Given
Q=20K0.5L0.5Q = 20 K^{0.5}L^{0.5}Q=20K0.5L0.5
MPL=dqdl=20×0.5×L−0.5×K0.5=10×L−0.5×K0.5MPL = \frac{dq}{dl}\\ = 20\times 0.5×L^{-0.5} ×K^{0.5}\\ =10×L^{-0.5} ×K^{0.5}MPL=dldq=20×0.5×L−0.5×K0.5=10×L−0.5×K0.5
MPK=dqdl=20×0.5×L0.5×K−0.5=10×L0.5×K−0.5MPK = \frac{dq}{dl}\\ = 20\times 0.5×L^{0.5} ×K^{-0.5}\\ =10×L^{0.5} ×K^{-0.5}MPK=dldq=20×0.5×L0.5×K−0.5=10×L0.5×K−0.5
Now expansion path = MPL/Price of labor= MPK/Price of capital
[10×L−0.5×K0.5]4=[10×L0.5×K−0.5]5K4=L5K=4L5\frac{[10×L^{-0.5} ×K^{0.5}]}{ 4} =\frac{ [ 10×L^{0.5}×K^{-0.5} ]}{5}\\ \frac{K}{4} =\frac{L}{5}\\ K=\frac{4L}{5}4[10×L−0.5×K0.5]=5[10×L0.5×K−0.5]4K=5LK=54L
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