<p>four things :</p>

<ol>
<li><p>The desired result is actually
<img src="https://latex.codecogs.com/gif.latex?%5Csigma%20%3D%20%5Cfrac%7B1%7D%7B1-p%20%7D" alt="equation"> which is the answer you are getting if you multiply the negative into the denominator( you can cross-check on <a href="https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution</a>)
. However, you have made the following mistakes. </p></li>
<li><p>NON CONSEQUENTIAL ERROR: Your first error is when you differentiate to get MP1 and MP2. In both these cases, the long term should have power : (1/p - 1 ). This term gets canceled hence doesn't affect the solution. </p>

<p><img src="https://latex.codecogs.com/gif.latex?MP_%7B1%7D%3D%5Cfrac%7B1%7D%7Bp%7D%28%28a_%7B1%7Dx_%7B1%7D%5E%7Bp%7D%20&plus;%20a_%7B2%7Dx_%7B2%7D%5E%7Bp%7D%29%29%5E%7B%5Cfrac%7B1%7D%7Bp%7D-1%7D%28pa_%7B1%7Dx_%7B1%7D%5E%7Bp-1%7D%29" alt="equation"></p></li>
<li><p>Further MRTS(marginal rate of technical sub) = MP1/MP2 and not - MP1/MP2. This is because MRTS is the absolute number of factor x2 that has to be given up to employ one more unit of factor x1 while keeping the production constant( moving along the same isoquant curve). MRTS is always taken as positive. This is because the negative sign of the slope simply implies the "giving up" which is implicit in the definition of opportunity cost. </p></li>
<li><p>However, in both of your mistakes, your error got canceled due to the symmetry of the numerator and denominator. Hence you are actually getting the right answer.  Your method is a little long. I am posting here the way I do it : </p></li>
</ol>

<p><a href="https://i.sstatic.net/3zbcD.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/3zbcD.jpg" alt="enter image description here"></a></p>

CC BY-SA 4.0
2020-03-16T22:51:53.893Z
Thanks man. Your method is way easier.
Mathias
CC BY-SA 4.0
2020-03-16T21:35:16.833Z
2020-03-16T21:35:16.833Z
Poorvi
<p>I have to find the EOS for the general CES-function defined as</p>

<p><span class="math-container">$$
y = (a_1x_1^p+a_2x_2^p)^{1/p}
$$</span></p>

<p>I first find <span class="math-container">$MP_1$</span> and <span class="math-container">$MP_2$</span> to find <span class="math-container">$MRTS$</span>. We have that</p>

<p><span class="math-container">$MP_1 = \frac{\partial y} {\partial x_1} = \frac{1}{p} (a_1x_1^p+a_2x_2^p)^{1/p} \cdot pa_1x_1^{p-1}$</span></p>

<p><span class="math-container">$MP_2 = \frac{\partial y} {\partial x_2} = \frac{1}{p} (a_1x_1^p+a_2x_2^p)^{1/p} \cdot pa_2x_2^{p-1}$</span></p>

<p>and thus </p>

<p><span class="math-container">$MTRS = - \frac{MP_1}{MP_2} = - \frac{a_1x_1^{p-1}}{a_2x_2^{p-1}}$</span></p>

<p>By definition EOS is given by</p>

<p><span class="math-container">$$
\sigma = \frac{d(x_2/x_1)}{dMTRS} \frac{MTRS}{(x_2/x_1)}
$$</span>
so to find <span class="math-container">$x_2/x_1$</span> I have done the following. Let <span class="math-container">$\theta = - \frac{a_1x_1^{p-1}}{a_2x_2^{p-1}}$</span> then we have
<span class="math-container">$$
\frac{1}{\theta} = - \frac{a_2x_2^{p-1}}{a_1x_1^{p-1}} \Rightarrow \frac{x_2^{p-1}}{x_1^{p-1}} = - \frac{1}{\theta} \frac{a_1}{a_2} \Rightarrow \frac{x_2}{x_1} = (- \frac{1}{\theta} \frac{a_1}{a_2})^{\frac{1}{p-1}}
$$</span>
Thus
<span class="math-container">$$
\sigma = \frac{d(- \frac{1}{\theta} \frac{a_1}{a_2})^{\frac{1}{p-1}}}{d \theta} \cdot \frac{\theta}{(- \frac{1}{\theta} \frac{a_1}{a_2})^{\frac{1}{p-1}}} = -\frac{1}{p-1}
$$</span>
which does not give the desired result that <span class="math-container">$\sigma = \frac{1}{p+1}$</span>. Where am I doing wrong?</p>

<elasticity-of-substitution>
How to find the elasticity of substitution for the general CES-function?
Mathias
# How to find the elasticity of substitution for the general CES-function?

