice versa) in the production process; and ν is a scale parameter (ν > 0) that indicates the type of returns to scale (increasing, constant, or decreasing). Show that when ν = 1, this function exhibits constant returns to scale. [Increase capital K and labor L each by a factor of λ, or K* = (λ)K and L* = (λ)L, and show that output Q also increases by a factor of λ, or Q* = (λ)(Q).]