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So I was solving a question which says find the range in which MRTS is diminishing $f(k,l) = 600k^2l^2-k^3l^3$ is the production function I got the MRTS = $ -(1200kl^2-3k^2l^3) \over 1200k^2l-3k^3l^2$
Now, i'm getting $\frac{\partial f'(k,l)}{\partial k} = \frac{l}{k^2}$ then range is $l <0 $ which doesn't seem to make sense. Is my answer correct or is there some mistake??
Fix the production function to a constant value $f_0$ so that
$$ f_0 = 600k^2l^2 - k^3 l^3 \tag{1} $$
This is a figure for different values of $f_0$ (isoquants)
Note that with increasing $k$ the rate of change of $l$ becomes smaller for a given $f$ and a fixed change in $k$. We can show this more formally by taking the derivative w.r.t $l$ at both sides
$$ 0 = 600\left(2kl^2 + 2k^2 l \frac{\partial l}{\partial k}\right) - \left(3k^2l^3 + 3k^3l^2\frac{\partial l}{\partial k}\right) \tag{2} $$
You can further simplify this to
$$ 0 = -3k l (-400 + k l)\left(l + k \frac{\partial l}{\partial k}\right) \tag{3} $$
The solutions are $k = 0$, $l = 0$, $kl = 400$ or
$$ \frac{\partial l}{\partial k} = -\frac{l}{k} \tag{4} $$
Which confirms that that increasing $k$ decreases the MRTS, in other words, the range in which the marginal diminishes is $k > 0$