(Asked by: username_0 on 2020-03-16)

I have to find the EOS for the general CES-function defined as

$$
y = (a\_1x\_1^p+a\_2x\_2^p)^{1/p}
$$

I first find $MP\_1$ and $MP\_2$ to find $MRTS$. We have that

$MP\_1 = \frac{\partial y} {\partial x\_1} = \frac{1}{p} (a\_1x\_1^p+a\_2x\_2^p)^{1/p} \cdot pa\_1x\_1^{p-1}$

$MP\_2 = \frac{\partial y} {\partial x\_2} = \frac{1}{p} (a\_1x\_1^p+a\_2x\_2^p)^{1/p} \cdot pa\_2x\_2^{p-1}$

and thus

$MTRS = - \frac{MP\_1}{MP\_2} = - \frac{a\_1x\_1^{p-1}}{a\_2x\_2^{p-1}}$

By definition EOS is given by

$$
\sigma = \frac{d(x\_2/x\_1)}{dMTRS} \frac{MTRS}{(x\_2/x\_1)}
$$
so to find $x\_2/x\_1$ I have done the following. Let $\theta = - \frac{a\_1x\_1^{p-1}}{a\_2x\_2^{p-1}}$ then we have
$$
\frac{1}{\theta} = - \frac{a\_2x\_2^{p-1}}{a\_1x\_1^{p-1}} \Rightarrow \frac{x\_2^{p-1}}{x\_1^{p-1}} = - \frac{1}{\theta} \frac{a\_1}{a\_2} \Rightarrow \frac{x\_2}{x\_1} = (- \frac{1}{\theta} \frac{a\_1}{a\_2})^{\frac{1}{p-1}}
$$
Thus
$$
\sigma = \frac{d(- \frac{1}{\theta} \frac{a\_1}{a\_2})^{\frac{1}{p-1}}}{d \theta} \cdot \frac{\theta}{(- \frac{1}{\theta} \frac{a\_1}{a\_2})^{\frac{1}{p-1}}} = -\frac{1}{p-1}
$$
which does not give the desired result that $\sigma = \frac{1}{p+1}$. Where am I doing wrong?

**Tags:** elasticity-of-substitution

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## 1 answers

### Answer by username_1 on 2020-03-16. Score: 3

four things :

1. The desired result is actually
   ![equation](https://latex.codecogs.com/gif.latex?%5Csigma%20%3D%20%5Cfrac%7B1%7D%7B1-p%20%7D) which is the answer you are getting if you multiply the negative into the denominator( you can cross-check on <https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution>)
   . However, you have made the following mistakes.
2. NON CONSEQUENTIAL ERROR: Your first error is when you differentiate to get MP1 and MP2. In both these cases, the long term should have power : (1/p - 1 ). This term gets canceled hence doesn't affect the solution.

   ![equation](https://latex.codecogs.com/gif.latex?MP_%7B1%7D%3D%5Cfrac%7B1%7D%7Bp%7D%28%28a_%7B1%7Dx_%7B1%7D%5E%7Bp%7D%20+%20a_%7B2%7Dx_%7B2%7D%5E%7Bp%7D%29%29%5E%7B%5Cfrac%7B1%7D%7Bp%7D-1%7D%28pa_%7B1%7Dx_%7B1%7D%5E%7Bp-1%7D%29)
3. Further MRTS(marginal rate of technical sub) = MP1/MP2 and not - MP1/MP2. This is because MRTS is the absolute number of factor x2 that has to be given up to employ one more unit of factor x1 while keeping the production constant( moving along the same isoquant curve). MRTS is always taken as positive. This is because the negative sign of the slope simply implies the "giving up" which is implicit in the definition of opportunity cost.
4. However, in both of your mistakes, your error got canceled due to the symmetry of the numerator and denominator. Hence you are actually getting the right answer. Your method is a little long. I am posting here the way I do it :

[![enter image description here](https://i.sstatic.net/3zbcD.jpg)](https://i.sstatic.net/3zbcD.jpg)

> **Comments:**
> **username_0**: Thanks man. Your method is way easier.  

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