The production function $f ( k , \ell ) = A \left( \alpha \ell ^ { - \rho } + ( 1 - \alpha ) k ^ { - \rho } \right) ^ { - \beta / \rho }$ can have increasing returns to scale or decreasing returns to scale -- but it cannot have initially increasing and eventually decreasing returns to scale.

Question

Quizplus · Accepted Answer

The given production function is a CES (Constant Elasticity of Substitution) function, which is characterized by constant returns to scale. It cannot exhibit initially increasing and eventually decreasing returns to scale because its returns to scale are determined by the parameter $\beta$, which is constant.

The Production Function f(k,ℓ)=A(αℓ−ρ+(1−α)k−ρ)−β/ρf ( k , \ell ) = A \left( \alpha \ell ^ { - \rho } + ( 1 - \alpha ) k ^ { - \rho } \right) ^ { - \beta / \rho }f(k,ℓ)=A(αℓ−ρ+(1−α)k−ρ)−β/ρ

The Production Function $f ( k , \ell ) = A \left( \alpha \ell ^ { - \rho } + ( 1 - \alpha ) k ^ { - \rho } \right) ^ { - \beta / \rho }$