<blockquote>
  <p>Is there anyone who can help me with this?</p>
</blockquote>

<p>Here it is. Equations 1-3, and 5-6 are obtained in preparation for the 2nd derivatives of V with respect to L and K.</p>

<p><a href="https://i.sstatic.net/XS7tf.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/XS7tf.jpg" alt="enter image description here"></a></p>

<p>Let me know if you have any questions.</p>

CC BY-SA 4.0
2018-11-28T14:13:37.783Z
First of all thank you very much for your extremely clear answer! :) Then, if you have a minute I would like to ask your help with another passage of the same paper that I did not fully understand. Further on (page 20), the author calculates the elasticity of substitution (σ) which is equal to (3.24):

$$\sigma =\frac{b}{1-\frac{c}{X}\frac{f}{f'}}$$

Up to here everything is clear. Then the author rewrite (3.24) in the following way, obtaining (3.25):

$$\sigma =\frac{b}{1-c\left ( 1+\frac{R}{X} \right )}$$

It is this last result that I cannot get. Thank you so much, I owe you a beer!
Alessandro
CC BY-SA 4.0
2018-11-28T18:43:34.680Z
I wish I could help on that one. Yesterday I just computed the derivatives without looking at the paper at issue, but I'll need to delve into it to get a sense of what *R* means. If I get the chance to do so, I will be happy to address (3.25).
Iñaki Viggers
CC BY-SA 4.0
2018-11-28T18:56:40.973Z
Sorry, I forgot to write that R is the marginal rate of substitution of L for K, so basically just:

$$R=-\frac{dK}{dL}=\frac{MPL}{MPK}$$

I tried to make this calculation, which according to the formulas of the previous comment if divided by X and then adding 1 should be exactly equal to 

$$\frac{f}{Xf'}$$

i.e. equal to 

$$\frac{Y}{X\frac{dY}{dX}}$$

But I can't get this latter equality :((
Alessandro
CC BY-SA 4.0
2018-11-27T22:55:13.070Z
2018-11-27T22:55:13.070Z
Iñaki Viggers
<p>I'm following a paper (Full text available <a href="https://lib.dr.iastate.edu/cgi/viewcontent.cgi?referer=https://www.google.es/&amp;httpsredir=1&amp;article=4405&amp;context=rtd" rel="nofollow noreferrer" title="link to the paper">here</a>!) where at some point <strong>(pag.17 and 20)</strong> the author get the following derivative:</p>

<p><span class="math-container">$$\frac{\partial V}{\partial L}=Y-X\frac{dY}{dX}=\alpha X^{-\frac{c}{b}}Y^{\frac{1}{b}}$$</span></p>

<p>where: <span class="math-container">$Y=\frac{V}{L}$</span> and <span class="math-container">$X=\frac{K}{L}$</span></p>

<p>Then, starting from this he calculates the partial derivative with respect
to L and the cross second-order partial derivative (the partial derivative with respect to K), whose results are shown below:</p>

<p><span class="math-container">$$\frac{\partial^2 V}{L^{2}}=-\frac{\alpha }{bL} X^{-\frac{c}{b}}Y^{\frac{1}{b}-1}\left ( X\frac{dY}{dX}-cY \right )$$</span></p>

<p><span class="math-container">$$\frac{\partial^2 V}{dKdL}=\frac{\alpha }{bL} X^{-\frac{c}{b}-1}Y^{\frac{1}{b}-1}\left ( X\frac{dY}{dX}-cY \right )$$</span></p>

<p>I was stuck trying to derive these latter derivatives. Is there anyone who can help me with this? Thank you so much!</p>

<microeconomics><mathematical-economics><elasticity-of-substitution>
Second order partial derivative and cross second-order partial derivative
Alessandro
# Second order partial derivative and cross second-order partial derivative

(Asked by: username_0 on 2018-11-27)

I'm following a paper (Full text available [here](https://lib.dr.iastate.edu/cgi/viewcontent.cgi?referer=https://www.google.es/&httpsredir=1&article=4405&context=rtd "link to the paper")!) where at some point **(pag.17 and 20)** the author get the following derivative:

$$\frac{\partial V}{\partial L}=Y-X\frac{dY}{dX}=\alpha X^{-\frac{c}{b}}Y^{\frac{1}{b}}$$

where: $Y=\frac{V}{L}$ and $X=\frac{K}{L}$

Then, starting from this he calculates the partial derivative with respect
to L and the cross second-order partial derivative (the partial derivative with respect to K), whose results are shown below:

$$\frac{\partial^2 V}{L^{2}}=-\frac{\alpha }{bL} X^{-\frac{c}{b}}Y^{\frac{1}{b}-1}\left ( X\frac{dY}{dX}-cY \right )$$

$$\frac{\partial^2 V}{dKdL}=\frac{\alpha }{bL} X^{-\frac{c}{b}-1}Y^{\frac{1}{b}-1}\left ( X\frac{dY}{dX}-cY \right )$$

I was stuck trying to derive these latter derivatives. Is there anyone who can help me with this? Thank you so much!

**Tags:** microeconomics, mathematical-economics, elasticity-of-substitution

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## 1 answers

### Answer by username_1 on 2018-11-27. Score: 1 | Accepted answer

> Is there anyone who can help me with this?

Here it is. Equations 1-3, and 5-6 are obtained in preparation for the 2nd derivatives of V with respect to L and K.

[![enter image description here](https://i.sstatic.net/XS7tf.jpg)](https://i.sstatic.net/XS7tf.jpg)

Let me know if you have any questions.

> **Comments:**
> **username_0**: First of all thank you very much for your extremely clear answer! :) Then, if you have a minute I would like to ask your help with another passage of the same paper that I did not fully understand. Further on (page 20), the author calculates the elasticity of substitution (σ) which is equal to (3.24):  $$\sigma =\frac{b}{1-\frac{c}{X}\frac{f}{f'}}$$  Up to here everything is clear. Then the author rewrite (3.24) in the following way, obtaining (3.25):  $$\sigma =\frac{b}{1-c\left ( 1+\frac{R}{X} \right )}$$  It is this last result that I cannot get. Thank you so much, I owe you a beer!  
> **username_1**: I wish I could help on that one. Yesterday I just computed the derivatives without looking at the paper at issue, but I'll need to delve into it to get a sense of what *R* means. If I get the chance to do so, I will be happy to address (3.25).  
> **username_0**: Sorry, I forgot to write that R is the marginal rate of substitution of L for K, so basically just:  $$R=-\frac{dK}{dL}=\frac{MPL}{MPK}$$  I tried to make this calculation, which according to the formulas of the previous comment if divided by X and then adding 1 should be exactly equal to   $$\frac{f}{Xf'}$$  i.e. equal to   $$\frac{Y}{X\frac{dY}{dX}}$$  But I can't get this latter equality :((  

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