Answer to Question #251849 in Economics of Enterprise for Gift nkusang'ombe

Question

Given production function=(k+2)(l+1),cost of capital=$4,wage=$6.money available to the firm =8130, how to find q*?

Answer 1

Production function : $q=(k+2)(l+1)$

$q=kl+k+2l+2$

Cost function: wage$\times$ labor+cost of capital$\times$ capital=Money available

$C:6l+4k=8130$

The firm would like to maximize the production that is the value of q with respect to its cost function:

We will use the lagrangian function:

Max q subject to C

$L=kl+k+2l+2-λ(6l+4k-8130)$

Differentiating L wrt k, l, and λ

$\frac{dL}{dl}=k+2-6λ=0\\ k+2=6λ....(1)\\ \frac{dL}{dk}=l+1-4λ=0\\ l+1=4λ....(2)\\ \frac{dL}{dλ}=6l+4k-8130=0\\ 6l+4k=8130....(3)\\$

 

$From (1) and (2)\\ \frac{(k+2)}{(l+1)}=\frac{3}{2}\\ \frac{2(k+2)}{3}=l+1\\ \frac{2k}{3}+\frac{4}{3}-1=l\\ \frac{2k}{3}+\frac{1}{3}=l\\ Putting\space this\space in \space (3)\\ 6l+4k=8130\\ 6(\frac{2k}{3}+\frac{1}{3})+4k=8130\\ 4k+2+4k=8130\\ 8k=8128\\ k=1016$

Putting the value of k in the relation to get l

$\frac{2k}{3}+\frac{1}{3}=l\\ 2k+1=3l\\ 2(1016)+1=3l\\ l=677.67$

To get the value of q, we will put the obtained optimal value of k and l in the production function:

$q=(k+2)(l+1) q=(1016+2)(677.67+1)\\ q=1018\times 678.67\\ q=690,886.06$