Question

suppose we are given a production function: q=[al^p +(1-a)k^p ]^1/p

Please show that MP_k=(1-a)(k/q)^p-1 and MP_l=a(l/q)^p-1

 

Please derive the RTS

 

Please derive the elasticity of substitution a.

 

Does this production function exhibit constant, increasing, or decreasing returns to scale?

Answer 1

Solution:

MP_k = $\frac{\partial Q} {\partial k}$

For example: Q = A L^β K^α

MP_L = A β L^β-1 K^α, and MP_K = A α L^β K^α-1

Therefore, MP_K = (1-a) k/q^p-1

MP_L = a(l$\div$ q)^p-1

 

MRTS = MP_L $\div$ MP_K = a(l $\div$ q)^p-1 $\div$ (1-a) k $\div$ q^p-1 = al^p-1$\div$ k^p-1q^p-p(1-a)

 

MRTS = = al^p-1 $\div$ k^p-1q^p-p(1-a)

 

Elasticity of substitution (a) = dIn(K$\div$ L)/dInMRTS

K/L = MRTS²

In(K$\div$ L) = InMRTS²

dIn(K$\div$ L)/dInMRTS = 2 = a

a = 2

 

The production function exhibit decreasing returns to scale.