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?
Solution:
MPk = "\\frac{\\partial Q} {\\partial k}"
For example: Q = A Lβ Kα
MPL = A β Lβ-1 Kα, and MPK = 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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