Answer to Question #267663 in Microeconomics for harry

Question #267663

suppose we are given a production function: q=[alp +(1-a)kp ]1/p


Please show that MPk=(1-a)(k/q)p-1 and MPl=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?


1
Expert's answer
2021-11-17T10:01:22-0500

Solution:

MPk = "\\frac{\\partial Q} {\\partial k}"

For example: Q = A Lβ Kα

MP= A β Lβ-1 Kα, and MP= A α Lβ Kα-1

Therefore, MPK = (1-a) k/qp-1

MPL = a(l"\\div" q)p-1

 

MRTS = MPL "\\div" MPK = a(l "\\div" q)p-1 "\\div" (1-a) k "\\div" qp-1 = alp-1"\\div" kp-1qp-p(1-a)

 

MRTS = = alp-1 "\\div" kp-1qp-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.


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