Question

8. Find the marginal-product functions for the Cobb-Douglas production func- tion y = A.X. XXX A>0,0<«; <1 for i = 1, 2, 3, 4

Answer 1

The marginal product can be found out by differentiating the production function with respect to that input keeping other inputs constant.

So, marginal product of x₁ is $\partial$y / $\partial$x₁ = A X $\alpha$₁ X x₁^{($\alpha$₁ - 1)} X x₂^$\alpha$₂ X x₃^$\alpha$₃ X x₄^$\alpha$₄ = A$\alpha$₁x₁^{($\alpha$₁ - 1)} x₂^$\alpha$₂ x₃^$\alpha$₃ x₄^$\alpha$₄

Marginal product of x₂ is $\partial$y / $\partial$x₂ = A X $\alpha$₂ X x₁^$\alpha$₁ X x₂^{($\alpha$₂ - 1)} X x₃^$\alpha$₃ X x₄^$\alpha$₄ = A$\alpha$₂x₁^$\alpha$₁ x₂^{($\alpha$₂ - 1)} x₃^$\alpha$₃ x₄^$\alpha$₄

Marginal product of x₁ is $\partial$y / $\partial$x₃ = A X $\alpha$₃ X x₁^$\alpha$₁ X x₂^$\alpha$₂ X x₃^{($\alpha$₃ - 1)} X x₄^$\alpha$₄ = A$\alpha$₃ x₁^$\alpha$₁x₂^$\alpha$₂ x₃^{($\alpha$₃ - 1)} x₄^$\alpha$₄

Marginal product of x₁ is $\partial$y / $\partial$x₄ = A X $\alpha$₄ X x₁^$\alpha$₁X x₂^$\alpha$₂ X x₃^$\alpha$₃ X x₄^{($\alpha$₄ - 1)} = A$\alpha$₄x₁^$\alpha$₁x₂^$\alpha$₂ x₃^$\alpha$₃ x₄^{($\alpha$₄ - 1)}