<p>Fix the production function to a constant value <span class="math-container">$f_0$</span> so that</p>

<p><span class="math-container">$$
f_0 = 600k^2l^2 - k^3 l^3 \tag{1}
$$</span></p>

<p>This is a figure for different values of <span class="math-container">$f_0$</span> (isoquants)</p>

<p><a href="https://i.sstatic.net/KCa1T.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/KCa1T.png" alt="enter image description here"></a></p>

<p>Note that with increasing <span class="math-container">$k$</span> the rate of change of <span class="math-container">$l$</span> becomes smaller for a given <span class="math-container">$f$</span> and a fixed change in <span class="math-container">$k$</span>. We can show this more formally by taking the derivative w.r.t <span class="math-container">$l$</span> at both sides</p>

<p><span class="math-container">$$
0 = 600\left(2kl^2 + 2k^2 l \frac{\partial l}{\partial k}\right) - \left(3k^2l^3 + 3k^3l^2\frac{\partial l}{\partial k}\right) \tag{2}
$$</span></p>

<p>You can further simplify this to</p>

<p><span class="math-container">$$
0 = -3k l (-400 + k l)\left(l + k \frac{\partial l}{\partial k}\right) \tag{3}
$$</span></p>

<p>The solutions are <span class="math-container">$k = 0$</span>, <span class="math-container">$l = 0$</span>, <span class="math-container">$kl = 400$</span> or </p>

<p><span class="math-container">$$
\frac{\partial l}{\partial k} = -\frac{l}{k} \tag{4}
$$</span></p>

<p>Which confirms that that increasing <span class="math-container">$k$</span> decreases the MRTS, in other words, the range in which the marginal diminishes is <span class="math-container">$k &gt; 0$</span></p>

CC BY-SA 4.0
2018-11-24T12:59:16.120Z
I know I got that, now to find the range of diminishing MRTS , if you differentiate mrts you get $\frac{l}{k^2}$ same thing what I got. But, what is the range of values such that mrts is diminishing? I'm getting $l<0$ which doesn't make sense
Sumukh Sai
CC BY-SA 4.0
2018-11-24T15:20:13.753Z
@SumukhSai Sorry I misread the question. I updated my answer, hopefully it makes more sense now
caverac
CC BY-SA 4.0
2018-11-24T11:01:12.813Z
2018-11-24T15:19:44.103Z
2018-11-24T15:19:44.103Z
caverac
<p>So I was solving a question which says find the range in which MRTS is diminishing
<span class="math-container">$f(k,l) = 600k^2l^2-k^3l^3$</span> is the production function
I got the MRTS = <span class="math-container">$ -(1200kl^2-3k^2l^3) \over 1200k^2l-3k^3l^2$</span></p>

<p>Now, i'm getting <span class="math-container">$\frac{\partial f'(k,l)}{\partial k} = \frac{l}{k^2}$</span> then range is <span class="math-container">$l &lt;0 $</span> which doesn't seem to make sense. 
Is my answer correct or is there some mistake?? </p>

<production-function>
Diminishing MRTS range
Sumukh Sai
# Diminishing MRTS range

(Asked by: username_0 on 2018-11-24)

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??

**Tags:** production-function


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## 1 answers

### Answer by username_1 on 2018-11-24. Score: 3 | Accepted answer

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)

[![enter image description here](https://i.sstatic.net/KCa1T.png)](https://i.sstatic.net/KCa1T.png)

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 $k > 0$

> **Comments:**
> **username_0**: I know I got that, now to find the range of diminishing MRTS , if you differentiate mrts you get $\frac{l}{k^2}$ same thing what I got. But, what is the range of values such that mrts is diminishing? I'm getting $l<0$ which doesn't make sense  
> **username_1**: @username_0 Sorry I misread the question. I updated my answer, hopefully it makes more sense now  

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