Suppose I have this function:
$q=f(k,l)=600k^2l^2-k^3l^3$
Then,
$f_l=1200k^2l-3k^3l^2$
$f_k=1200kl^2-3k^2l^3$
$f_{ll}=1200k^2-6k^3l$
$f_{kk}=1200l^2-6kl^3$
$f_{kl}=f_{lk}=2400kl-9k^2l^2$
Now,
If we assume that:
$f_l>0$
$f_k>0$
$f_{ll} <0$
$f_{kk} < 0$
$f_{kl}=f_{lk}>0$
$RTS = \left. -\dfrac{dk}{dl}\right|_{q=q_0}= \dfrac{f_l}{f_k}$
Then,
$\dfrac{dRTS}{dl}=\dfrac{d \left( \dfrac{f_1}{f_k} \right)}{dl} \lt 0$
when $200<kl<266$ and we say that RTS is diminishing when $kl$ is in this interval.
My question is:
I want to visually see an RTS that is diminishing and an RTS that is not diminishing. What functions and parameters do I plot?
My understanding of RTS is that it is the slope of the function $f(k,l)=c$ where $c$ is a constant. So I tried doing a contour plot for various levels of $c$ but I don't see any significant differences in the contours (all are still download sloping and slowly becomes flat). Here's the plot:
Thank you in advance for any help provided.