Question

61. Given the production function
Q = 30 K0.7 L0.5
and input prices r = 20 and w = 30, determine the expansion path.

Answer 1

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

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

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

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

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

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

$C=MpL\times w+MpK\times r$

$C=15K^\frac{7}{10}L^\frac{-1}{2}\times30+$ $21K^\frac{-3}{10}L^\frac{1}{2} \times20$

Therefore the Expansion Path (C)

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