Answer to Question #300642 in Microeconomics for Abel

Question

The production function of a firm is given by Q=4L^0.5K^0.5 suppose the cost of labour is birr 40 per unit and the cost of using capital is Birr 10 per unit.

A. Determine the amount of labour and capital that should be used in order to minimize the cost to product 40 units of output?

B. Calculate the minimum cost of producing 40 units?

Answer 1

$Q=4L^{0.5}K^{0.5}$

Isocost:

$C=40L+10K$

Fitting in the langragian equation:

$l=40L+10K-h(4L^{0.5}K^{0.5})$

$\frac{dl}{dL}=40-h2L^{-0.5}K^{0.5}....(i)$

$\frac{dl}{dK}=10-h2L^{0.5}K^{-0.5}....(ii)$

Equating the (i) with (ii)

$-h2L^{-0.5}K^{0.5}=-h2L^{0.5}K^{-0.5}$

$L^{-0.5}K^{0.5}=L^{0.5}K^{-0.5}$

$\frac{K^{0.5}}{L^{0.5}}=\frac{L^{0.5}}{K^{0.5}}$

Therefore $K=L$

Replacing K in $Q=4L^{0.5}K^{0.5}$

$40=4L^{0.5}L^{0.5}$

$40=4L^{2}$

$L=3.2$

$K=3.2$