Question #143886
61. Given the production function
Q = 30 K0.7 L0.5
and input prices r = 20 and w = 30, determine the expansion path.
1
Expert's answer
2020-11-17T07:28:16-0500

Q=30K710L12Q=30K^\frac{7}{10}L^\frac{1}{2}

obtain the partial derivative of Q wrt to K and L to find Marginal product of labour and marginal product of Capital (K)

dQdK=30710K310L12\frac{dQ}{dK}=30•\frac{7}{10}•K^\frac{-3}{10}L^\frac{1}{2}

== 21K310L1221K^\frac{-3}{10}L^\frac{1}{2}


dQdL=3012K710L12\frac{dQ}{dL}=30•\frac{1}{2}•K^\frac{7}{10}L^\frac{-1}{2}

=15K710L12=15K^\frac{7}{10}L^\frac{-1}{2}


C=MpL×w+MpK×rC=MpL\times w+MpK\times r

C=15K710L12×30+C=15K^\frac{7}{10}L^\frac{-1}{2}\times30+ 21K310L12×2021K^\frac{-3}{10}L^\frac{1}{2} \times20


Therefore the Expansion Path (C)

C=450K710L12+C=450K^\frac{7}{10}L^\frac{-1}{2} + 420K310L12420K^\frac{-3}{10}L^\frac{1}{2}


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