<p>I am calculating elasticity of substitution for the following production function:</p>
<p><span class="math-container">$$F(K,L) = A(aK^{-\gamma}+bL^{-\gamma})^{-\mu/\gamma}$$</span></p>
<p>where <span class="math-container">$A, a, b, \mu, \gamma$</span> are constants. <span class="math-container">$A, a, b, &gt; 0$</span>, <span class="math-container">$\mu \neq 0 $</span> and  <span class="math-container">$\gamma \neq 0; \gamma&gt;-1$</span>.</p>
<p>We learned in our course that to calculate elasticity of substitution we should first calculate marginal rate of substitution and then calculate elasticity with respect to <span class="math-container">$K/L$</span>. I calculated MRTS to be:</p>
<p><span class="math-container">$$MRTS = \frac{b}{a} \left(\frac{K}{L}\right)^{\gamma+1}$$</span></p>
<p>Then I try to calculate the elasticity of the above with respect to <span class="math-container">$K/L$</span> which gives me:</p>
<p><span class="math-container">$$EL_{MRTS_{K,L}} = \frac{d MRTS}{d (K/L)} \frac{K/L}{MRTS} \\ = \frac{b}{a} (\gamma+1)\left(\frac{K}{L}\right)^{\gamma} \left( \frac{K/L}{\frac{b}{a} \left(\frac{K}{L}\right)^{\gamma+1}}\right)  \\= \gamma +1$$</span></p>
<p>However, the textbook says the correct answer is <span class="math-container">$\frac{1}{\gamma+1}$</span>. I think I just made some small mistake somewhere because the answer provided by book is just an inverse of my answer. But I don't know where I am making the mistake.</p>
<p>Please could someone help me with this? Thanks.</p>

CC BY-SA 4.0
2020-11-13T08:49:09.257Z
Your calculation is correct just a small definitional issue. The definition of [Elasticity of substitution](https://en.wikipedia.org/wiki/Elasticity_of_substitution) is the inverse of what you have taken.
Dayne
CC BY-SA 4.0
2020-11-13T10:06:08.660Z
@Dayne consider expanding it into full answer
1muflon1
CC BY-SA 4.0
2020-11-13T13:57:42.493Z
Okay. Sure will do and try to enrich it further
Dayne
<microeconomics><mathematical-economics><production-function><elasticity-of-substitution>
Why is my elasticity of substitution wrong?
WilliamT
# Why is my elasticity of substitution wrong?

(Asked by: username_0 on 2020-11-12)

I am calculating elasticity of substitution for the following production function:

$$F(K,L) = A(aK^{-\gamma}+bL^{-\gamma})^{-\mu/\gamma}$$

where $A, a, b, \mu, \gamma$ are constants. $A, a, b, > 0$, $\mu \neq 0 $ and $\gamma \neq 0; \gamma>-1$.

We learned in our course that to calculate elasticity of substitution we should first calculate marginal rate of substitution and then calculate elasticity with respect to $K/L$. I calculated MRTS to be:

$$MRTS = \frac{b}{a} \left(\frac{K}{L}\right)^{\gamma+1}$$

Then I try to calculate the elasticity of the above with respect to $K/L$ which gives me:

$$EL\_{MRTS\_{K,L}} = \frac{d MRTS}{d (K/L)} \frac{K/L}{MRTS} \\ = \frac{b}{a} (\gamma+1)\left(\frac{K}{L}\right)^{\gamma} \left( \frac{K/L}{\frac{b}{a} \left(\frac{K}{L}\right)^{\gamma+1}}\right) \\= \gamma +1$$

However, the textbook says the correct answer is $\frac{1}{\gamma+1}$. I think I just made some small mistake somewhere because the answer provided by book is just an inverse of my answer. But I don't know where I am making the mistake.

Please could someone help me with this? Thanks.

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

> **Comments:**
> **username_1**: Your calculation is correct just a small definitional issue. The definition of [Elasticity of substitution](https://en.wikipedia.org/wiki/Elasticity_of_substitution) is the inverse of what you have taken.  
> **username_2**: @username_1 consider expanding it into full answer  
> **username_1**: Okay. Sure will do and try to enrich it further  

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## 0 answers

