Question

Consider a firm with the following production function q = 10 l^1/2 k^1/2. It aims to produce 1000 units of output and faces prices for labour and capital as follows: w =20, v =5

Answer 1

Having this kind of question, we can do the following;

a)      Form the Lagrangian Function for the the cost minimization problem.  

solution

production function, $q=10 l^{1/2} k^{1/2}$

total output $=1000units$

price of labor, $w=20$

price of capital, $v=5$

total cost$=wl+vk=20l+5k$

Minimizing cost subject to the production function by setting up Lagrange:

a) min $wl+vk$ $st$ $q$

$L=wl+vk+\lambda(q-10l^{1/2}k^{1/2}$

b)      Find the cost-minimizing bundle of labor and capital, (k*, l*). 

solution

Solving Lagrange: Differentiating and equating it to zero

$dL/dl=w-\lambda 5l^{-1/2}k^{1/2}=0$

$w=\lambda5l^{-1/2}k^{1/2}...............(1)$

$dL/dk=v-\lambda5k^{-1/2}l^{1/2}=0$

$v=\lambda5k^{-1/2}l^{1/2}............(2)$

$dL/d\lambda=q-10l^{1/2}k^{1/2}=0$

$q=10l^{1/2}k^{1/2}............(3)$

Dividing (1) and(2)

$w/v=k/l$

$k=l(w/v)$

Putting this in (3)

$q=10l^{1/2}(l(w/v))^{1/2}=10l(w/v)^{1/2}$

$l^{*}=[q(v/w)^{1/2}]/10$

Putting l in equation of k

$k=l(w/v)=(w/v)[q(v/w)^{1/2}]/10$

$k^{*}=[q(v/w)^{1/2}]/10$