Question

2 /The production function for the automotive and where M is energy and materials (based on Klein, 2003). What kind of returns to scale does this production function exhibit? What is the marginal product of materials? industry is q 10.27 K0.16M0.61 pa rts ,

Answer 1

In this case, production function is represented by a Cogg Douglas function.

$q=L^{0.27}K^{0.16}M^{0.61}$

Where each exponent is respective elasticity of parameter with respect to q.

Elasticity coefficient of parameter L=0.27

Elasticity coefficient parameter K=0.16

Elasticity coefficient parameter L=0.61

Sum of elasticity coefficients=0.27+0.16+0.61=1.04

In case of Cogg Douglas function,we know that if

Sum of elasticity coefficient is more than 1, it represents increasing returns to scale

Sum of elasticity coefficient is less than 1, it represents decreasing returns to scale

Sum of elasticity coefficient is equal to 1, it represents constant returns to scale

Given production represents the increasing returns to scale.

Marginal Product of L=MP_L

$MP_{L}=\frac{\partial q}{\partial L}=0.27L^{(0.27-1)}K^{0.16}M^{0.61}=0.27L^{-0.73}K^{0.16}M^{0.61}$

Marginal Product of K=MP_K

$MP_{K}=\frac{\partial q}{\partial K}=0.16L^{0.27}K^{0.16-1}M^{0.61}=0.16L^{0.27}K^{-0.84}M^{0.61}$

Marginal Product of M=MP_M

$MP_{M}=\frac{\partial q}{\partial M}=0.61L^{0.27}K^{0.16}M^{0.61-1}=0.61L^{0.27}K^{0.16}M^{-0.39}$