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I have problems with the following function in two variables.

$F(K,L)=10\sqrt K\sqrt L -rK-wL $, where $K>0, L>0$ and $p,r,w$ are positive parameters.

I have to maximize this function wrt. $K$ and $L$, but I can not solve it.

Hope someone can help.

frankfranfrank
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3 Answers3

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computing the partial derivatives with respect to $K,L$ $$\frac{\partial F}{\partial K}=10\frac{1}{2}K^{-1/2}\sqrt{L}-r$$ $$\frac{\partial F}{\partial L}=10\sqrt{K}\frac{1}{2}L^{-1/2}-w$$ and solve the system $$\frac{\partial F}{\partial K}=0$$ and $$\frac{\partial F}{\partial L}=0$$

Dr. Sonnhard Graubner
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Do the following change of variable: $u = \frac{r}{r+w}K$ and $v = \frac{w}{r+w}L$. Then you can rewrite the objective as: \begin{align} 10p\frac{r+w}{\sqrt{rw}}\sqrt{uv}-(r+w)(u+v)\leq 10p\frac{r+w}{\sqrt{rw}}\frac{u+v}{2}-(r+w)(u+v) = (\frac{5p}{\sqrt{rw}}-1)(r+w)(u+v), \end{align} where the last inequality is due to the relationship between geometric mean and arithmitic mean. Note that the inequality is tight and it could be reached by simply choosing $u=v$. Now it is easy to see that if $(\frac{5p}{\sqrt{rw}}-1)\leq 0$, then the max would be zero, by choosing $u=0$ and $v=0$. On the other hand when $(\frac{5p}{\sqrt{rw}}-1)>0$, then the max would be infinity $u=v\rightarrow +\infty$.

In the case where $5p = \sqrt{rw}$, the optimal value which is $0$ could be attained at any point where $u=v>0$.

Maziar Sanjabi
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Ok. I don't understand that. I can see that I forgot to include $p$. So I actually have:

$F(K,L)=10p\sqrt K\sqrt L -rK-wL $

If that makes any difference.

frankfranfrank
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  • First of all, you don't need to submit a answer when you are just trying to comment. Second, I have added the $p$ in the above answer. Third, could you explicitly say which part of the answer you don't understand? – Maziar Sanjabi Oct 24 '17 at 20:09
  • Ok. Does this mean that there can not be a solution, where K and L is positive but not infinite? – frankfranfrank Oct 24 '17 at 20:35
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    It happens only when $5p = \sqrt{rw}$, in which case the optimal value is $0$ and can be attained at any point where $u = v$. – Maziar Sanjabi Oct 24 '17 at 23:29