<p>So, this is an economics question but the problem I have is a pure math problem I guess. So I have the following equation:<a href="https://i.sstatic.net/OaInc.png" rel="nofollow noreferrer">f(x,y)</a></p>

<p>this function have the elasticity of substitution(EOS): 1/(1-beta). a,b are constants >0 and beta is also a constant, but is ≤1. So, the general equation for EOS is the following: σ =−(∂log(x/y)) / (∂log((∂f(x,y)/∂x)/(∂f(x,y)/∂y)))</p>

<p>From some calculus I got following result for the inner part of the denominator(without the log or derivative): <a href="https://i.sstatic.net/YjEXu.png" rel="nofollow noreferrer">(∂f(x,y)/∂x)/(∂f(x,y)/∂y)</a></p>

<p>For the numerator, I rewrite the expression to this(since it appear to be the case that we want to have x/y explicit):<a href="https://i.sstatic.net/CgsKN.png" rel="nofollow noreferrer">x/y</a></p>

<p>Taking the logarithm of this gives me following result:<a href="https://i.sstatic.net/Adhzz.png" rel="nofollow noreferrer">log(x/y)</a></p>

<p>Putting these together and insert in to the formula gives me:<a href="https://i.sstatic.net/88h7i.png" rel="nofollow noreferrer">σ</a>
(note that the index should not be i and j, but x and y, my fault)</p>

<p>Okay, so this is how far as I get. And what I don't get from here is the "∂"-part, what am I suppose to take the derivative off? With respect to what variable? Even though there can be som errors in my calculations leading to that expression I still don't understand how I can take the derivative of an expression like that. The main formula for me does not make sense.</p>

<p>Can someone please explain how I should continue?</p>

<p>(Sorry for a messy post, I could not figure out if there was an effective way of writing in calculations in the post so I went with pictures instead)</p>

<p>Kind regards, Fridein</p>

CC BY-SA 4.0
2019-10-05T18:09:10.103Z
Hi welcome to SE, in regards to writing calculations you can format in text using LaTeX. For example I can write $f(x,y) = \frac{x}{y}$
Brennan
CC BY-SA 4.0
2019-10-05T20:20:21.263Z
See [this](https://economics.meta.stackexchange.com/questions/1417/mathjax-basic-tutorial-and-quick-reference) for the LaTex/Mathjax commands.
Giskard
CC BY-SA 4.0
2019-10-08T01:40:32.710Z
If you don't want to learn $\LaTeX$, you could also paste images directly into SE posts.
Art
<microeconomics><mathematical-economics><elasticity><elasticity-of-substitution>
Elasticity of substitution
Fridein
# Elasticity of substitution

(Asked by: username_0 on 2019-10-05)

So, this is an economics question but the problem I have is a pure math problem I guess. So I have the following equation:[f(x,y)](https://i.sstatic.net/OaInc.png)

this function have the elasticity of substitution(EOS): 1/(1-beta). a,b are constants >0 and beta is also a constant, but is ≤1. So, the general equation for EOS is the following: σ =−(∂log(x/y)) / (∂log((∂f(x,y)/∂x)/(∂f(x,y)/∂y)))

From some calculus I got following result for the inner part of the denominator(without the log or derivative): [(∂f(x,y)/∂x)/(∂f(x,y)/∂y)](https://i.sstatic.net/YjEXu.png)

For the numerator, I rewrite the expression to this(since it appear to be the case that we want to have x/y explicit):[x/y](https://i.sstatic.net/CgsKN.png)

Taking the logarithm of this gives me following result:[log(x/y)](https://i.sstatic.net/Adhzz.png)

Putting these together and insert in to the formula gives me:[σ](https://i.sstatic.net/88h7i.png)
(note that the index should not be i and j, but x and y, my fault)

Okay, so this is how far as I get. And what I don't get from here is the "∂"-part, what am I suppose to take the derivative off? With respect to what variable? Even though there can be som errors in my calculations leading to that expression I still don't understand how I can take the derivative of an expression like that. The main formula for me does not make sense.

Can someone please explain how I should continue?

(Sorry for a messy post, I could not figure out if there was an effective way of writing in calculations in the post so I went with pictures instead)

Kind regards, username_0

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

> **Comments:**
> **username_1**: Hi welcome to SE, in regards to writing calculations you can format in text using LaTeX. For example I can write $f(x,y) = \frac{x}{y}$  
> **username_2**: See [this](https://economics.meta.stackexchange.com/questions/1417/mathjax-basic-tutorial-and-quick-reference) for the LaTex/Mathjax commands.  
> **username_3**: If you don't want to learn $\LaTeX$, you could also paste images directly into SE posts.  

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## 0 answers

