Question

Given production function: Q = L 3/4 . K1/4 Find out the optimal quantities of the two factors using Lagrangian method, if it is given that price of labor is Rs.6 and price of capital is Rs.3 and total cost is equal to Rs.120.

Answer 1

Minimize Lagrangian$=wL+rK+\lambda(Q-L^{3/4}K^{1/4})$

Now differentiating with respect to L and K, and putting equal to zero,

we get $d$ $Lagrangian/d$ $Labour = w+3/4\lambda K^{1/4}L^{-1/4}$

In the above two equations, put $w=\$6$ and $r=\$3$

and we get $2L=3K$

and we know $wL+rK=cost$

$6L+3K=120$ ,putting $2L=3K$

then $L=120/8=15$ and $k=10$

and for the capital we get $r+1/4\lambda L^{3/4}K^{-3/4}$