<p>I want to study how <span class="math-container">$n$</span> varies with <span class="math-container">$m$</span>. They have the following relations:
<span class="math-container">$$n^{-\frac{\sigma}{\alpha(\sigma-1)}}\left(\frac{\frac{f(\sigma-1)}{m}}{L-\frac{n\sigma f}{m}}\right)^{-\frac{1}{\alpha}}=n^{\frac{1}{1-\sigma}}\frac{\sigma}{\sigma-1}$$</span>
For more context, <span class="math-container">$n$</span> is the number of varieties and <span class="math-container">$m$</span> is the size of the economy. <span class="math-container">$\sigma,\alpha$</span> are elasticities. <span class="math-container">$f$</span> is the fixed cost. <span class="math-container">$L$</span> is the labor supplied.</p>
<p>How do I calculate <span class="math-container">$\frac{\partial n}{\partial m}$</span>?</p>

CC BY-SA 4.0
2024-09-17T02:48:10.057Z
Looks ugly. Have you tried implicit function theorem?
Herr K.
CC BY-SA 4.0
2024-09-17T14:02:17.710Z
@HerrK. Honestly? I don't know how that works.
Ludwig Gershwin
CC BY-SA 4.0
2024-09-17T15:33:01.537Z
Try to rewrite the equation into the form $F(m,n)=0$ (some terms can be simplified/canceled out). According to the [implicit function theorem](https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/dfi/t), if $F(m,n)=0$ and $\frac{\partial F}{\partial n}\ne0$, then 
\begin{equation}
\frac{\mathrm dn}{\mathrm dm} = -\frac{\partial F/\partial m}{\partial F/\partial n}
\end{equation}
Herr K.
<international-trade>
How to solve this comparative static?
Ludwig Gershwin
# How to solve this comparative static?

(Asked by: username_0 on 2024-09-17)

I want to study how $n$ varies with $m$. They have the following relations:
$$n^{-\frac{\sigma}{\alpha(\sigma-1)}}\left(\frac{\frac{f(\sigma-1)}{m}}{L-\frac{n\sigma f}{m}}\right)^{-\frac{1}{\alpha}}=n^{\frac{1}{1-\sigma}}\frac{\sigma}{\sigma-1}$$
For more context, $n$ is the number of varieties and $m$ is the size of the economy. $\sigma,\alpha$ are elasticities. $f$ is the fixed cost. $L$ is the labor supplied.

How do I calculate $\frac{\partial n}{\partial m}$?

**Tags:** international-trade
> **Comments:**
> **username_1**: Looks ugly. Have you tried implicit function theorem?  
> **username_0**: @HerrK. Honestly? I don't know how that works.  
> **username_1**: Try to rewrite the equation into the form $F(m,n)=0$ (some terms can be simplified/canceled out). According to the [implicit function theorem](https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/dfi/t), if $F(m,n)=0$ and $\frac{\partial F}{\partial n}\ne0$, then  \begin{equation} \frac{\mathrm dn}{\mathrm dm} = -\frac{\partial F/\partial m}{\partial F/\partial n} \end{equation}  

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## 0 answers

