<p>This function is a Cobb-Douglas type production function. This family of functions has the general form <span class="math-container">$ Q = f(x_1,x_2) = A \cdot x_1^a \cdot x_2^b $</span>. For your specific question, <span class="math-container">$ A = 1, x_1 = l, x_2 = k, a = 1, b = 0.5 $</span>. </p>

<p>These functions have an interesting property, which is handy to keep in mind (you can show this using the same logic you used in your answer)</p>

<p>i) if <span class="math-container">$ a+b &gt; 1 $</span>, then the functions exhibits increasing returns to scale;
ii) if <span class="math-container">$ a+b = 1 $</span>, then the functions exhibits constant returns to scale;
iii) if <span class="math-container">$a+b &gt; 1 $</span>, then the functions exhibits decreasing returns to scale.</p>

<p>Therefore, the function you presented has increasing returns to scale. Now, in regards to the marginal productivity of the factors, we have:</p>

<p>i) <span class="math-container">$ MP_L = \frac{\partial Q}{\partial L} = K^{0.5} $</span>. But we want to check whether <span class="math-container">$ MP_L $</span> is increasing or decreasing, so we take the second derivative of Q with respect to L: <span class="math-container">$ \frac{\partial^2 Q}{\partial L^2} = \frac{\partial MP_L}{\partial L} = 0 $</span>, thus L has constant marginal returns;</p>

<p>ii) Doing the same for K, we arrive at <span class="math-container">$ \frac{\partial MP_K}{\partial K} &lt; 0 $</span>, which means K has decreasing marginal returns.</p>

<p>This alone would eliminate all alternatives but <strong>d</strong>.</p>

<p>However, for the sake of completeness: </p>

<p><span class="math-container">$ MRTS_{K,L} = \frac{MP_L}{MP_K} $</span> </p>

<p>but we showed that <span class="math-container">$ MP_L $</span> is constant and <span class="math-container">$ MP_K $</span> is decreasing, thus <span class="math-container">$ MRTS_{K,L} $</span> is also decreasing.</p>

<p>If you still have any doubts, feel free to ask.</p>

<p>Kind regards,
Pedro.</p>

CC BY-SA 4.0
2019-05-20T23:31:32.730Z
2019-05-20T23:31:32.730Z
Pedro Cunha
<p>My work out shows constant MRTS and also increasing returns to scale. I thought the answer was C as I only found increasing marginal products of labour and capital. I really don't see how the answer is d.</p>

<p>I have double checked my working also, I don't think I made any mistakes...</p>

<p>Thanks<a href="https://i.sstatic.net/sEBjV.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/sEBjV.png" alt="enter image description here"></a>
<a href="https://i.sstatic.net/VJr4n.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/VJr4n.jpg" alt=""></a></p>

<microeconomics><production-function><marginal>
MRTS question involving production function
onetwothree
# MRTS question involving production function

(Asked by: username_0 on 2019-05-20)

My work out shows constant MRTS and also increasing returns to scale. I thought the answer was C as I only found increasing marginal products of labour and capital. I really don't see how the answer is d.

I have double checked my working also, I don't think I made any mistakes...

Thanks[![enter image description here](https://i.sstatic.net/sEBjV.png)](https://i.sstatic.net/sEBjV.png)
[![](https://i.sstatic.net/VJr4n.jpg)](https://i.sstatic.net/VJr4n.jpg)

**Tags:** microeconomics, production-function, marginal

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## 1 answers

### Answer by username_1 on 2019-05-20. Score: 0 | Accepted answer

This function is a Cobb-Douglas type production function. This family of functions has the general form $ Q = f(x\_1,x\_2) = A \cdot x\_1^a \cdot x\_2^b $. For your specific question, $ A = 1, x\_1 = l, x\_2 = k, a = 1, b = 0.5 $.

These functions have an interesting property, which is handy to keep in mind (you can show this using the same logic you used in your answer)

i) if $ a+b > 1 $, then the functions exhibits increasing returns to scale;
ii) if $ a+b = 1 $, then the functions exhibits constant returns to scale;
iii) if $a+b > 1 $, then the functions exhibits decreasing returns to scale.

Therefore, the function you presented has increasing returns to scale. Now, in regards to the marginal productivity of the factors, we have:

i) $ MP\_L = \frac{\partial Q}{\partial L} = K^{0.5} $. But we want to check whether $ MP\_L $ is increasing or decreasing, so we take the second derivative of Q with respect to L: $ \frac{\partial^2 Q}{\partial L^2} = \frac{\partial MP\_L}{\partial L} = 0 $, thus L has constant marginal returns;

ii) Doing the same for K, we arrive at $ \frac{\partial MP\_K}{\partial K} < 0 $, which means K has decreasing marginal returns.

This alone would eliminate all alternatives but **d**.

However, for the sake of completeness:

$ MRTS\_{K,L} = \frac{MP\_L}{MP\_K} $

but we showed that $ MP\_L $ is constant and $ MP\_K $ is decreasing, thus $ MRTS\_{K,L} $ is also decreasing.

If you still have any doubts, feel free to ask.

Kind regards,
Pedro.


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